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author:
- Stephan Rachel
- Eric Mascot
- Sagen Cocklin
- Matthias Vojta
- 'Dirk K. Morr'
bibliography:
- 'TSCtransport.bib'
title: |
Supplemental Information for:\
Quantized Charge Transport in Chiral Majorana Edge Modes
---
Keldysh Formalism {#sec:keldysh}
=================
Our starting point for the study of Shiba islands on a square lattice is the Hamiltonian $H=H_{\rm SC}+ H_{\rm tip}$ [@li-16nc12297] where $$\begin{aligned}
H_{\rm SC} =& -t \sum_{{\langle \bf r r^\prime}\rangle \sigma} \left( c_{{\bf r},\sigma}^\dag c^{{\phantom{\dagger}}}_{{\bf r^\prime},\sigma} + {\rm H.c.}\right)
- \mu \sum_{{\bf r}\sigma} c_{{\bf r},\sigma}^\dag c_{{\bf r},\sigma}
+ \Delta_s \sum_{\bf r} \left( c_{{\bf r},{\uparrow}}^\dag c^\dag_{{\bf r},{\downarrow}} + {\rm H.c.}\right) \notag\\
+&\,i \alpha \sum_{{\bf r}\sigma\sigma'}\left( c_{{\bf r},\sigma}^\dag \sigma_{\sigma\sigma'}^2 c^{{\phantom{\dagger}}}_{{\bf r}+{\hat x},\sigma'} - c_{{\bf r},\sigma}^\dag \sigma_{\sigma\sigma'}^1 c^{{\phantom{\dagger}}}_{{\bf r}+ {\hat y},\sigma'} + {\rm H.c.} \right) \notag\\
+& \,J \sum_{{\bf R}\sigma\sigma'} c_{{\bf R},\sigma}^\dag \sigma_{\sigma\sigma'}^3 c^{{\phantom{\dagger}}}_{{\bf R},\sigma'} -t_{{\rm tip}} \sum_{\sigma} \left( c_{{\bf r},\sigma}^\dag d^{{\phantom{\dagger}}}_{\sigma} + {\rm H.c.} \right) \ .
\label{ham-realspace}\end{aligned}$$ Here, $c_{{\bf r},\sigma}^\dag$ creates an electron at lattice site ${\bf r}$ with spin $\sigma$, and $\sigma^i$ are the spin Pauli matrices. $t$ and $\Delta_s$ are the hopping and pairing amplitudes of the superconductor, $\mu$ is the chemical potential, $\alpha$ denotes the Rashba spin-orbit coupling arising from the breaking of the inversion symmetry at the surface[@nadj-perge-14s602], and $J$ is the magnetic exchange coupling. The lattice sites $\{{\bf R}\}$ correspond to the sites of the magnetic adatoms. Moreover, $t_{{\rm tip}}$ is the amplitude for electrons tunneling from the STS tip into the superconductor, with $d^{{\phantom{\dagger}}}_{\sigma}$ annihilating an electron with spin $\sigma$ in the tip, and $H_{{\rm tip}}$ describes the electronic structure of the tip, as discussed below.
To investigate the transport properties of such magnetic islands, we employ the non-equilibrium Keldysh Green’s function formalism [@keldysh65jetp1018; @caroli-71jpc916], which allows us to compute not only the tunneling current between the STS tip and the superconductor, but also the currents flowing on the surface of the $s$-wave superconductor. We note that due to the Rashba spin orbit coupling, the flow of charge between two sites in the system can be accompanied by a spin flip. Within the Keldysh formalism, the spin-resolved current flowing between sites $i$ and $j$ therefore has to be generalized to $$I^{\sigma \sigma^\prime}_{ij} = -2\textit{g}_{\textit{s}}\dfrac{\textit{e}}{\hbar}\int_{-\infty}^{\infty} \dfrac{d\omega}{2\pi} {\rm Re} \left[ {\hat T}^{\sigma \sigma^\prime}_{ij} G^{<}(i,\sigma; j,\sigma^\prime; \omega)\right] ,
\label{eq:I}$$ where $G^{<}(i,\sigma; j,\sigma^\prime; \omega)$ is the full, normal lesser Green’s function that describes the propagation of an electron with spin $\sigma$ at site $i$ to an electron with spin $\sigma^\prime$ at site $j$. Here, the matrix ${\hat T}^{\sigma \sigma^\prime}$ describes the hopping between two nearest-neighbor sites, which is either given by $-t$ if $\sigma =\sigma^\prime = \uparrow,\downarrow$, or by the Rashba coupling $\alpha$ if $\sigma \not =\sigma^\prime$.
To compute $G^{<}(i,\sigma; j,\sigma^\prime; \omega)$, we first define the Matsubara Green’s function matrix in frequency space using the effective action $$\begin{aligned}
S =& \frac{1}{\beta} \sum_{\omega_n >0} \Psi^\dagger(i \omega_n) \hat{G}^{-1}(i\omega_n) \Psi(i \omega_n)
\label{eq:eff_action}\end{aligned}$$ where the spinor $\Psi^\dagger(i \omega_n)$ is defined via $$\begin{aligned}
\Psi^\dagger(i \omega_n) = \left(d^\dagger_{\uparrow}(i \omega_n) ,d^\dagger_{\downarrow}(i \omega_n), d_{\downarrow}(-i \omega_n), d_{\uparrow}(-i \omega_n), \ldots ,c^\dagger_{{\bf r},\uparrow}(i \omega_n) ,c^\dagger_{{\bf r},\downarrow}(i \omega_n), c_{{\bf r},\downarrow}(-i \omega_n),c_{{\bf r},\uparrow}(-i \omega_n), \ldots \right)\end{aligned}$$ where ${\bf r}$ is a site in the superconductor. $\hat{G}(i\omega_n)$ is obtained from the Dyson equation $$\begin{aligned}
\hat{G}(i\omega_n) = \left\{ \left[ \hat{g} (i\omega_n) \right]^{-1} - \hat{H}_0 \right\}^{-1}.\end{aligned}$$ Here, $\hat{H}_0$ is the Hamiltonian matrix defined using the Hamiltonian of Eq.(\[ham-realspace\]) via $$\begin{aligned}
H_{\rm SC} =& \frac{1}{2} \Psi^\dagger \hat{H}_0 \Psi
\label{eq:H_matrix}\end{aligned}$$ with spinor $$\begin{aligned}
\Psi^\dagger = \left(d^\dagger_{\uparrow}, d^\dagger_{\downarrow}, d_{\downarrow}, d_{\uparrow}, \ldots , c^\dagger_{{\bf r},\uparrow}, c^\dagger_{{\bf r},\downarrow}, c_{{\bf r},\downarrow}, c_{{\bf r},\uparrow}, \ldots \right) \ .\end{aligned}$$ Note that the factor of $1/2$ in Eq.(\[eq:H\_matrix\]) arises since we consider particle- and hole-like operators for both spin-projections in the definition of $\hat{G}(i\omega_n)$ and the spinor $\Psi^\dagger(i \omega_n)$ in Eq.(\[eq:eff\_action\]). Finally, $\hat{g} (i\omega_n)$ is the Green’s function matrix that represents decoupled and non-interacting sites in the system (see below).
To obtain the lesser Green’s function in Eq.(\[eq:I\]), we define lesser and retarded Green’s function matrices $\hat{G}^{<,r}$ in real space whose $({\bf r r'})$ elements are given by $\hat{G}^{<,r}_{\bf r r'}$, and employ the Dyson equations in frequency space
$$\begin{aligned}
\hat{G}^{<} &= \hat{G}^{r}\left[ \left(\hat{g}^{r} \right)^{-1} \hat{g}^{<}
\left( \hat{g}^{a} \right)^{-1} \right] \hat{G}^{a} \label{eq:fullGa} \\[5pt]
\hat{G}^{r} &= \left[ \left( \hat{g}^{r} \right)^{-1} - \hat{H}_0 \right]^{-1}
\label{eq:fullGb}\end{aligned}$$
Here, $\hat{g}^{x}$ $(x=r,a,<)$ are given by $$\hat{g}^{x}= \left(
\begin{array}{cc}
\hat{g}_{{\rm tip}}^{x} & 0 \\
0 & \hat{g}_{\rm SC}^{x}
\end{array}\right)$$ where $\hat{g}_{\rm SC}^{x}$ and $\hat{g}_{{\rm tip}}^{x}$ are the Green’s function matrices describing the $s$-wave superconductor and the Shiba island, and the tip, respectively. Moreover, $\hat{g}_{\rm SC}^{x}$ are diagonal matrices with elements
$$\begin{aligned}
g_0^r(\omega) &= \frac{1}{\omega + i \delta } \\[5pt]
g_0^<(\omega) &= -2i n_F(\omega) {\rm Im}\{ g_0^r(\omega)\}\end{aligned}$$
where $n_F(\omega)$ is the Fermi distribution function, in the superconductor. Moreover, $\hat{g}_{{\rm tip}}^{x}$ are diagonal matrices with elements
$$\begin{aligned}
g_{{\rm tip}}^r(\omega) &= -i \pi \\[5pt]
g_{{\rm tip}}^<(\omega) &= -2i n_F(\omega-eV) \ {\rm Im}\{ g_{{\rm tip}}^r(\omega)\}\end{aligned}$$
implying that the tip’s density of states is equal to unity and that we consider the wide band limit. Moreover, $e$ is the electron charge, and $V$ is the potential difference between the tip and the grounded superconductor. The spin-resolved local density of states, $N_\sigma({\bf r}, E)$ at site [**r**]{} and energy $E$ is obtained from Eq.(\[eq:fullGb\]) via $$N_\sigma({\bf r}, E=\hbar \omega) = -\frac{1}{\pi} {\rm Im} \{ {\hat G}^r_{\bf rr}(\omega)\} \ .$$
Changing the Ratio between Coherence Length and System Size
===========================================================
If the superconducting coherence length, $\xi$, is reduced in comparison to the system size, we expect that the edge modes are more strongly localized in the vicinity of the edges. To investigate the effects of a shorter coherence length on the physical properties of Shiba islands, we consider a set of parameters $(\alpha, \Delta_s, J)=(0.8, 1.2, 2.0)\,t$ which yields a coherence length $\xi \approx 1.25 a_0$, which is about 5 times smaller than the one considered in the main text. In Figs. \[figSI-1\](a)-(c), we present the resulting LDOS at the edge of the Shiba island in the ${{\cal C}}=-1$ $(\mu=-4t$), ${{\cal C}}=0$ $(\mu=-2t$), and ${{\cal C}}=2$ $(\mu=0$) phases, respectively.
![(a) - (c) Spin-resolved LDOS at the edge of the Shiba island in the ${{\cal C}}=-1$ $(\mu=-4t$), ${{\cal C}}=0$ $(\mu=-2t$), and ${{\cal C}}=2$ $(\mu=0$) phases, respectively, for a set of parameters $(\alpha, \Delta_s, J)=(0.8, 1.2, 2.0)\,t$. Total LDOS for the lowest-energy topological edge modes in the topological (d) ${{\cal C}}=-1$ $(\mu=-4t$) and (e) ${{\cal C}}=2$ $(\mu=0$) phases. Suppercurrents carried by the lowest-energy topological edge modes in the topological (f) ${{\cal C}}=-1$ $(\mu=-4t$) and (g) ${{\cal C}}=2$ $(\mu=0$) phases.[]{data-label="figSI-1"}](SI_Fig1_V1){width="13cm"}
In Figs. \[figSI-1\](d) and (e) we present the total LDOS for the lowest-energy topological edge modes in the topological ${{\cal C}}=-1$ $(\mu=-4t$) and ${{\cal C}}=2$ $(\mu=0$) phases. A comparison with Fig.1 in the main text reveals that due to the decreased superconducting coherence length, the edge modes are much more narrowly confined to the edge of the island. The same conclusion also holds for the spatial form of the supercurrents carried by the lowest-energy modes, shown in Figs. \[figSI-1\](f) and (g). As discussed in the main text, we find that the chirality of the supercurrents is determined by the sign of the Chern number ${{\cal C}}$. Independent of the ratio between $\xi$ and system size, we again find that the conductance is quantized in the topological phases.
Irregularly Shaped Shiba Islands {#sec:disorder}
================================
![Disordered Shiba island of magnetic adatoms with no rotational or mirror symmetries. (a) Energy-resolved LDOS in the ${{\cal C}}=-1$ phase $(\mu=-4t)$. (b) Spatial LDOS of the lowest-energy edge mode shown in (a). (c) Supercurrent carried by the lowest-energy edge mode shown in (a). (d) Differential conductance, $G$, and (e) modified Chern number ${\tilde {{\cal C}}}$, as a function of $\mu$. Parameters used: $(\alpha, \Delta_s, J)=(0.2, 0.3, 0.5)\,t$.[]{data-label="figSI-7"}](SI_Fig7_V1){width="15cm"}
In the main text, we considered a Shiba island that possesses the same spatial symmetries as the underlying lattice of the $s$-wave superconductor (mirror and discrete rotational symmetries). Unless such highly ordered islands can be experimentally created using atomic manipulation techniques (as in the case of molecular graphene[@moon-09nn167]), it is very likely that the experimental realization of Shiba islands will result in disordered or irregularly shaped islands. The question therefore arises to what extent the properties of the topological phases, such as their quantized conductance, are robust against deformations in the shape of the island (as long as the topological phase is not destroyed).
To investigate this question, we consider the irregularly-shaped magnetic island possessing no spatial symmetries shown in Fig.\[figSI-7\](b). Despite its irregular shape, we find that the electronic and transport properties of the Shiba island, associated with the topological phases, remain qualitatively and to a large extent quantitatively unchanged. In Fig. \[figSI-7\](a), we present the LDOS at the edge of the island in the ${{\cal C}}=-1$ phase, which, as shown in Fig. 1 of the main text, exhibit a series of equally (in energy) spaced edge modes. A plot of the spatially resolved LDOS in Fig. \[figSI-7\](b) for the lowest-energy edge mode in Fig. \[figSI-7\](a) reveals that the edge mode is still strongly localized along the edge of the island, but penetrates further into the island due to its reduced size \[cf. Fig. 1(b) of the main text\]. The same conclusion also holds for the spatial form of the supercurrent shown in Fig. \[figSI-7\](c) that is carried by the lowest-energy edge mode \[cf. Fig. 3(b) of the main text\]. The chirality of the supercurrents is the same as that for the circular Shiba island. Moreover, the differential conductance $G$ \[Fig. \[figSI-7\](d)\] as well as the modified Chern number ${\tilde {{\cal C}}}$ \[Fig. \[figSI-7\](e)\], show very similar dependence on the chemical potential as those of the circular island. This reflects the persistent topological nature of the irregular-shaped island, as evidenced by a concomitant quantized tunneling conductance. Our results demonstrate the robustness of the topological phases and their intrinsic properties against deformations in the shape of the island.
${\boldsymbol{I(V)}}$ Curve and Differential Tunnel Conductance ${\boldsymbol{G}}$
==================================================================================
![(a) Schematic picture of a finite Shiba lattice with STS tip. Differential conductance, $G=dI/dV$, and current-voltage dependence $I(V)$ for a finite Shiba lattice of size $N_x=N_y=41$ in the (b) ${{\cal C}}=2$ ($\mu=0$), (c) ${{\cal C}}=0$ ($\mu=-2t$), and (d) ${{\cal C}}=-1$ ($\mu=-4t$) phases. Here, $(\alpha, \Delta_s, J)=(0.2, 0.3, 0.5)\,t$ and $t_{{\rm tip}}=0.2t$. The light blue shaded area indicates the range of $\Delta E$. []{data-label="figf:figSI-2"}](SI_Fig2_V3){width="8cm"}
The main difference between the topological non-trivial (${{\cal C}}\not = 0$) and trivial (${{\cal C}}= 0$) phases lies in the presence of topological low-energy in-gap states (whose conductance is quantized) in the former, and the absence of these states in the latter. To characterize this difference, we consider the maximum differential conductance $G= \max_{\Delta V}\left[ dI({\bf r})/d V \right]$ in a narrow voltage range, $\Delta V$, around zero voltage (light blue area in Fig. \[figf:figSI-2\]). Here, ${\bf r}$ is the position in the superconductor where the electrons from the tip tunnel into. For concreteness, we consider a finite-sized Shiba lattice of size $N_x=N_y=41$. Moreover, we chose the range $\Delta V$ such that only a single edge mode with energy $E$ (which might be degenerate) lies within the energy window $0 \leq E \leq e \Delta V = \Delta E$, as shown in Figs. \[figf:figSI-2\](b)-(d). We note that in the thermodynamic limit of a Shiba lattice, the ${{\cal C}}=2$ phase possesses pairs of two degenerate edge modes. As a result, the conductance is twice as large as in the ${{\cal C}}=-1$ phase \[see Figs. \[figf:figSI-2\](b) and (d), and Fig. 2 of the main text\]. For any finite Shiba lattice or island, the degeneracy between the two modes is broken \[see also the discussion in SI Sec.\[sec:ribbon\]\]. However, the non-zero electronic hopping $t_{{\rm tip}}$ between the STS tip and the superconductor leads to an energy broadening of the edge modes, such that for sufficiently large $t_{\rm tip}$ this broken degeneracy in the ${{\cal C}}=2$ phase cannot any longer be resolved even for a finite-sized system, as the width of the peaks is given by $\sim t_{\rm tip}^2$. Thus the conductance is still twice as large as in the ${{\cal C}}=-1$ phase \[cf. Figs. \[figf:figSI-2\](b) and (d)\]. In contrast, due to the absence of low energy edge modes in the topological trivial phases, no states lie within the energy window $\Delta E$, leading to $G \approx 0$. The small, but non-zero value of $G$ \[see Fig. \[figf:figSI-2\](c)\] in the ${{\cal C}}=0$ phase at low bias arises from the non-zero electronic hopping $t_{{\rm tip}}$, and vanishes in the limit $t_{{\rm tip}}\rightarrow 0$ .
Real-Space Chern Number for Hybrid Systems {#sec:Chern}
==========================================
The topologically invariant Chern number ${{\cal C}}$ is conventionally computed in momentum space for a translationally invariant system using[@thouless-82prl405] $${{\cal C}}= \frac{1}{2\pi i} \int_{\rm BZ} {\rm tr}
\left\{ P_{\boldsymbol{k}} \left[ \partial_{k_1} P_{\boldsymbol{k}}, \partial_{k_2} P_{\boldsymbol{k}} \right] \right\} d\boldsymbol{k}
\label{eq:C_k}$$ where $P_{{\boldsymbol{k}}}$ is the $k$-decomposition of the projector $P$ onto the occupied states.
![Real-space Chern number (RSCN) as a function of $\mu$ for a fully covered $41\times 41$ Shiba lattice with periodic boundary conditions. Parameters used: $(\alpha, \Delta_s, J)=(0.2, 0.3, 0.5)\,t$ leading to $\mu_c^{(1)} = -0.4\,t$ and $\mu_c^{(2)}=-3.6\,t$.[]{data-label="figSI-3"}](SI_Fig3_V1){width="9cm"}
The question of how this expression needs to be modified for systems with a broken translational invariance, for example due to the presence of disorder, was first considered by Bellissard [*et al.*]{}[@bellissard-94jmp5373]. They derived a formulation of the Chern number in real space, which for a translationally invariant system, and in the thermodynamic limit, reproduces the results obtained from the momentum-space formulation in Eq.(\[eq:C\_k\]). Further progress was made by the pioneering work of Prodan [*et al.*]{}[@prodan-10prl115501; @prodan11jpa113001; @prodan17] who introduced an optimized real-space Chern number (RSCN), given by $$\begin{aligned}
{{\cal C}}=& 2\pi i
\sum_{\alpha} \langle 0,\alpha| P \left[-i [x_1,P], -i[x_2,P]\right]|0,\alpha\rangle
\label{eq:C_r}\end{aligned}$$ where $$\begin{aligned}
-i [x_i,P] = \sum_m c_m e^{-i m \boldsymbol{\Delta}_i \boldsymbol{x}} P e^{i m \boldsymbol{\Delta}_i \boldsymbol{x}}\ .\end{aligned}$$ Within this formulation, the Chern number converges exponentially fast to the thermodynamic limit, such that one obtains very good approximations to the thermodynamic limit already for small real-space systems with periodic boundary conditions. To demonstrate this, we consider a $(41 \times 41)$ Shiba lattice with periodic boundary conditions (this implies that the surface of the $s$-wave superconductor is fully covered by magnetic atoms), and present in Fig.\[figSI-3\] the resulting RSCN as a function of chemical potential $\mu$, for the same parameters as used in the main text. We see that the RSCN reproduces the expected quantization of the Chern number of the infinitely large system to high accuracy even for this rather small system size. As the formulation of RSCN assumes periodic boundary conditions, the question naturally arises of whether it can be applied to Shiba lattices with open boundary conditions (OBC), or to finite-size Shiba islands on the surface of an $s$-wave superconductor, even if the latter possesses periodic boundary conditions. In order to investigate the latter case, we plot in Fig.\[figSI-4\] the Chern number for a Shiba stripe (“stripe”, see inset), a Shiba island (“island”) and a fully covered system where an island of magnetic adatoms is missing (“hole”) as a function of coverage (the coverage is defined as the ratio of sites covered by magnetic adatoms and the total number of sites in the system). We find that for such finite magnetic islands, the use of periodic or open boundary conditions shows very little quantitative effect on the RSCN.
![Real-space Chern number (RSCN) as a function of coverage, the ratio of sites covered by magnetic adatoms and the total number of sites, for $\mu=0$ corresponding to the ${{\cal C}}=2$ phase. The lattice is a $41\times 41$ superconductor with periodic boundary conditions where a stripe of magnetic adatoms (blue) or an island of adatoms (red) is deposited. In addition, a covered system is considered where an island of adatoms is missing, yielding a hole (green). Parameters used: $(\alpha, \Delta_s, J)=(0.8, 1.2, 2.0)\,t$.[]{data-label="figSI-4"}](SI_Fig4_V1){width="12cm"}
For coverage between approximately 30 and 80 percent we obtain a linear dependence of the Chern number on coverage in all three cases. To understand this linear scaling, we note that the systems considered in Fig. \[figSI-4\] consist of a topologically non-trivial region (the magnetic Shiba island) with ${{\cal C}}\not= 0$ and a trivial region (the surrounding superconductor) with ${{\cal C}}=0$. Since the RSCN contains a summation over all lattice sites in the system (this can be seen when writing the projector $P_{{\boldsymbol{k}}}$ in real space), and not only a summation over sites that belong to the topological island, one can think of it as an averaged quantity: trivial regions yield a zero contribution to the Chern number, while non-trivial regions yield a finite contribution, resulting in the observed scaling of the RSCN with coverage of the non-trivial region. Therefore, to describe the topological nature of systems consisting of topologically trivial and non-trivial regions, we introduce a modified Chern number, $\tilde {{\cal C}}$, defined as $$\tilde {{\cal C}}\equiv \frac{{{\cal C}}}{{\rm coverage}}\ .$$ The modified Chern number is not a topological invariant in a strict sense, as it does not reach integer values as expected from an invariant. Nevertheless, the modified Chern number provides important insight into the topological phase of the Shiba islands, as follows from a plot of $\tilde {{\cal C}}$ as a function of $\mu$ shown in Fig.\[figSI-5\] for the Shiba island considered in the main text. This plot demonstrates that $\tilde {{\cal C}}$ (despite not being a strict topological invariant) retains features of a topological invariant: (i) it clearly distinguishes between phases with ${{\cal C}}\not= 0$ and ${{\cal C}}=0$. In particular, for those values of $\mu$ where for the Shiba lattice with PBC one finds ${{\cal C}}=0$, the Shiba island also possesses a RSCN that is strictly zero. (ii) The sign of $\tilde {{\cal C}}$ is in all cases in agreement with the PBC results. (iii) Even quantitatively $\tilde {{\cal C}}$ leads to reasonable results: for instance, for the Shiba island which would correspond to ${{\cal C}}=-1$ (${{\cal C}}=2$) in the thermodynamic limit, we find $\tilde {{\cal C}}\approx -0.8$ ($\tilde {{\cal C}}\approx 1.7$). We therefore conclude that the modified Chern number $\tilde {{\cal C}}$ is a valuable tool for the detection of topological phases.
![Real-space Chern number divided by coverage, $\tilde {{\cal C}}$, as a function of $\mu$ which reproduces the phase diagram of an infinitely large Shiba lattice[@li-16nc12297]. Results correspond to a Shiba island with a diameter of 30 atoms on a $41\times 41$ superconducting square lattice. In addition to the results for the Shiba island (red), we also present $\tilde {{\cal C}}={{\cal C}}$ for the fully covered system (coverage = 1) with periodic boundary conditions (black) and open boundary conditions (blue). Parameters used: $(\alpha, \Delta_s, J)=(0.2, 0.3, 0.5)\,t$.[]{data-label="figSI-5"}](SI_Fig5_V1){width="10cm"}
Lastly, we consider the effects of a finite-size Shiba lattices with open boundary conditions (in contrast to the periodic boundary conditions considered above). Here, we find that switching from periodic to open boundary conditions (while keeping the coverage at unity) leads to a suppression of the Chern number by approximately $25 \%$ for the considered system sizes and for all values of $\mu$, as shown in Fig. \[figSI-5\]. As mentioned before, for Shiba islands the differece between periodic and open boundary conditions is negligible.
Induced Spin-Triplet Correlations in the ${\boldsymbol{p}}$-wave Channel
========================================================================
![Induced superconducting triplet correlations $\langle c^\dagger_{{\bf r},\sigma} c^\dagger_{{\bf r^\prime},\sigma} \rangle$ for $\sigma =\uparrow, \downarrow$. The upper row shows the sign of the real part, the lower row the sign of the imaginary part of the correlations – positive (negative) sign is show in blue (red). The parameters are $(\alpha, \Delta_s, J) = (0.2, 0.3, 0.5)\,t$, and $\mu=-0.3t$ corresponding to the ${{\cal C}}=2$ phase. Shown are just quarters of the Shiba island with radius $R=15a_0$.[]{data-label="fig:fig3"}](SI_Fig6_V2){width="10cm"}
The combination of magnetic impurities, Rashba spin-orbit interaction, and $s$-wave superconductivity gives rise to the emergence of superconducting triplet correlations [@lutchyn-10prl077001; @oreg-10prl177002; @li-14prb235433]. To investigate the spatial form of these correlations, we consider the spin-triplet, equal-spin correlations on nearest-neighbor sites ${\bf r},{\bf r^\prime}$, as described by the correlation function $\langle c^\dagger_{{\bf r},\sigma} c^\dagger_{{\bf r^\prime},\sigma} \rangle$. These correlations describe superconducting pairing in the $p$-wave channel. In Fig. \[fig:fig3\] we present the sign of the real and imaginary parts of $\langle c^\dagger_{{\bf r},\sigma} c^\dagger_{{\bf r^\prime},\sigma} \rangle$ for $\sigma =\uparrow, \downarrow$ and $\mu=-0.3t$ inside the ${{\cal C}}=2$ phase. The correlations are predominantly real along the horizontal links, and imaginary along the vertical links, suggesting a $\pm p_x \pm i p_y$ orbital structure of the induced triplet correlations. A closer analysis of the relative sign of the real and imaginary parts reveals that the correlations inside the droplet are $-p_x + i p_y$ for $\langle c^\dagger_{{\bf r},\uparrow} c^\dagger_{{\bf r^\prime},\uparrow} \rangle$ and $p_x + i p_y$ for $\langle c^\dagger_{{\bf r},\downarrow} c^\dagger_{{\bf r^\prime},\downarrow} \rangle$, which reflect, as expected, the broken time-reversal symmetry of the system due to the presence of magnetic defects. However, outside the magnetic island, the correlations are $p_x - i p_y$ for $\langle c^\dagger_{{\bf r},\uparrow} c^\dagger_{{\bf r^\prime},\uparrow} \rangle$ and $p_x + i p_y$ for $\langle c^\dagger_{{\bf r},\downarrow} c^\dagger_{{\bf r^\prime},\downarrow} \rangle$, which preserve the system’s time-reversal symmetry. This implies that the nature of the induced triplet correlations changes between the interior and exterior of the droplet. We find that these relations between the signs of the induced triplet correlations hold both for the topologically trivial and non-trivial phases, despite the fact that in the topologically trivial phase there are no edge modes. The ${{\cal C}}=0$ phase therefore represents an example of a system that exhibits superconducting (chiral) triplet ($p$-wave) correlations, but no corresponding edge modes.
Results for Ribbon Geometry {#sec:ribbon}
===========================
One of the main objectives of this article is to predict which topological properties of Shiba lattices will persist down to small nanoscopic Shiba islands. To extrapolate between Shiba lattices with periodic boundary conditions (such as the one discussed in Fig. \[figSI-3\]) and finite-size Shiba islands with open boundary conditions, it is instructive to consider Shiba nano-ribbons – systems with cylinder geometry – which can be thought of a system which has PBC imposed along one and OBC imposed along another direction. Formally, one performs a Fourier transformation along the $x$ direction but remains in real space regarding the $y$ coordinate. Energy spectra can then be plotted with respect to the momentum quantum number $k_x \equiv k$. Moreover, due to OBC along the $y$ direction the system possesses edges and carries thus ${{\cal C}}$ edge modes (due to bulk-boundary correspondence), which can be studied as a function of $k$. Note that a ribbon carries $2{{\cal C}}$ edge modes, ${{\cal C}}$ per edge, while a system with OBC in $x$ and $y$ directions such as the Shiba island possesses only ${{\cal C}}$ edge modes.
The relation between the cylinder spectra shown in Fig.\[figSI-cyl\] for two different parameter sets and different topological phases and the corresponding Shiba island LDOS plots \[Fig.1 of the main paper\] is obvious, as one simply has to project all energy levels at different $k$ values onto each other in order to obtain the global (spatially integrated) DOS. The number of energy levels or energy peaks in the LDOS depends on the number of lattice sites. For the cylinder spectra $k$ is a free parameter and we can choose arbitrary discretizations, the spectra in Fig.\[figSI-cyl\] are shown for 150 $k$ values. Fig.\[figSI-cyl\] discloses an interesting detail: for the ${{\cal C}}=2$ phase, the two chiral edge modes are at different wave vectors. Of course the distinction via wavevectors becomes useless for the Shiba islands, but we see that the naive picture that the ${{\cal C}}>1$ chiral edge modes are like identical copies on top of each other is by no means justified. For the LDOS plots of the Shiba island we should keep in mind, that energy levels do not necessarily need to come in pairs for ${{\cal C}}=2$ and that the two dispersive Majorana modes might behave differently, in particular when the island shape is not symmetric (see Sec.\[sec:disorder\]) or dirt and imperfections are present such as in realistic situations.
![Cylinder spectra on a ribbon consisting of 100 unit cells. Top row: $(\alpha, \Delta_s, J)=(0.2, 0.3, 0.5)\,t$. Bottom row: $(\alpha, \Delta_s, J)=(0.8, 1.2, 2.0)\,t$. Systems with Chern numbers ${{\cal C}}=-1$, $0$, and $2$ correspond to $\mu/t=-4$, $-2$, and $0$, respectively.[]{data-label="figSI-cyl"}](SI_Fig8_V1)
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---
author:
- |
Maria Han Veiga, and Philipp Öffner, and Davide Torlo\
Institute of Mathematics, University of Zurich, Switzerland\
Michigan Institute of Data Science, University of Michigan, USA
bibliography:
- 'literature.bib'
title: 'DeC and ADER: Similarities, Differences and a Unified Framework'
---
Acknowledgements {#acknowledgements .unnumbered}
================
P. Öffner has been funded by the UZH Postdoc Grant. Davide Torlo is supported by ITN ModCompShock project funded by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 642768. M. Han Veiga acknowledges financial support from MIDAS.
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---
abstract: |
In this effort we exactly solve the fractional diffusion-advection equation for solar cosmic-ray transport proposed in [@LE2014] and give its [*general solution*]{} in terms of hypergeometric distributions. Also, we regain all the results and approximations given in [@LE2014] as [*particular cases*]{} of our general solution.
Keywords:
author:
- |
M.C.Rocca$^{1,2}$,A.R.Plastino$^{3}$,\
A.Plastino$^{1,2}$,A.L.De Paoli$^{1,2}$\
\
,\
\
\
\
\
\
title: '[**General Solution of a Fractional Diffusion-Advection Equation for Solar Cosmic-Ray Transport**]{}'
---
Introduction
============
There is a considerable body of evidence, from data collected by spacecrafts like [*Ulysses*]{} and [*Vayager 2*]{}, indicating that the transport of energetic particles in the turbulent heliospheric medium is superdiffusive [@PZ2007; @PZ2009]. Considerable effort has been devoted in recent years to the development of superdiffusive models for the transport of electrons and protons in the heliosphere [@SS2011; @TZ2011; @ZP2013]. This kind of transport regime exhibits a power-law growth of the mean square displacement of the diffusing particles, $\langle \Delta x^2 \rangle \propto
t^{\alpha}$, with $\alpha > 1$ (see, for instance, [@SZ97]). The special case $\alpha = 2$ is called ballistic transport. The limit case $\alpha \to 1$ corresponds to normal diffusion, described by the well-known Gaussian propagator. The energetic particles detected by the aforementioned probes are usually associated with violent solar events like solar flares. These particles diffuse in the solar wind, which is a turbulent environment than can be assumed statistically homogeneous at large enough distances from the sun [@PZ2007]. This implies that the propagator $P(x,x',t,t')$, describing the probability of finding at the space time location $(x,t)$ a particle that has been injected at $(x',t')$, depends solely on the differences $x-x'$ and $t-t'$. In the superdiffusive regime the propagator $P(x,x',t,t')$ is not Gaussian, and exhibits power-law tails. It arises as solution a non local diffusive process governed by an integral equation that can be recast under the guise of a diffusion equation where the well-known Laplacian term is replaced by a term involving fractional derivatives [@C95]. Diffusion equations with fractional derivatives have attracted considerable attention recently (see [@LTRLGS2014; @LSSEL2009; @RLEL2007; @stern; @LMASL2005] and references therein) and have lots of potential applications [@MK2000; @P2013]. In particular, the observed distributions of solar cosmic ray particles are often consistent with power-law tails, suggesting that a superdiffusive process is at work.
A proper understanding of the transport of energetic particles in space is a vital ingredient for the analysis of various important phenomena, such as the propagation of particles from the Sun to our planet or, more generally, the acceleration and transport of cosmic rays. The superdiffusion of particles in interplanetary turbulent environments is often modelled using asymptotic expressions for the pertinent non-Gaussian propagator, which have a limited range of validity. A first step towards a more accurate analytical treatment of this problem was recently provided by Litvinenko and Effenberger (LE) in [@LE2014]. LE considered solutions of a fractional diffusion-advection equation describing the diffusion of particles emitted at a shock front that propagates at a constant upstream speed $V_{sh}$ in the solar wind rest frame. The shock front is assumed to be planar, leading to an effectively one-dimensional problem. Each physical quantity depends only on the time $t$ and on the spatial coordinate $x$ measured along an axis perpendicular to the shock front. In the present contribution we re-visit the fractional diffusion-advection equation previously studied by LE, providing a closed analytical solution.
Formulation of the Problem
==========================
The authors of [@LE2014] advanced the equation $$\label{ep2.1}
\frac {\partial f} {\partial t}=\kappa\frac
{{\partial}^{\alpha} f} {\partial |x|^{\alpha}}+a\frac {\partial
f} {\partial x}+\delta(x),$$ where $t>0$ and $f(x,t)$ is the distribution function for solar cosmic-rays transport. Here the fractional spatial derivative is defined as $$\label{ep2.2}
\frac {{\partial}^{\alpha} f} {\partial
|x|^{\alpha}}= \frac {1} {\pi}\sin\left(\frac {\pi\alpha}
{2}\right)\Gamma(\alpha+1) \int\limits_0^{\infty}\frac
{f(x+\xi)-2f(x)+f(x-\xi)} {{\xi}^{\alpha+1}}\;d\xi.$$ (See [@LE2014] and references therein).
To solve this equation the authors use the Green function governed by the equation: $$\label{ep2.3}
\frac {\partial {\cal G}} {\partial t}=\kappa\frac
{{\partial}^{\alpha} {\cal G}} {\partial
|x|^{\alpha}}+\delta(x)\delta(t).$$ With this Green function, the solution of (\[ep2.1\]) can be expressed as $$\label{ep2.4}
f(x,t)=\int\limits_0^t {\cal G}(x+at^{'},
t^{'})\;dt^{'}.$$ In this work we obtain the solutions of Eqs. (\[ep2.1\]) and (\[ep2.3\]) using distributions as main tools [@guelfand1]. Also, we re-obtain all results and approximations obtained in [@LE2014], but as particular cases of our general solutions of (\[ep2.1\]) and (\[ep2.2\]).
For our task we use, as a first step, the solution obtained in [@LE2014] for the Green function through the use of the Fourier Transform given by $$\label{ep2.5}
\hat{{\cal G}}(k,t)=\frac {1} {2\pi}
\int\limits_{-\infty}^{\infty}{\cal G}(x,t) e^{-ikx}\;dx,$$ from which we obtain for $\hat{{\cal G}}$: $$\label{ep2.6}
\hat{{\cal G}}(k,t)=-\kappa|k|^{\alpha}\hat{{\cal
G}}(k,t)+\frac {1} {2\pi}\delta(t),$$ whose solution is $$\label{ep2.7}
\hat{{\cal G}}(k,t)=\frac {H(t)} {2\pi} e^{-\kappa
|k|^{\alpha}t},$$ where $H(t)$ is the Heaviside’s step function.
General Solution of the Equations
=================================
From (\[ep2.7\]) we have for $\hat{{\cal G}}$ $$\label{ep3.1}
\hat{{\cal G}}(k,t)=\frac {H(t)} {2\pi} e^{-\kappa
|k|^{\alpha}t}= \frac {H(t)} {2\pi}
\sum\limits_{n=0}^{\infty}\frac {(-1)^n{\kappa}^nk^{\alpha n} t^n}
{n!},$$ and, invoking the inverse Fourier transform $${\cal G}(x,t)=\frac {H(t)} {2\pi}
\int\limits_{-\infty}^{\infty}
e^{-\kappa |k|^{\alpha}t}e^{ikx}\;dk=$$ $$\label{ep3.2}
\frac {H(t)} {2\pi} \sum\limits_{n=0}^{\infty}\frac
{(-1)^n{\kappa}^n t^n} {n!} \left[\int\limits_0^{\infty}k^{\alpha
n}e^{ikx}\;dx + \int\limits_0^{\infty}k^{\alpha
n}e^{-ikx}\;dx\right].$$ Fortunately, we can find in the classical book of [@guelfand1] the results for the two integrals of (\[ep3.2\]). We obtain $$\label{ep3.3}
{\cal G}(x,t)=\frac {H(t)} {2\pi}
\sum\limits_{n=0}^{\infty}\frac {(-1)^n{\kappa}^n t^n} {n!}
\Gamma(\alpha n+1) \left[\frac {e^{i\frac {\pi} {2}(\alpha n+1)}}
{(x+i0)^{\alpha n + 1}} + \frac {e^{-i\frac {\pi} {2}(\alpha
n+1)}} {(x-i0)^{\alpha n + 1}} \right].$$ Using now (\[ep2.4\]) we have for $f$ $$f(x,t)=\int\limits_0^t
{\cal G}(x+at^{'}, t^{'})\;dt^{'},$$ so that one can write $$f(x,t)=\frac {1} {2\pi}
\sum\limits_{n=0}^{\infty}\frac {(-1)^n{\kappa}^n} {n!}
\Gamma(\alpha n+1)\times$$ $$\label{ep3.4}
\int\limits_0^t \left[\frac {e^{i\frac {\pi}
{2}(\alpha n+1)}} {(x+at^{'}+i0)^{\alpha n + 1}} + \frac
{e^{-i\frac {\pi} {2}(\alpha n+1)}} {(x+at^{'}-i0)^{\alpha n + 1}}
\right]t^{'n}\;dt^{'}.$$ According to Eq. (\[a1\]) of the Appendix, where $t>0$, we now obtain for $f$, invoking hypergeometric functions $F(\alpha n+1,2;3;z)$ and Beta functions ${\cal B}(1,n+1)$, $$f(x,t)=\frac {1} {2\pi}\sum\limits_{n=0}^{\infty}
\frac {(-1)^n{\kappa}^nt^{n+1}} {n!}\Gamma(\alpha n+1)
{\cal B}(1,n+1)\times$$ $$\left[\frac {e^{i\frac {\pi} {2}(\alpha n + 1)}}
{(x+i0)^{\alpha n + 1}}
F\left(\alpha n+1,n+1;n+2;-\frac {at} {x+i0}\right)+\right.$$ $$\label{ep3.5}
\left.\frac {e^{-i\frac {\pi} {2}(\alpha n + 1)}}
{(x-i0)^{\alpha n + 1}}
F\left(\alpha n+1,n+1;n+2;-\frac {at} {x-i0}\right)\right].$$ This is the general solution of Eq.(\[ep2.1\]) for the initial condition $f(x,0)=0$.
In the next section we will see that all results and approximations obtained in [@LE2014] can be regarded as particular cases of the general solution (\[ep3.5\]).
Weak Diffusion Approximation
============================
Following LE, we shall now consider a weak diffusion approximation. Within this approximation we can treat $\kappa$ as a small parameter and develop $f$ up to order one [@LE2014]. Thus, we can write: $$\label{ep4.1}
f(x,t)=f_0(x,t)+f_1(x,t),$$ where i) $f_0$ is independent of $\kappa$ and ii) in $f_1$ the corresponding power of $\kappa$ is unity.
Eq. (\[ep3.5\]) entails that we have, for $f_0$ ($n=0$ in (\[ep3.5\])), $$f_0(x,t)=\frac {it} {2\pi}\left[
(x+i0)^{-1}F\left(1,1;2;-\frac {at} {x+i0}\right)\right.-$$ $$\label{ep4.2}
\left.(x-i0)^{-1}F\left(1,1;2;-\frac {at}
{x-i0}\right)\right].$$ Recourse to the celebrated Tables of [@gra1] allows us to write $$\label{ep4.3}
F(1,1;2;-z)=\frac {1} {z}\ln(1+z),$$ and we obtain for $f_0$ $$\label{ep4.4}
f_0(x,t)=\frac {1} {a}[H(-x)-H(-x-at)=\frac {1} {2a}
[Sgn(x+at)-Sgn(x)].$$ When we take $n=1$ in (\[ep3.5\]), $f_1$ is defined as $$f_1(x,t)=-\frac {i\kappa t^2} {4\pi}\Gamma(\alpha+1)\left[
\frac {e^{i\frac {\pi} {2}}} {(x+i0)^{\alpha+1}}
F\left(\alpha+1,2;3;-\frac {at} {x+i0}\right)\right.$$ $$\label{ep4.5}
+\left.\frac {e^{-i\frac {\pi} {2}}}
{(x-i0)^{\alpha+1}} F\left(\alpha+1,2;3;-\frac {at}
{x-i0}\right)\right].$$ Now, from (\[a1\]) of Appendix we have, for the hypergeometric function, $$F(\alpha+1,2;3;z)=\frac {2} {\alpha(\alpha-1)z^2}
\left[1+\frac {\alpha z-1} {(1-z)^{\alpha}}\right],$$ so that, using this result, $f_1$ adopts the form $$f_1(x,t)=\frac {i\kappa\Gamma(\alpha-1)} {2\pi a^2}
\left\{(x+\alpha at)
\left[\frac {e^{i\frac {\pi} {2}\alpha}} {(x+at+i0)^{\alpha}}-
\frac {e^{-i\frac {\pi} {2}\alpha}} {(x+at-i0)^{\alpha}}\right]+
\right.$$ $$\label{ep4.6}
\left.\frac {e^{-i\frac {\pi} {2}\alpha}}
{(x-i0)^{\alpha-1}}- \frac {e^{i\frac {\pi} {2}\alpha}}
{(x+i0)^{\alpha-1}}\right\}.$$ Using at this point (\[ep4.1\]), (\[ep4.4\]), and (\[ep4.6\]), the final result for $f$, up to first order in $\kappa$, reads, invoking the sign function $Sgn(x)$, $$f(x,t)=\frac {1} {2a}[Sgn(x+at)-Sgn(x)]+$$ $$\frac {i\kappa\Gamma(\alpha-1)} {2\pi a^2}
\left\{(x+\alpha at)
\left[\frac {e^{i\frac {\pi} {2}\alpha}} {(x+at+i0)^{\alpha}}-
\frac {e^{-i\frac {\pi} {2}\alpha}} {(x+at-i0)^{\alpha}}\right]+
\right.$$ $$\label{ep4.7}
\left.\frac {e^{-i\frac {\pi} {2}\alpha}}
{(x-i0)^{\alpha-1}}- \frac {e^{i\frac {\pi} {2}\alpha}}
{(x+i0)^{\alpha-1}}\right\}.$$ From this expression for $f$, we will obtain all approximate results reported in [@LE2014]. Thus, for $x>0$ (\[ep4.7\]) becomes $$\label{ep4.8}
f(x,t)=\frac {\kappa\sin(\frac {\pi\alpha}
{2})\Gamma(\alpha-1)} {\pi a^2} \left[\frac {1} {x^{\alpha-1}}-
\frac {x+\alpha at} {(x+at)^{\alpha}}\right].$$ We have to distinguish two limiting cases. The first one is the asymptotic situation $x>>at$. In this case, $$\label{ep4.9}
f(x,t)=\frac {1} {2\pi} \sin(\frac {\pi\alpha} {2})
\Gamma(\alpha+1)\frac {\kappa t^2} {x^{\alpha+1}}.$$ The second case limiting case is $0<x<<at$. The corresponding expression for $f$ becomes $$\label{ep4.10}
f(x,t)=\frac {1} {\pi} \sin(\frac {\pi\alpha} {2})
\Gamma(\alpha-1)\frac {\kappa} {a^2}x^{1-\alpha}.$$
We consider signs now. When $x+at<0$, from (\[ep4.7\]) we have $$\label{ep4.11}
f(x,t)=\frac {\kappa\sin(\frac {\pi\alpha} {2})\Gamma(\alpha-1)}
{\pi a^2} \left[\frac {1} {|x|^{\alpha-1}}+
\frac {x+\alpha at} {|x+at|^{\alpha}}\right]$$ Again, two special cases must be considered. One is for $x<<-at$ for which $$\label{ep4.12}
f(x,t)=\frac {1} {2\pi} \sin(\frac {\pi\alpha} {2})
\Gamma(\alpha+1)\frac {\kappa t^2} {|x|^{\alpha+1}}.$$ The other special situation is $x<0$, $x+at>0$, $x>>-at$. Here,
$$\label{ep4.13}
f(x,t)=\frac {1} {a}+\frac {1} {\pi} \sin(\frac
{\pi\alpha} {2}) \Gamma(\alpha-1)\frac {\kappa}
{a^2}|x|^{1-\alpha}.$$
At this stage, we have re-obtained all approximations given in [@LE2014], but using a more general procedure. More specifically, all approximations have been obtained from only one relation: Eq. (\[ep4.7\]), which, in turn, is deduced from our general formula (\[ep3.5\]).
Change of Variables
===================
We assume that in the solar wind rest frame the particles’ transport is described by the fractional-diffusion equation with no advection term (that is, with $a=0$ in (\[ep2.1\])). The shock front (which started at $x_0 = -V_{sh} t_0$, moves with constant speed $V_{sh}$, and is regarded as highly localized in the $x$-coordinate) constitutes the source of the particles. Consequently, we have a fractional-diffusion equation with a uniformly moving Dirac’s delta source of the form $\delta(x -
V_{sh} t)$. In order to have a stationary delta source we need to perform an appropriate change of coordinates, re-casting our problem in a reference frame where the shock front is stationary. We also change the origin of time so that the source starts being active at $t=0$. In this new reference frame the transport equation has an advection term with advection velocity $a=
V_{sh}$, and a stationary source $\delta (0)$ that starts at $t=0$. After solving the diffusion-advection equation in this new frame (which is what we have done in the previous sections) we re-express the solution in terms of the original coordinates associated with the solar wind rest frame. This last step is succinctly described by the three correspondences $a\rightarrow
v_{sh}$, $t\rightarrow t+t_0$, and $x\rightarrow x-v_{sh}t$, after which Eq. (\[ep3.5\]) adopts the appearance $$f(x,t)=\frac {1} {2\pi}\sum\limits_{n=0}^{\infty}
\frac {(-1)^n{\kappa}^n(t+t_0)^{n+1}} {n!}\Gamma(\alpha n+1)
{\cal B}(1,n+1)\times$$ $$\left[\frac {e^{i\frac {\pi} {2}(\alpha n + 1)}}
{(x-v_{sh}t+i0)^{\alpha n + 1}}
F\left(\alpha n+1,n+1;n+2;-\frac {v_{sh}(t+t_0)} {x-v_{sh}t+i0}\right)+\right.$$ $$\label{ep5.1}
\left.\frac {e^{-i\frac {\pi} {2}(\alpha n + 1)}}
{(x-v_{sh}t-i0)^{\alpha n + 1}} F\left(\alpha n+1,n+1;n+2;-\frac
{v_{sh}(t+t_0)} {x-v_{sh}t-i0}\right)\right].$$ Accordingly, in the weak diffusion approximation (\[ep4.7\]) we have $$f(x,t)=\frac {1} {2v_{sh}}
[Sgn(x+v_{sh}t_0)-Sgn(x-v_{sh}t)]+$$ $$\frac {i\kappa\Gamma(\alpha-1)} {2\pi V_{sh}^2}
\left\{(x+(\alpha-1) v_{sh}t+v_{sh}t_0)
\left[\frac {e^{i\frac {\pi} {2}\alpha}} {(x+v_{sh}t_0+i0)^{\alpha}}-
\right.\right.-$$ $$\label{ep5.2}
\left.\left.\frac {e^{-i\frac {\pi} {2}\alpha}}
{(x+v_{sh}t_0-i0)^{\alpha}}\right]
+\frac {e^{-i\frac {\pi} {2}\alpha}} {(x-v_{sh}t-i0)^{\alpha}}-
\frac {e^{i\frac {\pi} {2}\alpha}} {(x-v_{sh}t+i0)^{\alpha}}\right\}$$
Conclusions
===========
We have provided an explicit analytical solution for and advection-diffusion equation involving fractional derivatives in the diffusion term. This equation governs the diffusion of particles in the solar wind injected at the front of a shock that travels at a constant upstream speed $V_{sh}$ in the solar wind rest frame. The shock is assumed to have a planar front, leading to a problem with an effective one dimensional geometry, where all the relevant quantities depend solely on time and on the coordinate $x$ measured along an axis perpendicular to the front.
We obtained the exact solution of the above mentioned equation in the $x$-configuration space (besides the associated formal solution in the $k$-space related to the previous one via a Fourier transform). Our solution allow us to obtain in a unified and systematic way all the relevant approximations that were previously discussed by LE, each one in a separated way.
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Appendix:Some properties of Hypergeometric Function {#appendixsome-properties-of-hypergeometric-function .unnumbered}
===================================================
Using data from [@gra2] we have $$\int\limits_0^t\frac {t^{'n}} {(x+at^{'}\pm i0)^{\alpha n +1}}\;dt^{'}=
\frac {t^{n+1}} {(x\pm i0)^{\alpha n+1}}B(1,n+1)\times$$ $$\label{a1}
F\left(\alpha n +1, n+1,n+2;-\frac {at} {x\pm
i0}\right).$$ Now we appeal to the transformation formula given in [@gra3] for the hypergeometric function $$F(\alpha+1,2;3;z)=\frac {2\Gamma(1-\alpha)} {\Gamma(2-\alpha)}
(-1)^{\alpha+1}z^{-\alpha-1}
F\left(\alpha+1,\alpha-1;\alpha;\frac {1} {z}\right)+$$ $$\label{a2}
\frac {2\Gamma(\alpha-1)} {\Gamma(\alpha+1)} z^{-2}
F\left(2,0;2-\alpha;\frac {1} {z}\right),$$ with $$\label{a3}
F(a,0;c;z)=1.$$ In these circumstance we obtain $$F(\alpha+1,2;3;z)=\frac {2\Gamma(1-\alpha)} {\Gamma(2-\alpha)}
(-1)^{\alpha+1}z^{-\alpha-1}
F\left(\alpha+1,\alpha-1;\alpha;\frac {1} {z}\right)+$$ $$\label{a4}
\frac {2\Gamma(\alpha-1)} {\Gamma(\alpha+1)}. z^{-2}$$ Now we invoke the transformation formula [@gra4] $$\label{a5}
F\left(\alpha+1,\alpha-1;\alpha;\frac {1} {z}\right)=
\left(1-\frac {1} {z}\right)^{-\alpha}F\left(-1,1;\alpha;\frac {1}
{z}\right),$$ or $$\label{a6}
F\left(\alpha+1,\alpha-1;\alpha;\frac {1} {z}\right)=
\frac {z^{\alpha}} {(z-1)^{\alpha}}\left(\frac {\alpha z-1}
{\alpha z}\right).$$ At this stage, we have, finally, $$\label{a7}
F(\alpha+1,2;3;z)=\frac {2} {\alpha(\alpha-1)z^2} \left[1+\frac
{\alpha z-1} {(1-z)^{\alpha}}\right].$$
|
---
author:
- 'Bernard Lidický[^1]'
- 'Kacy Messerschmidt[^2]'
- 'Riste Škrekovski[^3]'
bibliography:
- 'MainBase.bib'
title: 'Facial unique-maximum colorings of plane graphs with restriction on big vertices'
---
Introduction
============
In this paper, we consider simple graphs only. A graph is *planar* if it can be drawn in the plane so that no edges cross and a graph is called *plane* if it is drawn in such a way. A *coloring* is an assignment of colors to the vertices of a graph. For the purposes of this paper, we will use integers to represent colors. A coloring is *proper* if no two adjacent vertices are assigned the same color. In [@FabGor16], Fabrici and Göring proposed a new type of coloring. A *facial unique maximum coloring* (or *FUM-coloring*) is a proper coloring using positive integers such that each face $f$ has only one incident vertex that receives the maximum color on $f$. For a graph $G$, the minimum number of colors required for a FUM-coloring of $G$ is called the *FUM-chromatic number* of $G$, and is denoted ${\chi_{\mathrm{fum}}}(G)$.
One of the most important theorems in graph theory is the Four Color Theorem, which states that any planar graph has a proper coloring using at most four colors. Fabrici and Göring [@FabGor16] proposed the following strengthening of the Four Color Theorem.
\[conj:plane4\] Given any plane graph $G$, ${\chi_{\mathrm{fum}}}(G) \leq 4$.
This conjecture was disproven in the general case by the authors [@LidMesSkr17]. Fabrici and Göring [@FabGor16] proved that for any plane graph $G$, ${\chi_{\mathrm{fum}}}(G) \leq 6$, while Wendland [@Wen16] improved the upper bound to $5$. Andova, Lidický, Lužar, and Škrekovski [@AndLidLuzSkr17] proved that if $G$ is a subcubic or outerplane graph, ${\chi_{\mathrm{fum}}}(G) \leq 4$.
Recall that a *star* is a connected graph with at most one vertex with degree greater than 1 and a *star forest* is graph consisting of disjoint stars. The main result of this paper is the following strengthening of the subcubic result.
\[thm:starforest\] Let $G$ be a plane graph and $X = \{ v \in V(G) : d(v) \geq 4 \}$. If $G[X]$ is a star forest, then ${\chi_{\mathrm{fum}}}(G) \leq 4$.
We present the proof of Theorem \[thm:starforest\] in the next section. We use the precoloring extension method that was introduced by Thomassen [@Tho94] to show planar graphs are 5-choosable. This approach could give even a stronger result with distant crossings [@bib-DLM]. It could be also used for the list-coloring version [@bib-DLS; @bib-thomass3] of Grötzsch theorem, for example. The approach works by a clever induction. The proof is actually a stronger statement, which allows the induction to work. We state the stronger version of Theorem \[thm:starforest\] in the next section.
Proof of the main result
========================
We prove Theorem \[thm:starforest\] as a corollary of a slightly stronger result stated in Lemma \[lem:precolor\].
\[lem:precolor\] Let $G$ be a plane graph with a path $P$ on the outer face with at most two vertices. Let $X = \{ v \in V(G) : d (v) \geq 4 \} \cup \{ v \in V(P) : d (v) = 3 \}$ such that $G[X]$ is a star forest. Given any proper coloring $c$ of the vertices in $P$ using colors $\{ 1, 2, 3 \}$, there exists an extension of $c$ to $G$ using colors $\{ 1, 2, 3, 4 \}$ such that color $4$ does not appear on the outer face and all internal faces have a unique maximum color.
Suppose for contradiction that $G$ is a counterexample of minimum order. Let $C$ be the boundary walk of the outer face of $G$.
First we introduce a claim, which we refer to repeatedly when using the minimality of $G$. When we use the minimality of $G$, we color some subgraph $H$ or $G$. However, precoloring a vertex of degree 3 in $H$ may ruin the property that $X$ in $H$ is a star forest. The purpose of the following claim is that a vertex can be a newly precolored vertex in $H$ as long as it has degree at least 4 in $H$ or its degree in $G$ is strictly bigger than in $H$.
\[claim:subgraph\] Let $H$ be a proper subgraph of $G$ and $P'$ be a path on the outer face of $H$ with at most two vertices such that $d_H (v) < d_G (v)$ for all $v \in V(P') \setminus V(P)$ if $d_H(v) = 3$. Then any proper coloring $c'$ of the vertices in $P'$ using colors $\{ 1, 2, 3 \}$ can be extended to a coloring of $H$ using colors $\{ 1, 2, 3, 4 \}$ such that color $4$ does not appear on the outer face and all internal faces have a unique maximum color.
Let $X' = \{ v \in V(H) : d_H (v) \geq 4 \} \cup \{ v \in V(P') : d_H (v) = 3 \}$. If $X'$ is a star forest, then $H$ can be colored by the minimality of $G$. Suppose for contradiction that $X'$ is not a star forest. Then there must be a vertex $v$ that is in $X'$ but not in $X$. Since $d_H (v) \leq d_G (v)$, we cannot have $d_H (v) \geq 4$, otherwise $d_G (v) \geq 4$ and thus $v$ would be in $X$. So $v$ must be in the set $\{ v \in V(P') : d_H (v) = 3 \}$. This implies that either $d_G (v) \geq 4$ or $v \in P$. In either case $v \in X$, which is a contradiction.
\[claim:Fcycle\] $C$ is a cycle.
First note that $G$ has only one connected component incident to the outer face, otherwise Claim \[claim:subgraph\] could be used on each such component of $G$ separately. Note that if $G$ has no internal faces, then it is a tree and any proper coloring using $\{1,2,3\}$ works.
If every vertex in the outer face is incident to exactly two edges of the outer face, $C$ is a cycle. So suppose for contradiction that $G$ has a vertex $v$ incident with at least three edges in the outer face. Notice that $v$ is a cut vertex. Let $Y$ be the set of vertices consisting of $v$ and the vertices of the connected component of $G - v$ that intersects $P$, if such a component exists. If no component of $G-v$ intersects $P$, pick an arbitrary one. Let $Y'$ be the union of $Y$ and the set of all vertices that are drawn in the interior faces of $G[Y]$ in $G$. By the minimality of $G$, there exists a coloring $c_{Y'}$ of $G[Y']$. Let $Z = (V(G) \setminus Y') \cup \{ v \}$. Since $d_{G[Z]} (v) < d_G (v)$, by Claim \[claim:subgraph\] a precoloring of $v$ with $c_{Y'} (v)$ can be extended to a coloring $c_Z$ of $G[Z]$. Since $v$ has the same color in $c_{Y'}$ and $c_Z$, we can combine these two colorings into a coloring $c$ of $G$, a contradiction.
\[claim:chord\] $C$ does not have any chords.
Suppose for contradiction that $C$ has a chord $uv$. Let $Y$ be the set of vertices consisting of $u$ and $v$ and the vertices of the connected component of $G - \{ u, v \}$ that intersects $P$, if such a component exists. If no component of $G-\{u,v\}$ intersects $P$, pick an arbitrary one. Let $Y'$ be the union of $Y$ and the set of all vertices that are drawn in the interior faces of $G[Y]$ in $G$. By the minimality of $G$, there exists a coloring $c_{Y'}$ of $G[Y']$. Let $Z = (V(G) \setminus Y') \cup \{ u, v \}$. Since $d_{G[Z]} (v) < d_G (v)$ and $d_{G[Z]} (u) < d_G (u)$, by Claim \[claim:subgraph\] a precoloring of $u$ with $c_{Y'} (u)$ and $v$ with $c_{Y'} (v)$ can be extended to a coloring $c_Z$ of $G[Z]$. Since each $u$ and $v$ have the same color in $c_{Y'}$ and $c_Z$, we can combine these two colorings into a coloring $c$ of $G$, a contradiction. Therefore, $C$ does not have any chords.
\[claim:cycle\] $G$ is not a cycle.
Suppose for contradiction that $G$ is a cycle. If a vertex in $P$ receives color $3$, then color the rest of $G$ alternately with colors $1$ and $2$. Otherwise, pick a vertex not in $P$ to receive color $3$, then color the rest of the vertices alternately with $1$ and $2$. The interior face has only one vertex that receives color $3$, hence all interior faces have a unique maximum color, which is a contradiction.
\[claim:23-vertex\] $C - P$ does not contain a $2$- or $3$-vertex.
![A non-precolored $2$- or $3$-vertex $v$ on the outer face of $G$[]{data-label="fig:23-vertex"}](fig-23-vertex.pdf)
Suppose for contradiction that $C - P$ contains a $2$- or $3$-vertex $v$. If $v$ is a $2$-vertex, by Claims \[claim:chord\] and \[claim:cycle\] there must be a vertex $u$ that is on the outer face of $G - v$ but not in $C$, as shown in Figure \[fig:23-vertex\]. If $v$ is a $3$-vertex, let $u$ be the neighbor of $v$ outside of $C$. Let $Y = V(G) \setminus \{ u, v \}$. By the minimality of $G$, we can color $Y$ such that every internal face of $G[Y]$ has a unique maximum color and the vertices in the outer face receive colors from $\{1,2,3\}$. Coloring $u$ with $4$ gives a unique maximum color to every face incident with $u$. We now color $v$ with the color from $\{ 1, 2, 3 \}$ not used on either of its two neighbors in $C$ to complete the coloring and arrive at a contradiction.
By Claim \[claim:Fcycle\], $C$ cannot have any $1$-vertices. By Claim \[claim:23-vertex\], $C - P$ cannot have $2$- or $3$-vertices either. Hence each vertex in $C - P$ must have degree at least $4$. Since $V(C) \backslash V(P) \subseteq X$, $C - P$ can contain at most three vertices. Moreover, since $X$ is acyclic, $P$ is nonempty and contains at least one vertex of degree two.
\[claim:22444\] $C - P$ does not contain three vertices.
![An outer face consisting of two precolored vertices and three non-precolored vertices of degree at least $4$[]{data-label="fig:22444"}](fig-22444.pdf)
Suppose for contradiction that $C - P$ contains three vertices. Since neither of the vertices in $P$ can belong to $X$, $P$ must consist of one or two $2$-vertices, as shown in Figure \[fig:22444\]. By Claims \[claim:chord\] and \[claim:cycle\], there must exist a vertex $u$ that is on the outer face of $G - P$ but not in $C$. Let $Y = V(G) \setminus \{ p_1, p_2, u \}$. Color $v_2$ with the color used on $p_2$ and color $v_3$ with the color from $\{ 1, 2, 3 \}$ not used on $p_1$ or $v_2$. By Claim \[claim:subgraph\], we can extend the precoloring on $\{ v_2, v_3 \}$ to all of $G[Y]$ such that every internal face of $G[Y]$ has a unique maximum color. Since $v_2$ received the same color as $p_2$, the color that $v_1$ receives will not conflict with $p_2$ in $G$. Coloring $u$ with $4$ gives a unique maximum color to every face incident with $u$, thus completing the coloring of $G$ and producing a contradiction.
Notice that Claim \[claim:22444\] implies that $C$ is either a 4-cycle or a triangle.
\[claim:2-verticesP\] $P$ does not consist of two $2$-vertices.
![A precolored path $P$ consisting of two $2$-vertices[]{data-label="fig:2-verticesP"}](fig-2-verticesP.pdf)
Suppose for contradiction that $P$ consists of two $2$-vertices $p_1$ and $p_2$. Let $v_1$ and $v_2$ be the neighbors of $p_1$ and $p_2$, respectively, in $C - P$. Note that it is possible that $|V(C)| = 3$, in which case $v_1 = v_2$. Since $C - P$ has fewer than three vertices, either $v_1 = v_2$ or $v_1$ and $v_2$ are the only vertices in $C - P$ and are therefore adjacent along $C$. By Claims \[claim:chord\] and \[claim:cycle\], there must exist a vertex $u$ that is on the outer face of $G - P$ but not in $C$, as shown in Figure \[fig:2-verticesP\]. Let $Y = V(G) \setminus \{ p_1, p_2, u \}$. Color $v_1$ with a color from $\{ 1, 2, 3 \}$ not used on $p_1$ and color $v_2$ with a color from $\{ 1, 2, 3 \}$ not used on either $p_2$ or $v_1$. By Claim \[claim:subgraph\], we can extend the precoloring on $\{ v_1, v_2 \}$ to all of $G[Y]$ such that every internal face of $G[Y]$ has a unique maximum color. Coloring $u$ with $4$ gives a unique maximum color to every face incident with $u$, thus completing the coloring of $G$ and producing a contradiction.
\[claim:2444\] $C$ does not have four vertices.
![An outer face consisting of a precolored vertices and two non-precolored vertices of degree at least $4$[]{data-label="fig:2444"}](fig-2444.pdf)
Suppose for contradiction that $C$ has four vertices. Since $C - P$ has at most two vertices, $P$ consists of two vertices, exactly one of which, say $p_2$, is a $2$-vertex, as shown in Figure \[fig:2444\]. By Claims \[claim:chord\] and \[claim:cycle\], there must exist a vertex $u$ that is on the outer face of $G - P$ but not in $C$. Let $Y = V(G) \setminus \{ p_2, u \}$. Color $v_2$ with the color used on $p_2$. By Claim \[claim:subgraph\], we can extend the precoloring on $\{ p_1, v_2 \}$ to all of $G[Y]$ such that every internal face of $G[Y]$ has a unique maximum color. Since $v_2$ received the same color as $p_2$, the color that $v_1$ receives will not conflict with $p_2$ in $G$. Coloring $u$ with $4$ gives a unique maximum color to every face incident with $u$, thus completing the coloring of $G$ and producing a contradiction.
\[claim:2-vertexP\] $P$ does not have a $2$-vertex.
![A precolored $2$-vertex $p$ on the outer face of $G$[]{data-label="fig:2-vertexP"}](fig-2-vertexP.pdf)
Suppose for contradiction that $P$ has a $2$-vertex $p$. Let $v_1$ and $v_2$ be the neighbors of $p$ in $C$. By symmetry, we can assume that $v_1$ is not in $P$. If $|V(P)| = 1$, then $v_2 \in V(C) \setminus V(P)$ and thus has degree at least $4$. Since $X$ is a star forest, $v_1$ and $v_2$ must be the only vertices in $C - P$. If $|V(P)| = 2$, then $v_2 \in V(P)$ and $d (v_2) \geq 3$. Since $d (v_1) \geq 4$, it must be the only vertex in $C - P$. In either case, $v_1$ and $v_2$ must be adjacent along $C$. If $v_2$ is not in $V(P)$, color $v_2$ with a color from $\{ 1, 2, 3 \}$ not used on $p$. Next, color $v_1$ with a color from $\{ 1, 2, 3 \}$ not used on either $p$ or $v_2$. By Claims \[claim:chord\] and \[claim:cycle\], there must exist a vertex $u$ that is on the outer face of $G - v$ but not in $C$, as shown in Figure \[fig:2-vertexP\]. Let $Y = V(G) \setminus \{ u, p \}$. By Claim \[claim:subgraph\], we can extend the precoloring on $\{ v_1, v_2 \}$ to all of $G[Y]$ such that every internal face of $G[Y]$ has a unique maximum color. Coloring $u$ with $4$ gives a unique maximum color to every face of $G$ incident with $u$, thus completing the coloring and producing a contradiction.
By Claim \[claim:2-vertexP\], each vertex in $P$ must have degree at least $3$. Furthermore, by Claim \[claim:23-vertex\] each vertex in $C - P$ must have degree at least $4$. This means that every vertex in $C$ also belongs to $X$, a contradiction since $X$ is a star forest and $C$ is a cycle.
We are now ready to prove the main theorem of this paper.
Let $G$ be a plane graph and $X = \{ v \in V(G) : d (v) \geq 4 \}$ such that $G[X]$ is a star forest. Pick a vertex $v$ on the outer face of $G$ and apply Lemma \[lem:precolor\] to $G - v$. Color $v$ with $4$ to complete the coloring.
Conclusion
==========
Let $G$ be a plane graph and $X = \{ v \in V(G) : d(v) \geq 4 \}$. In our proof, we used the assumption that $X$ induces a star-forest when arguing before Claim \[claim:22444\] that $C$ is short. It might be possible to extend the proof to any acyclic graph.
If $G[X]$ is an acyclic graph, then ${\chi_{\mathrm{fum}}}(G) \leq 4$.
A special case of Theorem \[thm:starforest\] is that if $G[X]$ forms a matching, then ${\chi_{\mathrm{fum}}}(G) \leq 4$. We believe a stronger result will be true if the maximum degree in $G[X]$ is 2. If this is true, maybe it could be extended to maximum degree 3 in $G[X]$.
If $G[X]$ is a graph of maximum degree 2, then ${\chi_{\mathrm{fum}}}(G) \leq 4$.
The connected plane graph $H$ with ${\chi_{\mathrm{fum}}}(H) > 4$ found in [@LidMesSkr17] has minimum degree 4 and two vertices of degree five. The construction could be disconnected, which gives a 4-regular graph. It is not clear to us if adding the connectivity constraint gives ${\chi_{\mathrm{fum}}}= 4$ or not.
Is there a connected plane graph $G$ with maximum degree 4 with ${\chi_{\mathrm{fum}}}(G) > 4$?
Acknowledgment
==============
The research project has been supported by the bilateral cooperation between USA and Slovenia, project no. BI–US/17–18–013. R. Škrekovski was partially supported by the Slovenian Research Agency Program P1–0383. B. Lidický was partially supported by NSF grant DMS-1600390.
[^1]: Department of Mathematics, Iowa State University, USA. E-Mail: `lidicky@iastate.edu`
[^2]: Department of Mathematics, Iowa State University, USA. E-Mail: `kacymess@iastate.edu`
[^3]: Faculty of Information Studies, Novo mesto & University of Ljubljana, Faculty of Mathematics and Physics & University of Primorska, FAMNIT, Koper, Slovenia. E-Mail: `skrekovski@gmail.com`
|
---
author:
- 'A.A. Nucita, M. Guainazzi, A.L. Longinotti, M. Santos-Lleo, Y. Maruccia,'
- 'S. Bianchi'
date: 'Submitted: XXX; Accepted: XXX'
title: 'A characterization of the NGC 4051 soft X-ray spectrum as observed by [*XMM*]{}-Newton'
---
[ [Soft X-ray high resolution spectroscopy of obscured AGNs shows a complex soft $X$-ray spectrum dominated by emission lines of He and H-like transitions of elements from carbon to neon, as well as L-shell transitions due to iron ions.]{} [In this paper we characterize the [*XMM*]{}-Newton RGS spectrum of the Seyfert 1 galaxy NGC 4051 observed during a low flux state and infer the physical properties of the emitting and absorbing gas in the soft $X$-ray regime.]{} [X-ray high-resolution spectroscopy offers a powerful diagnostic tool because the observed spectral features strongly depend on the physical properties of matter (ionization parameter $U$, electron density $n_e$, hydrogen column density $N_H$), which in turn are tightly related to the location and size of the $X$-ray emitting clouds. We carried out a phenomenological study to identify the atomic transitions detected in the spectra. This study suggests that the spectrum is dominated by emission from a photoionized plasma. Then we used the photoionization code Cloudy to produce synthetic models for the emission line component and the warm absorber observed during phases of high intrinsic luminosity.]{} [The low state spectrum cannot be described by a single photoionization component. A multi-ionization phase gas with an ionization parameter in the range of $\log U \sim 0.63-1.90$ and a column density $\log N_H = 22.10-22.72$ cm$^{-2}$ is required, while the electron density $n_e$ remains unconstrained. A warm absorber medium is required by the fit with the parameters $\log U \sim 0.85$, $\log N_H = 23.40$ and $\log n_e \ut< 5$. The model is consistent with an $X$-ray emitting region at a distance $\ut> 5\times 10^{-2}$ pc from the central engine. ]{} ]{}
Introduction
============
It is commonly accepted that the center of active galaxies (Active Galactic Nuclei -AGNs) hosts a massive black hole (with a mass in the range $10^6-10^9$ M$_{\odot}$) accreting the surrounding material via the formation of a disk. How the energy released from the central engine interacts with the local environment and contributes to the history of the host galaxy is one of the crucial question of present astrophysical research. In this respect, while the mechanisms of energy output in the form of radiation and relativistic jets are quite well understood, it also seems that the outflowing winds have an important role in the overall energy budget. Although the origin of these winds is still controversial, at our present level of understanding the narrow-line regions, the inner part of an obscuring torus (@blustin2005) and the black hole accretion disk [@elvis2000] are all possible locations.
X-ray obscured AGNs (with an intrinsic column density $N_H\ut> 10^{22}$ cm$^{-2}$) are not completely dark in the soft X-ray band. High resolution [*XMM*]{}-Newton and Chandra observations revealed a complex spectrum dominated by emission lines from He-and H-like transitions of elements from carbon to neon as well as by L-shell transitions of [Fe<span style="font-variant:small-caps;">xvii</span>]{} to [Fe<span style="font-variant:small-caps;">xxi</span>]{} ions (@sakoa, @ali, @sambruna, @armentrout2007). This gas, which shows the signature of a photoionization process (@ali, @bianchi2007), is sometimes referred to as a warm mirror.
In unobscured AGNs a modification of the output energy spectrum may also occur as a consequence of absorption by a warm ionized gas along the line of sight. The properties of these so called warm absorbers can be summarized as follows: i) average ionization parameter in the range $\log \xi =0-3$, ii) total column density in the range $\log N_H = 21-22$ cm$^{-2}$, iii) outflow velocities of hundred of km s$^{-1}$ (see e.g. @blustin2005, but also @steenbrugge). Evidence of a multi-phase warm absorber gas was also recently reported for Mrk 841 (@longinotti2009).
In general, detecting warm mirror signatures is easier in sources in low flux states, because the emission features are not outshone by the continuum radiation. This was the case for the Seyfert 1 galaxy Mrk 335, whose soft X-ray spectrum resembled the spectra of obscured AGNs when the source was observed at low state (@longinotti2008), but does not show any evidence of a warm absorber in the high flux state (@longinotti2007).
The overall properties of the warm mirror (even if it is poorly constrained) and the warm absorber (as described above) are similar so that there is the possibility that they represent the same physical system. Conversely, the interplay between the warm absorber and warm mirror regimes is best studied in sources that display both components.
The source NGC 4051, a narrow-line Seyfert galaxy at the redshift of $0.00234$, was at the center of many past investigations in the X-ray band because it offers a unique laboratory where to test present theories and models about the physics of AGNs. The X-ray emission is characterized by rapid variations [@lamer2003; @ponti2006] sometimes showing periods of low activity (see @lawrence1987 and @uttley). Its power spectral distribution (PSD) in high state resembles the behavior of a galactic black hole system (@mchardy). At high X-ray flux, the spectrum of the galaxy is characterized by a power law with photon index $\Gamma\sim 1.8-2$ which becomes harder above 7 keV where a reflection component from cold matter has been observed. On long time-scales, the X-ray light curve of NGC 4051 shows low state flux periods of several months during which the spectrum in the energy range 2-10 keV becomes harder ($\Gamma \simeq 1$) and shows a strong iron $K\alpha$ line (as found by @guainazzi98 in Beppo-SAX data). A soft X-ray excess is also evident.
As reported by @warmabsorber, the high state X-ray spectrum of NGC 4051 in the soft band is a combination of continuum and emission line components. Curvature in the spectrum cannot be explained with simple models, i.e. a single power law or a black body, because an ionized absorber-emitter has to be taken into account as well. In this context, @krongold showed that the evolution in time of the properties of the warm absorber can constrain the physical parameters of the absorbing gas. In particular they find that at least two different ionization components are required with matter densities of $\simeq 10^6$ cm$^{-3}$ and $\ut> 10^7$ cm$^{-3}$, thus placing the warm absorber in the vicinity of the accretion-disk. Dynamical arguments permit us to infer that the warm absorber gas originates in a radiation-driven high-velocity outflow in accretion disk instabilities (@krongold).
On the other hand, as shown by [@pounds], the low state flux spectrum of NGC 4051 is dominated by narrow emission lines and radiative recombination continua (RRC) from hydrogenic and He-like carbon, oxygen, neon and nitrogen. To be specific, a fit to the identified RRCs yields a mean temperature for the emitting gas of $T\simeq 4\times 10^{4}$ K, which favors a scenario invoking a photoionization process. In this case, the soft X-ray spectrum of NFG 4051 in low state is similar to that observed for the prototype Seyfert 2 galaxy NGC 1068 (see @ali).
Below we do not repeat the analysis of the EPIC data but refer to @pounds for more details on the main results obtained in the energy band 0.3-10 keV. We only say that a comparison between the EPIC PN data for the 2001 and 2002 observations shows that the high state observation flux level is a factor $\sim 5$ greater with respect to the low state. Furthermore, the spectrum shows a gradual flattening of the continuum slope from 3 keV up to 6.4 keV. It was also noted that when the fit to the 0.3-10 keV band continuum is extrapolated down in the soft X-ray (0.3-3 keV) a strong excess appears in both the two observations, and as is clear from the RGS spectrum, it can be explained by a blending effect of fine structures (emission lines).
Here we first conducted a phenomenological study of the emission lines identified in the spectrum of NGC 4051 and compare our results with those known in literature. We further compared the RGS emission line spectrum with synthetic spectra generated with the photoionization code Cloudy 8 (@ferland). For this purpose we followed a similar approach as in [@armentrout2007] (to which we refer for more details) on NGC 4151.
The paper is structured as follows: in Sect. 2 we briefly describe the reduction of the [*XMM*]{}-Newton data set and describe our phenomenological analysis of the soft $X$-ray spectrum of NGC 4051. In Sects. 3 and 4 we give details on the Cloudy model developed and address some conclusions.
A phenomenological study of the low state of NGC 4051: data reduction and line identification
=============================================================================================
The source NGC 4051 ($\alpha=12^{h}03^{m}09.6^{s}$ and $\delta=44^{d}31^{m}53.0^{s}$) was observed by the [*XMM*]{}-Newton satellite on two occasions: on May 2001 for $\simeq 122$ ks and on November 2002 for $\simeq 52$ ks. While the former observation coincided with a period where the central engine was bright (with luminosity of $7\times10^{41}$ erg s$^{-1}$ in the $0.3-10$ keV, @pounds), the latter corresponded to a low state X-ray flux (corresponding to a luminosity of $1.5\times10^{41}$ erg s$^{-1}$ in the $0.3-10$ keV, @pounds) due to a low nuclear activity. This observation was conducted $\sim 20$ days after the onset of the low state (@pounds). Below we focus on the low state data analysis, because the warm absorber observed at high state was already well studied with physical models (@warmabsorber, @krongold, @steenbrugge). On the contrary, no attempt has yet been made to model the warm mirror in the low state with a self consistent physical model.
The ODF files (OM, MOS, PN and RGS) were processed with the [*XMM*]{}-Science Analysis System (SAS version $8$). Hence the raw data were reduced using SAS tasks with standard settings and the most update calibration files to produce the source and background spectra as well as the corresponding response matrices for the RGS cameras.
We used XSPEC 12.5.1 (@arnaud) for our quantitative analysis and adopted the cosmological parameters $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda}=0.73$ and $\Omega_{m}=0.27$.
To study the soft X-ray spectra of NGC 4051 in more detail, we then examined the first order spectra obtained by the [*XMM*]{}-Newton gratings. The spectral resolution of RGS in the first order spectrum is FWHM=72 mÅ and the calibration in wavelength is accurate up to 8 mÅ corresponding to FWHM$\simeq 620$ km s$^{-1}$ and $\Delta v\simeq 69$ km s$^{-1}$ at 35 Å(@xmmuserbook). Below we use the unbinned RGS 1 and RGS 2 spectra for the quantitative analysis. In Fig. \[spectrum\], we show the fluxed RGS spectrum in the wavelength range 5-38 Å.
As can be clearly seen, the 2002 RGS spectrum of NGC 4051 shows an unresolved continuum with a predominance of emission lines (see for comparison the RGS spectrum of NGC 1068 presented in @ali). By contrast, the high state RGS spectra of the same source show a higher continuum flux level with a pronounced curvature around $\sim 15$ Åand several absorption features [@pounds] typical of a warm absorber. Interestingly, the [N<span style="font-variant:small-caps;">vi</span>]{}, [O<span style="font-variant:small-caps;">vii</span>]{} and [Ne<span style="font-variant:small-caps;">ix</span>]{} forbidden lines are seen in both observations with the same flux level (@pounds).
The phenomenological spectral analysis follows the [*local fits*]{} method described in @bianchi2007. In particular, the unbinned spectra are divided in intervals of $\simeq 100$ channels wide and Gaussian profiles are used to account for all identified emission lines, with the line centroid energy as the only free parameter of the fit.
Analogously, free-bound transitions (i.e. Radiative Recombination Continua) were also modeled as Gaussian profiles with free line width. Best-fit values of these widths are reported in Table \[completarrc\] together with their errors. The local continuum was modeled as a power law with a fixed photon index $\Gamma = 1$ and free normalization. For line triplets and for emission lines close to free-bound transitions, the relative distance between the central energies was frozen to the value predicted by atomic physics.
We used the C-statistic as the estimator of the goodness of the performed fit (@cash). For any line under investigation, the emission feature was considered [*detected*]{} if, when repeating the fit without any Gaussian profile, we obtained a value of the C-statistic which differed from the previous one by at least 2.3, corresponding to the 68$\%$ confidence level (or, equivalently, 1 $\sigma$ for one interesting parameter) [@arnaud].
The results of the phenomenological fit to the emission lines are reported in Table \[completa\]. We recall that the fluxes of the identified emission lines were estimated integrating over the Gaussian line profiles with which the emission features are modeled. This operation resulted in flux values somehow higher (up to a factor of $\sim 1.5$) than those reported in @pounds even if the corresponding measured equivalent widths are fully consistent with the values quoted in the above mentioned paper. The origin of the discrepancy remains unknown.
Here we give the best-fit parameters for the transitions identified in the soft X-ray spectrum of NGC 4051. For those lines that were not identified (i.e. $\Delta C <$ 2.3), but which are part still of triplet features, we give the upper limits to the corresponding flux value. We detected no evidence for either inflows or outflows, with an upper limit on the velocity of $\sim 200$ km s$^{-1}$, consistent with that estimated by [@pounds]. This was calculated from the full width half maximum of the distribution of residual velocity as derived from the difference between the expected and the measured laboratory wavelengths (see Fig. \[redshift\] and also Table \[completa\]).
Line ID $\lambda_{exp} ({\rm {\AA}})$ $\lambda_{obs} ({\rm {\AA}})$ ${\rm v}~({\rm km s^{-1}})$ Flux ($\times 10^{-14}$ cgs) $\Delta {\rm C}$
----------------------------------------------------------------------- ------------------------------- ------------------------------- ----------------------------- ------------------------------ ------------------
[Ne<span style="font-variant:small-caps;">x</span>]{} Ly-$\alpha$ $12.134$ $12.13_{-0.13}^{+0.03}$ $-23_{-3100}^{+ 800}$ $1.5_{-1.4}^{+1.6}$ 3
[Ne<span style="font-variant:small-caps;">ix</span>]{} (r) $13.447$ $13.45_{-0.02}^{+0.02}$ $-7_{- 360}^{+ 400}$ $\ut< 0.81$ –
[Ne<span style="font-variant:small-caps;">ix</span>]{} (i) $13.550$ $13.55_{-0.02}^{+0.02}$ $23_{- 360}^{+ 400}$ $\ut< 0.77$ –
[Ne<span style="font-variant:small-caps;">ix</span>]{} (f) $13.700$ $13.70_{-0.02}^{+0.02}$ $-40_{- 360}^{+ 400}$ $4.7_{-1.2}^{+1.3}$ 50
[Fe<span style="font-variant:small-caps;">xvii</span>]{} 3s-2p $17.073$ $17.06_{-0.02}^{+0.04}$ $-135_{- 380}^{+ 710}$ $3.3_{-1.5}^{+1.4}$ 21
[O<span style="font-variant:small-caps;">viii</span>]{} Ly-$\alpha$ $18.969$ $18.98_{-0.02}^{+0.01}$ $150_{- 310}^{+ 190}$ $7.0_{-1.1}^{+1.2}$ 135
[O<span style="font-variant:small-caps;">vii</span>]{} (r) $21.600$ $21.58_{-0.01}^{+0.01}$ $-202_{-210}^{+ 150}$ $\ut< 0.80$ –
[O<span style="font-variant:small-caps;">vii</span>]{} (i) $21.790$ $21.80_{-0.20}^{+0.25}$ $194_{-2380}^{+3400}$ $7.2_{-1.6}^{+1.7}$ 74
[O<span style="font-variant:small-caps;">vii</span>]{} (f) $22.101$ $22.10_{-0.02}^{+0.03}$ $32_{-250}^{+ 400}$ $15.0_{-1.9}^{+2.1}$ 238
[N<span style="font-variant:small-caps;">vii</span>]{} Ly-$\alpha$ $24.781$ $24.79_{-0.02}^{+0.03}$ $198_{-220}^{+ 330}$ $2.0_{-0.8}^{+0.9}$ 17
[N<span style="font-variant:small-caps;">vi</span>]{} (r) $28.787$ $28.78_{-0.01}^{+0.02}$ $-104_{-150}^{+220}$ $\ut< 0.92$ –
[N<span style="font-variant:small-caps;">vi</span>]{} (i) $29.083$ $29.07_{-0.01}^{+0.02}$ $-94_{-150}^{+220}$ $1.3_{-0.6}^{+0.5}$ 6
[N<span style="font-variant:small-caps;">vi</span>]{} (f) $29.534$ $29.52_{-0.01}^{+0.02}$ $-103_{-150}^{+220}$ $3.6_{-0.8}^{+0.7}$ 38
[C<span style="font-variant:small-caps;">vi</span>]{} Ly-$\alpha$ $33.736$ $33.76_{-0.02}^{+0.03}$ $233_{-140}^{+ 240}$ $3.5_{-1.1}^{+1.3}$ 29
--------------------------------------------------------------- ------------------------------- ------------------------------- ----------------------------- ------------------------------ ------------------
Line ID $\lambda_{exp} ({\rm {\AA}})$ $\lambda_{obs} ({\rm {\AA}})$ ${\rm v}~({\rm km s^{-1}})$ Flux ($\times 10^{-14}$ cgs) $\Delta {\rm C}$
[O<span style="font-variant:small-caps;">vii</span>]{} RRC $16.771$ $16.78_{-0.07}^{+0.04}$ $220_{-1200}^{+800}$ $ 3.8_{-1.2}^{+1.3}$ 38
[C<span style="font-variant:small-caps;">v</span>]{} RRC $31.622$ $31.51_{-0.06}^{+0.10}$ $-1070_{-600}^{+800}$ $ 3.4_{-2.0}^{+2.0}$ 9
[C<span style="font-variant:small-caps;">vi</span>]{} RRC $25.304$ $25.19_{-0.04}^{+0.10}$ $-1250_{-500}^{+900}$ $ 6.4_{-3.0}^{+16.0}$ 24
--------------------------------------------------------------- ------------------------------- ------------------------------- ----------------------------- ------------------------------ ------------------
Radiative recombination continua (RRC)
--------------------------------------
The electron temperature $T_e$ can be inferred by studying the profiles of the radiative recombination continua (RRC). In the RGS spectrum of NGC 4051 the RRCs detected with $\Delta C \ut > 2.3$ correspond to [O<span style="font-variant:small-caps;">vii</span>]{}, [C<span style="font-variant:small-caps;">v</span>]{} and [C<span style="font-variant:small-caps;">vi</span>]{} (see Table \[completarrc\] for details). Expressing the temperature widths in eV to $kT_e$ (@lied), we estimate them to be $$\begin{array}{ll}
T_{O~VII}=\Big(5.0^{+4.0}_{-2.0}\Big)\times 10^{4}~{\rm K}~,\\ \\
T_{C~VI}=\Big(3.0^{+3.1}_{-1.3}\Big)\times 10^{4}~{\rm K},\\ \\
T_{C~V}=\Big(1.2^{+0.8}_{-0.8}\Big)\times 10^{4}~{\rm K},
\end{array}$$ respectively, so that the average gas temperature is $T_e = \Big(3.1^{+2.5}_{-1.6}\Big)\times 10^{4}~{\rm K}$, which agrees well with the result quoted by @pounds. It is to note that the low temperature found in this way is an indication that collisional ionization and excitation processes are negligible (@lied).
In this phenomenological analysis, we used Gaussian profiles to fit the RRC features, which are in principle asymmetric. Still we verified that the use of a more appropriate model, as e.g. [*redge*]{} in XSPEC, gives consistent results.
He-like triplet diagnostic
--------------------------
We detected the most intense lines of He-like ions in the range 5-35 Å. The transitions between the $n=2$ shell and the $n=1$ ground state shell as the resonance line (${\bf r:~}~1s^2$ $^1S_0$-$1s2p$ $^1P_1$), the two inter-combination lines (${\bf i:~}~1s$$^2$$^1S_0$-$1s2p$ $^3P_{2,1}$, often blended) and the forbidden line (${\bf f:~}$ $1s^2$$^1S_0$-$1s2s$$^3S_1$) were detected. As demonstrated by @porquet the relative emission strength of the r, i and f lines are good indicators of the physical conditions of density and temperature of the gas. Using standard notation we defined the ratios $R=f/i$, $L=r/i$ and $G=(f+i)/r$ (@porter2007). Figure \[ovii\] shows the triplet of the [O<span style="font-variant:small-caps;">vii</span>]{} complex (forbidden, inter-combination and resonance lines) locally fitted by a power law and three Gaussian. In this case, following the phenomenological fit approach described in the previous Section, we only had a measurement for the fluxes of the $f$ and $i$ components (see Table \[completa\]).
With the flux measurements quoted in Table \[completa\], the previous relations give $R=2.1^{+0.5}_{-0.6}$, $L=0.07\pm 0.06$ and $G= 47_{-39}^{+34}$. Analogously, for the NVI triplet we get $R=2.7_{-1.9}^{+1.2}$, $L\ut<0.69$ and $G\ut>5.4$ (poorly constrained because we only got an upper limit to the $r$ line flux value), respectively. For [Ne<span style="font-variant:small-caps;">ix</span>]{} we had a lower limit only on the $R(\ut> 5.6)$ ratio, while the ratio of the [O<span style="font-variant:small-caps;">viii</span>]{} Ly-$\alpha$ to the [O<span style="font-variant:small-caps;">vii</span>]{} forbidden intensity lines results in $0.47_{-0.13}^{+0.14}$. These line ratios are consistent with the results by @pounds.
Results of the phenomenological study: evidence of photoionized gas
-------------------------------------------------------------------
The results obtained from the phenomenological study allow us to highlight some considerations on the physical conditions of the $X$-ray emitting gas in NGC 4051. Indeed, according to the study of @porquet, a value of the G ratio higher than 4 is a strong indication of a photoionized gas. An estimate of the gas electron density $n_e$ can be done when the other two line ratios L and R are taken into account. In the particular case of the [O<span style="font-variant:small-caps;">vii</span>]{} triplet line ratios quoted above, the electron density is constrained to be $n_e \ut< 10^{10}$ cm$^{-3}$ for a pure photoionized gas (@porquet). Note however that the line intensities obtained from the phenomenological study described above do not account for a warm absorber, which is not taken into account in the model.
An additional constraint on the electron density value can be obtained noting that the [*XMM*]{}-Newton observation of NGC 4051 in its low state occurred $\sim 20$ days after the source entered this regime. Because the [O<span style="font-variant:small-caps;">vii</span>]{} triplet line intensity is consistent with that measured during high flux states (@pounds), it is believed that the recombination time of the [O<span style="font-variant:small-caps;">vii</span>]{} is larger than 20 days,thus implying (for a gas temperature of $\simeq 10^4$ K) a more stringent constraint on the electron density of $n_e \ut< 10^5$ $cm^{-3}$ (@pounds).
Fitting the spectra with the photoionization code Cloudy
========================================================
General properties of the model
-------------------------------
The results of the phenomenological analysis in Sect. 2 show that the bulk of the spectrum, measured by the RGS during the $X$-ray low state in NGC 4051, is dominated by photoionization as already suggested by other authors (see e.g. @pounds).
In this section we use the photoionization code Cloudy (@ferland) for modeling the overall spectrum of NGC 4051 in its low state assuming a plane parallel geometry with the central engine shining on the inner face of the cloud with a flux density depending on the ionization parameter $U$.
The spectrum produced by a photoionized nebula critically depends on the spectral energy distribution (SED) of the ionizing continuum. Below we adopt the AGN SED as in @korista. In a typical AGN the observed continuum can be well represented by a SED characterized by several components: a big blue bump with temperature $T_{BB}\simeq 10^{6}~K$ (1 Ryd), a power law with a low energy exponential cut-off in the infrared region at $kT_{IR}=0.01$ Ryd; in the $X$-ray band ($1.36$ eV-100 keV) the SED is well approximated by a power law with an exponential cut-off for energies lower than 1 Ryd; finally, for energies greater than 100 keV an exponential fall as $\propto \nu^{-2}$ is usually assumed . We also included in the modeling the cosmic microwave background so that the incident continuum has a non-zero intensity for long wavelengths. Hence the AGN spectrum is described by the law $$\label{sed}
F(\nu)=\nu^{\alpha_{UV}}e^{-\frac{h\nu}{kT_{BB}}}e^{-\frac{kT_{IR}}{h\nu}}+A\nu^{\alpha_{x}}e^{-\frac{1~{\rm Ryd}}{E({\rm Ryd})}}~,$$ where $\alpha_{UV}\simeq -0.50$, $\alpha_{x}$ is the spectral photon index and the constant $A$ is obtained requiring that $F(2 ~{\rm keV})/F(2500 ~{\rm \AA})\simeq 403.3^{\alpha_{ox}}$ [@korista1997], where $F(2 ~{\rm keV})$ and $F(2500~ {\rm \AA})$ are the flux densities at 2 keV and 2500 Å, respectively.
To determine $\alpha _x$ and $\alpha _{ox}$, we used the Epic data corresponding to the NGC 4051 high state observation. The resulting 0.2-10 keV energy band spectrum was fitted with a photoelectrically absorbed power law model within XSPEC, thus allowing us to measure $\alpha _x= 0.96\pm 0.05$ and $F(2 ~{\rm keV})=(2.91\pm0.01)\times 10^{-29}$ erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$, respectively.
From the OM instrument we estimated the aperture photometry of the target in the UVM2 filter (centered at $2310$ Å) obtaining a flux density of $F(2310 ~{\rm \AA})=(1.50\pm0.01)\times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$. From the flux densities at 2 keV and 2500 Å, the X-UV flux density ratio results in $\alpha _{ox}\simeq -1.14$.
Once the AGN SED (erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$) is known, it is straightforward to show that the number of hydrogen-ionizing photons $Q$[^1], the electron density $n_e$ and the dimensionless ionization parameter $U$ are related by $U=Q/4 \pi r^2 n_e c$ with $r$ the distance between the central engine and the innermost illuminated layer of the clouds. Here we require that integrating over the SED (between 13.6 eV and 13.6 keV) we get the ionizing luminosity $L_{ion}\simeq 4.1\times 10^{42}$ erg s$^{-1}$ [@warmabsorber]. In Fig. \[sed\] we compare the SED used in this paper (solid line) with that given in @warmabsorber. We recall that the dimensionless ionization parameter $U$ does not depend on the flux below $13.6$ eV. The two spectral energy distributions give rise to comparable integrated fluxes in the 0.3-10 keV energy band (within a few percent).
Note also that through the well known definitions of the ionizing luminosity $L_{ion}$, of the number of ionizing photons $Q$ (@ferland) and the used SED $F(\nu)$, it is possible to estimate a useful conversion relation between the dimensionless ionization parameter $U$ and the ionization parameter $\xi$ as given in @tarter, i.e. $\xi\simeq 20 U $ erg cm s$^{-1}$.
Fit to the RGS spectrum
-----------------------
Assuming the standard AGN continuum described above, we generated a grid of reflected spectra from a photoionized nebula, varying the ionization parameter $\log U$, the electron density $\log n_e$ and the total column density $\log N_H$. The free parameters spanned the ranges $\log U$=\[-1.0,4.0\]; $\log (n_e/{\rm cm^{3}})=[2,12]$ and $\log (N_H/{\rm cm^{2}})$=\[19,24\] in steps of 0.1 dex, respectively.
Initially we extracted the line intensities from these simulations for all lines detected in the soft spectrum of NGC 4051. Hence, we tried to determine the best model that can describe the RGS line spectrum in the whole $5-35$ Åband. Following the procedure described in @longinotti2008, we calculated the value of the merit function for each grid model $$\chi ^2 =\sum \frac{(I_c-I_o)^2}{\sigma_o ^2}~,$$ where $I_o$ is the intensity of each of the identified lines (with statistical error $\sigma_o$) and $I_c$ is the intensity as predicted by Cloudy, both normalized to the value of the [O<span style="font-variant:small-caps;">viii</span>]{} Ly-$\alpha$ line.
Minimizing the merit function quoted above gives a best-fit model ($\chi^2 \simeq 13.8$ with degrees of freedom $\nu=10$) corresponding to the parameter values $\log U=0.4$, $\log (n_e/{\rm cm^{3}})=4.4$ and $\log (N_H/{\rm cm^{2}})=21.8$. A quantitative measure of the fit goodness for the used model is given by the Chi-square Probability Function $Q(\chi^2,\nu)$ as defined in @press. If the single phase component model is the true representation of the data, the probability to obtain the observed $\chi^2$ value is as high as $Q\simeq 12\%$. In this case, the model consisting of a single ionization state can be statistically rejected.
We therefore investigated more complex models, including an additional warm mirror and one warm absorber covering the combination of emitting components. For this approach to be fruitful the constraints provided by the continuum shape are crucial. Below we will fit the whole RGS spectrum globally.
Global fit to the RGS spectrum
------------------------------
We generated additive and multiplicative fits tables (with the same grid of parameters as before) to account for both the emission and absorption features observed in the RGS spectrum, and imported them within XSPEC as described in @porter. Our final model can be described by the formula $ phabs*mtab(n_e,N_H,U)*\{\sum atab(n_e,N_H,U)\}$. Here, [*mtab*]{} and [*atab*]{} indicate the warm absorber component and the reflected component part of the spectrum depending on the electron density, hydrogen column density and ionization parameter, respectively. In the model, the redshift of each component is fixed to the cosmological value due to the lack of measurable velocity shifts from the phenomenological analysis (Fig. \[redshift\]), while all the other parameters are free to vary. In the fit procedure we fixed the column density of neutral hydrogen to the average value observed in the Galaxy along the line of sight to NGC 4051, i.e. $1.32\times10^{20}$ cm$^{-2}$ (@dickey).
The fit does not formally depend on values of $n_e\ut< 10^{9}$ cm$^{-3}$, which is expected because the ratios of the He-like triplets are insensitive to the electron density in this region of the space parameter ([@porquet]). Given the constraint on this parameter derived from the source time variability, we fixed its value to $10^{5}$ cm$^{-3}$ hereafter.
We recursively increased the number of Cloudy additive components until this operation resulted in a statistically significant improvement of the fit quality. We found that two emission and one absorption components are required to fit the data. In particular, the final model corresponds to a value of the C-statistic of 6300 with 5178 d.o.f. and the model parameters are given in Table 4. Conversely, when the warm absorber component is not taken into account the fit visibly worsens and converges to a C-statistic value of 9452 with 5182 d.o.f. In this case, the line intensities corresponding to the He-like transitions are not correctly estimated, with specifically the recombination line of the [O<span style="font-variant:small-caps;">vii</span>]{} triplet well over-estimated (see Fig. \[fig6\]).
We then extracted the line fluxes predicted by the best-fit model and compared them with the observations. In Figure \[fig4\], we show with filled squares the intensities of all observed (see Table \[completa\]) and simulated (triangles) lines once normalized to the [O<span style="font-variant:small-caps;">viii</span>]{} Ly-$\alpha$ flux. The lower panel of the same figure shows the residuals between observation and theory. Note that the simulation underestimates the contribution of the [Fe<span style="font-variant:small-caps;">xvii</span>]{} transitions, because the Cloudy database is inaccurate for the corresponding atomic parameters (see e.g. @bianchi2010). The normalized intensities of the observed lines as well as the Cloudy predictions are also reported in Table \[table3\] for clarity with the missing value of the [Fe<span style="font-variant:small-caps;">xvii</span>]{} transition.
Line ID $\lambda_{exp} ({\rm {\AA}})$ Observed ratio Cloudy predicted ratio
----------------------------------------------------------------------- ------------------------------- --------------------------------- ------------------------
[Ne<span style="font-variant:small-caps;">x</span>]{} Ly-$\alpha$ $12.134$ $0.2_{-0.2}^{+0.2}$ $ 0.2$
[Ne<span style="font-variant:small-caps;">ix</span>]{} (r) $13.447$ $ \ut<0.1$ $ 0.1$
[Ne<span style="font-variant:small-caps;">ix</span>]{} (i) $13.550$ $ \ut<0.1$ $0.2$
[Ne<span style="font-variant:small-caps;">ix</span>]{} (f) $13.700$ $ 0.7_{- 0.2}^{+ 0.2}$ $0.6$
[Fe<span style="font-variant:small-caps;">xvii</span>]{} 3s-2p $17.073$ $ 0.5_{-0.2}^{+0.2}$ $--$
[O<span style="font-variant:small-caps;">viii</span>]{} Ly-$\alpha$ $18.969$ $ 1.0_{-0.2}^{+0.2}$ $1.0$
[O<span style="font-variant:small-caps;">vii</span>]{} (r) $21.600$ $ \ut< 0.1$ $0.3$
[O<span style="font-variant:small-caps;">vii</span>]{} (i) $21.790$ $ 1.0_{- 0.2}^{+ 0.2}$ $0.6$
[O<span style="font-variant:small-caps;">vii</span>]{} (f) $22.101$ $ 2.2_{-0.3}^{+0.3}$ $2.5$
[N<span style="font-variant:small-caps;">vii</span>]{} Ly-$\alpha$ $24.781$ $ 0.3_{-0.1}^{+0.1}$ $0.2$
[N<span style="font-variant:small-caps;">vi</span>]{} (r) $28.787$ $ \ut<0.13$ $0.01$
[N<span style="font-variant:small-caps;">vi</span>]{} (i) $29.083$ $ 0.1_{- 0.1}^{+ 0.1}$ $0.01$
[N<span style="font-variant:small-caps;">vi</span>]{}(f) $29.534$ $ 0.5_{- 0.1}^{+ 0.1}$ $0.1$
[C<span style="font-variant:small-caps;">vi</span>]{} Ly-$\alpha$ $33.736$ $ 0.5_{-0.2}^{+0.2}$ $0.7$
In Table 4 we give the relevant quantities estimated from the fit procedure (i.e. $n_e$, $N_H$ and $U$) together with their respective errors at the 90$\%$ confidence level for one interesting parameter. The 5-35 Å low-state spectrum of NGC 4051 is plotted in Fig. \[spectrum\_all\] with the best-fit Cloudy model superimposed on the observed RGS 1 (red) and RGS 2 (black) data and residuals in the lower part of each panel.
Note that to estimate the covering factor of the source, we extracted the luminosity of the most prominent emission line, [O<span style="font-variant:small-caps;">vii</span>]{} (f), as predicted by Cloudy. For the two reflection components included in our model, we simulated the expected spectrum the SED described in Sect. 3.1 and fixing the electron density $n_e$ and hydrogen column density $N_H$ to the best-fit values given in Table 4. In addition, the SED was normalized to the ionizing luminosity $L_{ion}\simeq 4.1\times 10^{42}$ erg s$^{-1}$ [@warmabsorber]. Hence we fixed the distance from the source to the inner shell of the cloud to the lower values reported in Table 4 for each of the reflection components, i.e. $d\simeq 0.22$ pc and $d\simeq 0.05$ pc for the [*low*]{} and [*high*]{} component, respectively. After we defined the luminosity of the source, the emission line luminosities were predicted by Cloudy (see e.g. @hazy1).
With a redshift of 0.00234, the expected total intensity of the [O<span style="font-variant:small-caps;">vii</span>]{} (f) line is $\simeq 10.6\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$. Assuming a filling factor of unity, the ratio of the observed [O<span style="font-variant:small-caps;">vii</span>]{} (f) intensity (see Table \[completa\]) to the Cloudy expected value gives an estimate of the covering factor value, which turns out to be $\simeq 0.14$.
$\begin{array}{c@{\hspace{1in}}c}
\multicolumn{1}{l}{\mbox{\bf (a)}} &
\multicolumn{1}{l}{\mbox{\bf (b)}} \\ [-0.53cm]
\includegraphics[width=6cm, angle=-90]{5-10_new.ps}&
\includegraphics[width=6cm,angle=-90]{10-15_new.ps}\\ [0.8cm]
\multicolumn{1}{l}{\mbox{\bf (c)}} &
\multicolumn{1}{l}{\mbox{\bf (d)}} \\ [-0.53cm]
\includegraphics[width=6cm,angle=-90]{15-20_new.ps}&
\includegraphics[width=6cm,angle=-90]{20-25_new.ps}\\ [0.8cm]
\multicolumn{1}{l}{\mbox{\bf (e)}} &
\multicolumn{1}{l}{\mbox{\bf (f)}} \\ [-0.53cm]
\includegraphics[width=6cm,angle=-90]{25-30_new.ps}&
\includegraphics[width=6cm,angle=-90]{30-35_new.ps}\\ [0.8cm]
\end{array}$
------------------- ------------------------ ----------------------------------- ----------------------------------- ------------------ -----------------------------
[**$\log U$**]{} [**$\log (n_e/{\rm cm^{3}})$**]{} [**$\log (N_H/{\rm cm^{2}})$**]{} [**$d$**]{} (pc) [**$\Delta_{f=1}$**]{} (pc)
[**Low**]{} $0.63_{-0.03}^{+0.05}$ $\ut< 5$ $22.10_{-0.04}^{+0.30}$ $\ut> 0.22$ $\ut> 0.04$
[**High**]{} $1.90_{-0.10}^{+0.20}$ $\ut< 5$ $22.72_{-0.13}^{+0.25}$ $\ut> 0.05$ $\ut> 0.20$
[**Warm Abs.**]{} $0.85_{-0.02}^{+0.02}$ $\ut< 7$ $23.36_{-0.01}^{+0.01}$ $\ut> 0.02$ $\ut> 0.01$
------------------- ------------------------ ----------------------------------- ----------------------------------- ------------------ -----------------------------
\[tabellafinale\]
Discussion
==========
Most of the information on the physics and geometry of gas in AGNs is inferred by means of optical spectroscopy and imaging techniques with which it was shown that the AGN central high energy emission is the main source of ionizing photons with an occasional contribution from collisionally ionized plasma. In the last years X-rays observations acquired an important role in AGN studies particularly since Chandra showed the existence, at least for Seyfert 2 galaxies, of extended (a few kpc) X-ray emission [@bianchi2006] similarly to what was observed in the optical band. High resolution spectroscopy in the soft X-ray band ($0.2-2$ keV) confirms the overall scenario, and photoionization seems to be the dominant ionization mechanism which results in a spectrum characterized by recombination lines from He- and H-like transitions of C to Si elements and by Fe-L transitions. In this respect, X-ray high-resolution spectroscopy offers a powerful diagnostic tool because the observed spectral features strongly depend on the physical properties of matter (ionization parameter $U$, electron density $n_e$, hydrogen column density $N_H$ as well as size and location of the emitting clouds).
The Seyfert 1 object NGC 4051 shows a very rich emission line $X$-ray spectrum when observed in low-flux state. According to the analysis we conducted on the [*XMM*]{}-Newton RGS data, the observed soft $X$-ray features originate in a low-density photoionized gas. In order to constrain the physical properties of the photoionized gas, we simulated synthetic spectra via the Cloudy software (@ferland) and compared them to the RGS data with standard minimization techniques. We found that to describe the overall soft $X$-ray spectrum, at least a three-phase gas is required (two emission components and one warm absorbing component). Referring to the emission components respectively as [*low*]{} and [*high*]{} ionization components, our fit procedure gave us their physical properties. For the [*low*]{} component we have $\log U\simeq 0.63$, and $\log (N_H/{\rm cm^{2}})\simeq 22.10$ and for the [*high*]{} component we have $\log U\simeq 1.90$, and $\log (N_H/{\rm cm^{2}})\simeq 22.20$. Using Cloudy we get for the electron density $n_e$ an upper limit of $\log (n_e/{\rm cm^{3}}) \simeq 9$, which reduces to $\log (n_e/{\rm cm^{3}}) \simeq 5$ when the recombination time scale of [O<span style="font-variant:small-caps;">vii</span>]{} is taken into account. Even if the warm absorber gas seems to be required by our fit procedure, its parameters are poorly constrained. Thus it is characterized by $\log U\simeq 0.85$, $\log (N_H/{\rm cm^{2}})\simeq 23.36$, and $\log (n_e/{\rm cm^{3}}) \ut< 7$.
This technique was successfully applied before to the Seyfert 1 Mrk 335 and the Seyfert 2/starburst galaxy NGC 1365 (@longinotti2008 [@guainazzi2009]). The main difference is that NGC 4051 is characterized by a strong warm absorber component in the high flux state that is still affecting the spectrum even when the nuclear flux is attenuated. Indeed, we found out in our analysis of the low flux state data that the effect of the line of sight medium is not negligible, particularly not in the modeling of the resonance line of the [O<span style="font-variant:small-caps;">vii</span>]{} triplet (see Fig. \[fig6\]) which is close to several absorption features. For example, the resonance line could be weakened by the same line in absorption[^2] (see e.g. @krongold). Nonetheless, the physical parameters of the warm absorber cannot be well-constrained by the analysis of the low flux state data (see Table 4).
The average distance $r$ of each of the photoionized plasma-emitting components from the nuclear source can be estimated by the definition of the ionization parameter $U$ after normalizing to the ionization luminosity $L_{ion}$. However, our results are insensitive to values of the electron density $n_e$ lower than $10^5$ cm$^{-3}$. In this limit, we can only determine a lower limit of the $X$-ray-emitting gas location (Table 4).
The analysis carried out in this paper allowed us to identify two ionization states for the line emitting gas and one warm absorber medium. It is interesting to note that\
- The $X$-ray emitting region can be placed at a distance of $r\ut> 0.05$ pc.
Indeed, @warmabsorber found that the NGC 4051 $X$-ray narrow-line regions can be placed at a distance of the same order of magnitude. This was also confirmed by the Chandra ACIS-S images of the same galaxy (@uttley), which showed a size of the diffuse emission smaller than that of the optical narrow-line regions (30 -220 pc, @christopoulou1997), thus implying a clear separation between the $X$-ray and optical emissions.
This is also naturally expected as a consequence of projection effects: as shown by @schmitt2003, who studied a sample of 60 Seyfert galaxies with the Hubble Space Telescope, the Seyfert 1 narrow-line regions objects are more circular and compact than those in the Seyfert 2 galaxies, with the Seyfert 2 subsample characterized by more elongated shapes. This agrees well with the unified picture according to which the conical narrow-line region of a Seyfert 1 galaxy is observed close to the axis of symmetry, while that of a Seyfert 2 galaxy is observed from an orthogonal line of sight.
Furthermore, the scale-length found in this paper is consistent with the inner radius of the torus in NGC 4051 as determined by @blustin2005, i.e. $r\simeq 0.15$ pc.\
- The NGC 4051 low state warm absorber is poorly constrained but its existence is nevertheless required by the fit. In particular, we found a lower limit of the warm absorber distance $\simeq 0.02$ pc, i.e. at least a factor $10$ larger than that measured in the high state flux (@krongold).
Indeed, by using the long $XMM$-Newton exposure of NGC 4051 in its high flux state and studying the time evolution of the ionization states of the $X$-ray absorbers, @krongold were able to put severe constraints on the physical and geometrical properties of the warm absorber medium.
They specifically found that the warm absorber consists of two different ionization components which are located within $3.5$ lt-days (or $0.0029$ pc) from the central massive black hole. This result allowed the authors to exclude an origin in the dusty obscuring torus because the expected dust sublimation radius[^3] is at least one order of magnitude larger. Hence the authors suggested a model in which the black hole accretion disk is at the origin of a $X$-ray absorber wind, which forms a conical structure moving upward.
$\begin{array}{c@{\hspace{0.05in}}c@{\hspace{0.05in}}c}
\epsfxsize=3.55in \epsfysize=3.55in \epsffile{f8a.eps} &
\epsfxsize=3.55in \epsfysize=3.55in \epsffile{f8b.eps}
\end{array}$
If this is the correct picture, when the continuum source is switched off, the compact warm absorber might not be observed anymore during the low state flux of NGC 4051. Our analysis showed instead the existence of a more exterior $X$-ray absorber, which absorbs the soft $X$-ray photons emitted from sources (as for example the inner surface of the conical structure proposed by @krongold) located (in projection) at scales larger than the torus and/or the narrow-line regions. Remarkably, this could indicate the existence of a diffuse warm material filling the wind-generated cone.
Figure \[figcartoon\] gives a qualitative representation of the model. During the high state flux (left panel) a two ionization component warm absorber (here labeled as II) lying within a few l-days ($\simeq 0.003$ pc) from the accreting black hole was identified by @krongold. @warmabsorber found that the NGC 4051 $X$-ray narrow-line regions can be placed at a distance of $r>0.02$ pc, while the optical narrow-line regions are on the scale of tenth of parsec (@christopoulou1997). During the low state flux (right panel), the interior warm absorber might not be observed anymore since the central engine is switched off. A more exterior warm absorber (labeled as I) could now absorb the $X$-ray photons emitted from sources located on the scale larger than the torus and/or the narrow-line regions.
This paper is based on observations from [*XMM*]{}-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We are grateful to the anonymous referee for the suggestions that improved the paper a lot. AAN is grateful to R. Porter for help with the Cloudy code, to C. Gordon for solving a few issues with the XSPEC package and to Yair Krongold for many fruitful conversations while writing this paper. Our acknowledgements also to Marco Castelli for drawing the cartoon in Fig. 8.
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-Newton Users Handbook, 2009, Issue 2.7, Edited by Ness J.-U. et al.
[^1]: The default energy range used by the Cloudy code to evaluate the number of ionizing photons $Q$ is $1$ Ryd - $7.354\times 10^{6}$ Ryd, [@ferland].
[^2]: As noted by @sakob and @ali, the resonance line of the [O<span style="font-variant:small-caps;">vii</span>]{} triplet could be also enhanced by photoexcitation. Note however that this would also result in a boost of all the higher order resonance transitions of the H-like and He-like ions (Ly-$\beta$, Ly-$\delta$, He-$\beta$ and He-$\delta$), but this enhancement is not currently observed.
[^3]: The torus inner edge has to be at a distance larger than the dust sublimation radius $r_{sub}$. In the particular case of NGC 4051, @krongold found $r_{sub}\simeq 0.01$ pc.
|
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abstract: |
In high-dimensional settings, sparse structures are critical for efficiency in term of memory and computation complexity. For a linear system, to find the sparsest solution provided with an over-complete dictionary of features directly is typically NP-hard, and thus alternative approximate methods should be considered.
In this paper, our choice for alternative method is sparse Bayesian learning, which, as empirical Bayesian approaches, uses a parameterized prior to encourage sparsity in solution, rather than the other methods with fixed priors such as LASSO. Screening test, however, aims at quickly identifying a subset of features whose coefficients are guaranteed to be zero in the optimal solution, and then can be safely removed from the complete dictionary to obtain a smaller, more easily solved problem. Next, we solve the smaller problem, after which the solution of the original problem can be recovered by padding the smaller solution with zeros. The performance of the proposed method will be examined on various data sets and applications.
author:
-
-
title: 'Accelerated Sparse Bayesian Learning via Screening Test and Its Applications\'
---
=1
Sparse Bayesian learning, screening test, classification, signal reconstruction
Introduction
============
For a dynamic system with measurements of input and output signals, system identification is a statistical methodology for building a mathematical model which is powerful enough to describe the characteristics of the system. A classic method for the modeling is called the least squares (LS), to which the systematic treatment is available in many textbooks[@Rao][@Smith][@sys_id]. When the LS problems are ill-conditioned, regularization algorithms could be employed to seek optimal solutions. The regularization terms can take various forms, and thus leads to various variants of the regularized least squares. In this thesis, we focus on sparsity inducing regularization.
Finding the sparsest representation of a signal provided with an over-complete dictionary of features is an important problem in many cases, such as signal reconstruction, compressive sensing[@app_cs], feature selection[@app_fs], image restoration[@app_ir] and so on. The existing work includes a variety of algorithms. The traditional sparsity inducing regularization methods, including orthogonal matching pursuit (OMP)[@omp], basis pursuit (BP)[@BasisPursuit_1994], LASSO[@lasso_1996], usually prefer a fixed sparsity-inducing prior and perform a standard maximum a posterior probability (MAP)[@mle_map] estimation afterwards, thus they can be regarded as Bayesian methods. While in this thesis, we focus on sparse Bayesian learning. This Bayesian method uses a parameterized prior to encourage sparsity, where hyper-parameters are introduced to make the framework more flexible. It’s worth mentioning that as an empirical Bayesian method, sparse Bayesian learning has connections with the kernel-based regularization method (KRM)[@krm] and machine learning[@krm_ml]. When the kernel structure and hyper-parameters are defined specifically, KRM will become sparse Bayesian learning, as discussed in [@tac2014].
In real-world applications, the collected data sets often have large scales and high dimensions, which leads us to consider whether there’s a way to screen some features out before solving the high-dimensional problems. We name such operation as “screening test”. Based on the assumption of sparsity, screening aims to identify features that have zero coefficients and discard them from the optimization safely, therefore the computational burden can be reduced.
In this work, we will propose a screening test for sparse Bayesian learning, and then obtain an accelerated sparse Bayesian learning.
Our contribution can be summarized as follows:
1. We propose a screening test for sparse Bayesian learning, which achieves an acceleration in computation time without changing the original optimal solution of the original sparse Bayesian learning.
2. We examine this accelerated sparse Bayesian learning on various data sets and applications to verify that this method works well for real-world data and problems that can be modeled as linear systems.
And the rest of this note is organized as follows. In *Section 2*, we introduce sparse Bayesian learning to see its assumptions, methodology, and verify its equivalence to an iterative weighted convex $\ell_1$-minimization problem; in *Section 3*, we design a screening test for the iterative weighted convex $\ell_1$-minimization problem in *Section 2*. The screening test accelerates the computation for each iteration of the $\ell_1$-minimization problem, and thus speeds up the entire sparse Bayesian learning. We check its performance by simulations on two real-world data sets. Next in *Section 4*, we apply the accelerated sparse Bayesian learning method to do classification and verify the trade-off between acceleration and classification accuracy. In *Section 5*, we use the accelerated sparse Bayesian learning method to do source localization and denoising in astronomical imaging. In this application, not all parameters are linear to the response, thus sampling should be used to deal with the nonlinear ones, then sparse Bayesian learning can play its role. Finally in *Section 6*, we summarize the previous sections.
This work was typeset using LaTeX. All the simulations were preformed by Python and MATLAB.
Sparse Bayesian Learning
========================
In this section, we will first introduce linear regression model, and then explore how to find a sparse solution by sparse Bayesian learning (SBL). As the theoretical derivation for SBL has been discussed a lot in [@wipf_2003][@wipf_2004][@wipf_2007], our illustration will be mainly focused on how it can be transformed to a sequence of weighted convex $\ell_1$-minimization problems.
Problem Formulation
-------------------
The theory of regression aims at modeling relationships among variables and can be used for prediction. Linear regression is an approach to modeling the relationships as linear functions. More specifically, we consider a linear regression model as below: $$\label{eq:lm}
Y={\Phi}\theta+V$$ where ${Y\in{{\mathbb R}}^N}$ is the output, ${{\Phi}}$ = $[{\phi}_1, {\phi}_2, \ldots, {\phi}_n]\in {{\mathbb R}}^{N \times n}$ is the regression matrix made up of $n$ features $\phi_i$, $\phi_i\in\mathbb{R}^N$, $\theta\in{{\mathbb R}}^n$ is the parameter to be estimated, and $V\in{{\mathbb R}}^N$ is the noise vector, $V\sim\mathcal{N}(0,\lambda I_N)$, $\lambda\in{{\mathbb R}}_+$.
One way to estimate $\theta$ is to minimize the least squares (LS) criterion: $$\begin{aligned}
\hat{\theta}^{LS} = \mathop{\arg\min}_{\theta}||Y-\Phi\theta||_2^2=(\Phi^T\Phi)^{-1}\Phi^T Y\end{aligned}$$
When $\Phi\in{{\mathbb R}}^{N\times n}$ with $N\ll n$ is rank deficient, i.e., $\text{rank}(\Phi)<N$ or close to rank deficient, the LS estimate is said to be ill-conditioned. To handle this issue, the method of regularization could be considered: $$\begin{aligned}
\hat{\theta}^R = \mathop{\arg\min}_{\theta} ||Y-\Phi\theta||_2^2 + \gamma J(\theta) \end{aligned}$$ where $\gamma\in{{\mathbb R}}_+$ is called the regularization parameter, and $J(\theta)$ is called the regularization term. There’re many choices for $J(\theta)$ with respect to the prior of $\theta$, in this thesis, we focus on sparsity inducing regularization.
Given $\Phi$ and $Y$, to find a sparse $\theta$, we should solve the following problem: $$\label{eq:sp}
\begin{split}
\min_{{\theta} \in { {{\mathbb R}}} ^ { n }}&\ ||\theta||_0\\
\text{s.t.}&\ ||Y-{\Phi}\theta||_2^2 \leq \epsilon\\
\end{split}$$ where $\epsilon\geq0$ is a tuning parameter that controls the size of the data fit. The cost function to be minimized represents the $\ell_0$ norm of $\theta$, i.e., the number of non-zero elements in $\theta$. Note that problem (\[eq:sp\]) is combinatorial, which means solving it directly requires an exhaustive search over the entire solution space. For example, in the noise-free case where $V=0$, we have to deal with up to $\binom{n}{N}$ linear systems of size $N\times N$[@matrix_computations]. Consequently, approximation methods should be considered. Several approximation methods have been proposed, and one of the most widely used methods is a convex relaxation obtained by replacing the $\ell_0$-norm with the $\ell_1$-norm: $$\label{eq:sp_relax}
\begin{split}
\min_{{\theta} \in { {{\mathbb R}}} ^ { n }}&\ ||\theta||_1\\
\text{s.t.}&\ ||Y-{\Phi}\theta||_2^2 \leq \epsilon\\
\end{split}$$
The convex relaxation (\[eq:sp\_relax\]) is equivalent to LASSO[@lasso_1996]: $$\begin{aligned}
\label{lasso}
\min _ { {\theta} \in { {{\mathbb R}}} ^ { n } } \frac { 1 } { 2 } || { Y } - \Phi {\theta} || _ { 2 } ^ { 2 } + \lambda || {\theta} || _ { 1 }\end{aligned}$$ where $\lambda\geq 0$ is the regularization parameter.
Methodology
-----------
In this section, we will illustrate the methodology of sparse Bayesian learning (SBL). It was first proposed by Tipping[@tipping_2001], and then applied for signal reconstruction[@wipf_2003] and compressive sensing[@app_cs]. Compared with classic $\ell_1$-penalty methods like basis pursuit[@BasisPursuit_1994] and LASSO[@lasso_1996], sparse Bayesian learning outperforms them in many aspects, for which a reasonable explanation is that one can show SBL is equivalent to an iterative weighted convex $\ell_1$-minimization problem[@wipf_2007].
### Parameterized Prior
Sparse Bayesian learning[@wipf_2003] starts by assuming a Gaussian prior for the parameter $\theta$ as: $$\begin{aligned}
\theta\sim\mathcal{N}(0,\Gamma(\gamma))\end{aligned}$$ where $\Gamma(\gamma)=\text{diag}(\gamma)$, $\gamma\in\mathbb{R}_{+}^n$. We denote this prior of $\theta$ by $p(\theta;\gamma)$.
Based on the above assumption, sparse Bayesian learning tends to minimize a different cost function in the latent variable space, say $\gamma$-space, where $\gamma\in{{\mathbb R}}_+^n$ is a vector of $n$ non-negative hyper-parameters governing the prior variance of each unknown $\theta_i$. Since the likelihood $p(Y|\theta)$ is also Gaussian as defined in (\[eq:lm\]), the corresponding relaxed posterior will be Gaussian: $$\begin{aligned}
p(\theta|Y,\gamma)=\frac{p(Y|\theta)p(\theta;\gamma)}{\int p(Y|\theta)p(\theta;\gamma) d\theta}\end{aligned}$$
Suppose this Gaussian to be $p(\theta|Y,\gamma)=\mathcal{N}(\mu_{\theta},\Sigma_{\theta})$, we can obtain that: $$\label{mu_va_poster}
\begin{split}
\mu_{\theta}=& \Gamma\Phi^T(\lambda I_N +\Phi\Gamma\Phi^T)^{-1}Y \\
\Sigma_{\theta}=& (\Gamma^{-1}+\frac{1}{\lambda}\Phi^T\Phi)^{-1}
\end{split}
$$ where $\Gamma=\text{diag}(\gamma)$.
### Type-II Estimation
Mathematically, sparse Bayesian learning tends to select the optimal $\gamma$, say $\hat{\gamma}$, to be the most appropriate $\gamma$ to maximize $p(Y;\gamma)$, which leads to a type-II estimation[@mle2]: $$\label{opti_gamma}
\begin{split}
\hat{\gamma} &=\mathop{\arg\max}_{\gamma\succeq 0} p(Y;\gamma)\\
&=\mathop{\arg\max}_{\gamma\succeq 0}\int p(Y|\theta)p(\theta;\gamma)d\theta\\
&=\mathop{\arg\max}_{\gamma\succeq 0}\int p(Y|\theta) \prod_{i=1}^n \mathcal{N}(\theta_i|0,\gamma_i) d\theta
\end{split}$$
Then for the optimal $\hat{\gamma}$, we set a threshold $\epsilon>0$, such that when $\hat{\gamma}_i\leq \epsilon$, the corresponding $\theta_i$ will be $0$.
Define $\Sigma_{Y}\triangleq\lambda I_N +\Phi\Gamma\Phi^T$, then it can be proved that the optimal $\hat{\gamma}$ in (\[opti\_gamma\]) can be obtained by minimizing the following function with respect to $\gamma$: $$\begin{aligned}
\label{cost_gamma}
\text{Loss}(\gamma)=\log|\Sigma_{Y}|+Y^T\Sigma_{Y}^{-1}Y
\end{aligned}$$
This theorem indicates that we successfully turn the problem (\[lasso\]) in $\theta$-space into a new problem in $\gamma$-space with respect to the new cost function in (\[cost\_gamma\]).
### Hyper-parameter Estimation
Since $\log|\Sigma_{Y}|$ is concave in $\gamma$-space, then we can make use of its concave conjugate. Denote $\log|\Sigma_{Y}|$ as $h(\gamma)$, then we have its concave conjugate as: $$\begin{aligned}
h^*( \gamma_h)=\min_{\gamma} { \gamma_h}^T\gamma-h(\gamma)\end{aligned}$$ which indicates that we can also express $h(\gamma)$ as: $$\begin{aligned}
h(\gamma)=\min_{ \gamma_h} { \gamma_h}^T \gamma-h^*( \gamma_h)\end{aligned}$$
Then we obtain an auxiliary cost function for $\text{Loss}(\gamma)$ as: $$\begin{aligned}
\label{opt_aux}
\text{Loss}(\gamma, \gamma_h)\triangleq { \gamma_h}^T\gamma-h^*( \gamma_h)+Y^T\Sigma_{Y}^{-1}Y\end{aligned}$$ which should be an upper bound of $\text{Loss}(\gamma)$, i.e.: $$\begin{aligned}
\text{Loss}(\gamma, \gamma_h) \geq \text{Loss}(\gamma)\end{aligned}$$
For any fixed $\gamma$, this bound should be attained by minimizing $\text{Loss}(\gamma, \gamma_h)$ over $ \gamma_h$, indicating that we should choose this optimal value of $ \gamma_h$, denoted by $\gamma_{h_{\text{opt}}}$, as: $$\begin{aligned}
\label{opt_dual}
\gamma_{h_{\text{opt}}}=\nabla_{\gamma}(\log|\Sigma_{Y}|)=\text{diag}[\Phi^T\Sigma_{Y}^{-1}\Phi]\end{aligned}$$
Finally, we come to the algorithm for sparse Bayesian learning in [@wipf_2007]:
\[alg\_sbl\]
Initialize $ \gamma_h$;[^1]
Solve the following optimization problem: $$\begin{aligned}
\label{min_origi}
\gamma \leftarrow \mathop{\arg\min}_{\gamma} \text{Loss}_{ \gamma_h}(\gamma) \triangleq { \gamma_h}^T\gamma+Y^T\Sigma_{Y}^{-1}Y
\end{aligned}$$
Compute the new $ \gamma_h$ based on $\gamma$ according to (\[opt\_dual\]);
Repeat (2) and (3), until $\gamma$ converges to some $\gamma_{\text{opt}}$;
Then the optimal $\theta$, denoted by $\theta_{\text{opt}}$, will be obtained as: $\theta_{\text{opt}}= E[\theta|Y;\gamma_{\text{opt}}]=\Gamma_{\text{opt}}\Phi^T(\lambda I_N +\Phi\Gamma_{\text{opt}}\Phi^T)^{-1}Y$.
As for how to find the optimal $\gamma$ in step 2, we have the following lemma from [@wipf_2007]:
The optimal $\gamma$ in (\[min\_origi\]) can be obtained by solving a weighted convex $\ell_1$-regularized cost function: $$\begin{aligned}
\label{mini_l1}
\theta^{\text{tmp}}=\mathop{\arg\min}_{\theta}||Y-\Phi\theta||_2^2+2\lambda\sum\limits_{i=1}^n \sqrt{\gamma_{h_i}}|\theta_i|
\end{aligned}$$
And then we set $\gamma_i=\frac{|\theta_i^{\text{tmp}}|}{\sqrt{\gamma_{h_i}}}, i=1,\ldots,n$.
By solving a sequence of weighted convex $\ell_1$-minimization problems with respect to $\theta$, we obtain a sparse optimal solution of SBL, where the sparsity is induced by the weighted $\ell_1$ regularization term.
Screening Test for SBL
======================
Motivation
----------
Screening test aims to quickly identify the inactive features in $\Phi$ that have zero components in the optimal solution $\hat{{\theta}}\ (\text{i.e.}\ \hat{{\theta}}_i=0)$, and then remove them from the optimization without changing the optimal solution. Therefore, the computational cost and memory usage will be saved, especially when $N$ and $n$ are extremely large. For example, when we solve LASSO, the computational complexity of solving it by least angle regression[@lasso_sol] is $O(Nn \min\{N, n\})$.
In this section, we will design a screening test for sparse Bayesian learning. Let us first define the index set for the $n$ features $\phi_1,\ldots,\phi_n$ in $\Phi$ as $\mathcal{I}$, i.e. $\mathcal{I}=\{1,2,\ldots,n\}$, then screening test is to find a partition of $\mathcal{I}$ as: $$\begin{aligned}
\mathcal{I}=S \cup \overline{S}, S\cap \overline{S} = \emptyset\end{aligned}$$ where features indexed by $S$ are selected, while the rest features indexed by $\overline{S}$ are rejected.
After the screening, the size of original problem will be reduced. Instead of solving the original problem to obtain the solution $\hat{{\theta}}$ directly, we have an alternative way made up of the three steps below:
1. Do the screening to obtain the reduced problem;
2. Solve the reduced problem to obtain $\hat{\theta}_r$;
3. Recover $\hat{{\theta}}$ from $\hat{{\theta}}_r$ according to the partition $\mathcal{I}$.
At present, screening rules for LASSO have been explored a lot, which can be roughly divided into two categories: the heuristic screening methods[@screen_h1][@screen_h2] and the safe screening methods[@screen_s1][@screen_s2][@screen_s3]. The heuristic screening methods, as their name indicates, cannot ensure all the screened features really deserve. In other words, some features that have non-zero coefficients may be mistakenly discarded. However, if the screening is safe, then the reduced problem should be equivalent to the original one. In other words, when all the features indexed in $\overline{S}$ are reasonable to be rejected, the optimal solution $\hat{{\theta}}$ will not change. As for the efficiency of screening, there are two evaluation metrics that we’re interested in:
- The size of $\overline{S}$ as a fraction of $\mathcal{I}$, say the screening percentage:\
screening percentage$=\frac{\#{\overline{S}}}{\#\mathcal{I}}$.
- The total time taken to seek the partition $\mathcal{I}=S \cup \overline{S}$ and to solve the reduced problem relative to the time taken to solve the original problem directly without screening, say the speedup factor:\
speedup factor$=\frac{t_{\text{scr}} + t_{\text{red}}} {t_{\text{ori}} }\triangleq \frac{t_r}{t}$.
where $t_{\text{ori}}$ is the time to solve the original problem, $t_{\text{scr}}$ is the time to do screening, $t_{\text{red}}$ is the time to solve the reduced problem; and the notation shall be further simplified as $\frac{t_r}{t}$, where $t_r$ is the total time for the reduced case, $t$ is the same as $t_{\text{ori}}$.
Methodology
-----------
We try to design a screening test for the optimization problem in line $3$ of Algorithm \[alg\_sbl\]: $$\begin{aligned}
\label{lasso_revised}
\min _ { {\theta} \in { {{\mathbb R}}} ^ { n } } \frac { 1 } { 2 } || { Y } - \Phi {\theta} || _ { 2 } ^ { 2 } + \lambda\sum_{i=1}^{n}u_i^{(k)}|\theta_i|\end{aligned}$$ where the second term $\lambda \sum\limits_{i=1}^{n}u_i^{(k)}|\theta_i|$ is a weighted $\ell_1$-norm of $\theta$.
This is a LASSO-type problem. Based on the screening tests for LASSO[@Xiang2011][@wangjie][@XiangWR14], we will propose a safe screening test for (\[lasso\_revised\]), where the procedure including models, theorems, lemmas and so on, must be revised accordingly. To guarantee the accuracy and completeness of the thesis, we will go through all the details including proofs during the revision. Let’s start from the dual formulation first.
### Dual Formulation
By introducing $z=Y-\Phi\theta$ into (\[lasso\_revised\]), the primal problem becomes: $$\begin{aligned}
\label{primal_z}
\begin{split}
\min_{\theta\in{{\mathbb R}}^n}\ &\frac{1}{2}||z||_2^2+\lambda\sum_{i=1}^{n}u_i^{(k)}|\theta_i|\\
\text{s.t.}\ &z=Y-\Phi\theta
\end{split}\end{aligned}$$
Moreover, it can be proved that the dual problem of (\[lasso\_revised\]) should be: $$\begin{aligned}
\label{dual_u}
\begin{split}
\max_{\eta\in{{\mathbb R}}^N}\ &\frac{1}{2}||Y||^2-\frac{1}{2}||\eta-Y||^2\\
\text{s.t.}\ &| {\frac{\phi_i^T\eta}{\lambda u_i^{(k)}} } |\leq 1,i=1,\ldots,n
\end{split}\end{aligned}$$
Note that (\[primal\_z\]) is a convex problem with affine constraints. By Slater’s condition[@convex_opt], as long as the problem is feasible, strong duality will hold. Then we denote $(\hat{\theta},\hat{z})$ and $\hat{\eta}$ as optimal primal and dual variables, and make use of the Lagrangian again: $$\begin{aligned}
\mathcal{L}(\theta,z,\eta)=\frac{1}{2}||z||_2^2+\lambda \sum_{i=1}^{n}u_i^{(k)}|\theta_i|+\eta^T(Y-\Phi\theta-z)\end{aligned}$$
According to the Karush–Kuhn–Tucker (KKT) conditions[^2], we have: $$\begin{aligned}
\begin{split}
_{\theta}0\in\partial_{\theta} \mathcal{L}(\hat{\theta},\hat{z},\hat{\eta})&=-\Phi^T\hat{\eta}+\lambda u^{(k)}.*v,\\
&\text{where}\ ||{v}||_{\infty}\leq 1\ \text{and}\ {v}^T\hat{\theta}=||\hat{\theta}||_1;\\
\nabla_z \mathcal{L}(\hat{\theta},\hat{z},\hat{\eta}) &= \hat{z}-\hat{\eta}=0;\\
\nabla_{\eta} \mathcal{L}(\hat{\theta},\hat{z},\hat{\eta}) &= Y-\Phi\hat{\theta}-\hat{z}=0.
\end{split}\end{aligned}$$
By solving the equations above, we have: $$\begin{aligned}
Y=\Phi\hat{\theta}+\hat{\eta}\end{aligned}$$
And there exists a specific $\hat{v}\in\partial||\hat{\theta}||_1$ such that: $$u.*\hat{v}=\frac{{\Phi^T}{\hat{\eta}}}{\lambda},||\hat{v}||_{\infty}\leq1,\hat{v}^T\hat{\theta}=||\hat{\theta}||_1$$ which is equivalent to: $$|\frac{\phi_i^T\hat{\eta}}{ \lambda u_i^{(k)} }|\leq 1,i=1,2,...n$$
And we can further conclude: $$\label{nec_condi}
\begin{split}
\frac{\hat{\eta}^T\phi_i}{ \lambda u_i^{(k)} }=\left\{\begin{array}{ll}{\text{sign}(\hat{\theta}_i),} & {\text { if } \hat{\theta}_{i}\neq 0} \\ {[-1,1],} & {\text { if } \hat{\theta}_{i}=0}\end{array}\right.
\end{split}$$ which indicates the theorem below:
\[thm:suff\] $$|\frac{\hat{\eta}^T\phi_i}{\lambda u_i^{(k)}}|<1 \Rightarrow \theta_i=0, i=1,\ldots,n$$
### Region Test
Theorem \[thm:suff\] works as a sufficient condition to reject $\phi_i$: $$\begin{aligned}
|\frac{\hat{\eta}^T\phi_i}{\lambda u_i^{(k)}}|<1\end{aligned}$$
i.e.: $$\begin{aligned}
\max\{\frac{\hat{\eta}^T\phi_i}{\lambda u_i^{(k)}},-\frac{\hat{\eta}^T\phi_i}{\lambda u_i^{(k)}}\}< 1\end{aligned}$$
However, the optimal $\hat{\eta}$ is not available, which leads us to consider alternative methods. Region test is a good choice which works by bounding $\hat{\eta}$ in a region $\mathcal{R}$. Since there might be vectors other than $\hat{\eta}$ in $\mathcal{R}$, it will be harder[^3] for us to reject each $\phi_i$, therefore the sufficient condition will be relaxed. This relaxation can be expressed as a new theorem:
Suppose we find a region $\mathcal{R}$ such that $\hat{\eta}\in\mathcal{R}$, then: $$|\frac{{\eta}^T\phi_i}{\lambda u_i^{(k)}}|<1, \forall \eta\in\mathcal{R} \Rightarrow \theta_i=0, i=1,\ldots,n$$
Note that the optimal $\mathcal{R}$ in theory should be $\mathcal{R}=\{\hat{\eta}\}$. For convenience, we will define $\mu_{\mathcal{R}}(\phi_i)=\max\limits_{\eta\in\mathcal{R}} \frac{{\eta}^T\phi_i}{\lambda u_i^{(k)}}$, then the sufficient condition will become: $$\begin{aligned}
\label{criterion}
\max \{\mu_{\mathcal{R}}(\phi_i), \mu_{\mathcal{R}}(-\phi_i)\} <1 \Rightarrow \theta_i=0, i=1,\ldots,n\end{aligned}$$
Next, we will try to find an appropriate region $\mathcal{R}$. For the design of the region, the idea is quite similar to that of [@Xiang2011] and [@XiangWR14], however to guarantee the accuracy and completeness of the thesis, we will go through the construction of $\mathcal{R}$ from scratch.
[**Sphere Test**]{}
The simplest region is a sphere[@Xiang2011] decided by observing the objective function in (\[dual\_u\]). We notice that $\eta$, as the dual variable of $\theta$, is the projection of $Y$ on the feasible set: $$\mathcal{F}=\{ \eta:| {\frac{\phi_i^T\eta}{\lambda u_i^{(k)}} } |\leq 1,i=1,\ldots,n \}$$
If we can find a feasible point $\eta '$, then we will obtain a sphere to bound $\hat{\eta}$ with $\eta'$ as the sphere center. The sphere center can be chosen as: $$\begin{aligned}
\eta'=\frac{\lambda Y u_\text{min}^{(k)}}{\lambda_\text{max}}\end{aligned}$$ where $\lambda_\text{max}=\max\limits_{i} \phi_i^T Y$, $u_\text{min}^{(k)}=\min \{ u_1^{(k)},\ldots, u_n^{(k)} \}$. Then the sphere should be: $$\begin{aligned}
\label{sphere}
B(c,r)=\{\eta:||\eta-c||_2\leq r\}\end{aligned}$$ where $c=Y$, $r=||\eta'-Y||_2$. We can further compute the corresponding $\mu_{B}(\phi_i)$ as below: $$\begin{aligned}
\mu_{B}(\phi_i) &= \max\limits_{\eta\in B(c,r)}\frac{\eta^T\phi_i}{\lambda u_i^{(k)}}\\
&=\frac{1}{\lambda u_i^{(k)}}(\eta^T-c^T)\phi_i+\frac{c^T}{\lambda u_i^{(k)}}\phi_i\\
&\leq \frac{r}{\lambda u_i^{(k)}}||\phi_i||_2+\frac{c^T\phi_i}{\lambda u_i^{(k)}}\end{aligned}$$
Thus according to (\[criterion\]), the sphere test should be: $$\begin{aligned}
\label{sphere_test_u}
T_{B}(\phi_i)=\left\{\begin{array}{ll}{1,} & {\text { if }\left|c^{T} \phi_i\right|<\lambda u_i^{(k)}-r||\phi_i||_{2}} \\ {0,} & {\text { otherwise. }}\end{array}\right.\end{aligned}$$ where $1$ indicates $\theta_i$ is zero and $\phi_i$ can be rejected, $0$ indicates $\theta_i$ is non-zero and $\phi_i$ should be reserved.
[**Dome Test**]{}
Based on sphere test, we improve the region that bounds the optimal $\hat{\eta}$ by introducing a hyperplane[@Xiang2011], then the new region should be defined as: $$\mathcal{D}(c,r;n,h) = \{\eta:{n}^{T} {\eta} \leq h, ||{\eta}-c||_{2} \leq r\}$$ which shares the same $c,r$ as sphere test, however will further select a specific pair of $(n,h)$ among the $2n$ linear constraints (half spaces) in (\[dual\_u\]), where $n = \pm \frac{\phi_i}{||\phi_i||_2}$, $h=\frac{\lambda u_i^{(k)}}{||\phi_2||_2}$. With proper selection of $n$ and $h$, the selected hyperplane will cut into the sphere and bound the optimal $\hat{\eta}$ in a tighter region. We call such a region as “dome”.
For the selection of $n$ and $h$, we should define the following variables as preparations:
- $c_d$, the dome center on the hyperplane, for which the line passing $c$ and itself is in the direction $-n$;
- $\psi_d$, the fraction of the signed distance from $c$ to $c_d$ compared with the sphere radius $r$;
- $r_d$, the largest distance a point can move from $c_d$ within the dome and hyperplane.
These variables can be expressed in geometry as:
\[region\] ![Dome test[]{data-label="fig:4"}](region.jpg "fig:"){width="0.4\linewidth"}
By Euclidean geometry, the following relationships among these variables will be obtained: $$\begin{aligned}
\begin{array}{c}{\psi_{d}=\left({n}^{T} c-h\right) / r} \\ {{c}_{d} = {c}-\psi_{d} r {n}} \\ {r_{d}=r \sqrt{1-\psi_{d}^{2}}}\end{array}\end{aligned}$$
To ensure that $c_d$ is inside the sphere $B(c,r)$, we require $-1\leq\psi_{d}\leq 1$. Now, the optimal $\phi$, say $\phi_g$, should be the $\pm \phi_i$ that attains the smallest intersection of one half space and the sphere, thus $\psi_{d}$ should be maximized: $$\begin{aligned}
\label{phig_u}
\phi_g=\mathop{\arg\max}_{\{\pm\phi_i\}_{i=1}^n} \frac{ \phi_i^{T} c-\lambda u_i^{(k)}}{||\phi_i||_{2}}\end{aligned}$$
This optimal $i$ will be recorded as $i^*$ in discussion afterwards. As for $\mu(\phi_i)$ for dome, we have the following lemma revised from [@Xiang2011]:
\[lem:dt\] For a fixed dome $\mathcal{D}(c,r;n,h)$ satisfying $|\psi_{d}|\leq 1$, the corresponding $\mu_{\mathcal{D}}(\phi_i)$ should be: $$\begin{aligned}
\label{DT_criterion}
\mu_{\mathcal{D}}(\phi_i) = \frac{1}{\lambda u_i^{(k)}} [c^T\phi_i+M_1(n^T\phi_i,||\phi_i||_2)]
\end{aligned}$$ where $$M_1(t_1,t_2) = \\
\left\{\begin{array}{ll}{rt_2,} {\text { if } t_1<-\psi_dt_2} \\
{-\psi_{d}rt_1+r\sqrt{t_2^2-t_1^2}\sqrt{1-\psi_d^2},} {\text { if } t_1\geq-\psi_dt_2}\\
\end{array}\right.$$
Thus the dome test should be designed as:
\[thm:dt\] The screening test for a fixed dome $\mathcal{D}(c,r;n,h)$ should be: $$\begin{aligned}
T_{\mathcal{D}}(\phi_i) = \left\{
\begin{array}{ll}{1,} & { \text { if } V_l(n^T\phi_i,||\phi_i||_2)< c^T\phi_i <V_u(n^T\phi_i,||\phi_i||_2) } \\
{0,} & {\text { otherwise. }}\end{array}\right.
\end{aligned}$$ where $V_u(t_1,t_2)=\lambda u_i^{(k)}-M_1(t_1,t_2)$ and $V_l(t_1,t_2)=-V_u(-t_1,t_2)$.
[**Two Hyperplane Test**]{}
Based on the dome test, we try to introduce one more hyperplane to the region[@XiangWR14], which ensures a better bound for $\hat{\eta}$. However, it’s necessary to guarantee that the new intersection of a single sphere and two hyperplanes should be non-empty. For this purpose, we make use of the following lemma from [@XiangWR14]:
Let the sphere $B(c,r)$ and half space $(n,h)$ bound the dual optimal solution $\hat{{\eta}}$ with the dome $ \mathcal{D}(c,r;n,h) = \{\eta:{n}^{T} {\eta} \leq h, ||{\eta}-c||_{2} \leq r\} $ satisfying $0<\psi_{d}\leq 1$, then the new sphere $B(c_d,r_d)$, which is smaller than the original $B(c,r)$, is the circumsphere of the dome $\mathcal{D}$ and thus still bounds $\hat{{\eta}}$.
Based on this lemma, we can name $\phi_g$ as $\phi^{(1)}$, and then select a $\phi^{(2)}$ other than $\phi^{(1)}$. This $\phi^{(2)}$ should ensure the smallest intersection of $B(c_d,r_d)$ and one of the rest half spaces: $$\begin{aligned}
\label{phi2_u}
\phi^{(2)}=\mathop{\arg\max}_{\{\pm\phi_i\}_{i=1}^n \backslash\phi^{(1)}} \frac{ \phi_i^{T} c_d-\lambda u_i^{(k)}}{||\phi_i||_{2}}\end{aligned}$$
We call this optimal $i(i\neq i^*)$ as $j^*$, and now the intersection among one sphere and two hyperplanes should be non-empty. Therefore we can finally define the region denoted by $\mathcal{H}_2$ as: $$\begin{aligned}
\mathcal{H}_2 = \mathcal{H}_2(c,r;n_1,h_1,n_2,h_2)\end{aligned}$$ where $c=Y$, $r=||\eta'-Y||_2$, $n_1 = \frac{\phi^{(1)}}{||\phi^{(1)}||_2}$, $h_1=\frac{\lambda u_{i^*}^{(k)}}{||\phi^{(1)}||_2}$, $n_2 = \frac{\phi^{(2)}}{||\phi^{(2)}||_2}$, $h_2=\frac{\lambda u_{j^*}^{(k)}}{||\phi^{(2)}||_2}$. And we can express the region as the figure below:
\[region\_tht\] ![Two hyperplane test[]{data-label="fig:5"}](region_tht.jpg "fig:"){width="0.6\linewidth"}
As for the criterion $\mu(\phi_i)$ for $\mathcal{H}_2$, we revise the lemma in [@XiangWR14] to obtain:
Fix the region $\mathcal{H}_2(c,r;n_1,h_1;n_2,h_2)$, suppose $\psi_i$ satisfies $\left|\psi_i\right|\leq1,i=1,2$ and $\arccos \psi_1 + \arccos \psi_2\geq \arccos (n_1^T n_2)$, define: $$\begin{aligned}
h(x,y,z)=\sqrt{ (1-\tau^2)z^2+2\tau xy-x^2-y^2 }
\end{aligned}$$ where $\tau = n_1^T n_2$. Then for $\phi_i\in {{\mathbb R}}^N$, we have: $$\begin{aligned}
\label{THT_criterion}
\mu_{\mathcal{H}_2}(\phi_i) =\frac{1}{\lambda u_i^{(k)}}[ c^T\phi_i+M_2(n_1^T\phi_i,n_2^T\phi_i,||\phi_i||_2)]
\end{aligned}$$ where\
$$M_2(t_1,t_2,t_3) = \\
\left\{\begin{array}{ll}{rt_3,} {\text { if } (a)} \\
{-rt_1\psi_1+r\sqrt{t_3^2-t_1^2}\sqrt{1-\psi_1^2},} {\text { if } (b)}\\
{-rt_2\psi_2+r\sqrt{t_3^2-t_2^2}\sqrt{1-\psi_2^2},} {\text { if } (c)}\\
{ -\frac{r}{1-r^2}[(\psi_1-\tau\psi_2)t_1+(\psi_2-\tau\psi_1)t_2] }\\
{+\frac{r}{1-r^2}h(\psi_1,\psi_2,1)h(t_1,t_2,t_3), }\ {{otherwise}} \end{array}\right.$$ and conditions $(a),(b),(c)$ are given by: $$\begin{aligned}
(a)&\ t_1<-\psi_1t_3\ \&\ t_2<-\psi_2t_3;\\
(b)&\ t_1\geq-\psi_1t_3\ \&\ \frac{\left(t_{2}-\tau t_{1}\right)} {\sqrt{t_{3}^{2}-t_{1}^{2}}} < \frac{\left(-\psi_{2}+\tau \psi_{1}\right)} { \sqrt{1-\psi_{1}^{2}}};\\
(c)&\ t_2\geq-\psi_2t_3\ \&\ \frac{\left(t_{1}-\tau t_{2}\right)} {\sqrt{t_{3}^{2}-t_{2}^{2}}} < \frac{\left(-\psi_{1}+\tau \psi_{2}\right)} {\sqrt{1-\psi_{2}^{2}}};
\end{aligned}$$
Then the two hyperplane test can be designed as:
The two hyperplane test for the region $\mathcal{H}_2 = \mathcal{H}_2(c,r;n_1,h_1;n_2,h_2)$ is: $$T_{\mathcal{H}_2}(\phi_i) = \left\{\begin{array}{ll}{1,} & {\text { if } (a')} \\ {0,} & {\text { otherwise. }}\end{array}\right.$$ where condition $(a')$ is: $$V_{l}\left({n}_{1}^{T} \phi_i, {n}_{2}^{T} \phi_i,||\phi_i||_{2}\right)<c^{T} \phi_i<V_{u}\left({n}_{1}^{T} \phi_i, {n}_{2}^{T} \phi_i,||\phi_i||_{2}\right)$$ with $V_u(t_1,t_2,t_3)=\lambda u_i^{(k)}-M_2(t_1,t_2,t_3)$ and $V_l(t_1,t_2,t_3)=-V_u(-t_1,-t_2,t_3)$.
As for the computational complexity, the two hyperplane test requires $n$ triples of $(n_1^T\phi_i,n_2^T\phi_i,||\phi_i||_2)$ with the help of $u_i^{(k)}$, $i=1,2,\ldots,n$, thus the computational complexity should be $O(Nn)$. It’s also worth mentioning that if we continue increasing the number of hyperplanes, the region test should be more complicated, however will have the potential to reject more features since the region that bounds $\hat{\eta}$ should be tighter. Here we stop at $m=2$, and summarize the new algorithm which is similar to the THT algorithm in [@XiangWR14] as Algorithm \[alg\_wtht\], and name it as weighted-THT (W-THT):
\[alg\_wtht\]
$\phi^{\text{norm}}_i\leftarrow ||\phi_{i}||_2,i=1,\dots,n$;
$c\leftarrow Y$;(sphere center)
$\rho_i\leftarrow c^T\phi_{i},i=1,\dots,n$;
$\lambda_{\text{max}}\leftarrow {\text{max}_i\left|\rho_i\right|}$;
$\eta_{\mathcal{F}}=\frac {\lambda Y u_\text{min}^{(k)}} {\lambda_{\text{max}}}$;
$r\leftarrow||\eta_{\mathcal{F}}-c||_2$;(sphere radius)
$i_*\leftarrow \arg\max_i\frac{\left|\rho_i\right|-\lambda u_i^{(k)}}{\phi^{\text{norm}}_i}$;
$n_1\leftarrow \text{sign}(\rho_{i^*})\phi_{i_*}/{\phi^{\text{norm}}_{i_*}}$;
$h_1\leftarrow \lambda u_{i^*}^{(k)}/{\phi^{\text{norm}}_{i_*}}$;
$a\leftarrow n_1^Tc-h_1$;
$\sigma_i\leftarrow n_1^T\phi_i,i=1,\ldots,n$;
$t_i\leftarrow \rho_i-a\sigma_i,i=1,\ldots,n$;
$j_*\leftarrow \arg\max \limits_{i\neq i_*}\frac{\left|t_i\right|-\lambda u_i^{(k)}}{\phi^{\text{norm}}_i}$;
$n_2\leftarrow \text{sign}(\rho_{j^*})\phi_{j_*}/{\phi^{\text{norm}}_{j_*}}$;
$h_2\leftarrow \lambda u_{j^*}^{(k)} /{\phi^{\text{norm}}_{j_*}}$;
$\tau_i \leftarrow n_2^T\phi_{i}, i=1,\ldots,n$;
$v_i\leftarrow [V_l(\sigma_i,\tau_i,{\phi^{\text{norm}}_i})<\rho_i<V_u(\sigma_i,\tau_i,{\phi^{\text{norm}}_i})].$
In line $17$: for condition $a$, $[a]$ returns to $1$ (TRUE) if $x$ is true.
Simulation
----------
In this section, we conduct experiments to verify that the proposed sparse Bayesian learning with screening test does outperform in speed while keeping the optimal solution unchanged at the same time. To solve (\[lasso\_revised\]) we use CVX, a package for specifying and solving convex programs[@cvx_1][@cvx_2].
### Real-world Data Sets
The experiments are based on real-world data sets. These data sets often have complicated structures which will affect the performance of the screening, and we will model them as a linear system in (\[eq:lm\]). The two data sets we used are listed as below:
- MNIST handwritten image data (MNIST)[@mnist1].
MNIST is made up of $70,000$ images ($28\times 28$) as a record for handwritten digits. It has $60,000$ images in the training set and $10,000$ images in the testing set. We will vectorize all the images as $784$-dimensional vectors and scale them to unit norm. Then we randomly selected $10,000$ images in the training set to be the columns of our regression matrix $\Phi$ ($1,000$ for each digit), and randomly sample one target image from the testing set as $Y$. Therefore, we will do a simulation with $\Phi\in{{\mathbb R}}^{784\times 10000}$ and $Y\in\mathbb{R}^{784}$.
- New York Times bag-of-words data (NYTW)[@Lichman].
This data set can be downloaded from the UCI Machine Learning Repository. The raw data can be stored as a matrix which contains $300,000$ documents expressed as vectors with respect to a vocabulary of $102,660$ words. In this matrix, the element $(i,j)$ represents the number of occurrences of the $i$th word in the $j$th document. We will preprocess the raw data by randomly selecting $50,000$ documents and $5,000$ words to become the regression matrix $\Phi\in{{\mathbb R}}^{5000\times 50000}$; and the response $Y\in{{\mathbb R}}^{5000}$ will be the subset of randomly-chosen document column with respect to the $5,000$ words in the regression matrix.
### Results and Analysis
When it comes to the performance of the proposed method, we should set a metric for different data sets. A possible choice is to make use of $\lambda_{\text{max}}$. Recall that we define $\lambda_{\text{max}}=\max\limits_{i} \phi_i^T Y$ during the construction of sphere, then we can use the ratio $\lambda/\lambda_{\text{max}}$ as measure of regularization. The simulation results for MNIST with respect to screening percentage and time reduction are shown as the following two figures:
{width="0.6\linewidth"}
{width="0.6\linewidth"}
Moreover, to ensure the optimal solution doesn’t change, we can check whether the optimal solution changes by computing: $$\begin{aligned}
\text{max}\ |\theta_o-\theta_s|\end{aligned}$$ where $\theta_o$ is the solution without screening, $\theta_s$ is the solution with screening. And the maximum of these absolute values turns to be zero, which indicates the optimal solution doesn’t change.
Similarly, the two figures can also be plotted for NYTW as:
{width="0.55\linewidth"}
{width="0.55\linewidth"}
The two curves are a bit different from those of MNIST, while the tendencies are alike.
### Conclusions
In this section, we manage to speed up sparse Bayesian learning by screening test. As we can see in the figures, the acceleration will increase as $\lambda/\lambda_{\text{max}}$ goes larger, especially when $\lambda=\lambda_{\text{max}}$, the region $\mathcal{R}$ for the region test is nearly empty, thus almost all the features are rejected, which is is consistent with the our illustration in region test. What’s more, to verify the proposed sparse Bayesian learning does work smoothly without making damage to the original optimal solution, we also checked whether the two solutions are identical.
We should note that this acceleration is not so attractive when $\lambda/\lambda_{\text{max}}$ is too small, which is consistent with the performance of the THT in [@XiangWR14], this can be explained by observing the region $\mathcal{R}$. The smaller $\lambda$ is, the larger the sphere will be, thus the looser our bound will become.
What’s more, considering what $\lambda$ represents (the noise variance for the linear system), the larger it is, the noisier our system will be. For different data sets, the numerical performances of the proposed sparse Bayesian learning with screening test should be different, however it still can be concluded that the screening test is indeed safe and efficient.
By choosing $\lambda$ appropriately, the optimal solution with respect to the specific $\lambda$ will be obtained more efficiently without making too much damage to the accuracy. In other words, there is a trade-off between acceleration and accuracy.
Application to Classification Problem {#chapter05}
=====================================
In this section, we will apply the proposed method to do classification for real-world data sets. We will do classification for MNIST[@mnist1] data set, which we have used in the last section.
Introduction
------------
In the last section, we had a brief introduction for MNIST, and used it to verify the proposed sparse Bayesian learning with screening test does outperform in speed while keeping the optimal solution unchanged at the same time. However, the simulation in the last section is lacking in value of application, in other words, we only verified that screening test works for sparse Bayesian learning, but ignored the discussion on how the acceleration via screening test can make contributions to real-world applications.
Now we will do classification for MNIST by the proposed method to check its practical performance. The figure below provides some samples in MNIST indicating the images can be classified with respect to the digits $0,1,\ldots,9$:
{width="0.6\linewidth"}
This dataset is a popular tutorial for image classification in machine learning, for which lots of techniques and frameworks have been developed. The $70,000$ images ($60,000$ for training and $10,000$ for testing) of handwritten digits are in grayscale and share a resolution of $28\times 28$. What’s more, the numerical pixel values for the images are integers between $0$ and $255$.
Methodology
-----------
The simulation settings are similar to what we did in Section 4, we vectorize and scale the images in the data set to construct a linear system in (\[eq:lm\]). However, this time we will do classification by cross validation with respect to the optimal solution obtained for different $\lambda/\lambda_{\text{max}}$.
The methodology is shown as below:
1. To make the result more convincing, we will make use of Monte-Carlo method[@monte_carlo], which defines the first loop of size $N_1$.
2. Next, for each $N_1$, the same grid of $\lambda$ will be generated, the length of grid should be $N_2$, which is our second loop.
3. For each $N_1$ and the specific grid of $\lambda$, we randomly choose $N_3$ target images as a testing batch for $Y$, and find the sparse representations accordingly by the proposed method based on the $10,000$ images selected in $\Phi$, i.e.: $$\begin{aligned}
Y^{(i)} = \Phi\theta^{(i)} + V^{(i)}, i=1,\ldots,N_3
\end{aligned}$$ where $Y^{(i)}\in\mathbb{R}^{784}$ is the vectorized target image, $\Phi\in\mathbb{R}^{784\times 10000}$, $V^{(i)}\in\mathbb{R}^{784}$ is the unknown noise vector, and $\theta^{(i)}\in\mathbb{R}^{10000}$ is the parameter to be estimated.
4. Since the columns in the regression matrix $\Phi$ represent different handwritten digits, we can accumulate the elements in $\theta$, i.e., weights of the feature images, to decide the classification. Since the weights could be negative, so we will add up the absolute values of $\theta_i$: $$\begin{aligned}
ABS_k = \sum\limits_{\phi_i\text{ represents digit k}}|\theta_i|, k=0,\ldots,9
\end{aligned}$$ and then define the metric $prob_k$ as: $$\begin{aligned}
prob_k =\frac{ABS_k}{ \sqrt{\sum\limits_{i=1}^k ABS_k^2}}, k=0,\ldots,9
\end{aligned}$$ where $prob_k\in[0,1]$.
5. Decide the classification by the largest $prob_k$, and compare it with the truth.
6. For each value of $\lambda$, we should first gather $N_3$ classification results to obtain the classification accuracy for each Monte-Carlo simulation, and then compute the average accuracy with respect to $N_1$ Monte-Carlo simulations as overall accuracy. The overall accuracy should be with respect to the defined grid of $\lambda$. Standard error of the overall accuracy should be available as well.
Simulation Result
-----------------
We let $N_1=50,N_2=11,N_3=100$, i.e., the number of Monte-Carlo simulations is $50$, $\lambda$ is selected as $[0,0.1,\ldots,0.9,1.0]$, and $100$ images are considered in the testing batch.
To visualize the prediction, we can make use of color bar to display the value of $prob_k$. For example, we can check the prediction with respect to a small interval of $\lambda$ as below:
![Classification for MNIST - color bar[]{data-label="app1_bar"}](classification_mnist.png){width="0.95\linewidth"}
As we can see, for a fixed $\lambda$, the red line represents the true digit of the target image, while the color blocks represent the values of $prob_k$, and the colors are decided with respect to the color bar on the right side of the figure. In this figure, as $\lambda$ increases from zero, the prediction will be closer to the truth. However, this is only the case for a small interval of $\lambda$; also, it’s just one of the images in the testing batch, the overall accuracy should be computed based on $100$ testing images and $50$ Monte-Carlo simulations.
Based on all the simulations, finally we can obtain the classification accuracy with standard error as below:
{width="0.8\linewidth"}
This figure indicates that as $\lambda/\lambda_{\text{max}}$ goes larger, the accuracy for classification will decrease first, increase afterwards, and decrease again in the end. Even though in this simulation, we obtain the largest accuracy when $\lambda/\lambda_{\text{max}}\in(0,0.1)$, it’s still acceptable to sacrifice some accuracy to save computation time.
Conclusions
-----------
This section examines the performance of the proposed method on a classical data set for classification: MNIST, where the classification is decided by the scaled accumulation of weights. As Section 3 indicates, the acceleration by screening is not so attractive when $\lambda/\lambda_{\text{max}}$ is too small. So in this application, we have two goals:
- To make sure sparse Bayesian learning works for such kind of classification.
This is the minimum requirement, otherwise the acceleration will have no foundation.
- To explore whether significant acceleration can be achieved.
Even if sparse Bayesian learning works, we cannot make sure whether to use screening test is meaningful. If the classification accuracy crashes as $\lambda$ goes too large, then the acceleration will be unreasonable. We want to select a $\lambda$ that balances the acceleration and accuracy.
The simulation results indicate that our classification for MNIST can achieve both of the two goals successfully.
Application to Signal Reconstruction {#chapter06}
====================================
In this section, we will apply the proposed method to signal reconstruction in astronomical imaging. In signal reconstruction and image processing, provided with the prior knowledge that the signal (or image) has very few nonzero components, sparse Bayesian learning with screening test can be put into good use.
Astronomical images with many pixels can be represented by a series of point sources. To achieve source localization and denoising, we will model the signal as a linear combination of a set of features. We should also note that this framework is not limited to astronomical imaging, but can also be extended to other systems that can be modeled alike.
Problem Formulation
-------------------
In this application, the proposed method will be used for performing dictionary learning to determine an optimal feature set for reconstructing a signal representing light sources. The signal of multiple light sources to be constructed should be generated as linear combinations of single-source signals with Gaussian noise, and the performance of reconstruction will be evaluated according to scientific metrics.
First, we should introduce a fluorescence model as described in [@app2]. For a single source, the expected photon count depends on the choice of point spread function (PSF). Here we approximate a 3-dimensional PSF by a Gaussian distribution as below: $$\operatorname{PSF}(x, y, z)=\frac{1}{\sqrt{8 \pi^{3} \sigma_{x y}^{2} \sigma_{z}}} e^{-\frac{1}{2 \sigma_{x y}^{2}}\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]-\frac{\left(z-z_{0}\right)^{2}}{2 \sigma_{z}^{2}}}$$
Then the PSF must be integrated over the pixel area to become the expected photon count at each pixel: $$\label{app2_form}
\mu_{i j k} =I_{i j k} \Delta E_{x y}\left(x_{i}-x_{0}\right) \Delta E_{x y}\left(y_{j}-y_{0}\right) \Delta E_{z}\left(z_{k}-z_{0}\right)+b g$$ with $$\Delta E_{\mathrm{k}}(u)=\frac{1}{2}\left[\operatorname{erf}\left(\frac{u+\frac{1}{2}}{\sqrt{2} \sigma_{\mathrm{k}}}\right)-\operatorname{erf}\left(\frac{u-\frac{1}{2}}{\sqrt{2} \sigma_{\mathrm{k}}}\right)\right]$$ where $I$ is the intensity, $(x_i,y_j,z_k)\in{{\mathbb R}}^3$ are the pixel coordinates in unit of pixel, $(x_0,y_0,z_0)\in{{\mathbb R}}^3$ is the location of light source, $bg\in{{\mathbb R}}$ is the background intensity, $\operatorname{erf}(\cdot)$ is the error function encountered in integrating the normal distribution, and $\sigma_k$ including $\sigma_{x y}$ and $\sigma_z$ are the variances.
While in our application, we will reconstruct a blurred 2-dimensional target image with multiple sources based on a dictionary of single-source images (features), therefore the fluorescence model will degenerate to 2-dimensional accordingly. Then the PSF should be: $$\begin{aligned}
\operatorname{PSF}(x, y)=\frac{1}{\sqrt{4 \pi^{2} \sigma_{x y}^{2} }} e^{-\frac{1}{2 \sigma_{x y}^{2}}\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]}\end{aligned}$$ and the expected photon count for pixel $(i,j)$ should be: $$\begin{aligned}
\mu_{i j} =I_{i j} \Delta E_{x y}\left(x_{i}-x_{0}\right) \Delta E_{x y}\left(y_{j}-y_{0}\right) +b g\end{aligned}$$
Then we can generate a target image with $m$ light sources according to the following PSF: $$\begin{aligned}
PSF_{target}(x,y) = \sum_{i=1}^m \theta_i PSF_i(x,y)\end{aligned}$$ where $\theta_i\in\mathbb{R}_+$[^4] is the weight of the $i$th single source. An example for a target image with four light sources is shown as below:
![Target image[]{data-label="app2_tar"}](true.png){width="0.5\linewidth"}
As for features, they will be generated with the same resolution of the target image according to a dictionary of single-source PSFs. Four examples for features are listed as below:
![Examples of feature images[]{data-label="fig:res"}](1.png){width="3.0cm"}
\(a) Feature 1
![Examples of feature images[]{data-label="fig:res"}](2.png){width="3.0cm"}
\(b) Feature 2
![Examples of feature images[]{data-label="fig:res"}](3.png){width="3.0cm"}
\(c) Feature 3
![Examples of feature images[]{data-label="fig:res"}](4.png){width="3.0cm"}
\(d) Feature 4
Note that all the images used will be generated as $28\times 28$[^5], thus we can vectorize these images as $784$-dimensional vectors and construct the response $Y$ and regression matrix $\Phi$ in (\[eq:lm\]) as: $$\begin{aligned}
\label{app2_lm}
Y = \sum_{i=1}^n \theta_i \phi_i(x,y,p_i) + V = \Phi(x,y,p)\theta + V\end{aligned}$$ where $Y\in{{\mathbb R}}^N$ is the blurred target image to be reconstructed, $\Phi=[\phi_1,\ldots,\phi_n]\in{{\mathbb R}}^{N\times n}$ is the regression matrix made up of $n$ feature images, $(x,y)$ are pairwise coordinates of pixels with respect to the mesh grid based on $[1,2,\ldots,28]$ and $[1,2,\ldots,28]$, $\theta\in{{\mathbb R}}^n$ is the weights of features, $bg$ will be set to zero for convenience, which means we will generate $Y$ and $\Phi$ under the same background intensity, $V\sim\mathcal{N}(0,\lambda I_N)$ is the noise vector, $\lambda\in{{\mathbb R}}_+$. As for $p$, we have $p_i=(x_{0_i},y_{0_i},\sigma_{xy_i}),i=1,\ldots,n.$ Then the parameter set to be estimated, say $\Theta$, should be: $$\begin{aligned}
\label{param_set}
\Theta=(\theta,p)=(\theta_1,\ldots,\theta_n,p_1,\ldots,p_n)\in{{\mathbb R}}^{n+3n}\end{aligned}$$
Notice that $\Theta$ can be divided into two parts with respect to being linear to $Y$ or not: $\theta$ is the linear part, while $p$ is the nonlinear part. As discussed in Section 2 and Section 3, the occasions to use the proposed method should satisfy that the parameter to be estimated is linear to the response. Therefore, in the next section, we will try to find a reasonable $p$ by sampling.
Sampling
--------
In this section, we will decide $p$ by sampling. Sampling is a process used in statistical analysis, in which a specific number of observations are selected from a larger observation pool. Since $p$ includes $x_0,y_0,\sigma_{xy}$, the sampling is equivalent to finding the $n$ triples of parameters $(x_{0_i},y_{0_i},\sigma_{xy_i})$ that define the $n$ features in the regression matrix $\Phi$.
Theoretically, our sampling should be based on the prior of $p$. Even in the worst case where we have no idea how the light sources in the target image are distributed, we can still sample $p$ with respect to Gaussian distribution or uniform distribution. As $n$ goes larger, our samples should be able to cover more possible features, which will definitely influence the performance for reconstruction. In our simulation, we sample $10,000$ features to construct $\Phi$.
Since parameters in $p$ are obtained, multiple pairs of $(x,y)$ representing pixel units are known inputs, thus all features can be generated accordingly with respect to PSF; then we can finally model the problem as: $$\begin{aligned}
Y=\Phi\theta+V\end{aligned}$$ where $Y\in\mathbb{R}^{784}$ is the target vector, $\Phi\in\mathbb{R}^{784\times 10000}$ is the feature matrix, $\theta\in{{\mathbb R}}^{10000}$ is the parameter to be estimated, $V\in{{\mathbb R}}^{10000}$ is the noise vector in actual observations, and $V\sim\mathcal{N}(0,\lambda I_{784})$. Now the proposed method is applicable.
Note that unlike the classification in the last section, this time we will introduce the noise $V$ manually. Then the image in Figure \[app2\_tar\] will be blurred as:
![Blurred target image[]{data-label="app2_tar_blur"}](blur.png){width="0.5\linewidth"}
For the recovery of the blurred image, we have two goals to achieve:
- The first is source localization, which aims at recovering the true light sources in the generation of the target image. If some $\theta_i$ is non-zero, then the corresponding $\phi_i$ will be included in the sparse representation, then the light source center $(x_{0_i},y_{0_i})$ will show up in the reconstructed image. The performance will be evaluated with respect to a self-defined metric.
- The second is denoising. We want more information besides locations for light sources, which means we hope to recover the entire image efficiently. The performance will be evaluated with respect to a traditional metric and compared with built-in denoising methods in MATLAB.
Source Localization
-------------------
As the title indicates, source localization is the detection of the light sources in an image. After we obtain an optimal $\hat{\theta}$ by solving the linear system, we will be able to reconstruct the target image as: $$\begin{aligned}
\hat{Y}=\Phi\hat{\theta}\end{aligned}$$
If some $\hat{{\theta}}_i$ in $\hat{{\theta}}$ is non-zero, the corresponding feature image $\phi_i$ should be included in the reconstruction, and thus the light source $(x_{0_i},y_{0_i})$ will be detected. To make the simulation more convincing, $20$ different target images will be generated and the statistics will be averaged accordingly. The following figures show the average performance of this application. First, we check the performance of the proposed method with respect to screening percentage:
![Screening percentage - source localization[]{data-label="fig:6.4"}](screen_percentage_1.jpg){width="0.6\linewidth"}
The figure indicates that the screening percentage increases rapidly as $\lambda/\lambda_{\text{max}}$ becomes larger than 0.3. Next, we check the time reduction:
{width="0.6\linewidth"}
When $\lambda/\lambda_{\text{max}}$ is no larger than 0.3, the reduced time $t_r$ is even larger than the raw time $t$ without screening. Since the screening percentage is too low, little computation time will be saved while the screening will still consume extra time.
Finally, we observe the whole process that how the sparse solution converges to the true light sources. Four figures are provided below, where the green points are the true sources to be detected, the red circles represent the sparse representation we obtain.
When $\lambda$ is too small and sparsity is not enough:
{width="0.5\linewidth"}
When $\lambda$ is larger:
{width="0.5\linewidth"}
And $\lambda$ continues increasing:
{width="0.5\linewidth"}
The most proper $\lambda$ leads to the result below:
{width="0.5\linewidth"}
As we can see, the reconstructed signals based on the sparse representations are gathering around the true light sources gradually as $\lambda$ goes larger, even though the recovery is not completely accurate, it definitely provides us with significant information.
As for the accuracy for the detection, we choose a very popular evaluation metric used in the object detection: intersection over union (IoU)[@iou]. IoU, also known as Jaccard Index or Jaccard similarity coefficient, is a statistic used to measure the similarity and diversity of sample sets. It measures similarity between finite sample sets by computing the size of the intersection divided by the size of the union of the sample sets: $$\begin{aligned}
IoU = \frac{DetectionResult\bigcap GroundTruth}{DetectionResult\bigcup GroundTruth}\end{aligned}$$
However, as we can see, the definition of IoU is not enough when the number of detection results and ground truths are different. Therefore, we have to further define group-IoU. In case we mistake some bad detections as good ones, the group-IoU will be defined with respect to $m$ is larger than $n$ or not:
Suppose we have $m$ detection results and $n$ ground truths, then:
- When $m> n$, for each detection result, we compute the IoUs between this result and all the ground truths, select the largest one, and then use the average of the $m$ largest IoUs as the group-IoU.
- When $m\leq n$, for each ground truth, we compute the IoUs between this truth and all the detection results, select the largest one, and then use the average of the $n$ largest IoUs as the group-IoU.
Then we can use this group-IoU as IoU for our detection. It’s worth mentioning that in our codes, both the detection result and ground truth are defined as rectangulars in the same size, rather than what is shown in the four figures above. And the IoU for detection can be shown as:
![Intersection over union[]{data-label="fig:iou"}](iou_new2.png){width="0.7\linewidth"}
We notice that the tendency of IoU curve is more complicated compared with the curves in the previous figures. However, this doesn’t mean the characteristics for screening change. As the definition of group-IoU indicates, both numbers and locations of the detection results will influence the value of group-IoU. Therefore, even if we use such an IoU as criterion, the true performance may not be totally decided by IoU. For example, even though Figure \[fig:iou\] indicates that $\lambda/\lambda_{\text{max}}\in(0.1,0.2)$ guarantees a higher IoU, however when we check the detection results manually, the results for $\lambda/\lambda_{\text{max}}\in(0.5,0.7)$ look a lot better. Thus the defined group-IoU may not be crucial, but it does tell us some significant information.
What’s more, unlike the situation in the last section, this time we have numerical information for the noise $V$, therefore it’s natural for us to prefer selecting the optimal $\lambda$ as the true variance for noise $V$ in theory; however in practice, it’s completely possible that these two may differ.
Denoising
---------
Denoising is the task of removing noise from an image, which leads to our new goal, to pursue the similarity of the original image and the reconstructed image. We will still reconstruct the target image as: $$\begin{aligned}
\hat{Y}=\Phi\hat{\theta}\end{aligned}$$
However, different from source localization, this time we will focus on the similarity between the reconstructed image $\hat{Y}$ and the true image denoted by $Y_0$. The similarity will be quantified by PSNR[@psnr] (peak signal-to-noise ratio). The higher the PSNR is, the better our reconstructed image will be. PSNR is defined as below:
Suppose $I_1$ denotes the matrix data of the original image, $I_2$ denotes the matrix of the reconstructed image; and $m$ represents the number of rows in the images, $n$ represents the number of columns in the images; moreover, $MAX_{I_1}$ is the maximum intensity in our original image, then: $$\begin{aligned}
PSNR = 20 \log _{10}\left(\frac{M A X_{I_1}}{\sqrt{M S E}}\right)
\end{aligned}$$ where $MSE=\frac{1}{m n} \sum\limits_{i=1}^{m} \sum\limits_{j=1}^{n}(I_1(i, j)-I_2(i, j))^{2}$.
Since the simulation settings are almost identical to source localization, the screening percentage and time reduction for denoising should be the same as well. The only difference is that PSNR will work as a new criterion rather than IoU. The recovery performance for one of the target images is shown as below:
{width="2.5cm"}
{width="2.5cm"}
{width="2.5cm"}
The average accuracy with respect to PSNR is shown as:
{width="0.8\linewidth"}
In both theory and practice, we find that $\lambda/\lambda_{\text{max}}\in (0.5,0.6)$ yields a satisfying performance. Moreover, we compare the performance of the proposed method with traditional algorithms for denoising, for example, wavelet signal denoising method. During the simulation, we generate the data set with different noise variances. When the signal-to-noise ratio (SNR) is small, there’s no significant difference between wavelet signal denoising and sparse Bayesian learning with screening test; however, when SNR goes too large, sparse Bayesian learning with screening test will definitely outperforms the wavelet method, which is consistent with our conclusions in Section 3. The figure below shows the performances when SNR=$0.4$.
\[fig:group\]
![Comparison between SBL with screening test and wavelet denoising under high SNR[]{data-label="fig:compare2"}](2_1.png){width="3.0cm"}
Original image
![Comparison between SBL with screening test and wavelet denoising under high SNR[]{data-label="fig:compare2"}](2_2.png){width="3.0cm"}
Image with Gaussian noise
![Comparison between SBL with screening test and wavelet denoising under high SNR[]{data-label="fig:compare2"}](2_4.png){width="3.0cm"}
Reconstructed - wavelet
![Comparison between SBL with screening test and wavelet denoising under high SNR[]{data-label="fig:compare2"}](2_3.png){width="3.0cm"}
Reconstructed - SBL
Conclusions
-----------
In this section, the proposed method is applied to signal reconstruction in astronomical imaging. This application has two parts, one is source localization, the other is denoising.
Since the limitations of the proposed method still exist, the two goals mentioned in the conclusions of Section 3 should be inherited. And the simulation results indicate that we achieve both the two goals successfully. Moreover, the reconstruction performs especially well in high-SNR occasions.
What’s more, the methodology of this application is obviously more complicated than the application in the last section. That is because even though we manage to model the problem as a linear system, the parameter space $\Theta$ is not completely linear to the response $Y$, thus we have to use sampling as a pretreatment to deal with the non-linear part before using sparse Bayesian learning. Therefore the overall performance will not only rely on our proposed method, but also depend on the pretreatment.
As we said at the beginning of this section, this framework should not be limited to astronomical imaging, but can also be extended to other systems that can be modeled alike.
Conclusions
===========
As the era of big data is coming, the inter-discipline between traditional statistical methods and machine learning shall draw more and more attention continuously, and the needs for exploration on sparsity will persist as well.
In this work, to find a sparse solution $\theta$ to a linear system more efficiently, we apply screening test to sparse Bayesian learning, thus the new algorithm can inherit the characteristics of sparse Bayesian learning while achieving an acceleration at the same time, which indicates its potential to influence related fields.
In Section 2 and Section 3, we introduce the methodology of sparse Bayesian learning and design a screening test for it, then we examine the performance on two real-world data sets. Though the simulation shows a fairly good performance, we should admit some limitations of the proposed method listed as follows:
- The proposed method only works on sparse Bayesian learning that is equivalent to a weighted $\ell_1$-minimization problem, but cannot be used for all types of sparse Bayesian learning.
- According to the methodology, whether an efficient bound for $\hat{\eta}$ is chosen will definitely influence the performance, thus we have to admit that, both in theory and practice, the performance of Algorithm \[alg\_wtht\] is no better than that of the THT in [@XiangWR14], though it has extra advantages of SBL.
- Last but not least, both the screening ratios of THT and Algorithm \[alg\_wtht\] depend on $\lambda$ too much. The dependency cannot be totally eliminated in theory, however according to the simulation, to obtain a satisfying acceleration, the value of $\frac{\lambda}{\lambda_\text{max}}$ should be no smaller than $0.4$; considering what $\lambda_\text{max}$ represents, this value range of $\lambda$ may not be always acceptable.
In Section 4 and Section 5, we examine sparse Bayesian learning with screening test on two applications. One is classification, the other is signal reconstruction (source localization and denoisng). In these applications, we achieve our goals successfully and efficiently. Especially in the second application, we make it to formulate the problem as a linear system, even though the linear relationship does not hold with respect to the full parameter space $\Theta$. For such issue, we choose to estimate the nonlinear parameters by tricks like sampling. Consequently, we must be aware that the overall performance is decided not only by sparse Bayesian learning with screening test, but also the trick we use before sparse Bayesian learning. For example, the accuracy of sampling will definitely impact on the performance of reconstruction.
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[^1]: For example, we can initialize $ \gamma_h $ as: $\forall \gamma_{h_i}=1,i=1,\ldots,n$.
[^2]: Here we use the subgradient for $\theta$ because $\ell_1$ norm is not differentiable at the kink.
[^3]: Even if $\phi_i$ can make it to satisfy the sufficient condition, other vectors will possibly fail the condition.
[^4]: $\theta_i\geq 0$ because intensity cannot be negative.
[^5]: In this section, the figures to show the performance, including target image, blurred image, and reconstructed image, have been resized to a larger scale by interpolation method in MATLAB for better display.
|
---
abstract: |
Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If $X$ is such a lattice, those notions seem – at first glance – intimately related to the lattice operations on $X$. The last fifteen year, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces.
In fact if $X$ is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give rise to a non-trivial structure theory.
The purpose of this note is twofold: (i) We show that, on any pre-Riesz space, the structure of the space of all band projections is remarkably close to what we have in the case of vector lattices. In particular, this space is a Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already be a vector lattice. These criteria are coined in terms of disjointness and closely related concepts, and they mark how lattice-like the order structure of pre-Riesz spaces can get before the theory collapses to the vector lattice case.
address: 'Jochen Glück, Fakultät für Informatik und Mathematik, Universität Passau, Innstraße 33, D-94032 Passau, Germany'
author:
- Jochen Glück
bibliography:
- 'literature.bib'
title: 'On disjointness, bands and projections in partially ordered vector spaces'
---
Introduction {#section:introduction}
============
Disjointness {#disjointness .unnumbered}
------------
Two elements $x$ and $y$ of a vector lattice $X$ are called *disjoint* if ${\left\vert x \right\rvert} \land {\left\vert y \right\rvert} = 0$ – a notion that is well-motivated by the case where $X$ is one of the classical function spaces such as $L^p$. A straightforward generalisation to ordered vector spaces that are not lattices seems to be difficult at first glance, as there is no obvious replacement of the modulus of $x$ and $y$. Van Gaans and Kalauch, though, observed more than a decade ago [@Gaans2006] that one can circumvent this obstacle by noting that any two elements $x$ and $y$ of a vector lattice $X$ are disjoint if and only if ${\left\vert x+y \right\rvert} = {\left\vert x-y \right\rvert}$, and that this is in turn true if and only if the sets $\{x+y,-x-y\}$ and $\{x-y,y-x\}$ have the same set of upper bounds. The latter property clearly allows a generalisation to other ordered vector spaces, which gives rise to the following definition.
Let $(X,X_+)$ be an *ordered vector space*, by which we mean that $X$ is a real vector space and $X_+ \subseteq X$ is a non-empty subset of $X$ which satisfies $X_+ \cap (-X_+) = \{0\}$ and $\alpha X_+ + \beta X_+ \subseteq X_+$ for all scalars $\alpha,\beta \in [0,\infty)$ (we call $X_+$ the *positive cone* in $X_+$). To elements $x,y \in X$ are called *disjoint* if both sets $\{x+y,-x-y\}$ and $\{x-y,y-x\}$ have the same set of upper bounds in $X$. We use the notation $x \perp y$ to denote that $x$ and $y$ are disjoint. Note that $x \perp y$ if and only if $y \perp x$, and that $x \perp x$ if and only if $x = 0$.
If $x,y \in X$ are both *positive* – i.e. $x,y \in X_+$ – then one can prove that $x \perp y$ if and only if the infimum of $x$ and $y$ in $X$ exists and is equal to $0$; see [@Glueck2019 Proposition 2.1].
Disjoint complements and pre-Riesz spaces {#disjoint-complements-and-pre-riesz-spaces .unnumbered}
-----------------------------------------
Let $(X,X_+)$ be an ordered vector space and let $S \subseteq X$. The set $$\begin{aligned}
S^\perp := \{x \in X: \; x \perp s \text{ for all } s \in S\}\end{aligned}$$ is called the *disjoint complement* of $S$. We note that $S_1^\perp \supseteq S_2^\perp$ whenever $S_1$ and $S_2$ are two subsets of $X$ such that $S_1 \subseteq S_2$.
From the theory of vector lattices we would expect $S^\perp$ to always be a vector subspace of $X$ – but it turns out that one can construct examples of ordered spaces where this is not true (see for instance [@Gaans2006 Example 4.3]). On the other hand though, such counterexamples are somewhat pathological: in fact, one can show that $S^\perp$ is always a vector subspace of $X$ if the cone $X_+$ is *generating* in $X$ (i.e. $X = X_+ - X_+$) and $X$ is *Archimedean* (i.e. $nx \le y$ for all $n \in {\mathbb{N}}:= \{1,2, \dots\}$ implies $x \le 0$ whenever $x,y$ are two vectors in $X$).
There is also the slightly more general class of *pre-Riesz spaces* that is relevant in this context: an ordered vector space $(X,X_+)$ is called a pre Riesz space if for every non-empty finite set $A \subseteq X$ and every $x \in X$ the following implication is true: if the set of upper bounds of $x+A$ is contained in the set of upper bounds of $A$, then $x \in X_+$. This concept was introduced by van Haandel in [@Haandel1993 Definition 1.1(viii)].
If $(X,X_+)$ is a pre-Riesz space and $S \subseteq X$, then the disjoint complement $S^\perp$ is always a vector subspace of $X$; see [@Gaans2006 Corollary 2.2 and Section 3]. Moreover, we note that every pre-Riesz space has generating cone and that, conversely, every ordered vector space which has generating cone and is, moreover, Archimedean, is a pre-Riesz space [@Gaans2006 Theorem 3.3].
The theory of pre-Riesz spaces has undergone a considerable development over the last 15 year. References to papers that deal with bands and projection bands on pre-Riesz spaces are given at the beginning of Sections \[section:bands\] and \[section:band-projections\]. Further contributions to the theory of pre-Riesz spaces can be found in [@Gaans2010; @Kalauch2014; @Kalauch2014a; @Kalauch2019a] and, with a focus on operator theory, in [@Kalauch2018; @Kalauch2019c; @Kalauch2019d; @KalauchPreprint5]. The present state of the art in the theory of pre-Riesz spaces is presented in the recent monograph [@Kalauch2019].
Organisation of the paper {#organisation-of-the-paper .unnumbered}
-------------------------
In the rest of the introduction we recall a bit more terminology and a simple result about disjointness. In Section \[section:bands\] we recall how a band is defined in a pre-Riesz space, and we show a few elementary results about the structure of the set of all bands. In Section \[section:band-projections\] we discuss projection bands and band projections. We show, among other things, that the band projections on a pre-Riesz space constitute a Boolean algebra and that, under appropriate assumptions on the space, the intersection of arbitrarily many projection bands is again a projection band. In the final Section \[section:characterisations-of-vector-lattices\] we give various sufficient conditions for a pre-Riesz space to be a vector lattice; these conditions are related to several variations of the notion *disjointness*.
Setting the stage {#setting-the-stage .unnumbered}
-----------------
Throughout the rest of the paper, let $(X,X_+)$ be a pre-Riesz space.
By an *operator* on $X$ we always mean a linear map $X \to X$, and by a *projection* on $X$ we always mean a linear projection $X \to X$.
We use standard terminology and notation from the theory of ordered vector spaces (which has, to some extent, already been employed above). In particular, we write $x \le y$ (or $y \ge x$) for $x,y \in X$ if $y-x \in X_+$ and we note that the relation $\le$ is a partial order on $X$ which is compatible with the vector space structure. Elements of $X_+$ will be called *positive*. For $x,z \in X$ we denote the *order interval* between $x$ and $z$ by $$\begin{aligned}
[x,z] := \{y \in X: \; x \le y \le z\}.\end{aligned}$$ A linear map $T: X \to X$ is said to be *positive*, which we denote by $T \ge 0$, if $TX_+ \subseteq X_+$, and for two linear maps $S,T: X \to X$ we write $S \le T$ if $T-S \ge 0$.
For each vector subspace $V \subseteq X$ we set $V_+ := V \cap C_+$, and we say that $V$ has *generating cone* or that $V$ is *directed* if $V = V_+ - V_+$. The following simple proposition is quite useful.
\[prop:orthogonal-subspaces-by-positive-parts\] Let $V,W \subseteq X$ be vector subspaces of $X$ with generating cone. If $V_+ \perp W_+$, then $V \perp W$.
Here we use the notation $A \perp B$ for two subsets $A,B \subseteq X$ if $a \perp b$ for each $a \in A$ and each $b \in B$.
We use that orthogonal complements in pre-Riesz spaces are always vector subspaces. Since $V_+ \subseteq (W_+)^\perp$ we conclude $V \subseteq (W_+)^\perp$. The latter inclusion is equivalent to $W_+ \subseteq V^\perp$, which in turn implies $W \subseteq V^\perp$.
Bands {#section:bands}
=====
This section is in a sense prologue to our main results in Sections \[section:band-projections\] and \[section:characterisations-of-vector-lattices\]. We briefly recall some basics about bands (Subsection \[subsection:basics-on-bands\]), we show that the collection of all bands in $X$ is a complete lattice with respect to set inclusion (Subsection \[subsection:the-lattice-of-all-bands\]) and we briefly discuss how the sum of two bands can be computed under certain assumptions (Subsection \[subsection:the-sum-of-two-bands\]).
Bands in pre-Riesz spaces were first defined in [@Gaans2006 Section 5] and were further studied in [@Gaans2008; @Gaans2008a; @Kalauch2015; @Kalauch2019b].
Basics {#subsection:basics-on-bands}
------
For $S \subseteq X$ we use the notation $S^{\perp\perp} := (S^\perp)^\perp$, and we call this set the *bi-disjoint complement* of $S$. Of course, we always have $S \subseteq S^{\perp \perp}$.
A subset $B \subseteq X$ is called a *band* if $B = B^{\perp \perp}$. For every set $S \subseteq X$ the disjoint complement $S^\perp$ is a band [@Gaans2006 Proposition 5.5(ii)]. As a consequence, for each $S \subseteq X$ the bi-disjoint complement $S^{\perp \perp}$ is the smallest band in $X$ that contains $S$. Since $(X,X_+)$ is a pre-Riesz space, every band in $X$ is a vector subspace of $X$. Note that if $B$ is a band in $X$ and $0 \le x \le b$ for $x \in X$ and $b \in B$, then we also have $x \in B$.
In the classical theory of vector lattices, the concept of bands is of outstanding importance. For the convenience of the reader we recall a few examples of bands in vector lattices.
\[ex:bands-in-function-spaces\] (a) Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, let $p \in [1,\infty]$ and let $X = L^p(\Omega,\mu)$ with the standard cone. If $A \subseteq \Omega$ is a measurable set, then $$\begin{aligned}
B & = \{f \in X: \; \text{there is a representative of $f$ that vanishes a.\ e.\ on } A\} \\
& = \{f \in X: \; \text{every representative of $f$ vanishes a.\ e.\ on } A\}
\end{aligned}$$ is a band in $X$, and in fact all bands in $X$ are of this form.
\(b) Let $X = C([0,1])$ be the space of continuous real-valued functions on $K$ and let $0 \le a \le 1$. Then $$\begin{aligned}
B_a = \{f \in X: \; f \text{ vanishes on } [a,1]\}
\end{aligned}$$ is a band in $X$ (this example is further discussed in [@Schaefer1974 Example 5 on p.63]).
Interesting examples for bands in a non-lattice ordered pre-Riesz space can for instance be found in the space ${\mathbb{R}}^3$ ordered by the so-called *four ray cone*:
\[ex:four-ray-cone-bands\] Let $X = {\mathbb{R}}^3$ let $X_+ := \{\sum_{k=1}^4 \alpha_k v_k: \; \alpha_1,\dots,\alpha_4 \in [0,\infty)\}$, where $$\begin{aligned}
v_1 =
\begin{pmatrix}
1 \\ 0 \\ 1
\end{pmatrix},
\quad
v_2 =
\begin{pmatrix}
0 \\ 1 \\ 1
\end{pmatrix},
\quad
v_3 =
\begin{pmatrix}
-1 \\ 0 \\ 1
\end{pmatrix},
\quad
v_4 =
\begin{pmatrix}
0 \\ -1 \\ 1
\end{pmatrix}.
\end{aligned}$$ The cone $X_+$ is called the *four ray cone* in $\mathbb{R}^3$. All bands in $X$ are computed in [@Kalauch2019 Example 4.4.18]. Besides the two trivial bands $\{0\}$ and $X$ there are four directed bands – namely the lines spanned by $v_1,\dots,v_4$, respectively. Moreover, there exist two non-directed bands – namely the lines spanned by $$\begin{aligned}
\begin{pmatrix}
1 \\ 1 \\ 0
\end{pmatrix}
\quad \text{and} \quad
\begin{pmatrix}
1 \\ -1 \\ 0
\end{pmatrix},
\end{aligned}$$ respectively.
The lattice of all bands {#subsection:the-lattice-of-all-bands}
------------------------
The following proposition shows that the intersection of any collection of bands in $X$ is again a band.
\[prop:intersections-of-arbitrary-many-bands\] The intersection of arbitrarily many bands $B_i$ in $X$ (where the indices $i$ are taken from a – possibly empty – index set $I$) is again a band, and it is given by $$\begin{aligned}
\bigcap_{i \in I} B_i = \left(\bigcup_{i \in I}B_i^\perp\right)^\perp.
\end{aligned}$$
It suffices to prove the formula. If $x \in \bigcap_{i \in I} B_i$, then $x$ is disjoint to each set $B_i^\perp$, so $x$ is also disjoint to the union $\bigcup_{i \in I} B_i^\perp$. Conversely, fix $i_0 \in I$. Then $B_{i_0}^\perp \subseteq \bigcup_{i \in I} B_i^\perp$ and hence, $B_{i_0} = B_{i_0}^{\perp\perp} \supseteq \left(\bigcup_{i \in I}B_i^\perp\right)^\perp$.
It is an immediate consequence of this proposition that the set of all bands in $X$ is a complete lattice with respect to set inclusion; let us state this explicitly in the following corollary.
\[cor:lattice-of-bands\] Let $\operatorname{Bands}(X)$ denote the set of all bands in $X$, ordered by set inclusion. Then every subset of $\operatorname{Bands}(X)$ has a supremum and an infimum in $\operatorname{Bands}(X)$, i.e. $\operatorname{Bands}(X)$ is a complete lattice.
The sum of two bands {#subsection:the-sum-of-two-bands}
--------------------
Even in the case of vector lattices, the sum of two bands need not be a band, in general. Let us illustrate this by means of the following simple example.
\[ex:two-bands-whose-sum-is-not-a-band\] Let $X = C([-1,1])$ denote the space of all continuous real-valued functions on the interval $[-1,1]$ and endow this space with the standard cone. Then the sets $$\begin{aligned}
B = \{f \in X: \; f|_{[-1,0]} = 0\} \quad \text{and} \quad C = \{f \in X: \; f|_{[0,1]} = 0\}
\end{aligned}$$ are bands in $X$, but their sum $B + B = \{f \in X: \; f(0) = 0\}$ is not a band in $X$.
Another counterexample – in a non-lattice ordered pre-Riesz space – can be found in ${\mathbb{R}}^3$ endowed with the four ray cone from Example \[ex:four-ray-cone-bands\]. In this space, all non-trivial bands are one-dimensional; hence, the sum of two distinct non-trivial bands cannot be a band in this space.
If, however, the sum of two bands $B$ and $C$ is a band, then we can can compute it be means of the formula $B+C = (B^\perp \cap C^\perp)^\perp$; this is part of the following proposition.
\[prop:sum-of-two-bands\] Let $B, C \subseteq X$ be bands.
1. We have $B \cap C = (B^\perp + C^\perp)^\perp$.
2. We have $B+C = (B^\perp \cap C^\perp)^\perp$ if (and only if) $B+C$ is a band.
3. More generally than , we always have $B+C \subseteq (B+C)^{\perp \perp} = (B^\perp \cap C^\perp)^\perp$.
\(a) According to Proposition \[prop:intersections-of-arbitrary-many-bands\] we have $B \cap C = (B^\perp \cup C^\perp)^\perp$, and the latter set clearly contains $(B^\perp + C^\perp)^\perp$. On the other hand, if $x \in X$ is disjoint to $B^\perp \cup C^\perp$, then it is also disjoint to $B^\perp + C^\perp$ since the disjoint complement of $\{x\}$ is a vector subspace of $X$; this shows that we also have $(B^\perp \cup C^\perp)^\perp \subseteq (B^\perp + C^\perp)^\perp$.
\(c) It follows from (a) that $$\begin{aligned}
B^\perp \cap C^\perp = (B^{\perp \perp} + C^{\perp \perp})^\perp = (B+C)^\perp,
\end{aligned}$$ so $(B^\perp \cap C^\perp)^\perp = (B+C)^{\perp\perp}$.
\(b) This is an immediate consequence of (c).
The main point of the above proposition – and the reason for the title of this subsection – is assertion (b). Anyway, we chose to include assertion (a) in the same proposition in order to have an immediate comparison between (a) and (b).
We point out that the assumption of (b) that $B+C$ be a band is automatically satisfied of both $B$ and $C$ are projection bands; see Proposition \[prop:sum-of-projection-bands\] below. On the other hand, Example \[ex:two-bands-whose-sum-is-not-a-band\] shows that there are situations in which $B+C$ is not a band – and in this case the formula from Proposition \[prop:sum-of-two-bands\] necessarily fails.
Band projections {#section:band-projections}
================
Band projections (and, accordingly, projection bands) in pre-Riesz spaces are a main subject of study in [@KalauchPreprint4; @Glueck2019]. In this section we further develop their theory.
Basics {#subsection:basics-on-band-projections}
------
If $B$ is a band in $X$, then it intersects its orthogonal band $B^\perp$ only in $0$. However, the sum of $B$ and $B^\perp$ can be smaller than the entire space $X$, in general; this happens, for instance, in Example \[ex:two-bands-whose-sum-is-not-a-band\], where $C = B^\perp$.
We call a subset $B \subseteq X$ a *projection band* if $B$ is a band and if, in addition, $X = B \oplus B^\perp$. It is not difficult to see that a band $B$ is a projection band if and only if $B^\perp$ is a projection band. Every projection $B$ has generating cone according to [@Glueck2019 Proposition 2.5].
The notion of a projection band also gives rise to the following definition: a linear projection $P: X \to X$ is called a *band projection* if there exists a projection band $B \subseteq X$ such that $P$ is the projection onto $B$ along $B^\perp$. In other word, $P$ is a band projection if and only if $PX$ is a projection band and $\ker P$ equals the disjoint complement of $PX$.
The following proposition contains various characterisations of band projections.
\[prop:characterisation-of-band-projections\] For every linear projection $P: X \to X$ the following assertions are equivalent:
1. $P$ is a band projection.
2. $\ker P = (PX)^\perp$.
3. $PX = (\ker P)^\perp$.
4. $PX \perp \ker P$.
5. Both projections $P$ and ${I}- P$ are positive.
6. ${I}- P$ is a band projection.
“(i) $\Leftrightarrow$ (v)” This equivalence was proved in [@Glueck2019 Theorem 3.2].
“(i) $\Leftrightarrow$ (vi)” This equivalence follows from the fact that a band $B$ is a projection band if and only if $B^\perp$ is a projection band (alternatively, it follows immediately from the equivalence of (i) and (v)).
“(i) $\Rightarrow$ (ii)” and “(i) $\Rightarrow$ (iii)” These implications follow immediately from the definition of a band projection.
“(ii) $\Rightarrow$ (iv)” and “(iii) $\Rightarrow$ (iv)” These implications are obvious.
“(iv) $\Rightarrow$ (v)” Let $x \in X_+$. Then $Px$ and $({I}- P)x$ are disjoint and sum up to $x$, so it follows from [@Glueck2019 Proposition 2.4(a)] that $Px$ and $({I}-P)x$ are positive, too. This shows (v).
If $P$ is a band projection in $X$, then both the range and the kernel of $P$ are projection bands. In Corollary \[cor:band-projection-if-range-and-kernel-are-projection-bands\] below we will see that the converse implication is also true, which yields another characterisation of band projections.
We conclude this subsection with a few examples.
\[ex:examples-for-projection-bands\] (a) If $X$ is a Dedekind complete vector lattice, then every band in $E$ is a projection band [@Schaefer1974 Theorem II.2.10].
\(b) Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and let $p \in [1,\infty]$. The bands in $L^p(\Omega,\mu)$ are described in Example \[ex:bands-in-function-spaces\](a). Since $L^p(\Omega,\mu)$ is Dedekind complete, it follows from (a) that each of these bands is actually a projection band.
\(c) The bands $B_a$ in $C([0,1])$ from Example \[ex:bands-in-function-spaces\](b) are not projection bands unless $a = 0$ (see [@Schaefer1974 Example 5 on p.63]). More generally, it is not difficult to see that, for a compact Hausdorff space $K$, there are no non-trivial projection bands in $C(K)$ if $K$ is connected.
\(d) Let $X$ and $Y$ be two pre-Riesz spaces and endow the product space $Z := X \times Y$ with the product order (i.e. $Z_+ = X_+ \times Y_+$). Then $Z$ is a pre-Riesz space, too, and $X$ and $Y$ – which we identify with the subspaces $X \times \{0\}$ and $\{0\} \times Y$ of $Z$, respectively – are projection bands in $Z$. Indeed, we have $X^\perp = Y$, and vice versa.
On a related note, we will see in Theorem \[thm:projection-band-structure-of-finite-dimensional-pre-riesz-spaces\] below that every finite-dimensional pre-Riesz space can be written as the product of finitely many minimal projection bands.
\(e) If $X$ is a Banach lattice and we identify $X$ with a subspace of its bi-dual space $X''$ by means of evaluation, then $X$ is a band in $X''$ if and only if $X$ is a projection band if and only if $X$ is a so-called *KB-space*. This class of space includes all reflexive Banach lattices and all $L^1$-spaces (over arbitrary measure spaces). For details we refer for instance to [@Meyer-Nieberg1991 Section 2.4].
Example \[ex:examples-for-projection-bands\](e) can be extended to also include spaces that are not lattice-ordered. An ordered vector space $(Y,Y_+)$ is called an *ordered Banach space* if $Y$ carries a complete norm and $Y_+$ is closed. Note that the order in an ordered Banach space is always Archimedean. Hence, if $Y_+$ is, in addition, generating, then the ordered Banach space $(Y,Y_+)$ is a pre-Riesz space.
Throughout the rest of the paper, we will tactily use some important concepts from the theory of ordered Banach spaces – such as *normality* of cones and the fact the the dual of an ordered Banach space with generating cone is again an ordered Banach space. For details about the theory of ordered Banach spaces we refer the reader for instance to the monograph [@Aliprantis2007], in particular to Section 2.5 there.
\[ex:ordered-banach-space-band-in-bidual\] Assume that our pre-Riesz space $X$ is an ordered Banach space with normal cone. Then we can consider $X$ as a subspace of the bi-dual space $X''$ by means of evaluation.
There are interesting examples where $X$ is not a Banach lattice and not reflexive, but yet a projection band in $X''$. This is, for instance, the case if $X$ is the pre-dual of a von Neumann algebra; see [@Sakai1971 Proposition 1.17.7] or [@Takesaki1979 pp.126–127].
Ordered Banach space that are bands in their bi-dual were employed in [@GlueckWolffLB] to study the long-term behaviour of positive operator semigroups.
One can easily find examples where a pre-Riesz space $X$ does not contain any projection bands except for $\{0\}$ and $X$ itself. One situation of this type has already been discussed in Example \[ex:examples-for-projection-bands\](c) above. Here are two more examples.
\[ex:non-lattices-without-projection-bands\] (a) Let us endow $X = {\mathbb{R}}^3$ with the four ray cone $X_+$ from Example \[ex:four-ray-cone-bands\]. Then every non-trivial band $B$ in $X_+$ is one-dimensional, so there is no non-trivial projection band in $X$.
\(b) Assume that $X$ is a so-called *anti-lattice*, which means that any two vectors $x,y$ in $X$ have a supremum if and only if $x \ge y$ or $x \le y$. Then there are, according to [@Kalauch2019 Theorem 4.1.10(ii)], no non-trivial disjoint elements in $X_+$. Hence, there are no non-trivial projections bands in $X$.
We note that a classical example of an anti-lattice is the space of all self-adjoint bounded linear operators on a Hilbert space; this result goes back to Kadison [@Kadison1951 Theorem 6].
The Boolean algebra of band projections {#subsection:the-boolean-algebra-of-band-projections}
---------------------------------------
In this section we study the structure of the collection of all band projections on $X$. As in the vector lattice case, this collection turns out to be a boolean algebra (Theorem \[thm:band-projections-form-boolean-algebra\]).
We begin with the following proposition which shows that a band projection $Q$ dominates a band projection $P$ (in the sense of operators on the ordered vector space $X$) if and only of the range of $Q$ contains the range of $P$:
\[prop:domination-of-band-projections\] For two band projections $P$ and $Q$ on $X$ the following assertions are equivalent.
1. $PX \subseteq QX$.
2. $QP = P$.
3. $P \le Q$.
“(i) $\Leftrightarrow$ (ii)” This can immediately be checked to be true for all projections on arbitrary vector spaces.
“(ii) $\Rightarrow$ (iii)” We have $P = QP \le Q \cdot {I}= Q$.
“(iii) $\Rightarrow$ (ii)” We have $QP \le {I}\cdot P = P = P^2 \le QP$, so $QP = P$.
Next we describe the interaction of two arbitrary band projections on $X$ in a bit more detail; in particular, we prove that any two band projections commute.
\[prop:interaction-of-band-projections\] For two band projection $P$ and $Q$ on $X$ the following assertions hold:
1. $P$ leaves the range of $Q$ invariant, and vice versa.
2. $P$ and $Q$ commute.
3. The mapping $PQ = QP$ is a band projection, too.
4. We have $PQX = PX \cap QX$.
\(a) Let $0 \le x \in QX$. For each $0 \le z \in (QX)^\perp$ it follows from $0 \le Px \le x$ that $Px \perp z$ (see [@Glueck2019 Proposition 2.2]); since the positive cone in the projection band $(QX)^\perp$ is generating in $(QX)^\perp$, we conclude that $Px \perp (QX)^\perp$.
Now we also use that the positive cone in the projection band $QX$ is generating in $QX$, which implies that $Px \perp (Qx)^\perp$ for each $x \in Qx$. Hence, $Px \in (Qx)^{\perp \perp} = QX$ for each $x \in QX$, which shows that $P$ leaves $QX$ invariant. By interchanging the roles of $P$ and $Q$ we also obtain that $Q$ leaves $PX$ invariant.
\(b) It follows from (a) that $Q$ leaves both $PX$ and $({I}- P)X$ invariant. Thus, $$\begin{aligned}
PQP = QP \qquad \text{and} \qquad PQ({I}- P) = 0.
\end{aligned}$$ The second equality is equivalent to $PQP = PQ$ which yields, in conjunction with the first equality, $QP = PQ$.
\(c) Clearly, $0 \le PQ \le {I}\cdot {I}= {I}$, so it remains to show that $PQ$ is a projection. Since $P$ leaves $QX$ invariant, we know that $QPQ = PQ$, so $(PQ)^2 = P(QPQ) = P(PQ) = PQ$.
\(d) “$\supseteq$” For $x \in PX \cap QX$ we have $x = Px = PQx \in PQX$.
“$\subseteq$” If $x \in PQX$, then $x = PQx \in PX$ and $x = QPx \in QX$.
We point out that assertion (d) in the above proposition is in fact true for all commuting projection $P$ and $Q$ on an arbitrary vector space.
As a consequence of the fact that any two band projections commute we obtain the following proposition which shows, in particular, that the sum of two projection bands is a projection band (and a formula for such a sum can thus be found in Proposition \[prop:sum-of-two-bands\](b) above).
\[prop:sum-of-projection-bands\] Let $P$ and $Q$ be band projections on $X$. Then $P+Q-PQ$ is a band projection, too, and its range coincides with the set $PX + QX$. In particular, the sum of two projection bands is a projection band.
Since $P$ and $Q$ commute, a direct computation shows that $P+Q-PQ$ is a projection. Moreover, $$\begin{aligned}
P+Q-PQ = P + Q({I}- P),
\end{aligned}$$ and the latter mapping is clearly positive and dominated by $P + {I}({I}-P) = {I}$. Thus, $P+Q-PQ$ is a band projection.
Obviously, the range of $P+Q-PQ$ is contained in the vector space sum $PX + QX$. The converse inclusion follows from the formula $$\begin{aligned}
Px+Qy = (P+Q-PQ)(Px + Qy)
\end{aligned}$$ which holds for all $x,y \in X$.
Now we can prove that the set of all band projections on $X$ is a Boolean algebra. Recall (for instance from [@Schaefer1974 Definition II.1.1]) that a *Boolean algebra* is a non-empty partially ordered set $A$ with the following properties:
(a) For all $x,y \in A$ the infimum $x \land y$ and the supremum $x \lor y$ exist (i.e. $A$ is a *lattice*).
(b) The lattice operations $\land$ and $\lor$ are *distributive*, i.e. we have $$\begin{aligned}
(x \lor y) \land z = (x \land z) \lor (y \land z)
\end{aligned}$$ for all $x,y,z \in A$ (this is equivalent to assuming that $(x \land y) \lor z = (x \lor z) \land (y \lor z)$ for all $x,y,z \in A$, see [@Birkhoff1967 Theorem 9 on p.11]).
(c) There exists a smallest element $0$ and a largest element $1$ in $A$.
(d) $A$ is *complemented*, i.e. for each $x \in A$ there exists a so-called *complement* $x^c \in A$ such that $$\begin{aligned}
x \land x^c = 0 \quad \text{and} \quad x \lor x^c = 1.
\end{aligned}$$
We note that, in a Boolean algebra $A$, the complement of each element is uniquely determined; this follows from [@Birkhoff1967 Theorem 10 on p.12].
\[thm:band-projections-form-boolean-algebra\] Let $\operatorname{BandPr}(X)$ denote the set of all band projections on $X$, ordered by the usual order of positive operators on $X$. Then $\operatorname{BandPr}(X)$ is a Boolean algebra with smallest element $0$ and largest element ${I}$. The lattice operations $\land$ and $\lor$ on this Boolean algebra are given by $$\begin{aligned}
P \land Q = PQ \quad \text{and} \quad P \lor Q = P+Q - PQ
\end{aligned}$$ for all band projections $P$ and $Q$, and the complement is given by $$\begin{aligned}
P^c = {I}- P
\end{aligned}$$ for each band projection $P$.
In the proof we make use of the facts established in the propositions above; in particular we will frequently – and often tacitly – use that $P \le Q$ for two band projections $P$ and $Q$ if and only if $PX \subseteq QX$.
We first show that $\operatorname{BandPr}(X)$ is a lattice with respect to its given order, and that the lattice operators are given by the formulas in the theorem. Let $P,Q \in \operatorname{BandPr}(X)$.
It follows from Proposition \[prop:interaction-of-band-projections\](d) that $PQ$ is a lower bound of $P$ and $Q$. If $R \in \operatorname{BandPr}(X)$ is another lower bound of $P$ and $Q$, then $RX \subseteq PX$ and $RX \subseteq QX$, so $RX \subseteq PQX$, again by Proposition \[prop:interaction-of-band-projections\](d); hence, $R \le PQ$. This proves that $P$ and $Q$ have infimum $PQ$ in $\operatorname{BandPr}(X)$.
On the other hand, $P+Q-PQ$ is an upper bound of $P$ and $Q$ according to Proposition \[prop:sum-of-projection-bands\]. If $R \in \operatorname{BandPr}(X)$ is another upper bound of $PX$ and $QX$, then $RX \supseteq PX \cup QX$, hence $RX \supseteq PX + QX$ and thus, it follows again from Proposition \[prop:sum-of-projection-bands\] that $R \ge P+Q-PQ$. This proves that $P$ and $Q$ have supremum $P+Q-PQ$ in $\operatorname{BandPr}(X)$.
In particular, $\operatorname{BandPr}(X)$ is a lattice. The fact that it is even a distributive lattice, i.e., that the distributive law $$\begin{aligned}
(P \lor Q) \land R = (P \land R) \lor (Q \land R)
\end{aligned}$$ is satisfied for all band projections $P,Q,R$, can now be checked by a straightforward computation that uses the formula for $\land$ and $\lor$ established above.
Clearly, $\operatorname{BandPr}(X)$ has the smallest element $0$ and the largest element ${I}$, and for every band projection $P$, the projection $Q := {I}- P$ satisfies $P \land Q = PQ = 0$ and $P \lor Q = P + Q - PQ = P+Q = {I}$; hence, $I-P$ is the complement of any $P \in \operatorname{BandPr}(X)$ and $\operatorname{BandPr}(X)$ is indeed a Boolean algebra.
Let $\operatorname{PrBands}(X)$ denote the set of all projection bands in $X$, ordered via set inclusion. The mapping $$\begin{aligned}
\varphi: \operatorname{BandPr}(X) & \to \operatorname{PrBands}(X), \\
P & \mapsto PX
\end{aligned}$$ is an order isomorphism between the partially ordered sets $\operatorname{BandPr}(X)$ and $\operatorname{PrBands}(X)$. In particular, $\operatorname{PrBands}(X)$ is a Boolean algebra with infimum and supremum given by $$\begin{aligned}
B \land C = B \cap C \quad \text{and} \quad B \lor C = B+C
\end{aligned}$$ for all projections bands $B,C$ in $X$ and with the complement operation given by $$\begin{aligned}
B^c = B^\perp
\end{aligned}$$ for each projection band $B$ in $X$.
The mapping $\varphi$ is surjective by definition of the notions “projection band” and “band projection”, and it is injective since every band projection $P$ is uniquely determined by its range $PX$. It follows from Proposition \[prop:domination-of-band-projections\] that $\varphi$ and its inverse map $\varphi^{-1}$ are monotone. Thus, $\operatorname{PrBands}(X)$ is indeed a Boolean algebra and $\varphi$ is an isomorphism between the boolean algebras $\operatorname{BandPr}(X)$ and $\operatorname{PrBands}(X)$.
The formulae for the lattice operations on $\operatorname{PrBands}(X)$ now follow from Propositions \[prop:interaction-of-band-projections\] and \[prop:sum-of-projection-bands\], and the formula for the complement follows from the fact that $({I}- P)X = \ker P = (PX)^\perp$ for each band projection $P$.
The intersection of arbitrarily many projection bands {#subsection:the-intersection-of-arbitrarily-many-projection-bands}
-----------------------------------------------------
According to Proposition \[prop:interaction-of-band-projections\], the intersection of finitely many projection bands is again a projection band. In general, this is no longer true for infinitely many projections bands (not even in the case of Banach lattices) as the following simple example shows:
Consider the compact space $K = [-1,0] \cup \{\frac{1}{n}: \; n \in {\mathbb{N}}\}$ and the Banach lattice $C(K)$ of continuous real-valued functions on $K$.
For each $n \in {\mathbb{N}}$ the set $B_n := \{f \in C(K): \; f(x) = 0 \text{ for all } x \ge \frac{1}{n}\}$ is a projection band in $C(K)$. However, the intersection $$\begin{aligned}
\bigcap_{n \in {\mathbb{N}}} B_n & = \{f \in C(K): \; f(x) \\
& = 0 \text{ for all } x > 0\} = \{f \in C(K): \; f(x) = 0 \text{ for all } x \ge 0\}
\end{aligned}$$ is not a projection band in $C(K)$.
However, in a Dedekind complete vector lattice every band is a projection band and hence, the intersection of arbitrarily many projection bands is still a projection band.
Motivated by this we show in this subsection that the intersection of arbitrarily many projection bands in a Dedekind complete pre-Riesz space is again a projection band. Here, we call our pre-Riesz space $X$ *Dedekind complete* if the supremum $\sup A$ exists in $X$ for every non-empty upwards directed set $A \subseteq X$ that is bounded above.
Assume for a moment that $X$ is Dekekind complete. If $(x_j)$ and $(y_j)$ are decreasing nets in $X$ (with the same index set) that are bounded below, then the net $(x_j)$ has an infimum $x$ (we write $x_j \downarrow x$ for this), the net $(y_i)$ has an infimum $y$, and it is not difficult to show that the sum $(x_j + y_j)$ has infimum $x+y$; similarly, for $\lambda \in [0,\infty)$ the net $(\lambda x_j)$ has infimum $\lambda x$.
\[thm:decreasing-net-of-band-projections\] Assume that $X$ is Dedekind complete and let $(P_j)$ be a net of band projections on $X$ such that $P_j \le P_i$ (equivalently: $P_jX \subseteq P_iX$) whenever $j \ge i$. Then there exists a band projection $P_0$ on $X$ with the following two properties:
1. We have $P_j x \downarrow P_0x$ for each $x \in X_+$.
2. $P_0X = \bigcap_{j} P_jX$.
First we define a mapping $P_0: X_+ \to X_+$ by means of $P_0x = \inf_j P_j x$ for each $x \in X_+$. By the remarks we made before the theorem, $P_0$ is linear in the sense that $P_0(\alpha x + \beta y) = \alpha P_0 x + \beta P_0 y$ for all $x,y \in X_+$ and all $\alpha,\beta \in [0,\infty)$. As $X_+$ is generating in $X$, we can extend $P_0$ to a (uniquely determined) linear map – that we again denote by $P_0$ – from $X$ to $X$. For each $x \in X_+$ we have $0 \le P_0 x \le x$.
Let us show next that $P_0$ is a projection; to this end, it suffices to consider $x \in X_+$ and show that $P_0^2x = P_0x$. For each index $j$ we have $0 \le P_0x \le P_j x$, so $P_0x \in P_jX$ and hence, $P_j(P_0x) = P_0x$. This shows that $P_0(P_0x) = P_0x$, so $P_0^2 = P_0$. Consequently, $P_0$ is a band projection that has property (a). Let us now show (b).
“$\subseteq$” Let $x \in P_0X$. Then we can write $x$ as $x = y-z$, where $y$ and $z$ are positive vectors in $P_0X$. Then $0 \le y = P_0y \le P_j y$; hence, $y \in P_jX$ for each index $j$, and likewise for $z$. Thus, $x = y-z \in P_jX$ for each $j$.
“$\supseteq$” Let $x \in \bigcap_j P_j X$. We decompose $x$ as $x = y-z$ for $y,z \in X_+$. For each $j$ we have $x = P_j x = P_jy - P_j z$, so $$\begin{aligned}
x \le P_jy \qquad \text{and} \qquad -x \le P_j z.
\end{aligned}$$ Consequently, $x \le P_0y$ and $-x \le P_0 z$, so $x \in [-P_0z, P_0y]$, which proves that $x \in P_0X$.
In order to derive from Theorem \[thm:decreasing-net-of-band-projections\] that the intersection of an arbitrary – maybe non-directed – collection of projection bands is still a projection band, we need the following lemma (which is true on every pre-Riesz space, be it Dedekind complete or not).
\[lem:product-of-band-projections-as-infimum\] Let $P_1,\dots,P_n$ be band projections on $X$ and let $x \in X_+$.
1. If $z \in X$ and $z \le P_1x$, …, $z \le P_nx$, then also $z \le P_1 \cdots P_n x$.
2. We have $P_1\cdots P_n x = \inf\{P_1x,\dots,P_n x\}$.
\(a) We first note that, if $z \le Px$ for a band projection $P$, then $({I}- P)z \le 0$. Now we prove the assertion be induction over $n$. For $n=1$ the assertion is obvious, so assume that it has already been proved for some $n \in {\mathbb{N}}$. If $P_{n+1}$ is another band projections such that $z \le P_{n+1}x$, then $$\begin{aligned}
z = ({I}- P_{n+1})z + P_{n+1}z \le P_{n+1}z \le P_{n+1}P_1 \cdots P_n x = P_1 \cdots P_{n+1}x.
\end{aligned}$$
\(b) Clearly, $P_1\cdots P_nx$ is a lower bound of $\{P_1x,\dots,P_n x\}$, and according to (a) it is also the greatest lower bound of this set.
In the proof of the following corollary we only need assertion (a) of the lemma. We included assertion (b) in the lemma anyway since we think it is interesting in its own right.
\[cor:intersection-of-infinitely-many-projection-bands\] Assume that $X$ is Dedekind complete. Then the intersection of arbitrarily many projection bands in $X$ is again a projection band. More precisely, if ${\mathcal{P}}$ is a set of band projections on $X$, then there exists a (unique) band projection $P_0$ on $X$ with range $\bigcap_{P \in {\mathcal{P}}} PX$; if ${\mathcal{P}}$ is non-empty, then we have $$\begin{aligned}
P_0x = \inf\{Px: \; P \in {\mathcal{P}}\}
\end{aligned}$$ for each $x \in X_+$
We may assume that ${\mathcal{P}}$ is non-empty. Let $\hat {\mathcal{P}}$ denote the set of all finite products of elements from ${\mathcal{P}}$. Then it is easy to see that $\bigcap_{P \in {\mathcal{P}}}PX = \bigcap_{P \in \hat {\mathcal{P}}} PX$. Moreover, $\hat {\mathcal{P}}$ is directed by the converse of the usual order $\le$ on linear operators (since $\hat {\mathcal{P}}$ is closed with respect to taking finite products). Thus, $(P)_{P \in \hat {\mathcal{P}}}$ is a decreasing net of band projections, so Theorem \[thm:decreasing-net-of-band-projections\] shows the existence of a band projection $P_0$ on $X$ such that $P_0X = \bigcap_{P \in \hat{\mathcal{P}}} PX$.
It remains to prove the formula for $P_0x$, so let $x \in X_+$. By Theorem \[thm:decreasing-net-of-band-projections\](a) we have $$\begin{aligned}
P_0x = \inf\{Px: \; P \in \hat {\mathcal{P}}\}.
\end{aligned}$$ Clearly, $P_0x$ is a lower bound of $\{Px: \; P \in {\mathcal{P}}\}$, so let $z \in X$ be another lower bound of this set. Then $z$ is, according to Lemma \[lem:product-of-band-projections-as-infimum\](a), also a lower bound of $\{Px: \; P \in \hat {\mathcal{P}}\}$, and hence $z \le P_0x$.
If $X$ is Dedekind complete, then it follows from Corollary \[cor:intersection-of-infinitely-many-projection-bands\] that, for every set $S \subseteq X$, there exists a smallest projection band that contains $S$. This projection band can, however, be much larger than the band generated by $S$, as the following example shows:
Let $X = \mathbb{R}^3$, let $X_+$ be the four ray cone from Example \[ex:four-ray-cone-bands\] and let $v_1$ by the vector introduced in that example. The span of $\{v_1\}$ is a band, but according to Example \[ex:non-lattices-without-projection-bands\](a) there are no non-trival projection bands in $X$.
Hence, the band generated by $v_1$ equals $\operatorname{span}\{v_1\}$, while the projection band generated by $\{v_1\}$ equals $X$.
Another characterisation of band projections {#subsection:another-characterisation-of-band-projections}
--------------------------------------------
The following propositions shows that if two projections bands $B$ and $C$ have trivial intersection, than we automatically have $B \perp C$.
\[prop:projection-band-with-trivial-intersection\] For two band projections $P$ and $Q$ on $X$ the following assertions are equivalent.
1. $PQ = 0$
2. $PX \cap QX = \{0\}$.
3. $PX \perp QX$.
“(i) $\Leftrightarrow$ (ii)” This equivalence follows from Proposition \[prop:interaction-of-band-projections\](d) (and is thus true for arbitrary commuting projections on every vector space).
“(i) $\Rightarrow$ (iii)” According to Proposition \[prop:orthogonal-subspaces-by-positive-parts\] it suffices to show that $PX_+ \perp QX_+$, so let $x \in PX_+$ and $y \in QX_+$. In order to show that $x \perp y$ it is necessary and sufficient to prove that $x$ and $y$ have infimum $0$ in $X$. Obviously, $0$ is a lower bound of $x$ and $y$, so let $b$ be another lower bound of those vectors. We then have $$\begin{aligned}
Pb \le Py = PQy = 0 \quad \text{and} \quad Qb \le Qx = QPx = 0,
\end{aligned}$$ so $(P+Q)b \le 0$. On the other hand, we know from Proposition \[prop:sum-of-projection-bands\] that $P+Q$ is a band projection (since $PQ = 0$), so ${I}- (P+Q)$ is positive. Hence, $$\begin{aligned}
({I}- (P+Q))b \le (1 - (P+Q))x = x - Px - Qx = -Qx = 0.
\end{aligned}$$ Consequently, $b = ({I}- (P+Q))b + (P+Q)b \le 0$. This proves that $x$ and $y$ indeed have infimum $0$.
“(iii) $\Rightarrow$ (ii)” For each $x \in PX \cap QX$ we have $x \perp x$, so $x = 0$.
We remark that the implication “(iii) $\Rightarrow$ (ii)” in Proposition \[prop:projection-band-with-trivial-intersection\] remains true if $PX$ and $QX$ are replaced with arbitrary bands (over even arbitrary vector subspaces) in $X$. However, the converse implication fails for general bands, even if they are assumed to be directed. We illustrated this, again, in the space $\mathbb{R}^3$ endowed with the four ray cone.
\[ex:non-disjoint-bands-with-trivial-intersection\] Let $X = \mathbb{R}^3$, let $X_+$ denote the four ray cone from Example \[ex:four-ray-cone-bands\], and let $v_1,\dots, v_4$ denote the four vectors defined in the same example.
Then $B_1 := \operatorname{span}\{v_1\}$ and $B_2 := \operatorname{span} \{v_2\}$ are bands in $X$ that intersect only in $0$. However, we do not have $B_1 \perp B_2$ since $v_1$ is not disjoint to $v_2$. To see this, consider the vector $$\begin{aligned}
w =
\begin{pmatrix}
1 \\ 1 \\ 0
\end{pmatrix}.
\end{aligned}$$ Then $v_1-w = v_4 \in X_+$ and $v_2-w = v_3 \in X_+$. Hence, $w$ is a lower bound of both $v_1$ and $v_2$. On the other hand, $w$ is not an element of the negative cone $-X_+$. Thus, $0$ is not the greatest lower bound of $v_1$ and $v_2$, so $v_1 \not\perp v_2$.
As a consequence of Proposition \[prop:projection-band-with-trivial-intersection\] we obtain another characterisation of band projections.
\[cor:band-projection-if-range-and-kernel-are-projection-bands\] For every linear projection $P: X \to X$ the following assertions are equivalent:
1. $P$ is a band projection.
2. Both $PX$ and $\ker P$ are projection bands.
“(i) $\Rightarrow$ (ii)” If $P$ is a band projection, then $PX$ is a projection band by definition, and hence $\ker = (PX)^\perp$ is also a projection band.
“(ii) $\Rightarrow$ (i)” If $PX$ and $\ker P$ are projection bands, than there exist band projections $Q_1,Q_2: X \to X$ such that $Q_1X = PX$ and $Q_2X = \ker P$. Since $Q_1X \cap Q_2X = \{0\}$, it follows from Proposition \[prop:projection-band-with-trivial-intersection\] that $Q_1X \perp Q_2X$, i.e., $PX \perp \ker P$. According to Proposition \[prop:characterisation-of-band-projections\] this implies that $P$ is a band projection.
We note that the implication “(ii) $\Rightarrow$ (i)” in Corollary \[cor:band-projection-if-range-and-kernel-are-projection-bands\] does not remain true, in general, if we replace “projection bands” in (ii) with “bands”. More precisely, we have the following situation:
(a) There exists a (*weakly pervasive*, see Definition \[def:pervasive-and-weakly-pervasive\]) pre-Riesz space $X$ and two bands $B$ and $C$ in $X$ such that $X = B \oplus C$, but $C \not= B^\perp$. A concrete example of this situation can be found in [@KalauchPreprint4 Example 19]; it is, however, important to observe that one of the bands is not directed in this example.
(b) If $X$ is weakly pervasive and $X = B \oplus C$ for two directed bands – or, more generally, two directed ideals – $B$ and $C$, then it is shown in [@KalauchPreprint4 Theorem 18] that $B$ and $C$ are projections bands and $B = C^\perp$.
(c) If $X$ is even *pervasive* (see Definition \[def:pervasive-and-weakly-pervasive\]), then the implication mentioned in (b) remains true even if $B$ and $C$ are only ideals in $X$ (which are not a priori assumed to be directed); this is shown in [@KalauchPreprint4 Theorem 18].
(d) Now, let $X$ be a general pre-Riesz space and let $X = B\oplus C$ for two directed bands – or, more generally, directed ideals – $B$ and $C$. It seems to be open whether this implies $C = B^\perp$.
Characterisations of vector lattices {#section:characterisations-of-vector-lattices}
====================================
In this section we give various criteria for a pre-Riesz space to actually be a vector lattice. All these criteria are in some way related to disjointness. We note that, in the important special case where $X$ is finite dimensional and Archimedean, several sufficient criteria for $X$ to be a vector lattice are known. It suffices, for instance, if $X$ has the Riesz decomposition property (see for instance [@Aliprantis2007 Corollary 2.48]) or if $X$ is pervasive [@KalauchPreprint4 Theorem 38]. In Corollary \[cor:finite-dimensional-weakly-pervasive-spaces\] below we give a simultaneous generalisation of those two results.
Criteria in terms of one-dimensional projection bands {#subsection:criteria-in-terms-of-one-dimensional-projection-bands}
-----------------------------------------------------
In this subsection we prove that a finite-dimensional pre-Riesz space is automatically a vector lattice if their exist sufficiently many projection bands in it. We begin with the following proposition about linear independence.
\[prop:disjoint-vectors-are-linearly-independent\] Let $m \in {\mathbb{N}}$ and let $x_1,\dots,x_m \in X \setminus \{0\}$ be pairwise disjoint. Then the tuple $(x_1,\dots,x_m)$ is linearly independent.
For $m = 1$ the assertions is obvious, and we next show it for $m = 2$. So let $\alpha_1 x_1 + \alpha_2 x_2 = 0$ for real numbers $\alpha_1,\alpha_2$. Since the sum of the disjoint vectors $\alpha_1 x_1$ and $\alpha_2 x_2$ is both positive and negative, it follows from [@Glueck2019 Proposition 2.4(a)] that both vectors $\alpha_1 x_1$ and $\alpha_2 x_2$ are both positive and negative, and thus $0$. This implies that $\alpha_1 = \alpha_2 = 0$ since $x_1,x_2 \not= 0$ by assumption.
Now assume that the assertion has been proved for a fixed integer $m \ge 2$ and let $\alpha_1, \dots, \alpha_{m+1} \in {\mathbb{R}}$ such that $\sum_{k=1}^{m+1} \alpha_k x_k = 0$. Since the vectors $\sum_{k=1}^m \alpha_k x_k$ and $x_{m+1}$ are disjoint and linearly dependent, it follows from the case $m=2$ considered above that $\sum_{k=1}^m \alpha_k x_k = 0$. Using that the assertion has already been proved for the number $m$, we conclude that $\alpha_1 = \dots = \alpha_m = 0$. Finally, we observe that $\alpha_{m+1}x_{m+1} = 0$, so $\alpha_{m+1} = 0$ since $x_{m+1} \not= 0$.
\[thm:n-distinct-band-projections-with-rank-1\] Assume that $n = \dim X < \infty$. If there exist (at least) $n$ distinct band projections of rank $1$ on $X$, then $X$ is an Archimedean vector lattice, i.e. $X$ is linearly order isomorphic to ${\mathbb{R}}^n$ with the standard cone.
We may assume that $n \not= 0$. Let $P_1,\dots,P_n$ denote $n$ distinct band projections of rank $1$. Then we have $P_k P_jX = \{0\}$ for $j \not= k$. Indeed, if we assumed $\dim (P_k P_jX) = 1$, then $P_kP_j X = P_kX = P_jX$ – which would imply $P_k = P_j$ since band projections are uniquely determined by their range.
Since each projection band is spanned by its positive elements, each space $P_kX$ is spanned by a vector $x_k > 0$. According to Proposition \[prop:projection-band-with-trivial-intersection\] the vectors $x_1,\dots,x_n$ are pairwise disjoint. Hence, they are linearly independent by Proposition \[prop:disjoint-vectors-are-linearly-independent\]. Since $\dim X = n$, this implies that the vectors $x_1,\dots,x_n$ span $X$.
It follows from Proposition \[prop:projection-band-with-trivial-intersection\] that $P_k P_j = 0$ whenever $j \not= k$, so $$\begin{aligned}
(P_1+\dots+P_n)x_k = x_k \qquad \text{for each } k \in \{1,\dots,n\};
\end{aligned}$$ hence, $P_1 + \dots + P_n = {I}$, and we conclude that the linear mapping $$\begin{aligned}
J: \; Y:= P_1X \times \dots \times P_nX \ni (z_1,\dots,z_n) \to z_1+\dots+z_n \in X
\end{aligned}$$ is a bijection. Each $P_kX$ is an ordered space with respect to the order inherited from $X$, and as such it is isomorphic to ${\mathbb{R}}$ with the cone $[0,\infty)$. If we endow $Y$ with the product order, then $Y$ is isomorphic to ${\mathbb{R}}^n$ with the standard cone, and the mapping $J$ is an order isomorphism between $Y$ and $X$, which proves the assertion.
There is a certain conceptual similarity between the above proof and the approach taken in [@KalauchPreprint4 Section 6] to prove [@KalauchPreprint4 Theorem 38]: the authors of [@KalauchPreprint4] prove that every finite dimensional Archimedean pervasive pre-Riesz space is actually a vector lattice by considering atoms in such a space and by showing that if atoms $a_1,\dots,a_m$ in a pervasive pre-Riesz space are pairwise linearely independent, then the entire system $(a_1,\dots,a_m)$ is linearly independent. Our usage of Proposition \[prop:disjoint-vectors-are-linearly-independent\] and of rank-$1$ band projections in the above proof is somewhat reminiscent of this approach (as it is easy to see that the range of a rank-$1$ band projection is always spanned by an atom).
As a simple consequence of Theorem \[thm:n-distinct-band-projections-with-rank-1\] we obtain the following numerical bound on the number of rank-$1$ band projections in $X$:
\[cor:n-distinct-one-dimensional-projection-bands\] Assume that $n = \dim X < \infty$. Then there exists at most $n$ distinct band projections of rank $1$ on $X$.
If there were strictly more than $n$ distinct band projections of rank $1$, then Theorem \[thm:n-distinct-band-projections-with-rank-1\] would imply that $X$ is isomorphic to ${\mathbb{R}}^n$ with the standard cone – but on this space there exist precisely $n$ distinct rank-$1$ band projections, so we arrive at a contradiction.
Criteria in terms of the number of projection bands {#subsection:criteria-in-terms-of-the-number-of-projection-bands}
---------------------------------------------------
If our space $X$ is finite-dimensional and has closed cone, then it follows from [@Kalauch2019 Theorem 4.4.26] that there exist only finitely many bands in $X$. In particular, the Boolean algebra $\operatorname{PrBands}(X)$ is finite, so we conclude that the number of projection bands in $X$ is a power $2^m$ of $2$. In the following we are going to prove a bit more: we will not assume $X_+$ to be closed a priori, we will show that we always have $m \le \dim X$, and that equality holds if and only if $X$ is an Archimedean vector lattice.
Let us start with the following slightly more sophisticated version of Proposition \[prop:disjoint-vectors-are-linearly-independent\].
\[prop:disjoint-vectors-are-linearly-independent-more-sophisticated\] Let $A_1, \dots, A_m \subseteq X$ be subsets of $X$ such that $A_i \perp A_j$ whenever $i\not= j$. For each $j \in \{1,\dots,m\}$, let $(x_{j,1},\dots,x_{j,n_j})$ be a linearly independent system of vectors in $A_j$. Then the entire system $$\begin{aligned}
(x_{1,1},\dots,x_{1,n_1}, \dots, x_{m,1}, \dots, x_{m,n_m})
\end{aligned}$$ is linearly independent.
First we note that, for any two distinct indices $i,j \in \{1,\dots,m\}$, we have $\operatorname{span}(A_i) \perp A_j$ and hence $\operatorname{span}(A_i) \perp \operatorname{span}(A_j)$ since $X$ is a pre-Riesz space. Now assume that $$\begin{aligned}
\sum_{j=1}^m \sum_{k=1}^{n_j} \lambda_{j,k} x_{j,k} = 0
\end{aligned}$$ for scalars $\lambda_{j,k} \in {\mathbb{R}}$. We define vectors $y_j = \sum_{k=1}^{n_j} \lambda_{j,k} x_{j,k} \in \operatorname{span}(A_j)$ for $j \in \{1,\dots,m\}$. Thus, the vectors $y_1,\dots,y_m$ are pairwise disjoint. Since $$\begin{aligned}
y_1 + \dots + y_m = 0,
\end{aligned}$$ it follows from Proposition \[prop:disjoint-vectors-are-linearly-independent\] that one of theses vectors is $0$, and inductively we then derive that actually all vectors $y_1,\dots,y_m$ are $0$.
Now, fix $j \in \{1,\dots,m\}$. Since $0 = y_j = \sum_{k=1}^{n_j} \lambda_{j,k} x_{j,k}$, we conclude from the linear independence of the system $(x_{j,1},\dots,x_{j,n_j)}$ that $\lambda_{j,1} = \dots = \lambda_{j,n_j} = 0$. This proves the assertion.
A second ingredient that we need is the following simple observation about band projections.
\[lem:sum-of-band-projections\] Let $P_1,\dots,P_m: X \to X$ be band projections and assume that $P_i P_j = 0$ whenever $i\not= j$. Then $$\begin{aligned}
P_1 + \dots + P_m
\end{aligned}$$ is also a band projection.
The assumptions clearly imply that $P_1 + \dots + P_m$ is a positive projection. Next, we show by induction over $m$ that $$\begin{aligned}
{I}- (P_1 + \dots + P_m) = ({I}- P_1) ({I}- P_2) \cdots ({I}- P_m).
\end{aligned}$$ For $m = 1$ this is obvious, so assume that it has been proved for some fixed $m \in {\mathbb{N}}$ and consider now one more band projection $P_{m+1}$ such that $P_jP_{m+1} = 0$ for all $j \in \{1,\dots,m\}$. Then $$\begin{aligned}
({I}- P_1) ({I}- P_2) \cdots ({I}- P_m)({I}- P_{m+1}) & = \big({I}- (P_1 + \dots + P_m)\big)({I}- P_{m+1}) \\
& = {I}- (P_1 + \dots + P_{m+1}),
\end{aligned}$$ as claimed. We thus conclude that ${I}- (P_1 + \dots + P_m)$ is positive, too.
Now we can prove the first main result of this subsection.
\[thm:only-finitely-many-band-projections\] If $n = \dim X < \infty$, the following assertions hold:
1. The number of band projections on $X$ is equal to $2^m$ for some $m \in \{0,\dots,n\}$.
2. We have $m = n$ if and only if $X$ is an Archimedean vector lattice.
It is hardly surprising that the proof of Theorem \[thm:only-finitely-many-band-projections\] below is strongly related to the Boolean algebra structure of the set of all projection bands. However, we cannot rely on this Boolean structure alone since we want to relate the number $m$ to the dimension of $X$ – i.e., we need to take the linear structure of the underlying space into account.
We may assume throughout the proof that $n \not= 0$.
\(a) *Step 1:* Within this proof, let us call a band $B$ minimal if it is non-zero and if it does not contain any non-zero projection band except itself. Since $X$ is finite dimensional, every non-zero projection band contains a minimal projection band. Let ${\mathcal{M}}$ denote the set of all minimal projection bands in $X$. If $B,C \in {\mathcal{M}}$ are two distinct projection bands, then $B \cap C = \{0\}$; indeed, $B \cap C$ is a projection band that is contained in both $B$ and $C$. Hence, if it were non-zero, we would have $B \cap C = B$ and $B \cap C = C$, so $B = C$.
Consequently, $B \perp C$ for any two distinct $B,C \in {\mathcal{M}}$ by Proposition \[prop:projection-band-with-trivial-intersection\]. It thus follows from Proposition \[prop:disjoint-vectors-are-linearly-independent\] that there exist at most $n$ distinct minimal projection bands in $X$; we enumerate them as $B_1,\dots,B_m$ (where $1 \le m \le n$), and we denote the corresponding band projections by $P_1,\dots,P_m$.
Since $P_iP_j = 0$ whenever $i \not= j$, it follows from Lemma \[lem:sum-of-band-projections\] that $P_1 + \dots + P_m$ is a band projection. Actually, this band projection coincides with ${I}$, since otherwise the range of the complementary band projection ${I}- (P_1 + \dots + P_m)$ would contain one of the minimal projection bands $B_1,\dots,B_m$, which is a contradiction. Hence, $$\begin{aligned}
P_1 + \dots + P_m = {I}.
\end{aligned}$$ Consequently, $B_1 + \dots + B_m = X$.
*Step 2:* Next we note that, for each projection band $C$ in $X$ and each $k \in \{1,\dots,m\}$ we have either $B_j \subseteq C$ or $B_j \cap C = \{0\}$; this is a consequence of the minimality of $B_j$. Hence, for every band projection $P$ on $X$ and every $j \in \{1,\dots,m\}$ we have either $PP_j = P_j$ or $P P_j = 0$.
Thus, for every band projection $P$ on $X$ we have $$\begin{aligned}
P = \sum_{j\in I_P} P_j,
\end{aligned}$$ where $I_P := \big\{j \in \{1,\dots,m\}: \, P_jP \not= 0\big\}$. Conversely, we note that the sum $P_I := \sum_{j\in I} P_j$ is, for any $I \subseteq \{1,\dots,m\}$, a band projection (according to Lemma \[lem:sum-of-band-projections\]), and the set $I$ is uniquely determined by this sum (since it is the set of all $k$ such that $P_kP \not= 0$). This proves that there exist exactly $2^m$ band projections on $X$, and we have already observed above that $m \le n$. We have thus proved (a)
\(b) Assume now that $m = n$.
For every $j \in \{1,\dots,m\}$ we now choose a basis $(x_{j,1},\dots, x_{j,n_j})$ of the space $B_j$. It follows from Proposition \[prop:disjoint-vectors-are-linearly-independent-more-sophisticated\] that the system $$\begin{aligned}
(x_{1,1},\dots,x_{1,n_1}, \dots, x_{m,1}, \dots, x_{m,n_m})
\end{aligned}$$ is linearly independent. Hence, $n_1 + \dots + n_m \le n$. As $m = n$, it follows that none of the numbers $n_j$ can be larger than $1$, so each of the $n$ projection bands $B_1,\dots, B_m = B_n$ is one-dimensional. Theorem \[thm:n-distinct-band-projections-with-rank-1\] thus shows that $X$ is an Archimedean vector lattice.
Conversely, if $X$ is an Archimedean vector lattice, then it is isomorphic to ${\mathbb{R}}^n$ with the standard cone, so there exist indeed $2^n$ band projections on $X$, so $m=n$.
Step 1 in the proof of Theorem \[thm:only-finitely-many-band-projections\] also provides us with another interesting insight into the structure of finite-dimensional pre-Riesz spaces. The facts that $P_iP_j = 0$ for any two distinct $i,j \in \{1,\dots,m\}$ and that $P_1 + \dots + P_m = {I}$ imply that the mapping $$\begin{aligned}
X & \to B_1 \times \dots \times B_m, \\
x & \mapsto (P_1x,\dots,P_mx)\end{aligned}$$ is an isomorphism of order vector spaces, where $B_1 \times \cdot \times B_m$ is endowed with the product order. Moreover, we note that each of the projection bands $B_j$ has generating cone, so it is itself a pre-Riesz space according to [@Kalauch2019 Corollary 2.2.7].
We also observe that none of the pre-Riesz spaces $B_j$ contains a non-trivial projection band. Indeed, if $Q: B_j \to B_j$ is a band projection, then $QP_j: X \to X$ is a band projection with the same range as $Q$; by the minimality if $B_j$ this implies that this range is either $\{0\}$ or $B_j$. We thus have the following structure result, which is the second main result of this subsection.
\[thm:projection-band-structure-of-finite-dimensional-pre-riesz-spaces\] Assume that $1 \le \dim X < \infty$. Then there exists a number $m \in \{1,\dots,\dim X\}$ such that $X$ is isomorpic (as an ordered vector space) to the product of $m$ non-zero pre-Riesz spaces none of which contains a non-trivial projection band.
Criteria in the class weakly pervasive spaces {#subsection:criteria-in-the-class-of-weakly-pervasive-spaces}
---------------------------------------------
Assume for a moment that $X$ is finite dimensional with closed cone. In [@KalauchPreprint4 Theorem 38] it was shown that if $X$ is pervasive (see Definition \[def:pervasive-and-weakly-pervasive\] below), then $X$ is in fact a vector lattice. The same is true if $X$ is assumed to have the Riesz decomposition property instead if being pervasive (see for instance [@Aliprantis2007 Corollary 2.49]). These observations suggest to study the following two questions:
(a) Since the Riesz decomposition property and the property of beging pervasive are logically independent for general pre-Riesz spaces (see [@Kalauch2019a Example 13] and [@Malinowski2018 Example 23]), it is natural to seek for a simultaneous generalisations of the two above mentioned results.
(b) The fact that the Riesz decomposition property implies that $X$ is a vector lattice is actually not only true in finite-dimensional spaces (with closed cone), but for instance also for the more general case of reflexive ordered Banach spaces with generating cone [@Aliprantis2007 Corollary 2.48]. This suggests that searching for sufficient criteria for infinite dimensional spaces to be a vector lattice is a worthwhile endeavour.
In this subsection we pursue both goals outlined above. As before, we assume that $X$ is a general pre-Riesz space.
Two vectors $x,y \in X_+$ are called *D-disjoint* if $[0,x] \cap [0,y] = \{0\}$. For a more detailed discussion of this notion and of its origin, we refer to [@Kalauch2019 Section 4.1.3] and [@Katsikis2006 Defintion 8 and Proposition 9].
Every two disjoint elements in $X$ are clearly D-disjoint, but the converse implication is not true, in general; this can, for instance, again be seen by considering the four ray cone in ${\mathbb{R}}^3$:
\[ex:D-disjoint-does-not-imply-disjoint\] Let $X = \mathbb{R}^3$ and let $X_+$ denote the four ray cone from Example \[ex:four-ray-cone-bands\]; let $v_1$ and $v_2$ denote the vectors given in the same example. According to Example \[ex:non-disjoint-bands-with-trivial-intersection\] the vectors $v_1$ and $v_2$ are not disjoint. However, both elements $v_1$ and $v_2$ are so-called *atoms* in $X$ (see [@KalauchPreprint4 Definition 27 and Proposition 28] or Subsection \[subsection:criteria-in-terms-of-other-concepts-of-disjointness\] below), so it follows that $v_1$ and $v_2$ are $D$-disjoint.
Hence, disjointness of two vectors $x,y \in X_+$ is, in general, a much stronger property than D-disjointness. There are, however, spaces in which both notions coincide; this gives rise to part (a) of the following definition.
\[def:pervasive-and-weakly-pervasive\]
(a) The pre-Riesz space $X$ is called *weakly pervasive* if any two D-disjoint vectors in $X_+$ are automatically disjoint.
(b) The pre-Riesz space $X$ is called *pervasive* if for every $b \in X$ such that $b \not\le 0$ there exists $x \in X_+ \setminus \{0\}$ such that every positive upper bound of $b$ is also an upper bound of $x$.
The concept of a weakly pervasive pre-Riesz space was coined in [@Kalauch2019a Definition 8 and Lemma 9]. The usual definition of a prevasive pre-Riesz space in the literature is somewhat different and employs the Riesz completion of $X$ (see [@Kalauch2019 Definition 2.8.1]). However, this definition is equivalent to the one given above according to [@Kalauch2019a Theorem 7].
If one uses that two vectors $x,y \in X_+$ are disjoint if and only if they have infimum $0$, it is easy to show that every pervasive pre-Riesz space is also weakly pervasive. Moreover, every vector lattice is pervasive and hence weakly pervasive.
We also note that every pre-Riesz space with the Riesz decomposition property is weakly pervasive [@Kalauch2019a Proposition 11]; hence, weakly pervasive spaces are a simultaneous generalisation of pervasive pre-Riesz spaces and pre-Riesz spaces with the Riesz decomposition property.
Let us give a simple criterion in order to check that several function spaces are prevasive.
\[prop:subspaces-of-continuous-functions-which-are-weakly-pervasive\] Let $\Omega \subseteq {\mathbb{R}}^d$ be open, let $\Omega \subseteq L \subseteq \overline{\Omega}$ and let $X$ be a directed vector subspace of $C(L)$ (where $C(L)$ denotes the space of all real-valued continuous functions on $L$). Then $X$ is a pre-Riesz space; if, in addition, $X$ contains all test functions on $\Omega$, then $X$ is weakly pervasive.
As $C(L)$ as Archimedean, so is $X$, and since $X_+$ is generating in $X$ by assumption, it follows that $X$ is a pre-Riesz space.
Now assume that $X$ contains all test functions on $\Omega$. Let $b \in X$ and $b \not\le 0$. Since $C(L)$ is a vector lattice, we can take the positive part $b^+$ in $C(L)$. This is a non-zero positive continuous function on $L$, so there exists a positive non-zero test function $x$ on $\Omega$ such that $x \le b^+$. We note that $x \in X$ by assumption. Now, if $u \in X_+$ is an upper bound of $b$ in $X$, then it is also an upper bound of $b^+$ in $C(L)$. Hence, $u \ge x$.
As a consequence of the above proposition we obtain, for instance, the following examples of pervasive spaces.
\[ex:list-of-weakly-pervasive-spaces\] (a) Let $\emptyset \not= \Omega \subseteq {\mathbb{R}}^d$ be open and bounded and let $k \in \mathbb{N}_0$. Then the space $C_b^k(\overline{\Omega})$ of functions that are $k$-times continuously differentiable on $\Omega$ and whose partial derivatives up to order $k$ all have a continuous extension to $\overline{\Omega}$ is pervasive.
\(b) Let $\emptyset \not= \Omega \subseteq {\mathbb{R}}^d$ be open and bounded with Lipschitz boundary, let $p \in [1,\infty]$ and $k \in {\mathbb{N}}$ such that $kp > d$. Then the positive cone in the Sobolev space $W^{k,p}(\Omega)$ is closed (with respect to the usual Sobolev norm), and it is also generating (see [@Arendt2009 Examples 2.3(c) and (d)]); hence, $W^{k,p}(\Omega)$ is a pre-Riesz space. Moreover, $W^{k,p}(\Omega)$ is also pervasive since it embeds into $C(\Omega)$ and since it contains all test functions on $\Omega$.
In [@Kalauch2019a Example 13] one can find an example of a pre-Riesz space that is not pervasive, but has the Riesz decomposition property and is thus weakly pervasive.
We now prove the main result of this subsection; it gives a sufficient criterion for a weakly pervasive pre-Riesz space to already be a vector lattice.
\[thm:weakly-pervasive-monotonically-complete-spaces-are-lattices\] Assume that every totally ordered subset of $X$ that is bounded from above has a supremum. If $X$ is weakly pervasive, then $X$ is a lattice.
If suffices to show that any two positive elements in $X$ have an infimum, so let $x,y \in X_+$. It follows from Zorn’s lemma and from the assumption on $X$ that the set $[0,x] \cap [0,y]$ has a maximal element $a$. Let us show that $[0,x-a] \cap [0,y-a] = \{0\}$: if $z$ is an element of this set, then $0 \le z \le x-a$ and $0 \le z \le y-a$, so $0 \le z+a \le x$ and $0 \le z+a \le y$. Hence, $z+a$ is an element of $[0,x] \cap [0,y]$ that dominates $z$; it thus follows from the maximality of $z$ that $a = 0$.
As $X$ is weakly pervasive, this implies that the positive vectors $x-a$ and $y-a$ are disjoint, i.e. they have infimum $0$. Consequently, $x$ and $y$ also have an infimum (namely $a$).
In the context of ordered Banach spaces, the following proposition gives sufficient criteria for the first assumption of Theorem \[thm:weakly-pervasive-monotonically-complete-spaces-are-lattices\] to be satisfied.
\[prop:regularity-properties-of-ordered-banach-spaces\] Assume that our pre-Riesz space $X$ is an ordered Banach space. Consider the following assertions:
1. The cone $X_+$ is normal and the space $X$ is reflexive.
2. The cone $X_+$ is normal and $X$ is a projection band in its bi-dual (compare Example \[ex:ordered-banach-space-band-in-bidual\]).
3. Every order interval in $X$ is weakly compact.
4. $X$ is the dual space of an ordered Banach space $Y$ such that $Y$ has generating cone.
5. The norm is additive on $X_+$ (i.e. $\lVert x+y\rVert = \lVert x \rVert + \lVert y \rVert$ for all $x,y \in X_+$).
6. Every increasing norm bounded net in $X_+$ is norm convergent.
7. The cone $X_+$ is normal and every increasing net in $X_+$ that is bounded from above is norm convergent.
8. Every non-empty upwards directed set in $X$ that is bounded above has a supremum (i.e., in the terminology of Subsection \[subsection:the-intersection-of-arbitrarily-many-projection-bands\], $X$ is Dedekind complete).
9. Every non-empty totally ordered set in $X$ that is bounded above has a supremum.
Then the following implications hold:\
“(i) $\Rightarrow$ (ii)” This is obvious.
“(i) $\Rightarrow$ (iv)” This is obvious.
“(ii) $\Rightarrow$ (iii)” This was proved in [@GlueckWolffLB Proposition 2.6].
“(ii) $\Rightarrow$ (vi)” The proof of this implication has already been sketched in [@GlueckWolffLB Remark 6.2]; we give a few more details here:
Let $P: X'' \to X''$ be the band projection with range $X$ and let $(x_j)$ be an increasing and norm-bounded net in $X_+$. Then $(x_j)$ converges to a vector $x'' \in X''_+$ with respect to the weak${}^*$-topology. We have $Px'' \le x''$. On the other hand, $Px'' \ge Px_j = x_j$ for each index $j$, which implies that $Px'' \ge x''$. We have thus shown that $Px'' = x''$, i.e., $x := x''$ is an element of $X$.
The increasing net $(x_j)$ converges weakly to $x$, so it follows from [@Schaefer1999 Theorem V.4.3] that $(x_j)$ actually converges in norm to $x$.
“(iii) $\Rightarrow$ (vii)” Assertion (iii) implies that every order interval in $X$ is bounded; hence, the cone $X_+$ is normal. Now, let $(x_j)$ is an increasing net in $X_+$ that is bounded above. Then $(x_j)$ is contained in an order interval. Hence, $(x_j)$ is weakly convergent and therefore also norm convergent according to [@Schaefer1999 Theorem V.4.3].
“(iv) $\Rightarrow$ (viii)” Let $A \subseteq X$ be an upwards directed set that is bounded above. Then the increasing net $(a)_{a \in A}$ is weak${}^*$-convergent to an element $x \in X$, and one readily checks that $x$ is the supremum of $A$.
“(v) $\Rightarrow$ (vi)” Let $(x_j) \subseteq X_+$ be an increasing norm bounded net. We show that this net is Cauchy and thus norm convergent. To this end, set $\alpha := \sup_j \|x_j\| \in [0,\infty)$ and let $\varepsilon > 0$. Choose $j_0$ such that $\|x_{j_0}\| \ge \alpha - \varepsilon$. For all indices $j \ge j_0$ we then obtain $$\begin{aligned}
\alpha \ge \|x_j\| = \|x_{j_0}\| + \|x_j - x_{j_0}\| \ge \alpha - \varepsilon + \|x_j - x_{j_0}\|
\end{aligned}$$ so $\|x_j - x_{j_0}\| \le \varepsilon$. This proves that $(x_j)$ is indeed Cauchy.
“(vi) $\Rightarrow$ (vii)” As every increasing norm-bounded sequence in $X_+$ is norm convergent, it follows that $X_+$ is normal; see for instance [@Aliprantis2007 Theorem 2.45]. Hence, every increasing net in $X_+$ which is bounded above is also norm bounded and thus norm convergent according to (vi).
“(vii) $\Rightarrow$ (viii)” Let $D \subseteq X$ be an arbitrary upwards directs set which is bounded above by a vector $u \in X$. Choose $b \in X_+$ such that $b+D$ intersects the positive cone $X_+$. Then $\tilde D := X_+ \cap (b+D)$ is an upwards directed set, too, and $\tilde D$ is bounded above by $u+b$. Hence, the increasing net $(x)_{x \in \tilde D}$ converges to a vector $y \in X$. Clearly, $y$ is the supremum of $\tilde D$, and thus it is also the supremum of $b+D$ (here we used again that $b+D$ is directed). Therefore, $D$ has the supremum $y-b$.
“(viii) $\Rightarrow$ (ix)” This is obvious.
The author does not know whether the converse implication “(ix) $\Rightarrow$ (viii)” in Proposition \[prop:regularity-properties-of-ordered-banach-spaces\] is true.
As a consequence of Theorem \[thm:weakly-pervasive-monotonically-complete-spaces-are-lattices\] we observe that if our pre-Riesz space $X$ is a weakly pervasive ordered Banach space and satisfies at least one of the assertions (i)–(ix) in Proposition \[prop:regularity-properties-of-ordered-banach-spaces\], then $X$ is actually a vector lattice. Since every finite-dimensional Banach space is reflexive and every generating closed cone in such a space is normal, we obtain in particular the following corollary.
\[cor:finite-dimensional-weakly-pervasive-spaces\] Let $X$ be finite dimensional and assume that $X_+$ is closed. If $X$ is weakly pervasive, then $X$ is a vector lattice (and thus isomorphic to $\mathbb{R}^n$ with the standard order).
We note once again that, for the special case where $X$ is prevasive, Corollary \[cor:finite-dimensional-weakly-pervasive-spaces\] has been recently proved in [@KalauchPreprint4 Theorem 38]. Let us remark of few further consequences of Theorem \[thm:weakly-pervasive-monotonically-complete-spaces-are-lattices\] in conjuction with Proposition \[prop:regularity-properties-of-ordered-banach-spaces\].
(a) Recall that several examples of ordered Banach spaces that are pervasive (and hence weakly pervasive) are listed in Examples \[ex:list-of-weakly-pervasive-spaces\]. Theorem \[thm:weakly-pervasive-monotonically-complete-spaces-are-lattices\] and Proposition \[prop:regularity-properties-of-ordered-banach-spaces\] show that such examples have to satisfy many restrictions if we do not want to end up in the category of vector lattices.
(b) If our pre-Riesz space $X$ is an ordered Banach space with normal cone, then the dual space $X'$ also has generating cone and is thus a pre-Riesz space. Proposition \[prop:regularity-properties-of-ordered-banach-spaces\] shows that every non-empty totally ordered set in $X'$ that is bounded above has a supremum. Hence, if $X'$ is weakly pervasive, it follows from Theorem \[thm:weakly-pervasive-monotonically-complete-spaces-are-lattices\] that $X'$ is in fact a vector lattice, and thus we conclude in turn that $X$ has the Riesz-decomposition property [@Aliprantis2007 Theorem 2.47].
Hence, the dual space $X'$ of an ordered Banach space $X$ with normal (and generating) cone cannot be weakly pervasive unless $X$ itself has the Riesz decomposition property. This suggests that the property “weakly pervasive” is not particularly well-behaved with respect to duality (at least not in the category of ordered Banach spaces).
Criteria in terms of other concepts of disjointness {#subsection:criteria-in-terms-of-other-concepts-of-disjointness}
---------------------------------------------------
Recall that weakly pervasive spaces are precisely those pre-Riesz spaces in which any two D-disjoint elements of the positive cone are automatically disjoint. In this context is is interesting to observe that, in general pre-Riesz spaces, there exists an intermediate concept between disjointness and D-disjointness; this is the content of the following proposition.
\[prop:intermediate-between-disjoint-and-d-disjoint\] For all $x,y \in X_+$ we have the following implications: $$\begin{aligned}
x\text{ and } y \text{ are disjoint} \quad \Rightarrow \quad [-x,x] \cap [-y,y] = \{0\} \quad \Rightarrow \quad x\text{ and } y \text{ are D-disjoint.}
\end{aligned}$$
Assume first that $x$ and $y$ are disjoint. If $f \in [-x,x] \cap [-y,y]$, then $f$ is a lower bound of $x$ and $y$, so $f \le 0$. Moreover, $-f$ is also a lower bound of $x$ and $y$, so $-f \le 0$. Hence, $f= 0$. The second implication is obvious.
We will see in Example \[ex:disjointness-implication-cannot-be-reversed\] below that none of the two implications in Propositions \[prop:intermediate-between-disjoint-and-d-disjoint\] can be reversed in general pre-Riesz spaces. Before we give this example, we need a small auxiliary result.
We recall from [@KalauchPreprint4 Definition 27] that an element $a \in X_+ \setminus \{0\}$ is called an *atom* in $X$ if every vector $x \in [0,a]$ is a multiple of $a$; equivalently, the order interval $[0,a]$ equals the line segment $\{\lambda a: \; \lambda \in [0,1]\}$.
\[lem:symmetric-line-segments-for-atom\] Let $a$ be an atom in $X$. Then the order interval $[-a,a]$ equals the line segment $\{\lambda a: \; \lambda \in [-1,1]\}$.
Each $x \in [-a,a]$ can be written as $$\begin{aligned}
x = \frac{a+x}{2} - \frac{a-x}{2},
\end{aligned}$$ where both $\frac{a+x}{2}$ and $\frac{a-x}{2}$ are elements of $[0,a]$; hence, we have $[-a,a] = [0,a] - [0,a]$ (for this, we did not use that $a$ is an atom). Since $[0,a]$ is the line segment $\{\lambda a: \; \lambda \in [0,1]\}$, this implies the assertion.
\[ex:disjointness-implication-cannot-be-reversed\] Let $X = {\mathbb{R}}^3$, let $X_+$ denote the four ray cone from Example \[ex:four-ray-cone-bands\], and let $v_1,\dots,v_4 \in X$ be the vectors from that example.
(a) Let $w = v_1+v_2$ and $\tilde w = v_3 + v_4$. Then $w$ and $\tilde w$ are $D$-disjoint, but the set $[-w,w] \cap [-\tilde w, \tilde w]$ is non-zero since it contains the vector $(1, -1, 0)^T$.
(b) The order interval $[-v_1,v_1]$ is precisely the line segment $\{\lambda v_1: \; \lambda \in [-1,1]\}$; this follows from Lemma \[lem:symmetric-line-segments-for-atom\] since $v_1$ is an atom in $X$ (which in turn follows from [@KalauchPreprint4 Proposition 28]). Similarly, the order interval $[-v_2,v_2]$ is the line segment $\{\lambda v_2: \; \lambda \in [-1,1]\}$.
We thus conclude that $[-v_1,v_1] \cap [-v_2,v_2] = \{0\}$. Yet, we have seen in Example \[ex:non-disjoint-bands-with-trivial-intersection\] that $v_1$ and $v_2$ are not disjoint.
We now consider pre-Riesz spaces in which, for all $x,y \in X_+$, the property $[-x,x] \cap [-y,y] = \{0\}$ implies that $x \perp y$. This property of a pre-Riesz space is (at least formally) weaker than being weakly pervasive. In finite dimensions, though, this property still suffices to conclude that a pre-Riesz space with closed cone is a vector lattice; we prove this in the following theorem.
\[thm:weaker-version-of-weakly-pervasive-implies-vector-lattice\] Let $X$ be finite dimensional and assume that $X_+$ is closed. Suppose that all vectors $x,y \in X_+$ that satisfy $[-x,x] \cap [-y, y] = \{0\}$ are disjoint. Then $X$ is a vector lattice.
For the proof we need the notion of an *extreme ray*. Let $a \in X_+ \setminus \{0\}$. If the half ray $\{\lambda a: \; \lambda \in [0,\infty)\}$ is a face of $X_+$, then we call this half ray an extreme ray of $X_+$. We note that $\{\lambda a: \; \lambda \in [0,\infty)\}$ is an extreme ray of $X_+$ if and only if $a$ is an atom in $X$ ([@KalauchPreprint4 Proposition 28]). If $X$ is finite dimensional and non-zero and $X_+$ is closed, then $X_+$ is the convex hull of its extreme rays.
Set $n := \dim X$; we may assume that $n \ge 1$. Let $E$ denote the set of all extreme rays of $X_+$ and for each $R \in E$, choose a non-zero vector $x_R \in R$. Then the set $\{x_R: \, R \in E\}$ spans $X$, so $E$ has at least $n$ elements.
On the other hand, each point $x_R$ is an atom in $X$, so for any two distinct rays $R,S \in E$ we have $[-x_R,x_R] \cap [-x_S, x_S] = \{0\}$ according to Lemma \[lem:symmetric-line-segments-for-atom\]. Thus, it follows from the assumption that $x_S \perp x_R$ for any two distinct rays $R,S \in E$. Hence, we conclude from Proposition \[prop:disjoint-vectors-are-linearly-independent\] that the family of vectors $(x_R)_{R \in E}$ is linearly independent. Hence, $E$ has exactly $n$ elements. This proves that the positive cone $X_+$ is generated by exactly $n$ extreme rays, so $X$ is a vector lattice.
Acknowledgements {#acknowledgements .unnumbered}
----------------
It is my pleasure to thank Anke Kalauch, Helena Malinowski and Onno van Gaans for various suggestions and discussions which helped me to considerably improve the paper.
The paper was originally motivated by the question how many bands and projection bands can exist in a finite dimensional pre-Riesz space – a question I first became aware of during a plenary talk of Anke Kalauch at the *Positivity X* conference that took place in July 2019 in Pretoria, South Africa. I am indebted to the organisers of the conference for financially supporting my participation.
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abstract: 'This paper is concerned with the study of the nonlinear stability of the contact discontinuity of the Navier-Stokes-Poisson system with free boundary in the case where the electron background density satisfies an analogue of the Boltzmann relation. We especially allow that the electric potential can take distinct constant states at boundary. On account of the quasineutral assumption, we first construct a viscous contact wave through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system. The analysis is based on the techniques developed in [@DL] and an elementary $L^2$ energy method.'
author:
- 'Shuangqian Liu[^1],Haiyan Yin[^2],Changjiang Zhu[^3]'
title: '[**Stability of contact discontinuity for the Navier-Stokes-Poisson system with free boundary**]{}'
---
[**Key words.**]{} viscous contact discontinuity, quasineutral Euler equations, stability, free boundary.
[**AMS subject classifications.**]{} 35B35, 35Q35, 82D10.
Introduction
============
The problem
-----------
The dynamics of the charged particles in the collisional dusty plasma can be described by the Navier-Stokes-Poisson (denoted as NSP in the sequel) system [@GSK]. The one-dimensional NSP system in the Eulerian coordinates takes the form of $$\begin{aligned}
\label{NSPe}
&&\left\{\begin{aligned}
& {\partial}_{t}\rho+{\partial}_{x}(\rho u)=0,\\
& {\partial}_{t}( \rho u)+{\partial}_{x}(\rho u^{2}+p)
=\rho{\partial}_{x}\phi+\mu {\partial}_{x}^{2}u,\\
&{\partial}_{t}W+{\partial}_{x}(Wu+pu)=\rho u{\partial}_{x}\phi+\mu {\partial}_{x}(u{\partial}_{x}u)+\kappa{\partial}_{x}^{2}{\theta},\\
&{\partial}_{x}^{2}\phi=\rho-\rho_{e}(\phi).
\end{aligned}\right.\end{aligned}$$ The unknown functions $\rho$, $u$ and ${\theta}$ stand for the density, velocity and absolute temperature of ions, respectively, while $\mu>0$ is the viscosity coefficient and $\kappa>0$ is the heat conductivity coefficient. $W$ stands for the total energy of the ions, taking the following form: $$\begin{aligned}
\label{1.2}
W=\frac{\rho u^{2}}{2}+\frac{p}{\gamma-1},\end{aligned}$$ where $\gamma>1$ is the adiabatic exponent. $p$ is the pressure which is given by $$\begin{aligned}
\label{1.3}
p=R\rho{\theta}=A\rho^{\gamma}e^{\frac{\gamma-1}{R}S},\end{aligned}$$ where $S$ is the entropy and $A,$ $R$ are both positive constants. The self-consistent electric potential $\phi=\phi(x,t)$ is induced by the total charges through the Poisson equation. The density $\rho_e=\rho_e(\phi)$ of electrons in depends only on the potential in the sense of an analogue of the so-called Boltzmann relation, cf. [@Ch; @GP]. Specifically, through the paper we suppose that
$({\mathcal{A}})$
: $\rho_e(\phi): (\phi_m,\phi_M)\to (\rho_m,\rho_M)$ is a smooth function with $$\rho_m=\inf\limits_{\phi_m<\phi<\phi_M} \rho_e(\phi),\quad \rho_M=\sup\limits_{\phi_m<\phi<\phi_M} \rho_e(\phi),$$ satisfying the following two assumptions:
$(\mathcal {A}_1)$ $\rho_e(0)=1$ with $0\in (\phi_m,\phi_M)$;
$(\mathcal {A}_2)$ $\rho_e(\phi)>0$, $\rho'_e(\phi)<0$ for each $\phi\in (\phi_m,\phi_M)$.
The assumption $(\mathcal {A}_1)$ just means that the electron density has been normalized to be unit when the potential is zero, since the electric potential in can be up to an arbitrary constant. The sign of the first derivative of the function ${\rho}_e(\phi)$ in the assumption $({\mathcal{A}}_2)$ plays a crucial role in our analysis, it is to be further clarified later on, see , etc.
An important example satisfying $(\mathcal {A})$ can be given as $$\label{D-dene}
\rho_e(\phi)=\left[1-\frac{{\gamma}_e-1}{{\gamma}_e} \frac{\phi}{A_e}\right]^{\frac{1}{{\gamma}_e-1}}, \quad \phi_m=-\infty,\quad\phi_{M}=\frac{{\gamma}_e}{{\gamma}_e-1}A_e,$$ with ${\gamma}_e\geq 1$ and $A_e>0$ being constants. Note that $\rho_e(\phi)\to e^{-\frac{\phi}{A_e}}$ and $\phi_M\to +\infty$ as ${\gamma}_e\to1^+$, which corresponds to the classical Boltzmann relation. In fact, can be formally deduced from the momentum equation of the isentropic Euler-Poisson system for the fluid of electrons with the adiabatic exponent ${\gamma}_e$ under the zero-limit of electron mass, namely, $
{\partial}_x\left(A_e\rho_e^{{\gamma}_e}\right)=-\rho_e{\partial}_x\phi.
$
In this paper, we consider the system in the part $+\infty>x\geq x(t)$, where $x=x(t)$ is a free boundary with the following dynamical boundary conditions $$\label{BCe}
\frac{dx(t)}{dt}=u(x(t),t),\ \ x(0)=0,\ (p-\mu\partial_{x} u)\mid_{x=x(t)}=p_{-},\ \theta(x(t),t)=\theta_{-},\ \phi(x(t),t)=\phi_{-}.$$ We also assume $\phi$ satisfies the boundary condition at far field: $$\begin{aligned}
\label{pfc}
\lim_{x\rightarrow+\infty}\phi(x,t)=\phi_{+}.\end{aligned}$$ The initial data is given by $$\label{id.e}
(\rho,u,\theta)(x,0)=(\rho_{0},u_{0},\theta_{0})(x),\ \
\lim_{x\rightarrow+\infty}(\rho_{0},u_{0},\theta_{0})(x)=(\rho_{+},u_{+},\theta_{+}).$$ Here $\rho_{+}>0$, $\theta_{\pm}>0$, $p_->0$, $u_+$ and $\phi_{\pm}$ are assumed to be constant states. Also, ${\rho}_0(x)>0$ is supposed, so that the ions flow has no vacuum state. In addition, we of course assume ${\theta}_0(x)$ satisfies the compatibility condition and $\phi$ satisfies the the quasineutral condition at far field, i.e. $$\label{cqa.e}
\theta_{0}(0)=\theta_{-},\ \rho_{e}(\phi_{+})=\rho_{+}.$$ Our main purpose concerns the large time behavior of solutions to , , and , to explore this, it is more convenient to use the Lagrangian coordinates. That is, consider the coordinate transformation: $$\label{1.5}
x\Rightarrow\int_{x(t)}^x\rho(y,t)dy,\ \ t\Rightarrow t.$$ We still denote the Lagrangian coordinates by $(x,t)$ for simplicity of notation. Noticing that $$\int_{x(t)}^{x}\rho(y,t)dy\rightarrow+\infty,\ \textrm{as}\ x\rightarrow+\infty,
$$ one sees that , , and can be transformed as the problem with fixed boundary in the form of $$\begin{aligned}
\label{NSPl}
&&\left\{\begin{aligned}
&{\partial}_{t}v-{\partial}_{x}u=0,\ \ x>0,\ t>0,\\
& {\partial}_{t} u+{\partial}_{x}p
=\frac{{\partial}_{x}\phi}{v}+\mu {\partial}_{x}\left(\frac{{\partial}_{x}u}{v}\right),\ \ x>0,\ t>0,\\
&\frac{R}{\gamma-1}{\partial}_{t}{\theta}+p{\partial}_{x}u=\mu\frac{({\partial}_{x}u)^{2}}{v}+\kappa{\partial}_{x}\left(\frac{{\partial}_{x}{\theta}}{v}\right),\ \ x>0,\ t>0,\\
&{\partial}_{x}\left(\frac{{\partial}_{x}\phi}{v}\right)=1-v\rho_{e}(\phi),\ \ x>0,\ t>0,\\
\end{aligned}\right.\end{aligned}$$ with boundary condition $$\label{BCl}
\theta(0,t)=\theta_{-}, \ \left(p-\mu\frac{\partial_{x}
u}{v}\right)(0,t)=p_{-},\ \phi(0,t)=\phi_{-},\ \lim_{x\rightarrow+\infty}\phi(x,t)=\phi_{+},\ t\geq0,$$ and the initial data $$\label{idl}
(v,u,\theta)(x,0)=(v_{0},u_{0},\theta_{0})(x),\ \ \ x\geq0, \ \ \ \
\
\lim_{x\rightarrow+\infty}(v_{0},u_{0},\theta_{0})(x)=(v_{+},u_{+},\theta_{+}).$$ Here $v=1/\rho$ stands for the specific volume. Moreover, $$\label{cqa}
{\theta}_0(0)={\theta}_-\ \textrm{and}\ v_{+}=\frac{1}{\rho_{e}(\phi_{+})}$$ hold according to .
Quasineutral Euler equations and contact waves
----------------------------------------------
In order to study the large time behavior of the solution $[v(x,t),u(x,t),{\theta}(x,t),\phi(x,t)]$ to the initial boundary value problem , and , we expect that $[v(x,t),u(x,t),{\theta}(x,t),\phi(x,t)]$ tends time-asymptotically to viscous contact wave to the Riemann problem on the quasineutral Euler system $$\begin{aligned}
\label{MEt}
\left\{
\begin{array}{clll}
\begin{split}
&{\partial}_tv-{\partial}_x u=0,\\
&{\partial}_t u+{\partial}_xp=\frac{{\partial}_x\phi}{v},\\
&\frac{R}{\gamma-1}{\partial}_{t}\theta+p{\partial}_xu=0,\\
&1/v=\rho_e(\phi),
\end{split}
\end{array}
\right.\end{aligned}$$ with Riemann initial data given by $$\label{MEtid}
[v, u, \theta](x,0)= \left\{\begin{array}{rll}[v_-,u_{-}, \theta_-],&\ \ x<0,\\[2mm]
[v_+,u_{+}, \theta_+],&\ \ x>0.
\end{array}
\right.$$ According to [@Da; @S], one sees that the Riemann problem and admits a contact discontinuity solution $$\begin{aligned}
\label{con.dis}
\begin{split}
\left[v^{CD}, u^{CD}, {\theta}^{CD},\phi^{CD}\right](x,t)=\left\{\begin{array}{rll}[v_-,u_-, \theta_-, \phi_{-}],&\ \ x<0,\\[2mm]
[v_+,u_+, \theta_+, \phi_{+}],&\ \ x>0,
\end{array}
\right.
\end{split}\end{aligned}$$ on the condition that $$\label{cd.con}
\begin{split}
u_-=u_+,\ \
p_{-}{\overset{\mbox{\tiny{def}}}{=}}p(v_-,\theta_-)=p_{+}+p^{\phi}(v_+)-p^{\phi}(v_-),
\end{split}$$ where $$\label{cd.con2}
p_{+}=p(v_+,\theta_+),\
\phi_{\pm}=\rho_{e}^{-1}(1/v_{\pm}) \ \textrm{and}\ p^{\phi}=p^{\phi}(v)=\int^{v}\frac{1}{\varrho^{3}\rho_{e}'(\rho_{e}^{-1}(\frac{1}{\varrho}))}d\varrho.$$
On the other hand, due to the dissipation effect of the NSP system , a viscous contact wave $\left[v^{cd}, u^{cd}, {\theta}^{cd},\phi^{cd}\right]$ corresponding to the contact discontinuity $\left[v^{CD}, u^{CD}, {\theta}^{CD},\phi^{CD}\right]$ defined as can be constructed as follows. We first denote $p^{cd}=p^{cd}(v^{cd},{\theta}^{cd})=\frac{R{\theta}^{cd}}{v^{cd}}$. Since the quasineutral pressure $p^{cd}+p^{\phi}$ for the profile $\left[v^{cd}, u^{cd}, {\theta}^{cd},\phi^{cd}\right]$ is expected to be almost constant, we set $$\label{p.cs}
p_{-}=p^{cd}+\int_{v_-}^{v^{cd}}\frac{1}{\varrho^{3}\rho_{e}'(\rho_{e}^{-1}(\frac{1}{\varrho}))}d\varrho.$$ Noticing that $\frac{{\partial}p^{cd}}{{\partial}v^{cd}}<0$ and $\rho_{e}'(\cdot)<0$, from which and and the implicit function theorem, we see that there exists a differentiable function $f({\theta}^{cd})$ such that $$\label{r.vta}
v^{cd}=f({\theta}^{cd}), \ v_\pm=f({\theta}_\pm),$$ provided that $|{\theta}_+-{\theta}_-|$ is suitably small. Furthermore, by a direct calculation, it follows that $$\label{d.f}
\begin{split}
f'(\theta^{cd})=\frac{R}{p^{cd}-\frac{1}{(v^{cd})^{2}\rho_{e}'(\phi^{cd})}}>0.
\end{split}$$ We now rewrite the leading part of $\eqref{NSPl}_3$ (the third equation of ) as $$\label{tac.eqn}
\begin{split}
\frac{R}{\gamma-1}{\partial}_{t}{\theta}^{cd}+p^{cd}{\partial}_{x}u^{cd}=\kappa{\partial}_{x}\left(\frac{{\partial}_{x}{\theta}^{cd}}{v^{cd}}\right).
\end{split}$$ With and in hand, we further conjecture that $\left[v^{cd}, u^{cd}, {\theta}^{cd}\right]$ satisfies $$\begin{aligned}
\label{pro.eqn}
\left\{
\begin{array}{clll}
\begin{split}
&{\partial}_{t}v^{cd}- {\partial}_{x}u^{cd}=0,\ v^{cd}=f({\theta}^{cd}),\\
&\frac{R}{\gamma-1}{\partial}_{t}{\theta}^{cd}+p^{cd}{\partial}_{x}u^{cd}=\kappa{\partial}_{x}\left(\frac{{\partial}_{x}{\theta}^{cd}}{v^{cd}}\right),\\
&{\theta}^{cd}(0,t)=\theta_{-},\ {\theta}^{cd}(+\infty,t)=\theta_{+},\ v^{cd}(0,t)=v_-,\ v^{cd}(+\infty,t)=v_+.
\end{split}
\end{array}
\right.\end{aligned}$$ By virtue of , we obtain a nonlinear diffusion equation as follows: $$\label{tac.eqn2}
\begin{split}
{\partial}_{t}{\theta}^{cd}=\frac{\kappa}{g(\theta^{cd})}{\partial}_{x}\left(\frac{{\partial}_{x}{\theta}^{cd}}{f(\theta^{cd})}\right),\
\ {\theta}^{cd}(0,t)=\theta_{-},\ \ {\theta}^{cd}(+\infty,t)=\theta_{+},
\end{split}$$ where $g(\theta^{cd})=\frac{R}{\gamma-1}+p^{cd}f'(\theta^{cd})>0.$ Applying the same argument as in [@APKo], one sees that admits a unique self similarity solution $\theta^{cd}(\xi)$, $\xi=\frac{x}{\sqrt{1+t}}$. Additionally, it turns out that $\theta^{cd}$ is a monotone function, increasing if $\theta_{+}>\theta_{-}$ and decreasing if $\theta_{+}<\theta_{-}$, and more importantly, one can show that there exists some positive constant $\overline{\delta}$, such that for $\delta=|\theta_{+}-\theta_{-}|\leq \overline{\delta}, $ $\theta^{cd}$ satisfies $$\label{tac.pt}
\begin{split}
(1+t)\left|\partial^{2}_{x}\theta^{cd}\right|+(1+t)^{\frac{1}{2}}\left|\partial_{x}\theta^{cd}\right|+\left|\theta^{cd}-\theta_{\pm}\right|
\leq &C\delta e^{-\frac{c_{1}x^{2}}{1+t}}, \ \ \textrm{as}\ \
x\rightarrow +\infty,
\end{split}$$ where $c_{1}$ is some positive constant. After ${\theta}^{cd}$ and $v^{cd}$ are obtained, we now define $\left[u^{cd},\phi^{cd}\right]$ as follows $$\begin{aligned}
\label{pro.eqn2}
\left\{
\begin{array}{clll}
\begin{split}
&\phi^{cd}={\rho}_e^{-1}(1/v^{cd}),\\
&u^{cd}=u_+-{\kappa}\int_x^{+\infty}\frac{f'({\theta}^{cd})}{g({\theta}^{cd})}{\partial}_x\left(\frac{{\partial}_x{\theta}^{cd}}{f({\theta}^{cd})}\right)dx
\\&\quad=u_{+}+\frac{\kappa
f'(\theta^{cd})}{g(\theta^{cd})f(\theta^{cd})}\partial_{x}\theta^{cd}
+\kappa\int_{x}^{+\infty}\frac{(\partial_{x}\theta^{cd})^{2}}{f(\theta^{cd})}\left(\frac{f'}{g}\right)'(\theta^{cd})
dx,\\[2mm]
&\phi^{cd}(0,t)=\phi_-,\ \phi^{cd}(+\infty,t)=\phi_+,\ u^{cd}(+\infty,t)=u_+.
\end{split}
\end{array}
\right.\end{aligned}$$ It should be noted that $\phi_{\pm}={\rho}_e^{-1}(1/v_\pm)$, and $u^{cd}(0,t)$ may not equal to $u_+$.
In view of , , and , it is straightforward to compute that $\left[v^{cd},u^{cd},\theta^{cd},\phi^{cd}\right]$ satisfies $$\left\|\left[v^{cd}-v^{CD},u^{cd}-u^{CD},\theta^{cd}-{\theta}^{CD},\phi^{cd}-\phi^{CD}\right]\right\|_{L^{p}({\mathbb{R}}_{+})}
=O\left(\kappa^{\frac{1}{2p}}\right)(1+t)^{\frac{1}{2p}},\ \ p\geq1,$$ which implies the viscous contact wave $\left[v^{cd},u^{cd},\theta^{cd},\phi^{cd}\right](x,t)$ constructed in and approximates the contact discontinuity solution $\left[v^{CD}, u^{CD}, {\theta}^{CD},\phi^{CD}\right]$ to the quasineutral Euler system in $L^{p}$ norm, $p\geq1$ on any finite time interval as the heat conductivity coefficients $\kappa$ tends to zero. Moreover, we see that the viscous contact wave $\left[v^{cd},u^{cd},\theta^{cd},\phi^{cd}\right](x,t)$ solves the Navier-Stokes-Poisson system time asymptotically, that is, $$\begin{aligned}
\label{1.23*x}
\left\{
\begin{array}{clll}
\begin{split}
&{\partial}_{t}v^{cd}- {\partial}_{x}u^{cd}=0,\\[2mm]
&{\partial}_{t} u^{cd}+{\partial}_{x}p^{cd} =\frac{{\partial}_{x}\phi^{cd}}{v^{cd}}+\mu
{\partial}_{x}\left(\frac{{\partial}_{x}u^{cd}}{v^{cd}}\right)+{\mathcal{R}}_{1},\\[2mm]
&\frac{R}{\gamma-1}{\partial}_{t}{\theta}^{cd}+p^{cd}{\partial}_{x}u^{cd}=\mu\frac{({\partial}_{x}u^{cd})^{2}}{v^{cd}}
+\kappa{\partial}_{x}\left(\frac{{\partial}_{x}{\theta}^{cd}}{v^{cd}}\right)+{\mathcal{R}}_{2},\\[2mm]
&{\partial}_{x}\left(\frac{{\partial}_{x}\phi^{cd}}{v^{cd}}\right)=1-v^{cd}\rho_{e}(\phi^{cd})+{\mathcal{R}}_{3},
\end{split}
\end{array}
\right.\end{aligned}$$ where $$\label{1.24*x}
\begin{split}
{\mathcal{R}}_{1}=&{\partial}_{t}\left(\frac{\kappa
f'(\theta^{cd})}{g(\theta^{cd})f(\theta^{cd})}\partial_{x}\theta^{cd}
+\int^{+\infty}_{x}\frac{\kappa(\partial_{x}\theta^{cd})^{2}}{f(\theta^{cd})}\left(\frac{f'}{g}\right)'(\theta^{cd})
dx\right)-\mu\partial_{x}{\partial}_t\left[\ln \left(f(\theta^{cd})\right)\right]\\
=&O(\delta)(1+t)^{-\frac{3}{2}}e^{-\frac{c_{1}x^{2}}{1+t}}, \
\mbox{as}\ \ x\rightarrow +\infty,
\end{split}$$ $$\label{1.25*x}
\begin{split}
{\mathcal{R}}_{2}=-\mu\frac{\left(f'(\theta^{cd})\partial_{t}\theta^{cd}\right)^{2}}{f(\theta^{cd})}=O(\delta)(1+t)^{-2}e^{-\frac{c_{1}x^{2}}{1+t}},
\ \mbox{as}\ \ x\rightarrow +\infty,
\end{split}$$ and $$\label{1.26*x}
\begin{split}
{\mathcal{R}}_{3}={\partial}_{x}\left(\frac{{\partial}_{x}\phi^{cd}}{v^{cd}}\right)=O(\delta)(1+t)^{-1}e^{-\frac{c_{1}x^{2}}{1+t}},
\ \mbox{as}\ \ x\rightarrow +\infty.
\end{split}$$
Main results
------------
Now we are in a position to state our main results.
\[main.res.\] For any given $[v_+,u_+,{\theta}_+, p_-]$ with $v_+>0$ and ${\theta}_+>0$, suppose that $[v_-,u_-,{\theta}_-]$ satisfies , $\phi_\pm={\rho}_e^{-1}(v_\pm)$ with $\phi_\pm\in (\phi_m,\phi_M)$, and the function ${\rho}_e(\cdot)$ satisfies the assumption $({\mathcal{A}})$. Let $\left[v^{cd},u^{cd},{\theta}^{cd},\phi^{cd}\right](x,t)$ be the viscous contact wave defined in and with strength $\delta=|\theta_{+}-\theta_{-}|.$ There exist positive constants $\epsilon_{0}>0$ and $C_0>0$, such that if $\left[v_{0}(x)-v^{cd}(x,0),u_{0}(x)-u^{cd}(x,0)\right]\in H^1$, $\left[{\theta}_{0}(x)-{\theta}^{cd}(x,0)\right]\in H^1_0$ and $$\label{p.ID}\left\|\left[v_{0}(x)-v^{cd}(x,0),u_{0}(x)-u^{cd}(x,0),{\theta}_{0}(x)-{\theta}^{cd}(x,0)\right]\right\|_{H^1}
+{\delta}\leq
\epsilon_{0},$$ then the initial boundary value problem , and admits a unique global solution $[v,u,{\theta},\phi](x,t)$ satisfying $\left[v-v^{cd},u-u^{cd}\right]\in C(0,+\infty; H^1)$, $\left[{\theta}(x)-{\theta}^{cd},\phi-\phi^{cd}\right]\in C(0,+\infty; H_0^1)$ and $$\label{main.eng}
\sup\limits_{t\geq0}\left\|\left[v-v^{cd},u-u^{cd},{\theta}-{\theta}^{cd},\phi-\phi^{cd}\right]\right\|_{H^1}
\leq C_0{\epsilon}_0^{2/3}.$$ Moreover, it holds that $$\label{sol.lag}
\begin{split}
\lim_{t\rightarrow+\infty}\sup_{x\in
{\mathbb{R}}_+}\left|\left[v-v^{cd},u-u^{cd},{\theta}-{\theta}^{cd},\phi-\phi^{cd}\right]\right|=0.
\end{split}$$
From a physical point of view, the motion of the ion-dust plasma (cf. [@KG; @GSK]), the self-gravitational viscous gaseous stars (cf. [@Chan]) and the charged particles in semiconductor devices (cf. [@MRC]) can be governed by the NSP system. On the other hand, the NSP system at the fluid level can be justified by taking the hydrodynamical limit of the Vlasov-type Boltzmann equation by the Chapman-Enskog expansion, cf. [@CC; @Gr; @G; @GJ]. In recent years, there have been a great number of mathematical studies of the NSP system. In what follows, we only mention some of them related to our interest. Ducomet [@Du] obtained the existence of nontrivial stationary solutions with compact support and proved the dynamical stability related to a free-boundary value problem for the three-dimensional NSP system in the case that the background profile is vacuum. Donatelli [@D] established the global existence of weak solutions to the Cauchy problem with large initial data. Recently, Ding-Wen-Yao-Zhu [@DWYZ-09] proved the global existence of weak solutions to the one dimensional isentropic NSP system with density-dependent viscosity and free boundary. Donatelli-Marcati [@DM] studied the quasineutral limit by using some dispersive estimates of Strichartz type. We point out that some nonexistence result of global weak solutions was also obtained in Chae [@Chae]. Zhang-Fang [@ZF] studied the large-time behavior of the spherically symmetric NSP system with degenerate viscosity coefficients and with vacuum in three dimensions. Jang-Tice [@JT] investigated the linear and nonlinear dynamical instability for the Lane-Emden solutions of the NSP system in three dimensions under some condition on the adiabatic exponent. Tan-Yang-Zhao-Zou [@TYZZ] established the global strong solution to the one-dimensional non-isentropic NSP system with large data for density-dependent viscosity. In the case when the background profile is strictly positive, the global existence and convergence rates for the three-dimensional NSP system around a non-vacuum constant state were studied by Li-Matsumura-Zhang [@LMZ], Zhang-Li-Zhu [@ZLZ-11] and Hsiao-Li [@HL] through carrying out the spectrum analysis. We point out that Duan [@D-NSM] also used the method of Green’s function to obtain the large time behaviors of the more complex Navier-Stokes-Maxwell system.
Another interesting and challenging problem is to study the stability of the NSP system on half space, to the best of our knowledge, there are very few results in this line. Duan-Yang [@DY] recently proved the stability of rarefaction wave and boundary layer for outflow problem on the two-fluid NSP system. The convergence rate of corresponding solutions toward the stationary solution was obtained in Zhou-Li [@ZL]. We remark that due to the techniques of the proof, it was assumed in [@DY] that all physical parameters in the model must be unit, which is obviously impractical since ions and electrons generally have different masses and temperatures. One important point used in [@DY] is that the large-time behavior of the electric potential is trivial and hence the two fluids indeed have the same asymptotic profiles which are constructed from the Navier-Stokes equations without any force instead of the quasineutral system. Duan-Liu [@DL] then improved the results of [@DY] in the sense that all physical constants appearing in the model can be taken in a general way, and the large-time profile of the electric potential is nontrivial on the basis of the quasineutral assumption. For the investigations in the stability of the rarefaction wave of the related models, see also [@DL2] for the study of the more complicated Vlasov-Poisson-Boltzmann system with more general background profile.
When there is no self-consistent force, the NSP system reduces to the well-known Navier-Stokes equations. It is known that there have been extensive investigations on the stability of wave patterns, namely, shock wave, rarefaction wave, contact discontinuity and their compositions, in the context of gas dynamical equations and related kinetic equations. Among them, we only mention [@Go; @HLM; @HXY; @JQZ; @KT; @KZ; @LX; @LYYZ; @LY; @M; @MN86; @MN92; @MN-S; @PLM; @Y] and reference therein. Moreover, we would also point out some previous works only related to the current work. Huang-Mastumura-Shi [@HMS-04] proved the stability of contact discontinuity of compressible Navier-Stokes equations with free boundary for the ideal polytropic gas through the construction of viscous contact wave profiles, the key observation in [@HMS-04] is that the asymptotic profile of the temperature ${\theta}$ satisfies a nonlinear diffusion equation, which can be solved by the technique developed in [@APKo; @HLiu], and later on Huang-Mastumura-Xin [@HMX-05] and Huang-Li-Mastumura [@HLM] established the stability of the contact waves of the Cauchy problem. Recently Huang-Wang-Zhai [@HWZ-10] extended the results in [@HMS-04] to the general gas, however, for the Cauchy problem, it still remains an interesting open problem to generalize the results in [@HMX-05; @HLM] for the general gas.
In this paper, we intend to study the stability of the contact wave of the NSP system with free boundary. Motivated by [@DL] and [@HMS-04], we first construct the nontrivial asymptotic profiles of the quasineural Euler equations, it should be noted that the background density ${\rho}_e(\phi)$ satisfying assumption $({\mathcal{A}})$ allows that the asymptotic profile of the electrical potential can be distinct at the boundary. Then we perform the elementary energy estimates to the perturbative equations to obtain the global existence and the large time behaviors. Compared to the classical Navier-Stokes system without any force, the main difficulty in the proof for the NSP system is to treat the estimates on the terms caused by the potential function $\phi$. Precisely, the delicate term $\left(\frac{{\partial}_x\phi}{v}-\frac{{\partial}_x\phi^{cd}}{v^{cd}}\right)\psi$ can not be directly controlled, as in [@DL], the key point to overcome the difficulty is to use the good dissipative property from the Poisson equation by expanding ${\rho}_e(\phi)$ around the asymptotic profile up to the third-order. In addition, it is shown [@DL] that the sign of the first derivative of the rarefaction profile of the velocity and the good time decay properties of the smooth rarefaction profiles are important to the [*a priori*]{} estimate. Thus compared with [@DL] in which the stability of the rarefaction wave of the NSP system is proved, a new difficulty will arise, that is, the critical term ${\int_{0}^{T}\int_{{\mathbb{R}}_{+}}}\psi^{2}({\partial}_x\theta^{cd})^{2}dxdt$ is beyond control, unlike that of [@HLM], we need to pay extra effort to take care of the terms involving the self-consistent force, and it can be seen that the assumption $({\mathcal{A}}_2)$ plays an essential role to obtain the desired estimates, see Lemma \[key.es\] for the details.
The rest of the paper is arranged as follows. In the main part Section 2, we give the [*a priori*]{} estimates on the solutions of the perturbative equations. The proof of Theorem \[main.res.\] is concluded in Section 3. In the Appendix, we present the details that are left in the proofs of the previous sections for completeness of the paper.
[*Notations.*]{} Throughout this paper, we denote a generally large constant by $C$, which may vary from line to line. For two quantities $a$ and $b$, $a\thicksim b$ means $\frac{1}{C}a \leq b \leq C a $. $L^p =
L_x^p(\mathbb{{\mathbb{R}}_{+}}) \ (1 \leqslant p \leqslant \infty)$ denotes the usual Lebesgue space on ${\mathbb{R}}_+=[0,+\infty]$ with its norm $ \|\cdot\|_ {L^p}$, and for convenient, we write $ \| \cdot \| _{ L^2} = \| \cdot \|$. We also use $H^{k}$ $(k\geq0)$ to denote the usual Sobolev space with respect to $x$ variable on ${\mathbb{R}}_+$. $C([0,T ]; H^k) (k\geq0)$ denotes the space of the continuous functions on the interval $[0, T ]$ with values in $H^{k}$. We use $ (\cdot, \cdot )$ to denote the inner product over the Hilbert space $ L^{2}$. $[f_1,f_2]\in H^1$ means $f_1\in H^1$ and $f_2\in H^1$, and so on so forth.
The a priori estimates
======================
In order to study the stability of contact wave of the initial boundary value problem , and , that is, to prove Theorem \[main.res.\], we first define the perturbation as $$[\varphi, \psi, \zeta, \sigma](x,t)=\left[v-v^{cd},u-u^{cd}, {\theta}-{\theta}^{cd},
\phi-\phi^{cd}\right](x,t).$$ Then $[\varphi, \psi, \zeta, \sigma](x,t)$ satisfies $$\begin{aligned}
&&{\partial}_t\varphi-{\partial}_x\psi=0,\label{pv}\\
&&{\partial}_t \psi+{\partial}_x
p-{\partial}_xp^{cd}=\left(\frac{{\partial}_x\phi}{v}-\frac{{\partial}_x\phi^{cd}}{v^{cd}}\right)+\mu{\partial}_x\left(\frac{{\partial}_x
\psi}{v}\right)+F,\label{pu}\\
&&\frac{R}{\gamma-1}{\partial}_t\zeta+p{\partial}_x u-p^{cd}{\partial}_x u^{cd}=\kappa
{\partial}_x\left(\frac{{\partial}_x{\theta}}{v}-\frac{{\partial}_x{\theta}^{cd}}{v^{cd}}\right)+G,\label{pta}\\
&&v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)=-\varphi+v\left[1-v^{cd}\rho_{e}\left(\sigma+\phi^{cd}\right)\right]
-v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right),\label{pp}\\
&&\left(p(v,\theta)-\mu\frac{{\partial}_x u}{v}\right)(0,t)=p_{-},\
\zeta(0,t)={\sigma}(0,t)={\sigma}(+\infty,t)=0,\label{p.BC}\\
&&\left[\varphi,\psi,\zeta](x,0) =[\varphi_0,\psi_0,
\zeta_0\right](x)
\notag\\
&&\qquad\qquad\quad\quad=\left[v_0(x)-v^{cd}(x,0),u_0(x)-u^{cd}(x,0),{\theta}_0(x)-{\theta}^{cd}(x,0)\right],\label{p.id}\end{aligned}$$ where $x\geq0$, $t\geq0$, $F=-\partial_{t}u^{cd}+\mu{\partial}_x(\frac{{\partial}_x u^{cd}}{v})$ and $G=\mu\frac{({\partial}_x u)^{2}}{v}.$ We note that the structural identity will be of extremal importance for the later proof.
The local existence of , and can be established by the standard iteration argument cf. [@HMS-04] and hence will be skipped in the paper. To obtain the global existence part of Theorem \[main.res.\], it suffices to prove the following [*a priori*]{} estimates. For results in this direction, we have
\[ape\]Assume all the conditions listed in Theorem \[main.res.\] hold. Let $[\varphi,\psi,\zeta,\sigma]$ be a solution to the initial boundary value problem , , , , and on $0\leq t\leq T$ for some positive constant T. There are constants ${\delta}>0$, ${\epsilon}_0>0$ and $C>0$, such that if $[\varphi,\psi]\in C(0,T; H^1)$, $[\zeta,\sigma]\in C(0,T; H_0^1)$ and $$\label{aps}
\begin{split}
\sup_{0\leq t\leq
T}\left\|[\varphi,\psi,\zeta,\sigma](t)\right\|_{H^1}+{\delta}\leq {\epsilon}_{0},
\end{split}$$ then the solution $[\varphi,\psi,\zeta,\sigma](x,t)$ satisfies $$\label{eng.p1}
\begin{split}
\sup_{0\leq t\leq
T}&\left\|[\varphi,\psi,\zeta,\sigma](t)\right\|_{H^1}^2+\int_{0}^{T}\|{\partial}_x\varphi\|^2
+\|{\partial}_x\left[\psi,\zeta,\sigma\right]\|_{H^1}^2dt\\
\leq&
C\delta+C\left\|[\varphi_{0},\psi_{0},\zeta_{0}]\right\|^{4/3}_{H^1}.
\end{split}$$
We divide it by the following three steps.
**Step 1.** The zero-order energy estimates. Multiplying , and by $-R{\theta}^{cd}\left(\frac{1}{v}-\frac{1}{v^{cd}}\right)$, $\psi$ and $\zeta\theta^{-1}$, respectively, then taking the summation of the resulting equations, we obtain $$\label{zs}
\begin{split}
{\partial}_{t}&\left(\frac{1}{2}\psi^{2}+R\theta^{cd}\Phi\left(\frac{v}{v^{cd}}\right)
+\frac{R}{\gamma-1}\theta^{cd}\Phi\left(\frac{\theta}{\theta^{cd}}\right)\right)
+\mu\frac{({\partial}_{x}\psi)^{2}}{v}\\&+\frac{\kappa}{v\theta}({\partial}_{x}\zeta)^{2}+H_{x}+Q_{1}+Q_{2}=F\psi+\frac{\zeta}{\theta}G
+\underbrace{\left(\frac{{\partial}_x\phi}{v}-\frac{{\partial}_x\phi^{cd}}{v^{cd}}\right)\psi}_{I_{1}},
\end{split}$$ where $$\label{4.1}
\begin{split}
\Phi(s)=s-1-\ln s,
\end{split}$$ $$\label{4.2}
\begin{split}
H=\left(p-p^{cd}\right)\psi-\mu\frac{\psi{\partial}_x\psi}{v}-\kappa\frac{\zeta}{\theta}\left(\frac{{\partial}_x\theta}{v}-\frac{{\partial}_x\theta^{cd}}{v^{cd}}\right),
\end{split}$$ $$\label{4.3}
\begin{split}
Q_{1}=-R{\partial}_t\theta^{cd}\Phi\left(\frac{v}{v^{cd}}\right)-p^{cd}{\partial}_tv^{cd}\left(2-\frac{v}{v^{cd}}-\frac{v^{cd}}{v}\right)
+\frac{R}{\gamma-1}{\partial}_t\theta^{cd}
\Phi\left(\frac{\theta^{cd}}{\theta}\right)+\frac{\zeta}{\theta}\left(p-p^{cd}\right){\partial}_xu^{cd},
\end{split}$$ and $$\label{4.4}
\begin{split}
Q_{2}=-\kappa\frac{{\partial}_x\theta}{\theta^{2}v}\zeta{\partial}_x\zeta-\kappa\frac{\varphi{\partial}_x\zeta}{\theta
vv^{cd}}{\partial}_x\theta^{cd}+\kappa\frac{\zeta\varphi{\partial}_x\theta}{\theta^{2}
vv^{cd}}{\partial}_x\theta^{cd}.
\end{split}$$ Let us now consider the most delicate term $I_1$ on the right hand side of . The key technique to handle $I_1$ is to use the good dissipative property of the Poisson equation by expanding $\rho_{e}(\sigma+\phi^{cd})$ around the asymptotic profile up to the third-order. Only in this way, we can observe some new cancelations and obtain the higher order nonlinear terms.
With the aid of and , one has $$\label{I1}
\begin{split}
I_1=&-\frac{{\partial}_x\psi\sigma}{v}
+\frac{\psi{\partial}_x\varphi\sigma}{v^2}+\frac{\psi{\partial}_xv^{cd}\sigma}{v^2}
+\frac{\psi{\partial}_xv^{cd}\varphi}{(v^{cd})^3v\rho_{e}'(\phi^{cd})}+{\partial}_x\left(\frac{\sigma\psi}{v}\right)\\
=&\underbrace{-{\partial}_t\left[-v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)+v\left(1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right)
-v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right]\sigma
v^{-1}}_{I_{1,1}}
\\
&\underbrace{+{\partial}_x\left[-v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)+v\left(1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right)
-v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right]\psi\sigma v^{-2}}_{I_{1,2}}\\
&\underbrace{+\left[-v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)+v\left(1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right)
-v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right]\psi{\partial}_xv^{cd}(v^{cd})^{-3}[v\rho_{e}'(\phi^{cd})]^{-1}}_{I_{1,3}}\\
&\underbrace{+\frac{\psi{\partial}_xv^{cd}\sigma}{v^2}}_{I_2}+{\partial}_x\left(\frac{\sigma\psi}{v}\right).
\end{split}$$ To deal with the lower order terms involving $1-v^{cd}\rho_{e}(\sigma+\phi^{cd})$, we first get from the Taylor’s formula with an integral remainder that $$\label{taylor}
1-v^{cd}\rho_{e}(\sigma+\phi^{cd})=-v^{cd}\rho_{e}'(\phi^{cd})\sigma-\frac{v^{cd}\rho_{e}''(\phi^{cd})}{2}\sigma^{2}
\underbrace{-v^{cd}\int_{\phi^{cd}}^{\phi}\rho_{e}'''(\varrho)\frac{(\phi-\varrho)^{2}}{2}d\varrho}_{I_0}.$$ By virtue of , we then compute $I_{1,1}$, $I_{1,2}$ and $I_{1,3}$ as follows: $$\label{I1.1}
\begin{split}
I_{1,1}=&-\frac{1}{2}{\partial}_t\left(\frac{v^{cd}}{v^2}({\partial}_x\sigma)^2\right)
-{\partial}_t\left(\frac{{\partial}_x\sigma}{v}\sigma{\partial}_x\left(\frac{v^{cd}}{v}\right)\right)
-\frac{1}{2}{\partial}_t\left(\frac{v^{cd}}{v^2}\right)({\partial}_x\sigma)^2\\
&+{\partial}_x\left(\frac{v^{cd}}{v}\right)\frac{{\partial}_x\sigma}{v}{\partial}_t\sigma
-v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)\sigma{\partial}_t(v^{-1})
+\frac{1}{2}{\partial}_t\left(v^{cd}\rho_{e}'(\phi^{cd})\sigma^2\right)
\\&+\frac{1}{3}{\partial}_t\left(v^{cd}\rho_{e}''(\phi^{cd})\sigma^3\right)-
{\partial}_t I_0\sigma
\underbrace{+\frac{{\partial}_tv}{v}v^{cd}\rho_{e}'(\phi^{cd})\sigma^2}_{I_3}
+\frac{1}{2}{\partial}_{t}\left(v^{cd}\rho_{e}'(\phi^{cd})\right)\sigma^2
\\&+\frac{1}{2}\frac{{\partial}_tv}{v}v^{cd}\rho_{e}''(\phi^{cd})\sigma^3
+\frac{1}{6}{\partial}_{t}\left(v^{cd}\rho_{e}''(\phi^{cd})\right)\sigma^3
-\frac{{\partial}_tv}{v}I_0\sigma
+{\partial}_t\left(v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right)\sigma
v^{-1}\\&+{\partial}_x{\partial}_t\left(\frac{v^{cd}\sigma{\partial}_x\sigma}{v^{2}}\right)-{\partial}_x\left(\frac{v^{cd}{\partial}_t\sigma{\partial}_x\sigma}{v^{2}}\right),
\end{split}$$ $$\label{I1.2}
\begin{split}
I_{1,2}=&\frac{v^{cd}}{v^3}{\partial}^2_x\sigma{\partial}_x\sigma\psi
+\frac{v^{cd}}{v}{\partial}^2_x\sigma \sigma {\partial}_x(\psi v^{-2})
+v^{cd}{\partial}_x\sigma {\partial}_x(v^{-1}){\partial}_x(\sigma \psi v^{-2})\\
&\underbrace{-v^{cd}\rho_{e}'(\phi^{cd})\frac{{\partial}_x\sigma
\sigma\psi}{v}-
v^{cd}\rho_{e}'(\phi^{cd})\frac{{\partial}_xv\sigma^2\psi}{v^2}-{\partial}_{x}\left(v^{cd}\rho_{e}'(\phi^{cd})\right)\frac{\sigma^2\psi}{v}}_{I_4}
-v^{cd}\rho_{e}''(\phi^{cd})\frac{{\partial}_x\sigma\sigma^2\psi}{v}
\\&-\frac{1}{2}v^{cd}\rho_{e}''(\phi^{cd})\frac{{\partial}_xv\sigma^3\psi}{v^2}
-\frac{1}{2}{\partial}_{x}\left(v^{cd}\rho_{e}''(\phi^{cd})\right)\frac{\sigma^3\psi}{v}
+\frac{{\partial}_xI_0\sigma\psi}{v} +\frac{{\partial}_xvI_0\sigma\psi}{v^2}\\
& +v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right){\partial}_x\left(\psi\sigma
v^{-2}\right)-{\partial}_x\left(v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi}{v}\right)\psi\sigma
v^{-2} \right),
\end{split}$$ $$\label{I1.3}
\begin{split}
I_{1,3}=&-{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)
\frac{\psi{\partial}_xv^{cd}}{v(v^{cd})^{2}\rho_{e}'(\phi^{cd})}
\underbrace{-\frac{{\partial}_xv^{cd}\sigma\psi}{(v^{cd})^2}}_{I_5}
\underbrace{-\frac{1}{2}\frac{\rho_{e}''(\phi^{cd}){\partial}_xv^{cd}\sigma^2\psi}{(v^{cd})^2\rho_{e}'(\phi^{cd})}}_{I_6}\\
&+\frac{{\partial}_xv^rI_0\psi}{(v^{cd})^3\rho_{e}'(\phi^{cd})}
-{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\frac{\psi{\partial}_xv^{cd}}{v(v^{cd})^2\rho_{e}'(\phi^{cd})}.
\end{split}$$ Note that $I_l$ $(2\leq l\leq 6)$ can not be directly controlled. To overcome this difficulty, we first get from and that $$\begin{split}
(I_2+I_5)+I_6+I_4=&-\frac{{\partial}_xv^{cd}\sigma\psi\varphi(v+v^{cd})}{(vv^{cd})^2}
-\frac{\rho_{e}''(\phi^{cd}){\partial}_xv^{cd}\sigma^2\psi}{2\rho_{e}'(\phi^{cd})(v^{cd})^2}
-v^{cd}\rho_{e}'(\phi^{cd})\frac{{\partial}_x\sigma
\sigma\psi}{v}\\&-\frac{{\partial}_xv\sigma^2\psi}{v^2}v^{cd}\rho_{e}'(\phi^{cd})
-\left(v^{cd}\rho_{e}'(\phi^{cd})\right)_{x}\frac{\sigma^2\psi}{v}
\\=&\frac{{\partial}_xv^{cd}\sigma^2\psi}{(v^{cd})^2}\left[v^{cd}\rho_{e}'(\phi^{cd})-\frac{\rho_{e}''(\phi^{cd})}{2\rho_{e}'(\phi^{cd})}\right]
-\frac{{\partial}_x\sigma\sigma\psi}{v}v^{cd}\rho_{e}'(\phi^{cd})
+\frac{{\partial}_xv^{cd}\sigma^2\psi}{v} \frac{\rho_{e}''(\phi^{cd})}{v^{cd}\rho_{e}'(\phi^{cd})}\\
&-\frac{{\partial}_xv\sigma^2\psi}{v^2}v^{cd}\rho_{e}'(\phi^{cd})+\frac{1}{2}\frac{{\partial}_xv^{cd}\sigma^3\psi(v+v^{cd})}{vv^{cd}}\rho_{e}''(\phi^{cd})
-\frac{{\partial}_xv^{cd}\sigma\psi I_0(v+v^{cd})}{v(v^{cd})^2}\\
&+\frac{{\partial}_xv^{cd}\sigma\psi{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)(v+v^{cd})}{v^2v^{cd}}
+\frac{{\partial}_xv^{cd}\sigma\psi{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)(v+v^{cd})}{v^2v^{cd}},
\end{split}$$ which is further equal to $$\label{sumJ}
\begin{split}
&\underbrace{\frac{1}{2}\frac{{\partial}_x\psi\sigma^2}{v}v^{cd}\rho_{e}'(\phi^{cd})}_{I_7}
+\sigma^2\psi
v^{cd}\rho_{e}'(\phi^{cd}){\partial}_x\left(\frac{1}{v}-\frac{1}{v^{cd}}\right)+\frac{1}{2}\frac{\sigma^2\psi\varphi}{v^{2}}{\partial}_xv^{cd}
\left[\rho_{e}'(\phi^{cd})-\frac{v\rho_{e}(\phi^{cd})\rho_{e}''(\phi^{cd})}{v^{cd}\rho_{e}'(\phi^{cd})}\right]
\\
&-\frac{1}{2}\frac{\sigma^2\psi{\partial}_x\varphi}{v^{2}}v^{cd}\rho_{e}'(\phi^{cd})
+\frac{1}{2}\frac{{\partial}_xv^{cd}\sigma^3\psi(v+v^{cd})}{vv^{cd}}\rho_{e}''(\phi^{cd})
-\frac{{\partial}_xv^{cd}\sigma\psi I_0(v+v^{cd})}{v(v^{cd})^2}\\
&+\frac{{\partial}_xv^{cd}\sigma\psi{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)(v+v^{cd})}{v^2v^{cd}}
+\frac{{\partial}_xv^{cd}\sigma\psi{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)(v+v^{cd})}{v^2v^{cd}}-{\partial}_x\left(\frac{\psi\sigma^2}{2v}v^{cd}\rho_{e}'(\phi^{cd})\right).
\end{split}$$ For $I_3$ and $I_7$, it follows from $\eqref{NSPl}_1$, and that $$\label{J3pJ7}
\begin{split}
I_3+I_7=&\frac{3{\partial}_t\varphi\sigma^2}{2v}v^{cd}\rho_{e}'(\phi^{cd})
+\frac{{\partial}_xu^{cd}\sigma^2}{v}v^{cd}\rho_{e}'(\phi^{cd})\\
=&{\partial}_t\left(\frac{3\varphi\sigma^2}{2v}v^{cd}\rho_{e}'(\phi^{cd})\right)
-\frac{3}{2}v^{cd}\rho_{e}'(\phi^{cd})\varphi{\partial}_{t}\left(\frac{\sigma^2}{v}\right)\\
&-\frac{3}{2}{\partial}_{t}\left(v^{cd}\rho_{e}'(\phi^{cd})\right)\frac{\varphi\sigma^2}{v}
+\frac{{\partial}_xu^{cd}\sigma^2}{v}v^{cd}\rho_{e}'(\phi^{cd}).
\end{split}$$ Plugging , , , , and into , integrating the resulting identity with respect to $x$ over ${\mathbb{R}}_+$, and using $(\mathcal
{A}_{2})$, we thus arrive at $$\label{zs2}
\begin{split}
\frac{d}{dt}&\int_{{\mathbb{R}}_{+}}\left(\frac{1}{2}\psi^{2}+R\theta^{cd}\Phi\left(\frac{v}{v^{cd}}\right)
+\frac{R}{\gamma-1}\theta^{cd}\Phi\left(\frac{\theta}{\theta^{cd}}\right)
+\frac{v^{cd}}{2}|\rho_{e}'(\phi^{cd})|\sigma^2+\frac{v^{cd}}{2v^2}({\partial}_x\sigma)^2\right)dx
\\&+\frac{d}{dt}\int_{{\mathbb{R}}_{+}}\frac{{\partial}_x\sigma}{v}\sigma{\partial}_x\left(\frac{v^{cd}}{v}\right)dx
-\frac{3}{2}\frac{d}{dt}\int_{{\mathbb{R}}_{+}}v^{cd}\rho_{e}'(\phi^{cd})\frac{\varphi\sigma^2}{v}dx
-\frac{1}{3}\frac{d}{dt}\int_{{\mathbb{R}}_{+}}v^{cd}\rho_{e}''(\phi^{cd})\sigma^3dx
\\&+\mu\int_{{\mathbb{R}}_{+}}\frac{({\partial}_{x}\psi)^{2}}{v}dx+\int_{{\mathbb{R}}_{+}}\frac{\kappa}{v\theta}({\partial}_{x}\zeta)^{2}dx
\\=&-\int_{{\mathbb{R}}_{+}}Q_{1}dx
-\int_{{\mathbb{R}}_{+}}Q_{2}dx+\int_{{\mathbb{R}}_{+}}F\psi dx
+\int_{{\mathbb{R}}_{+}}\frac{\zeta}{\theta}G dx
+\widetilde{H}(0,t)+\sum\limits_{l=1}^{31}{\mathcal{I}}_l,
\end{split}$$ where $$\label{78uiH}
\begin{split}
\widetilde{H}=\widetilde{H}(x,t)=&(p-p^{cd})\psi-\mu\frac{\psi{\partial}_x\psi}{v}
-\kappa\frac{\zeta}{\theta}\left(\frac{{\partial}_x\theta}{v}-\frac{{\partial}_x\theta^{cd}}{v^{cd}}\right)
-\frac{\sigma\psi}{v}-{\partial}_t\left(\frac{v^{cd}\sigma{\partial}_x\sigma}{v^{2}}\right)\\&+\frac{v^{cd}{\partial}_t\sigma{\partial}_x\sigma}{v^{2}}
+v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi}{v}\right)\psi\sigma v^{-2}
+\frac{\psi\sigma^2}{2v}v^{cd}\rho_{e}'(\phi^{cd}),
\end{split}$$ and $$\begin{aligned}
\left\{\begin{array}{rll}
\begin{split}
&{\mathcal{I}}_1=\int_{{\mathbb{R}}_{+}}{\partial}_x\left(\frac{v^{cd}}{v}\right)\frac{{\partial}_x\sigma}{v}{\partial}_t\sigma
dx,\ \ \
{\mathcal{I}}_2=-\int_{{\mathbb{R}}_{+}}v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)\sigma{\partial}_t(v^{-1})dx,\\
&{\mathcal{I}}_3=-\int_{{\mathbb{R}}_{+}}{\partial}_t I_0\sigma dx,\ \ \
{\mathcal{I}}_4=-\frac{1}{2}\int_{{\mathbb{R}}_{+}}{\partial}_t\left(\frac{v^{cd}}{v^2}\right)({\partial}_x\sigma)^2dx,\\
&{\mathcal{I}}_5=-\int_{{\mathbb{R}}_{+}}\frac{{\partial}_tv}{v}I_0\sigma dx,\ \ \
{\mathcal{I}}_6=\int_{{\mathbb{R}}_{+}}{\partial}_t\left(v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right)\sigma
v^{-1}dx,
\\
&{\mathcal{I}}_7=\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v^3}{\partial}^2_x\sigma{\partial}_x\sigma\psi dx,\ \ \
{\mathcal{I}}_8=\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v}{\partial}^2_x\sigma\sigma{\partial}_x(\psi v^{-2})dx,\\
&{\mathcal{I}}_{9}=\int_{{\mathbb{R}}_{+}}v^{cd}{\partial}_x\sigma{\partial}_x(v^{-1}){\partial}_x(\sigma\psi
v^{-2})dx,\ \ \
{\mathcal{I}}_{10}=\frac{1}{2}
\int_{{\mathbb{R}}_{+}}{\partial}_t(v^{cd}\rho_{e}'(\phi^{cd}))\sigma^2dx,\\
&{\mathcal{I}}_{11}=\frac{1}{2}\int_{{\mathbb{R}}_{+}}\frac{{\partial}_tv}{v}v^{cd}\rho_{e}''(\phi^{cd})\sigma^3dx,\ \ \
{\mathcal{I}}_{12}=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}I_0\psi}{(v^{cd})^3\rho_{e}'(\phi^{cd})}dx,\\
&{\mathcal{I}}_{13}=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_x v I_0\sigma\psi}{v^2}dx,\ \ \
{\mathcal{I}}_{14}=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xI_0\sigma\psi}{v}dx,\\
&{\mathcal{I}}_{15}=-\int_{{\mathbb{R}}_{+}}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)
\frac{\psi{\partial}_xv^{cd}}{v(v^{cd})^{2}\rho_{e}'(\phi^{cd})}dx,\ \ \
{\mathcal{I}}_{16}=\int_{{\mathbb{R}}_{+}}v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right){\partial}_x\left(\psi\sigma
v^{-2}\right)dx,\\
\end{split}
\end{array}\right.\end{aligned}$$ $$\begin{aligned}
\left\{\begin{array}{rll}
\begin{split}
&{\mathcal{I}}_{17}=-\int_{{\mathbb{R}}_{+}}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\frac{\psi{\partial}_xv^{cd}}{v(v^{cd})^2\rho_{e}'(\phi^{cd})}
dx,\ \ \
{\mathcal{I}}_{18}=\frac{1}{6}\int_{{\mathbb{R}}_{+}}{\partial}_{t}\left(v^{cd}\rho_{e}''(\phi^{cd})\right)\sigma^3dx,\\
&{\mathcal{I}}_{19}=-\frac{1}{2}\int_{{\mathbb{R}}_{+}}v^{cd}\rho_{e}''(\phi^{cd})\frac{{\partial}_xv\sigma^3\psi}{v^2}dx,\
\ \ \
{\mathcal{I}}_{20}=-\frac{1}{2}\int_{{\mathbb{R}}_{+}}{\partial}_{x}\left(v^{cd}\rho_{e}''(\phi^{cd})\right)\frac{\sigma^3\psi}{v}dx,\\
&{\mathcal{I}}_{21}=-\int_{{\mathbb{R}}_{+}}v^{cd}\rho_{e}''(\phi^{cd})\frac{{\partial}_x\sigma\sigma^2\psi}{v}dx,\
\ \ \
{\mathcal{I}}_{22}=\int_{{\mathbb{R}}_{+}}\sigma^2\psi v^{cd}\rho_{e}'(\phi^{cd}){\partial}_x\left(\frac{1}{v}-\frac{1}{v^{cd}}\right)dx,\\
&{\mathcal{I}}_{23}=\frac{1}{2}\int_{{\mathbb{R}}_{+}}\frac{\sigma^2\psi\varphi}{v^{2}}{\partial}_xv^{cd}
\left[\rho_{e}'(\phi^{cd})-\frac{v\rho_{e}(\phi^{cd})\rho_{e}''(\phi^{cd})}{v^{cd}\rho_{e}'(\phi^{cd})}\right]dx,\\
&{\mathcal{I}}_{24}=-\frac{1}{2}\int_{{\mathbb{R}}_{+}}\frac{\sigma^2\psi{\partial}_x\varphi}{v^{2}}v^{cd}\rho_{e}'(\phi^{cd})dx,\
\ \ \
{\mathcal{I}}_{25}=\frac{1}{2}\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}\sigma^3\psi(v+v^{cd})}{vv^{cd}}\rho_{e}''(\phi^{cd})dx,\\
&{\mathcal{I}}_{26}=-\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}\sigma\psi
I_0(v+v^{cd})}{v(v^{cd})^2}dx,\ \ \
{\mathcal{I}}_{27}=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}\sigma\psi{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)(v+v^{cd})}{v^2v^{cd}}dx,\\
&
{\mathcal{I}}_{28}=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}\sigma\psi{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)(v+v^{cd})}{v^2v^{cd}}dx,\
\ \ \
{\mathcal{I}}_{29}=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xu^{cd}\sigma^2}{v}v^{cd}\rho_{e}'(\phi^{cd})dx,\\
&{\mathcal{I}}_{30}=-\frac{3}{2}\int_{{\mathbb{R}}_{+}}{\partial}_{t}\left(v^{cd}\rho_{e}'(\phi^{cd})\right)\frac{\varphi\sigma^2}{v}dx,\
\ \
{\mathcal{I}}_{31}=-\frac{3}{2}\int_{{\mathbb{R}}_{+}}v^{cd}\rho_{e}'(\phi^{cd})\varphi{\partial}_{t}\left(\frac{\sigma^2}{v}\right)dx.
\end{split}
\end{array}\right.\end{aligned}$$ We now turn to estimate the right hand side of term by term. It should be noted that the following Poincaré type inequalities play an important role in our computations: $$\label{p.ine}
\begin{split}
|\zeta(x,t)|\leq x^{\frac{1}{2}}\|{\partial}_x\zeta\|,\ \
|\varphi(x,t)|\leq |\varphi(0,t)|+x^{\frac{1}{2}}\|{\partial}_x\varphi\|, \
|\sigma(x,t)|\leq x^{\frac{1}{2}}\|{\partial}_x\sigma\|.
\end{split}$$ From and Lemma \[es.tap\], one can further obtain $$\begin{aligned}
\label{p.ine2}
\left\{\begin{array}{rll}
&{\displaystyle}{\int_{{\mathbb{R}}_{+}}}\varphi^{2}\left((\partial_{x}\theta^{cd})^{2}+|\partial^{2}_{x}\theta^{cd}|\right)dx
\leq
C\delta^2\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t}+C\delta^2\|\partial_{x}\varphi\|^{2},\\[4mm]
&{\displaystyle}{\int_{{\mathbb{R}}_{+}}}(\zeta^{2}+\sigma^{2})\left((\partial_{x}\theta^{cd})^{2}+|\partial^{2}_{x}\theta^{cd}|\right)dx
\leq
C\delta^2\|\partial_{x}[\zeta,\sigma]\|^{2},
\end{array}\right.\end{aligned}$$ where the following Sobolev inequality is also used: $$\label{sob.}
|h(x)|\leq\sqrt{2}\|h\|^{1/2}\|{\partial}_xh\|^{1/2}\ \textrm{for}\ h(x)\in H^1({{\mathbb{R}}_+}).$$ By applying , Lemma \[pbd\], the [*a priori*]{} assumption , Cauchy-Schwarz’s inequality with $0<\eta<1$ and Sobolev’s inequality , we obtain the estimates for terms involving $Q_1$ and $Q_2$ as follows: $$\label{Q1}
\begin{split}
\left|\int_{{\mathbb{R}}_{+}}Q_{1}dx\right|\leq C
\int_{{\mathbb{R}}}(\varphi^{2}+\zeta^{2})\left((\partial_{x}\theta^{cd})^{2}+|\partial^{2}_{x}\theta^{cd}|\right)dx
\leq
C\delta\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t}+C\delta\|\partial_{x}[\varphi,\zeta]\|^{2},
\end{split}$$ $$\label{Q2}
\begin{split}
\left|\int_{{\mathbb{R}}_{+}}Q_{2}dx\right|\leq&
(C\epsilon_0+\eta)\|{\partial}_x\zeta\|^2+C_{\eta}
\int_{{\mathbb{R}}_{+}}(\varphi^{2}+\zeta^{2})(\partial_{x}\theta^{cd})^{2}dx
\\ \leq&
(C\epsilon_0+\eta)\|{\partial}_x\zeta\|^2+C_\eta\delta^2\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t}+C_\eta\delta^2\|\partial_{x}[\varphi,\zeta]\|^{2}.
\end{split}$$ For the terms involving $F$ and $G$, noticing that $$|{\partial}_tu^{cd}|=O(1){\delta}(1+t)^{-\frac{3}{2}} e^{-\frac{c_{1}x^{2}}{1+t}},\ |{\partial}^2_xu^{cd}|=|{\partial}_x{\partial}_tv^{cd}|=O(1){\delta}(1+t)^{-\frac{3}{2}} e^{-\frac{c_{1}x^{2}}{1+t}},\ \textrm{as}\ x\rightarrow+\infty,$$ we get from Cauchy-Schwarz’s inequality that $$\label{F.es}
\begin{split}
\left|\int_{{\mathbb{R}}_{+}}F\psi dx\right|\leq& \int_{{\mathbb{R}}_{+}}|{\partial}_tu^{cd}\psi| dx+C\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_xu^{cd}\psi\right| dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xu^{cd}{\partial}_xv^{cd}\psi\right| dx+
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xu^{cd}{\partial}_x\varphi\psi\right| dx\\
\leq&
C{\delta}(1+t)^{-1-{\alpha}}\|\psi\|^2+C{\delta}(1+t)^{-3/2+{\alpha}}+C{\delta}\|{\partial}_x\varphi\|^2,
\end{split}$$ where $0<{\alpha}<1/2$, and $$\label{G.es}
\begin{split}
\left|\int_{{\mathbb{R}}_{+}}\frac{\zeta}{\theta}G dx\right|\leq C\|\zeta\|_{\infty}\|{\partial}_x
u\|^{2} \leq
C\epsilon_{0}\|{\partial}_x\psi\|^2+C{\epsilon}_0(1+t)^{-\frac{3}{2}}.
\end{split}$$ We next compute the term $\widetilde{H}(0,t)$ arising from the boundary. Since $\zeta(0,t)={\sigma}(0,t)=0$, $\left|\widetilde{H}(0,t)\right|$ can be reduced to $$\label{H.r}
\begin{split}
\left| \frac{R{\theta}_-\varphi(0,t)}{v(0,t)v_-}\psi(0,t)+\mu\left(\frac{\psi{\partial}_x\psi}{v}\right)(0,t)\right|,
\end{split}$$ which is further dominated by $$\label{H.es}
\begin{split}
C&|\varphi(0,t)\psi(0,t)|+C|(\partial_{t}\varphi)(0,t)\psi(0,t)|
\\ \leq& C|\varphi_{0}(0)|e^{-\frac{p_{-}}{\mu}t}|\psi(0,t)|
\leq C\|\varphi_{0}(x)\|_{H^1}\|\psi\|^{1/2}\|{\partial}_x\psi\|^{1/2}e^{-\frac{p_{-}}{\mu}t}
\\ \leq& C\|\varphi_{0}(x)\|^{4/3}_{H^1}e^{-\frac{p_{-}}{\mu}t}+C{\epsilon}^2_0\|{\partial}_x\psi\|^2,
\end{split}$$ according to Lemma \[pbd\], Sobolev’s inequality and Young’s inequaity.
In order to estimate ${\mathcal{I}}_l$ $(1\leq l\leq 31)$, we first calculate $$\begin{gathered}
\label{tIR}
I_0\sim \sigma^3,\ \ {\partial}_t I_0=-v^{cd}{\partial}_t\phi\int_{\phi^{cd}}^{\phi}(\phi-\varrho)\rho_{e}'''(\varrho)
d\varrho
+\frac{1}{2}\sigma^2{\partial}_t\phi^{cd}v^{cd}\rho_{e}'''(\phi^{cd})-{\partial}_tv^{cd}
\int_{\phi^{cd}}^{\phi}\frac{(\phi-\varrho)^{2}}{2}\rho_{e}'''(\varrho)
d\varrho
\\ \sim {\partial}_t\phi \sigma^2+{\partial}_tv^{cd} \sigma^2+{\partial}_tv^{cd}\sigma^3= {\partial}_t\sigma \sigma^2
+2{\partial}_tv^{cd}\sigma^2+{\partial}_tv^{cd}\sigma^3,\end{gathered}$$ and similarly, $$\label{xIR}
{\partial}_x I_0\sim{\partial}_x\sigma \sigma^2
+2{\partial}_xv^{cd}\sigma^2+{\partial}_xv^{cd}\sigma^3.$$ In addition, from and , it follows $$\label{ptsi}
\|{\partial}_t\sigma\|^2+\|{\partial}_t{\partial}_x\sigma\|^2 \leq
C\|{\partial}_x\psi\|^2+C{\epsilon}_0\left\|\left[{\partial}_x\varphi,{\partial}^2_x\psi,
{\partial}_x\sigma,{\partial}^2_x\sigma\right]\right\|^2 +C\delta(1+t)^{-\frac{3}{2}}.$$ For the sake of completeness, the proof of is given in the appendix. With , and in hand, we now employ , Cauchy-Schwarz’s inequality with $0<\eta<1,$ Sobolev’s inequality and Lemma \[es.tap\] repeatedly to present the following estimates: $$\label{R1}
|{\mathcal{I}}_1|\leq
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x[v^{cd},v]{\partial}_x\sigma{\partial}_t\sigma\right|dx \leq
C {\epsilon}_0\left\|\left[{\partial}_x\sigma,{\partial}^{2}_x\sigma,{\partial}_t\sigma\right]\right\|^2,$$ $$\label{R2}
\begin{split}
|{\mathcal{I}}_2|\leq C&\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_x\sigma\sigma{\partial}_x[\psi,
u^{cd}]\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma\sigma{\partial}_x[\varphi,v^{cd}]{\partial}_x[\psi, u^{cd}]\right|dx
\leq C
{\epsilon}_0\left\|{\partial}_x\left[\varphi,\psi,\sigma,{\partial}_x\sigma\right]\right\|^2,
\end{split}$$ $$\label{R3}
\begin{split}
|{\mathcal{I}}_3|&+|{\mathcal{I}}_5|+|{\mathcal{I}}_{11}|+|{\mathcal{I}}_{18}|\\
\leq&C\int_{{\mathbb{R}}_{+}}\left|\partial_{x}u^{cd}\sigma^{3}\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|\sigma^{3}[\partial_{t}\sigma,\partial_{x}\psi]\right|
dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\psi\sigma^4\right|dx
+C\int_{{\mathbb{R}}}\left|{\partial}_xu^{cd}\sigma^4\right|dx\\
\leq& C{\epsilon}_0\left\|\left[{\partial}_t\sigma,{\partial}_x\sigma,{\partial}_x\psi\right]\right\|^2,
\end{split}$$ $$\label{R4}
\begin{split}
|{\mathcal{I}}_4|\leq C\int_{{\mathbb{R}}_{+}}|{\partial}_t[v^{cd},v]||{\partial}_x\sigma|^2dx \leq
C{\epsilon}_0\left\|{\partial}_x[\sigma,{\partial}_x\sigma]\right\|^2,
\end{split}$$ $$\label{R7}
\begin{split}
|{\mathcal{I}}_6|\leq&\int_{{\mathbb{R}}_{+}}\left|{\partial}_t{\partial}_xv^{cd}\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\sigma
v^{-1}\right|dx
+\int_{{\mathbb{R}}_{+}}\left|{\partial}_tv^{cd}\left(\frac{{\partial}_x\phi^{cd}}{v}\right){\partial}_x\sigma
v^{-1}\right|dx
\\&+\int_{{\mathbb{R}}_{+}}\left|{\partial}_tv^{cd}\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\sigma {\partial}_x (v^{-1})\right|dx
+\int_{{\mathbb{R}}_{+}}\left|{\partial}_xv^{cd}{\partial}_t\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\sigma
v^{-1}\right|dx
\\&+\int_{{\mathbb{R}}_{+}}\left|v^{cd}{\partial}_t\left(\frac{{\partial}_x\phi^{cd}}{v}\right){\partial}_x\sigma v^{-1}\right|dx
+\int_{{\mathbb{R}}_{+}}\left|v^{cd}{\partial}_t\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\sigma {\partial}_x (v^{-1})\right|dx\\
\leq &C
\delta(1+t)^{-\frac{3}{2}}+C{\epsilon}_0\|{\partial}_x[\varphi,\psi,{\partial}_x\sigma]\|^2,
\end{split}$$ $$\label{R8}
\begin{split}
|{\mathcal{I}}_7|\leq C\|\psi\|_{L^\infty} (\|{\partial}_x\sigma\|^2+\|{\partial}_x^2\sigma\|^2)
\leq C {\epsilon}_0\|{\partial}_x[\sigma,{\partial}_x\sigma]\|^2,
\end{split}$$ $$\label{R9}
\begin{split}
|{\mathcal{I}}_8|\leq &C\int_{{\mathbb{R}}_{+}}|{\partial}^2_x\sigma\sigma{\partial}_x\psi|dx+
C\int_{{\mathbb{R}}_{+}}|{\partial}^2_x\sigma\sigma\psi{\partial}_x\varphi|dx+C\int_{{\mathbb{R}}_{+}}|{\partial}^2_x\sigma\sigma\psi{\partial}_xv^{cd}|dx\\
\leq &C {\epsilon}_0\|{\partial}_x[\psi, \varphi,{\partial}_x\sigma]\|^2+C\epsilon_{0}
\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\sigma^{2}dx
\leq C {\epsilon}_0\|{\partial}_x[\psi, \varphi,{\sigma},{\partial}_x\sigma]\|^2,
\end{split}$$ $$\label{R10}
\begin{split}
|{\mathcal{I}}_{9}|\leq& C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma
{\partial}_xv{\partial}_x\sigma\psi\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma{\partial}_xv\sigma{\partial}_x\psi\right|dx+
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma{\partial}_xv\sigma\psi{\partial}_xv\right|dx
\\
\leq& C\|{\partial}_x{\sigma}\|_{H^1}\|\psi\|_{H^1}\|{\partial}_x\sigma\|\|{\partial}_xv\|
+C\|{\partial}_x{\sigma}\|_{H^1}\|{\sigma}\|_{H^1}\|{\partial}_x\psi\|\|{\partial}_xv\|
+C\|{\partial}_x{\sigma}\|_{H^1}\|{\sigma}\|_{H^1}\|\psi\|_{H^1}\|{\partial}_x\varphi\|^2
\\&+C\|{\partial}_x{\sigma}\|\|{\partial}_x\varphi\|\|{\sigma}\|_{H^1}\|\psi\|_{H^1}\|{\partial}_xv^{cd}\|_{H^1}
+C\|\psi\|_{H^1}\left(\|{\partial}_x{\sigma}\|^2+\|{\sigma}({\partial}_xv^{cd})^2\|^2\right)
\\ \leq&
C{\epsilon}_0\|{\partial}_x[\sigma,{\partial}_x\sigma, \varphi, \psi]\|^2,
\end{split}$$ $$\label{R101}
\begin{split}
|{\mathcal{I}}_{10}|+|{\mathcal{I}}_{29}|+|{\mathcal{I}}_{30}|\leq C&\int_{{\mathbb{R}}_{+}}\left|
{\partial}_xu^{cd}\sigma^{2}\right|dx \leq
C\int_{{\mathbb{R}}_{+}}\left((\partial_{x}\theta^{cd})^{2}+|\partial^{2}_{x}\theta^{cd}|\right)\sigma^{2}dx\leq C{\delta}\|{\partial}_x{\sigma}\|^2,
\end{split}$$ $$\label{R101}
\begin{split}
|{\mathcal{I}}_{12}|&+|{\mathcal{I}}_{13}|+|{\mathcal{I}}_{14}|+|{\mathcal{I}}_{19}|+|{\mathcal{I}}_{20}|+|{\mathcal{I}}_{25}|+|{\mathcal{I}}_{26}|\\
\leq& C\int_{{\mathbb{R}}_{+}}\left|
{\partial}_xv^{cd}\sigma^{3}\psi\right|dx+C\int_{{\mathbb{R}}_{+}}\left|
{\partial}_x[\varphi,\sigma]\sigma^{3}\psi\right|dx\\
\leq& C{\epsilon}_0\|{\partial}_x[\sigma,\varphi]\|^2+C{\epsilon}_0
\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\sigma^{2}dx
\leq C{\epsilon}_0\|{\partial}_x[\sigma,\varphi]\|^2,
\end{split}$$ $$\label{R16}
\begin{split}
|{\mathcal{I}}_{15}|
\leq&\eta\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_x\sigma\right|^2dx+C_\eta\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma{\partial}_x\varphi\right|^2dx+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma{\partial}_xv^{cd}\right|^2dx
\\ \leq& (C{\epsilon}_0+\eta)\|{\partial}_x[\sigma,{\partial}_x{\sigma}]\|^2+C_\eta\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dx,
\end{split}$$ $$\label{R15}
\begin{split}
|{\mathcal{I}}_{16}|\leq&C\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_xv^{cd}{\partial}_x\sigma\psi\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_xv^{cd}\sigma{\partial}_x\psi\right|dx\\
&+C\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_xv^{cd}[{\partial}_x\varphi,{\partial}_xv^{cd}]\sigma\psi\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xv^{cd}[{\partial}_x\varphi,{\partial}_xv^{cd}]{\partial}_x\sigma\psi\right|dx
\\&+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xv^{cd}[{\partial}_x\varphi,{\partial}_xv^{cd}]\sigma{\partial}_x\psi\right|dx+
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xv^{cd}[{\partial}_x\varphi,{\partial}_xv^{cd}]^2\sigma\psi\right|dx\\
\leq& C{\epsilon}_0\|{\partial}_x[\sigma,\varphi,
\psi]\|^2+C\delta(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{R18}
\begin{split}
|{\mathcal{I}}_{17}|+|{\mathcal{I}}_{27}|\leq&C\int_{{\mathbb{R}}_{+}}\left|({\partial}_xv^{cd})^{2}{\partial}_xv\psi\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}^2_xv^{cd}{\partial}_xv^{cd}\psi\right|dx
\\ \leq&
C{\epsilon}_0\|{\partial}_x\varphi\|^2+C\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dx+C{\delta}^2(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{R18}
\begin{split}
|{\mathcal{I}}_{21}|+|{\mathcal{I}}_{22}|+|{\mathcal{I}}_{23}|+|{\mathcal{I}}_{24}|\leq&C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x[\sigma,\varphi]\sigma^2\psi\right|dx+C\int_{{\mathbb{R}}_{+}}\left|\sigma^2\psi\varphi{\partial}_xv^{cd}\right|dx\\
\leq& C{\epsilon}_0\|{\partial}_x[\sigma,\varphi]\|^2+C{\epsilon}_0
\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\varphi^{2}dx
\\ \leq& C{\epsilon}_0\|{\partial}_x[\sigma,\varphi]\|^2+
C\delta^2\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t},
\end{split}$$ $$\label{R16}
\begin{split}
|{\mathcal{I}}_{28}|\leq&C\int_{{\mathbb{R}}_{+}}\left|{\partial}^{2}_x\sigma{\partial}_xv^{cd}\sigma\psi\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma{\partial}_xv{\partial}_xv^{cd}\psi\sigma\right|dx\\
\leq& C{\epsilon}_0\|{\partial}_x[\sigma,{\partial}_x\sigma,\varphi]\|^2+C{\epsilon}_0
\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\sigma^{2}dx\leq C{\epsilon}_0\|{\partial}_x[\sigma,{\partial}_x\sigma,\varphi]\|^2.
\end{split}$$ For the last term ${\mathcal{I}}_{31}$, applying again, one can see that $$\label{I22}
\begin{split}
{\mathcal{I}}_{31}
=&-\frac{3}{2}\int_{{\mathbb{R}}_{+}}v^{cd}\rho_{e}'(\phi^{cd}){\partial}_t\left(\frac{\sigma^2}{v}\right)
\left(-v^{cd}{\partial}_x\left(\frac{{\partial}_x\sigma}{v}\right)
+v\left(1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right)
-v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right)dx,
\end{split}$$ which implies $$\label{I22}
\begin{split}
\left|{\mathcal{I}}_{31}
-\frac{d}{dt}\int_{{\mathbb{R}}_{+}}\left(v^{cd}\rho_{e}'(\phi^{cd})\right)^{2}\sigma^3dx\right|
\leq C{\epsilon}_0\left\|\left[{\partial}_x\psi,{\partial}_x\varphi,{\partial}_t\sigma,
{\partial}_x\sigma,{\partial}_x^2\sigma\right]\right\|^2 +C\delta(1+t)^{-\frac{3}{2}}.
\end{split}$$ Let us now define ${\sigma}_0(x)={\sigma}(x,0)=\phi(x,0)-\phi^{cd}(x,0)$. From the Poisson equation , it follows that for any $t\geq0$ $$\begin{aligned}
\|{\sigma}(t)\|_{H^1}^2&\leq &C\|\varphi(t)\|^2 +C\left\|{\partial}_x^2 \phi^{cd}(t)\right\|^2+C\left\|\left({\partial}_x \phi^{cd}\right)^2(t)\right\|^2
\leq C\|\varphi(t)\|^2 +C{\delta}^2,\end{aligned}$$ and hence in particular, $$\label{si.ides}
\|\sigma_0\|_{H^1}^2\leq C\|\varphi_0\|^2 +C{\delta}^2.$$
We now conclude from , , , , , , , and the above estimates on ${\mathcal{I}}_{l}$ $(1\leq l\leq 31)$ that $$\label{zeng.p3}
\begin{split}
\left\|[\psi,\varphi,\zeta]\right\|^2&+\left\|\sigma\right\|^2_{H^1}+{\epsilon}_0\left\|\partial_{x}\varphi\right\|^2
+\int_{0}^{T}\|{\partial}_x\left[\psi,\zeta\right]\|^2dt\\
\leq& C\left\|[\psi_{0},\zeta_{0}]\right\|^2+C\left\|\varphi_{0}\right\|^{4/3}_{H^1}
+(C{\epsilon}_0+\eta)\int_{0}^{T}\|{\partial}_x[\varphi,{\partial}_x\psi,\sigma,{\partial}_x\sigma]\|^2dt
\\&+C_\eta\int_{0}^{T}\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dxdt
+C\delta,
\end{split}$$ for suitably small ${\epsilon}_0>0$, ${\delta}>0$ and $\eta>0$.
[**Step 2.**]{} [*Dissipation of ${\partial}_x[\varphi,\sigma,{\partial}_x\sigma]$.*]{}
We first differentiate with respect to $x$, to obtain $$\label{dpp}
{\partial}_t{\partial}_x\varphi-{\partial}^2_x\psi=0.$$ Then multiplying , and by ${\partial}^{2}_x\sigma$, $-v{\partial}_x\varphi$ and $\mu{\partial}_x\varphi$, respectively, and integrating the resulting equalities with respect to $x$ over ${\mathbb{R}}_{+}$, one has $$\label{pp.ip1}
\begin{split}
\int_{{\mathbb{R}}_{+}}&\frac{v^{cd}}{v}({\partial}^2_x\sigma)^{2}dx
+\int_{{\mathbb{R}}_{+}}vv^{cd}|\rho_{e}'(\phi^{cd})|({\partial}_x\sigma)^{2}dx\\
=&\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v^2}{\partial}_xv{\partial}_x\sigma{\partial}^2_x\sigma
dx-\int_{{\mathbb{R}}_{+}}{\partial}_xv\left[1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right]{\partial}_x\sigma
dx
+\int_{{\mathbb{R}}_{+}}v{\partial}_x[v^{cd}\rho_{e}'(\phi^{cd})]\sigma{\partial}_x\sigma dx\\
&+\int_{{\mathbb{R}}_{+}}v{\partial}_x\left[\frac{v^{cd}\rho_{e}''(\phi^{cd})}{2}\sigma^{2}\right]{\partial}_x\sigma
dx+\int_{{\mathbb{R}}_{+}}{\partial}_x\varphi{\partial}_x\sigma
dx+\varphi(0,t){\partial}_x{\sigma}(0,t)-\int_{{\mathbb{R}}_{+}}v{\partial}_xI_0{\partial}_x\sigma
dx
\\&
-\int_{{\mathbb{R}}_{+}}v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right){\partial}^{2}_x\sigma
dx,
\end{split}$$ $$\label{pu.ip1}
\begin{split}
-&\int_{{\mathbb{R}}_{+}}{\partial}_t \psi v{\partial}_x\varphi dx-\int_{{\mathbb{R}}_{+}}({\partial}_x
p-{\partial}_xp^{cd})v{\partial}_x\varphi dx+\int_{{\mathbb{R}}_{+}}{\partial}_x\varphi{\partial}_x\sigma dx
+\int_{{\mathbb{R}}_{+}}\left(\frac{{\partial}_x\phi^{cd}}{v}-\frac{{\partial}_x\phi^{cd}}{v^{cd}}\right)v{\partial}_x\varphi dx\\
&=-\int_{{\mathbb{R}}_{+}}\mu{\partial}^2_x \psi{\partial}_x\varphi dx
-\int_{{\mathbb{R}}_{+}}\mu{\partial}^2_x u^{cd}{\partial}_x\varphi
dx-\mu\int_{{\mathbb{R}}_{+}}{\partial}_x(v^{-1}){\partial}_x uv{\partial}_x\varphi
dx+\int_{{\mathbb{R}}_{+}}{\partial}_tu^{cd}v{\partial}_x\varphi dx,
\end{split}$$ and $$\label{pv.ip1}
\int_{{\mathbb{R}}_{+}}\mu\left({\partial}_t{\partial}_x\varphi-{\partial}^2_x\psi\right){\partial}_x\varphi
dx=0.$$ The summation of , and further implies $$\label{sum.ip1}
\begin{split}
-\frac{d}{dt}&\int_{{\mathbb{R}}_{+}}\psi v{\partial}_x\varphi dx
+\frac{\mu}{2}\frac{d}{dt}\int_{{\mathbb{R}}_{+}}({\partial}_x\varphi)^{2}dx+\int_{{\mathbb{R}}_{+}}p^{cd}({\partial}_x\varphi)^{2}dx
\\&+\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v}({\partial}^2_x\sigma)^{2}dx
+\int_{{\mathbb{R}}_{+}}vv^{cd}|\rho_{e}'(\phi^{cd})|({\partial}_x\sigma)^{2}dx\\
=&\underbrace{-\int_{{\mathbb{R}}_{+}}\psi{\partial}_tv{\partial}_x\varphi dx}_{J_1}\underbrace{- \int_{{\mathbb{R}}_{+}}\psi
v{\partial}_t{\partial}_x\varphi dx}_{J_2}\underbrace{+\int_{{\mathbb{R}}_{+}}R{\partial}_x
\left[\frac{\zeta}{v}\right]v{\partial}_x\varphi dx}_{J_3}
\underbrace{-\int_{{\mathbb{R}}_{+}}R\varphi{\partial}_x
\left[\frac{{\theta}^{cd}}{vv^{cd}}\right]v{\partial}_x\varphi dx}_{J_4}\\
&\underbrace{-\int_{{\mathbb{R}}_{+}}\left(\frac{{\partial}_x\phi^{cd}}{v}-\frac{{\partial}_x\phi^{cd}}{v^{cd}}\right)v{\partial}_x\varphi
dx}_{J_5} \underbrace{-\int_{{\mathbb{R}}_{+}}\mu{\partial}^2_x u^{cd}{\partial}_x\varphi
dx}_{J_6}\underbrace{-\mu\int_{{\mathbb{R}}_{+}}{\partial}_x(v^{-1}){\partial}_x uv{\partial}_x\varphi dx}_{J_7}\\
&\underbrace{+\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v^2}{\partial}_xv{\partial}_x\sigma{\partial}^2_x\sigma
dx}_{J_8}\underbrace{-\int_{{\mathbb{R}}_{+}}{\partial}_xv\left[1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right]{\partial}_x\sigma
dx}_{J_9}\underbrace{+\varphi(0,t){\partial}_x{\sigma}(0,t)}_{J_{10}}\\& \underbrace{-\int_{{\mathbb{R}}_{+}}v{\partial}_xI_0{\partial}_x\sigma dx}_{J_{11}}
\underbrace{+\int_{{\mathbb{R}}_{+}}v{\partial}_x\left[v^{cd}\rho_{e}'(\phi^{cd})\right]\sigma{\partial}_x\sigma
dx}_{J_{12}}\underbrace{+\int_{{\mathbb{R}}_{+}}v{\partial}_x\left[\frac{v^{cd}\rho_{e}''(\phi^{cd})}{2}\sigma^{2}\right]{\partial}_x\sigma
dx}_{J_{13}}\\&\underbrace{+\int_{{\mathbb{R}}_{+}}{\partial}_tu^{cd}v{\partial}_x\varphi
dx}_{J_{14}}\underbrace{-\int_{{\mathbb{R}}_{+}}v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right){\partial}^2_x\sigma
dx}_{J_{15}}.
\end{split}$$ We now turn to compute $J_l$ $(1\leq l\leq 15)$ term by term. For brevity, we directly give the following computations: $$\begin{split}
|J_{1}|\leq&C\int_{{\mathbb{R}}_{+}}\left|\psi{\partial}_x\psi{\partial}_x\varphi\right|
dx+C\int_{{\mathbb{R}}_{+}}\left|\psi{\partial}_xu^{cd}{\partial}_x\varphi\right|dx\leq
C{\epsilon}_0\|{\partial}_x[\psi,\varphi]\|^2+C\delta(1+t)^{-3/2},
\end{split}$$ $$\begin{split}
|J_{2}|\leq&\left|\psi(0,t)v(0,t)({\partial}_t\varphi)(0,t)\right|+\left|\int_{{\mathbb{R}}_{+}}v({\partial}_x\psi)^{2}dx\right|+
\left|\int_{{\mathbb{R}}_{+}}\psi{\partial}_xv{\partial}_x\psi dx\right|\\
\leq&C\|\varphi_{0}\|_{H^1}^{\frac{4}{3}}e^{-\frac{p_{-}}{\mu}t}+
C{\epsilon}_0\|{\partial}_x[\psi,\varphi]\|^2+C\|{\partial}_x\psi\|^2
+C\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dx,
\end{split}$$ $$\begin{split}
|J_{3}|+|J_{4}|+|J_{5}| &\leq
(\eta+C{\epsilon}_0)\|{\partial}_x\varphi\|^2+C_{\eta}\|{\partial}_x\zeta\|^2
+C_{\eta}\int_{{\mathbb{R}}_{+}}(\varphi^{2}+\zeta^{2})({\partial}_x\theta^{cd})^{2}dx\\
&\leq(\eta+C{\epsilon}_0)\|\partial_{x}[\varphi,\zeta]\|^2+C_{\eta}\|{\partial}_x\zeta\|^2
+C\delta\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t},
\end{split}$$ $$\begin{split}
|J_{6}| \leq C\delta\|{\partial}_x\varphi\|^2+C\delta(1+t)^{-5/2},
\end{split}$$ $$\begin{split}
|J_{7}|\leq& C\int_{{\mathbb{R}}_{+}}({\partial}_x\varphi)^{2}|{\partial}_x\psi| dx
+C\int_{{\mathbb{R}}_{+}}({\partial}_x\varphi)^{2}|{\partial}_xu^{cd}|
dx+C\int_{{\mathbb{R}}_{+}}|{\partial}_xv^{cd}{\partial}_xu^{cd}{\partial}_x\varphi| dx+
\int_{{\mathbb{R}}_{+}}|{\partial}_xv^{cd}{\partial}_x\psi{\partial}_x\varphi|dx\\
\leq&C{\epsilon}_0\|{\partial}_x[\psi,{\partial}_x\psi,\varphi]\|^2
+C\delta(1+t)^{-2},
\end{split}$$ $$|J_{8}|+|J_{9}| \leq C{\epsilon}_0\|{\partial}_x[\varphi,{\partial}_x\sigma,\sigma]\|^2,$$ $$\begin{split}
|J_{10}|\leq
C{\epsilon}_0\|{\partial}_x\sigma\|_{H^1}^2+C\|\varphi_{0}\|_{H^1}^{\frac{4}{3}}e^{-\frac{4p_{-}}{3\mu}t},
\end{split}$$ $$\begin{split}
|J_{11}|+|J_{12}|+|J_{13}|+|J_{14}|+|J_{15}|\leq
C{\epsilon}_0\|{\partial}_x[\varphi,\sigma,{\partial}_x{\sigma}]\|^2+C\delta(1+t)^{-3/2}.
\end{split}$$ Substituting the above estimations for $J_l$ $(1\leq l\leq 15)$ into , letting $\eta>0$ be suitably small and combing , we obtain $$\label{sum.ip2}
\begin{split}
\left\|[\psi,\varphi,\zeta]\right\|^2&+\left\|\sigma\right\|^2_{H^1}+
\|{\partial}_x\varphi\|^2
+\int_{0}^{T}\|{\partial}_x\sigma\|^2_{H^1}dt
+\int_{0}^{T}\|{\partial}_x\left[\varphi,\psi,\zeta\right]\|^2dt
\\
\leq&C{\epsilon}_0\int_{0}^{T}\|{\partial}^{2}_x\psi\|^2dt+C\delta+
C\left\|[\psi_{0},\zeta_{0}]\right\|^2+C\left\|\varphi_{0}\right\|^{4/3}_{H^1}
+C\int_{0}^{T}\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dxdt.
\end{split}$$ [**Step 3.**]{} [*Higher order energy estimates.*]{}
Multiplying by $-{\partial}^{2}_x\psi$, and integrating the resultant equality with respect to $x$ over ${\mathbb{R}}_{+}$, one has $$\label{sum.ip3}
\begin{split}
\frac{1}{2}\frac{d}{dt}&\int_{{\mathbb{R}}_{+}}({\partial}_x\psi)^{2}dx+\mu\int_{{\mathbb{R}}_{+}}\frac{({\partial}^{2}_x\psi)^{2}}{v}dx
\\
=&\underbrace{-\int_{{\mathbb{R}}_{+}}\left(\frac{{\partial}_x\phi}{v}-\frac{{\partial}_x\phi^{cd}}{v^{cd}}\right){\partial}^{2}_x\psi
dx}_{J_{16}}\underbrace{+\int_{{\mathbb{R}}_{+}}{\partial}_x(p-p^{cd}){\partial}^{2}_x\psi
dx}_{J_{17}}\underbrace{+\mu\int_{{\mathbb{R}}_{+}}\frac{{\partial}_x\psi{\partial}_x\varphi}{v^{2}}{\partial}^{2}_x\psi
dx}_{J_{18}}\\&\underbrace{+\mu\int_{{\mathbb{R}}_{+}}\frac{{\partial}_x\psi{\partial}_xv^{cd}}{v^{2}}{\partial}^{2}_x\psi
dx}_{J_{19}}\underbrace{-\int_{{\mathbb{R}}_{+}}F{\partial}^{2}_x\psi
dx}_{J_{20}}\underbrace{-({\partial}_t\psi{\partial}_t\varphi)(0,t)}_{J_{21}}.
\end{split}$$ To obtain the estimates for $J_l$ $(16\leq l\leq 21)$, we use Cauchy-Schwarz’s inequality with $0<\eta<1$, Sobolev’s inequlity and repeatedly to perform the calculations as follows: $$\label{3.2}
\begin{split}
|J_{16}|\leq&
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xv^{cd}\varphi{\partial}^{2}_x\psi\right|dx+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x\sigma{\partial}^{2}_x\psi\right|dx\\
\leq&
(C\delta+\eta)\|{\partial}_x[\varphi,{\partial}_x\psi]\|^{2}+C_{\eta}\|{\partial}_x\sigma\|^{2}+C\delta\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t},
\end{split}$$ $$\label{3.3}
\begin{split}
|J_{17}|\leq&
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_x[\zeta,\varphi]{\partial}^{2}_x\psi\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|[\zeta,\varphi]{\partial}_x[\varphi,v^{cd}]{\partial}^{2}_x\psi\right|dx\\
\leq&
(C{\epsilon}_0+\eta)\|{\partial}_x[\varphi,{\partial}_x\psi]\|^{2}+C_{\eta}\|{\partial}_x[\zeta,\varphi]\|^{2}+C\delta\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t},
\end{split}$$ $$\label{3.4}
\begin{split}
|J_{18}|+|J_{19}|\leq C{\epsilon}_0\|{\partial}_x[\psi,{\partial}_x\psi]\|^{2},
\end{split}$$
$$\label{3.6}
\begin{split}
|J_{20}|\leq&
C\int_{{\mathbb{R}}_{+}}\left|{\partial}^{2}_xu^{cd}{\partial}^{2}_x\psi\right|dx+
C\int_{{\mathbb{R}}_{+}}\left|{\partial}_xu^{cd}{\partial}_xv{\partial}^{2}_x\psi\right|dx+C\int_{{\mathbb{R}}_{+}}\left|{\partial}_tu^{cd}{\partial}^{2}_x\psi\right|dx\\
\leq&
C\delta\|{\partial}_x[\varphi,{\partial}_x\psi]\|^{2}+C\delta(1+t)^{-\frac{5}{2}}.
\end{split}$$
For the last term $J_{21}$, in light of Lemma \[pbd\], we have $$\label{J21}
\begin{split}
J_{21}=-({\partial}_t\psi{\partial}_t\varphi)(0,t)=-{\partial}_t[(\psi{\partial}_t\varphi)(0,t)]+\varphi_{0}(0)\frac{(p_{-})^{2}}{\mu^{2}}\psi(0,t)
e^{-\frac{p_{-}}{\mu}t},
\end{split}$$ furthermore, it follows that $$\label{J211}
\begin{split}
|(\psi{\partial}_t\varphi)(0,T)|\leq C\varphi_{0}(0)\psi(0,T)
e^{-\frac{p_{-}}{\mu}T}\leq C{\epsilon}_0\|\varphi_{0}\|_{H^1}e^{-\frac{p_{-}}{\mu}T},
\end{split}$$ and $$\label{J212}
\begin{split}
|(\psi{\partial}_t\varphi)(0,0)|\leq C|\psi_{0}(0)\varphi_0(0)|\leq C(\|\psi_0\|^2_{H^1}+\|\varphi_{0}\|^2_{H^1}).
\end{split}$$ By virtue of , and and carrying out the similar calculations as , we thereby obtain $$\label{3.6}
\begin{split}
\left|\int_{0}^{T}J_{21}dt\right| \leq C{\epsilon}_0\int_{0}^{T}\|{\partial}_x\psi\|^2dt+C\|\varphi_{0}\|_{H^1}^{\frac{4}{3}}+C\|\psi_0\|^2_{H^1}+C{\epsilon}_0\|\varphi_{0}\|_{H^1}.
\end{split}$$
Plug the above estimations for $J_l$ $(16\leq l\leq 21)$ into , and recall and , then choose ${\epsilon}_0>0$, $\delta>0$ and $\eta>0$ suitably small, to derive $$\label{sum.ip4}
\begin{split}
\left\|[\psi,\varphi,\zeta]\right\|^2&+\left\|\sigma\right\|^2_{H^1}
+\|{\partial}_x\varphi\|^2+\|{\partial}_x\psi\|^2\\
&+\int_{0}^{T}\|{\partial}_x\sigma\|^2_{H^1}dt
+\int_{0}^{T}\|{\partial}_x\left[\varphi,\psi,\zeta\right]\|^2dt+\int_{0}^{T}\|{\partial}^{2}_x\psi\|^{2}dt\\
\leq& C\delta+
C\left\|\zeta_{0}\right\|^2+C\left\|[\psi_{0},\varphi_{0}]\right\|^{4/3}_{H^1}
+C\int_{0}^{T}\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dxdt.
\end{split}$$ Similarly, multiplying by $-{\partial}^{2}_x\zeta$, and integrating the resulting equality over ${\mathbb{R}}_{+}$, we obtain $$\label{sum.ip5}
\begin{split}
\frac{R}{2(\gamma-1)}&\frac{d}{dt}\int_{{\mathbb{R}}_{+}}({\partial}_x\zeta)^{2}dx+\kappa\int_{{\mathbb{R}}_{+}}\frac{({\partial}^{2}_x\zeta)^{2}}{v}dx
\\
=&\int_{{\mathbb{R}}_{+}}(p{\partial}_xu-p^{cd}{\partial}_xu^{cd}){\partial}^{2}_x\zeta
dx+\kappa\int_{{\mathbb{R}}_{+}}\frac{{\partial}_x\zeta{\partial}_x\varphi}{v^{2}}{\partial}^{2}_x\zeta
dx+\kappa\int_{{\mathbb{R}}_{+}}\frac{{\partial}_x\zeta{\partial}_xv^{cd}}{v^{2}}{\partial}^{2}_x\zeta
dx\\&+\kappa\int_{{\mathbb{R}}_{+}}\partial_{x}\left(\frac{\varphi\partial_{x}\theta^{cd}}{vv^{cd}}\right)
{\partial}^{2}_x\zeta dx-\int_{{\mathbb{R}}_{+}}G{\partial}^{2}_x\zeta
dx,
\end{split}$$ where we have used boundary condition $\zeta(0,t)=0.$ The right hand side of can be handled as $J_{l}$ $(16\leq l\leq 21)$, the details of which we omit, therefore one can get from and that $$\label{sum.ip6}
\begin{split}
\left\|[\psi,\varphi,\zeta]\right\|_{H^1}^2&+\left\|\sigma\right\|^2_{H^1}
+\int_{0}^{T}\|{\partial}_x\sigma\|^2_{H^1}dt
+\int_{0}^{T}\|{\partial}_x\left[\varphi,\psi,\zeta\right]\|^2dt+\int_{0}^{T}\|{\partial}^{2}_x[\psi,\zeta]\|^{2}dt\\
\leq& C\delta+
C\left\|[\varphi_{0},\psi_{0},\zeta_0]\right\|^{4/3}_{H^1}
+C\int_{0}^{T}\int_{{\mathbb{R}}_{+}}\psi^{2}({\partial}_x\theta^{cd})^{2}dxdt.
\end{split}$$ Finally, letting ${\delta}>0$ small enough, combing and in Lemma \[key.es\], we obtain as desired, this completes the proof of Proposition \[ape\].
Global existence and large time behavior
========================================
We are now in a position to complete the proof of Theorem 1.1.
In view of the energy estimates obtained in Proposition \[ape\], one sees that $$\label{eng.p2}
\begin{split}
\sup_{0\leq t\leq
T}&\left\|[\varphi,\psi,\zeta,\sigma](t)\right\|_{H^1}^2
\leq
C\delta+C\left\|[\psi_{0},\zeta_{0},\varphi_{0}]\right\|^{4/3}_{H^1}.
\end{split}$$ Notice that ${\delta}>0$ is a parameter independent of ${\epsilon}_0$. By letting ${\delta}>0$ be small enough, the global existence of the solution of the Cauchy problem , , , , and then follows from the standard continuation argument based on the local existence (cf. [@HMS-04]) and the [*a priori*]{} estimate . Moreover, implies . Our intention next is to prove the large time behavior as . For this, we first justify the following limits: $$\label{latm1}
\lim\limits_{t\rightarrow+\infty}\left\|{\partial}_x[\varphi,\psi,\zeta](t)
\right\|_{L^2}^2= 0,$$ and $$\label{latm2}
\lim\limits_{t\rightarrow+\infty}\left\|{\partial}_x\sigma(t)\right\|^2
= 0.$$ To prove and , we get from , , , and that $$\label{latm3}
\begin{split}
\int_{0}^{+\infty}\left|\frac{d}{dt}\left\|{\partial}_x[\varphi,\psi,\zeta]\right\|^2\right|dt
=&2\int_{0}^{+\infty}\left|\left({\partial}_t{\partial}_x[\varphi,\psi,\zeta],{\partial}_x[\varphi,\psi,\zeta]\right)\right|dt
\\ \leq& C+C\int_{0}^{+\infty}\left\|{\partial}_x\left[\varphi,\psi,\zeta,{\sigma},{\partial}_x\left[\psi,\zeta,{\sigma}\right]\right]\right\|^2dt<+\infty.
\end{split}$$ On that other hand, , and yield $$\label{latm4}
\int_{0}^{+\infty}\left|\frac{d}{dt}\left\|{\partial}_x\sigma\right\|^2\right|dt
=2\int_{0}^{+\infty}\left|\left({\partial}_t{\partial}_x\sigma,{\partial}_x\sigma\right)\right|dt<+\infty.$$ Consequently, , together with gives and . Then follows from , and Sobolev’s inequality . This ends the proof of Theorem \[main.res.\].
Appendix
========
In this appendix, we will give some basic results used in the paper. The first lemma is borrowed from [@HMS-04].
\[es.tap\] Let ${\theta}^{cd}$ satisfy , for $|\theta_{+}-\theta_{-}|=\delta$, it holds that $$\label{1.29gsg8*x}
\begin{split}
\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{4}dx\leq
C\delta^{4}(1+t)^{-\frac{3}{2}},\ \
\int_{{\mathbb{R}}_{+}}(\partial^{2}_{x}\theta^{cd})^{2}dx\leq
C\delta^{2}(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{1.29g8ko*x}
\begin{split}
\int_{{\mathbb{R}}_{+}}(\partial^{3}_{x}\theta^{cd})^{2}dx\leq
C\delta^{2}(1+t)^{-\frac{5}{2}},\ \
\int_{{\mathbb{R}}_{+}}x\left((\partial_{x}\theta^{cd})^{2}+|\partial^{2}_{x}\theta^{cd}|\right)dx
\leq C\delta.
\end{split}$$
Next is the key observation from the boundary condition .
\[pbd\] It holds that $$\label{pvbd}
\varphi(0,t)=\varphi_{0}(0)e^{-\frac{p_{-}}{\mu}t}.$$
Since $\partial_{x}u^{cd}(0,t)=0,$ from it follows that $$\label{resultqa12e}
\begin{split}
\frac{R\theta_{-}}{v_{-}+\varphi(0,t)}-\mu\frac{\partial_{t}\varphi(0,t)}{v_{-}+\varphi(0,t)}=p_{-},
\ \ t>0,
\end{split}$$ which implies $$\label{pvbd2}
\partial_{t}\varphi(0,t)=-\frac{p_{-}}{\mu}\varphi(0,t).$$ follows from and the compatibility condition $\varphi(0,0)=\varphi_{0}(0)$. This ends the proof of Lemma \[pbd\].
We now give the following estimates concerning the delicate term $\displaystyle{\int_{0}^{T}\int_{{\mathbb{R}}_{+}}}(\partial_{x}\theta^{cd})^{2}\psi^{2}dx
dt$.
\[key.es\] Assume all the conditions listed in Proposition \[ape\] hold, then for any $0\leq T\leq +\infty$, there exists an energy functional ${\mathcal{E}}(\varphi,\psi,\zeta)$ with $$|{\mathcal{E}}(\varphi,\psi,\zeta)|\leq C{\delta}^2\|[\varphi,\psi,\zeta]\|^2,$$ such that the following energy estimate holds $$\label{c.eng}
\begin{split}
{\mathcal{E}}(\varphi,\psi,\zeta)(T)+\int_{0}^{T}\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\psi^{2}dx dt
\leq C\delta+C\delta\left\|\varphi_{0}\right\|^{2}_{H^1}
+C\delta\int_{0}^{T}\|\partial_{x}[\varphi,\psi,\zeta,\sigma,\partial_{x}\sigma\|^{2}dt.
\end{split}$$
Define $$w=\int_{0}^{x}(\partial_{y}\theta^{cd})^{2}dy.$$ It is easy to check that $$\label{w.es}
\begin{split}
\|w(\cdot,t)\|_{\infty}\leq C\delta^{2} (1+t)^{-\frac{1}{2}},\ \
\|\partial_{t}w(\cdot,t)\|_{\infty}\leq
C\delta^{2}(1+t)^{-\frac{3}{2}}.
\end{split}$$ From and , it follows that $$\label{si.ep}
\begin{split}
\sigma=-\frac{\varphi}{vv^{cd}\rho_{e}'(\phi^{cd})}
\underbrace{-\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}\left[v^{cd}{\partial}_x
\left(\frac{{\partial}_x\sigma}{v}\right)+\left(\frac{v^{cd}\rho_{e}^{''}(\phi^{cd})}{2}\sigma^{2}-I_0\right)v
+v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right]}_{{\mathcal{M}}}.
\end{split}$$ On the other hand, can be rewritten as $$\label{pp.rw}
\begin{split}
{\partial}_t \psi+{\partial}_x\left(\frac{R\zeta-p^{cd}\varphi-\sigma}{v}\right)
=-{\partial}_x\left(\frac{1}{v}\right)\sigma-\frac{\varphi{\partial}_x\phi^{cd}}{vv^{cd}}+\mu{\partial}_x\left(\frac{{\partial}_x
\psi}{v}\right)+F.
\end{split}$$ Substituting into , one has $$\label{pp.rw2}
\begin{split}
{\partial}_t
\psi+{\partial}_x\left(\frac{R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi}{v}\right)
={\partial}_x\left(\frac{{\mathcal{M}}}{v}\right)-{\partial}_x\left(\frac{1}{v}\right)\sigma-\frac{\varphi{\partial}_x\phi^{cd}}{vv^{cd}}+\mu{\partial}_x\left(\frac{{\partial}_x
\psi}{v}\right)+F.
\end{split}$$ Multiplying by $\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]
vw,$ integrating the resulting equation over ${\mathbb{R}}_{+}$ leads to $$\label{c.eng2}
\begin{split}
\frac{1}{2}&\int_{{\mathbb{R}}_{+}}\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]^{2}(\partial_{x}\theta^{cd})^{2}dx\\
=&\frac{d}{dt}\int_{{\mathbb{R}}_{+}}\psi\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx\underbrace{-\int_{{\mathbb{R}}_{+}}
\psi\partial_{t}\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx}_{{\mathcal{K}}_1}\\
&\underbrace{-\int_{{\mathbb{R}}_{+}}\psi\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]\partial_{t}vwdx}_{{\mathcal{K}}_2}
\underbrace{-\int_{{\mathbb{R}}_{+}}\psi\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]v\partial_{t}wdx}_{{\mathcal{K}}_3}\\
&\underbrace{-\int_{{\mathbb{R}}_{+}}\frac{\partial_{x}v}{v}\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]^{2}wdx}_{{\mathcal{K}}_4}
\underbrace{+\mu\int_{{\mathbb{R}}_{+}}\frac{\partial_{x}\psi}{v}\partial_{x}
\left[\left(R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right)vw\right]dx}_{{\mathcal{K}}_5}\\
&\underbrace{-\int_{{\mathbb{R}}_{+}}F\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx}_{{\mathcal{K}}_6}
\underbrace{-\int_{{\mathbb{R}}_{+}}\frac{\partial_{x}\varphi}{v^{2}}
\sigma\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx}_{{\mathcal{K}}_7}
\\&\underbrace{-\int_{{\mathbb{R}}_{+}}\frac{\partial_{x}v^{cd}}{v^{2}}
\sigma\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx}_{{\mathcal{K}}_8}
\underbrace{+\int_{{\mathbb{R}}_{+}}\frac{{\mathcal{M}}}{v}\partial_{x}\left[\left(R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right)vw\right]dx}_{{\mathcal{K}}_9}
\\&\underbrace{+\int_{{\mathbb{R}}_{+}}\frac{\varphi\partial_{x}\phi^{cd}
}{vv^{cd}}\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx}_{{\mathcal{K}}_{10}}.
\end{split}$$ We now turn to compute ${\mathcal{K}}_{l}$ $(1\leq l\leq 10)$ term by term. For the delicate term ${\mathcal{K}}_1$, it can be rewritten as $$\label{ck1}
\begin{split}
{\mathcal{K}}_{1} =&-\int_{{\mathbb{R}}_{+}}\psi\partial_{t}\left(R\zeta-p^{cd}\varphi\right)vwdx-
\int_{{\mathbb{R}}_{+}}\psi\partial_{t}\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}\varphi\right)vwdx\\
=&-(\gamma-1)\int_{{\mathbb{R}}_{+}}\psi
vw\partial_{t}\left(\frac{R}{\gamma-1}\zeta+p^{cd}\varphi\right)dx+\gamma\int_{{\mathbb{R}}_{+}}\psi
vwp^{cd}\partial_{x}\psi dx\\&+\gamma\int_{{\mathbb{R}}_{+}}\psi
vw\partial_{t}p^{cd}\varphi dx-
\frac{1}{2}\int_{{\mathbb{R}}_{+}}\partial_{x}(\psi^{2})\left(\frac{w}{v^{cd}\rho_{e}'(\phi^{cd})}\right)dx-
\int_{{\mathbb{R}}_{+}}\psi\varphi\partial_{t}\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}\right)vwdx\\
=&\underbrace{(\gamma-1)\int_{{\mathbb{R}}_{+}}\psi
w\left(R\zeta-p^{cd}\varphi\right)\left(\partial_{x}u^{cd}+\partial_{x}\psi\right)dx}_{{\mathcal{K}}_{1,1}}
\underbrace{+\kappa(\gamma-1)\int_{{\mathbb{R}}_{+}}\frac{v^{cd}\partial_{x}\zeta-\partial_{x}\theta^{cd}\varphi}{vv^{cd}}\partial_{x}\left(\psi
vw\right)dx}_{{\mathcal{K}}_{1,2}}\\& \underbrace{-(\gamma-1)\int_{{\mathbb{R}}_{+}}\psi
vwGdx}_{{\mathcal{K}}_{1,3}}\underbrace{-\frac{\gamma}{2}\int_{{\mathbb{R}}_{+}}p^{cd}\partial_{x}vw\psi^{2}dx}_{{\mathcal{K}}_{1,4}}
\underbrace{-\frac{\gamma}{2}\int_{{\mathbb{R}}_{+}}\partial_{x}p^{cd}vw\psi^{2}dx}_{{\mathcal{K}}_{1,5}}\underbrace{+\gamma\int_{{\mathbb{R}}_{+}}\psi
vw\partial_{t}p^{cd}\varphi dx}_{{\mathcal{K}}_{1,6}}\\&+
\int_{{\mathbb{R}}_{+}}\left(\frac{1}{2v^{cd}\rho_{e}'(\phi^{cd})}-\frac{\gamma}{2}p^{cd}v\right)\psi^{2}(\partial_{x}\theta^{cd})^{2}dx
\underbrace{+\frac{1}{2}\int_{{\mathbb{R}}_{+}}\psi^{2}\partial_{x}\left(\frac{1}{v^{cd}\rho_{e}'(\phi^{cd})}\right)wdx}_{{\mathcal{K}}_{1,7}}
\\&\underbrace{-\int_{{\mathbb{R}}_{+}}\psi\varphi\partial_{t}\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}\right)vwdx}_{{\mathcal{K}}_{1,8}},
\end{split}$$ where in the third identity we have used $$\label{5.7}
\begin{split}
\frac{R}{\gamma-1}{\partial}_t\zeta+p^{cd}{\partial}_t\varphi=-\frac{R\zeta-p^{cd}\varphi}{v}\left(\partial_{x}u^{cd}+\partial_{x}\psi\right)
+\kappa{\partial}_x\left(\frac{v^{cd}{\partial}_x\zeta-{\partial}_x\theta^{cd}\varphi}{vv^{cd}}\right)+G,
\end{split}$$ which is derived from and .
Since $\rho_{e}'(\phi^{cd})<0$ according to the assumption $({\mathcal{A}})_2$, further implies $$\label{ps.ck1}
0<-\int_{{\mathbb{R}}_{+}}\left(\frac{1}{2v^{cd}\rho_{e}'(\phi^{cd})}-\frac{\gamma}{2}p^{cd}v\right)\psi^{2}(\partial_{x}\theta^{cd})^{2}dx
=-{\mathcal{K}}_1+\sum\limits_{l=1}^{8}{\mathcal{K}}_{1,l}.$$ To compute ${\mathcal{K}}_{1,l}$ $(1\leq l\leq 8)$ and ${\mathcal{K}}_{l}$ $(2\leq l\leq 10)$, by applying , , Cauchy-Schwarz’s inequality, Sobolev’s inequality , Young’s inequality and Lemmas \[pbd\] and \[es.tap\], we directly address the following estimates: $$\label{5.8}
\begin{split}
|{\mathcal{K}}_{1,1}|+|{\mathcal{K}}_{1,6}|+|{\mathcal{K}}_{1,8}|+|{\mathcal{K}}_{2}|\leq& C\int_{{\mathbb{R}}_{+}}|\psi
w|(|{\partial}_xu^{cd}|+|{\partial}_t\theta^{cd}|)(|\zeta|+|\varphi|)dx+C\int_{{\mathbb{R}}_{+}}|\psi
w{\partial}_x\psi|(|\zeta|+|\varphi|)dx
\\ \leq& C\|w{\partial}_t\theta^{cd}\|_{L^\infty}\|[\varphi,\psi,\zeta]\|^2
+C{\delta}\|{\partial}_x\psi\|^{2}+\frac{C}{{\delta}}\int_{{\mathbb{R}}_{+}}w^2\psi^2[\varphi,\zeta]^2dx
\\ \leq& C\delta\|{\partial}_x\psi\|^{2}+C\delta{\epsilon}_0(1+t)^{-\frac{3}{2}}+\frac{C}{{\delta}^2}\|w\|^4_{L^\infty}\|\psi\|^2\|[\varphi,\zeta]\|^4
\\ \leq& C\delta\|{\partial}_x\psi\|^{2}+C\delta{\epsilon}_0(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{5.9}
\begin{split}
|{\mathcal{K}}_{1,2}|+|{\mathcal{K}}_{5}|\leq&C\int_{{\mathbb{R}}_{+}}\left(|\partial_{x}\varphi|+|\partial_{x}\zeta|+|\partial_{x}\theta^{cd}\varphi|\right)
\left(|\partial_{x}\psi w|+\left|\psi(\partial_{x}\theta^{cd})^{2}\right|+|\psi w\partial_{x}v|\right)dx
\\ \leq&
C\delta\|{\partial}_x[\varphi,\psi,\zeta]\|^{2}+C\delta(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{5.10}
\begin{split}
|{\mathcal{K}}_{1,3}|\leq& C\int_{{\mathbb{R}}_{+}}\left|\psi
w\left((\partial_{x}u^{cd})^{2}+(\partial_{x}\psi)^{2}\right)\right|dx
\leq C\int_{{\mathbb{R}}_{+}}\psi^2
w^2(\partial_{x}u^{cd})^{2}dx+C\int_{{\mathbb{R}}_{+}}(\partial_{x}u^{cd})^{2}dx
+C
\delta\|{\partial}_x\psi\|^{2}
\\ \leq & C
\delta\|{\partial}_x\psi\|^{2}+C\delta(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{5.11}
\begin{split}
|{\mathcal{K}}_{1,4}|+|{\mathcal{K}}_{1,5}|+|{\mathcal{K}}_{1,7}|&\leq
C\int_{{\mathbb{R}}_{+}}\left|\partial_{x}\theta^{cd}w\psi^{2}\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|\partial_{x}\varphi w\psi^{2}\right|dx
\\& \leq
C\delta\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\psi^{2}dx+C\delta\|{\partial}_x[\psi,\varphi]\|^{2}+C\delta(1+t)^{-2},
\end{split}$$ $$\label{5.12}
\begin{split}
|{\mathcal{K}}_{3}|\leq
C\int_{{\mathbb{R}}_{+}}\left|\psi\left(|\zeta|+|\varphi|\right)\partial_{t}w\right|dx\leq
C\delta{\epsilon}_0(1+t)^{-\frac{3}{2}},
\end{split}$$ $$\label{5.11}
\begin{split}
|{\mathcal{K}}_{4}|+|{\mathcal{K}}_{10}|&\leq
C\int_{{\mathbb{R}}_{+}}\left|\partial_{x}\theta^{cd}w\left(\zeta^{2}+\varphi^{2}\right)\right|dx
+C\int_{{\mathbb{R}}_{+}}\left|\partial_{x}\varphi
w\left(\zeta^{2}+\varphi^{2}\right)\right|dx
\\ \leq& C{\delta}(1+t)^{-1/2}\int_{{\mathbb{R}}_{+}}|\partial_{x}\theta^{cd}||x|\left(\|{\partial}_x\zeta\|^{2}+\|{\partial}_x\varphi\|^{2}+|\varphi(0,t)|^2\right)dx
\\&+C{\delta}\|{\partial}_x\varphi,\zeta\|^{2}+\frac{C}{{\delta}}\|w\|_{L^\infty}^2\left(\|\zeta\|\|\partial_{x}\zeta\|
+\|\varphi\|\|\partial_{x}\varphi\|\right)\|[\varphi,\zeta]\|^2
\\ \leq&
C\delta\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t}+C\delta\|{\partial}_x[\varphi,\zeta]\|^{2}+C\delta(1+t)^{-2},
\end{split}$$ $$\label{5.13}
\begin{split}
|{\mathcal{K}}_{6}|\leq&
C\delta^2(1+t)^{-\frac{1}{2}}\|F\|_{L^{1}}\left(\|\zeta\|_{\infty}+\|\varphi\|_{\infty}\right)
\leq
C\delta^2(1+t)^{-1}\left(\|\zeta\|^{\frac{1}{2}}\|\partial_{x}\zeta\|^{\frac{1}{2}}
+\|\varphi\|^{\frac{1}{2}}\|\partial_{x}\varphi\|^{\frac{1}{2}}\right)
\\ \leq& C\delta\|{\partial}_x[\varphi,\zeta]\|^{2}+C\delta(1+t)^{-\frac{4}{3}},
\end{split}$$ $$\label{5.14}
\begin{split}
|{\mathcal{K}}_{7}|\leq&
C\int_{{\mathbb{R}}_{+}}\left|\partial_{x}\varphi\sigma(|\zeta|+|\varphi|)w\right|dx
\leq C\delta(1+t)^{-\frac{1}{2}}\left(\|\zeta\|_{\infty}+\|\varphi\|_{\infty}\right)\|\partial_{x}\varphi\|\|\sigma\|\\
\leq& C\delta(1+t)^{-\frac{1}{2}}\left(\|\zeta\|^{\frac{1}{2}}\|\partial_{x}\zeta\|^{\frac{1}{2}}
+\|\varphi\|^{\frac{1}{2}}\|\partial_{x}\varphi\|^{\frac{1}{2}}\right)\|\partial_{x}\varphi\|\|\sigma\|
\\ \leq& C\delta\|\partial_{x}[\varphi,\zeta]\|^{2}+C\delta(1+t)^{-2},
\end{split}$$ $$\label{5.15}
\begin{split}
|{\mathcal{K}}_{8}|
\leq C\int_{{\mathbb{R}}_{+}}|\partial_{x}\theta^{cd}w||{\sigma}|\left(|\zeta|+|\varphi|\right)dx
\leq C\delta\|\varphi_{0}\|_{H^1}^{2}e^{-\frac{p_{-}}{\mu}t}
+C\delta\|\partial_{x}[\varphi,\zeta,\sigma]\|^{2},
\end{split}$$ $$\label{5.16}
\begin{split}
|{\mathcal{K}}_{9}|\leq&
C\int_{{\mathbb{R}}_{+}}\left[|\partial^{2}_{x}\sigma|+|\partial_{x}\sigma\partial_{x}v|+|\sigma^{2}|
+|\partial_{x}v^{cd}\partial_{x}v|\right]\left[(|\partial_{x}\zeta|+|\partial_{x}\varphi|)|w|+(|\zeta|+|\varphi|)
((\partial_{x}\theta^{cd})^{2}+|\partial_{x}v
w|)\right]dx\\
\leq&C\delta\|\partial_{x}[\varphi,\zeta,\sigma,\partial_{x}\sigma]\|^{2}+C\delta(1+t)^{-\frac{3}{2}}.
\end{split}$$ We now plug the above estimates for ${\mathcal{K}}_{1,l}$ with $1\leq l\leq 8$ into to obtain $$\label{ps.ck11}
\begin{split}
&\left|{\mathcal{K}}_1-\int_{{\mathbb{R}}_{+}}\left(\frac{1}{2v^{cd}\rho_{e}'(\phi^{cd})}-\frac{\gamma}{2}p^{cd}v\right)\psi^{2}(\partial_{x}\theta^{cd})^{2}dx\right|
\\& \qquad\leq C\delta\|{\partial}_x[\varphi,\psi,\zeta]\|^{2}+C\delta(1+t)^{-\frac{3}{2}}
+C\delta\int_{{\mathbb{R}}_{+}}(\partial_{x}\theta^{cd})^{2}\psi^{2}dx.
\end{split}$$ Next by substituting the estimates for ${\mathcal{K}}_{l}$ $(2\leq l\leq 10)$ and into and integrating the resulting equality with respect to time over $[0,T]$, one has $$\label{c.eng3}
\begin{split}
-\int_{{\mathbb{R}}_{+}}&\psi\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx
\\&+\int_{0}^{T}\int_{{\mathbb{R}}_{+}}\left\{\frac{1}{2}\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]^{2}
+\left(-\frac{1}{2v^{cd}\rho_{e}'(\phi^{cd})}+\frac{\gamma}{2}p^{cd}v-C\delta\right)\psi^{2}\right\}(\partial_{x}\theta^{cd})^{2}dxdt\\
\leq&C\delta+C{\delta}{\epsilon}_0+C\delta\left\|\varphi_{0}\right\|^{2}_{H^1}+
C\delta\int_{0}^{T}\|\partial_{x}[\varphi,\psi,\zeta,\sigma,\partial_{x}\sigma\|^{2}dt,
\end{split}$$ for suitably small ${\delta}>0$ and ${\epsilon}_0>0$.
Let us now define $$\label{fe.def}
{\mathcal{E}}(\varphi,\psi,\zeta)=-\int_{{\mathbb{R}}_{+}}\psi\left[R\zeta+\left(\frac{1}{vv^{cd}\rho_{e}'(\phi^{cd})}-p^{cd}\right)\varphi\right]vwdx,$$ then follows from and , this ends the proof of Lemma \[key.es\].
Finally we give the detailed proof of .
Taking the inner product of ${\partial}_t\eqref{pp}$ with ${\partial}_t{\sigma}$ with respect to $x$ over ${\mathbb{R}}_+$, one has $$\label{ptsi.es}
\begin{split}
&\underbrace{\int_{{\mathbb{R}}_{+}}{\partial}_t\left(\frac{v^{cd}}{v}\right){\partial}^2_x\sigma{\partial}_t\sigma
dx}_{{\mathcal{J}}_1}\underbrace{+
\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v}{\partial}_t{\partial}^2_x\sigma{\partial}_t\sigma dx}_{{\mathcal{J}}_2}
\underbrace{- \int_{{\mathbb{R}}_{+}}{\partial}_t\left(\frac{v^{cd}}{v^2}{\partial}_xv\right){\partial}_x\sigma
{\partial}_t\sigma dx}_{{\mathcal{J}}_3}\underbrace{-\int_{{\mathbb{R}}_+}
\frac{v^{cd}}{v^2}{\partial}_xv{\partial}_t{\partial}_x\sigma{\partial}_t\sigma dx}_{{\mathcal{J}}_4}
\\&\quad=-\int_{{\mathbb{R}}_{+}}{\partial}_t\varphi{\partial}_t\sigma
dx+\int_{{\mathbb{R}}_{+}}{\partial}_tv\left(1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right){\partial}_t\sigma
dx
\\&\qquad+\int_{{\mathbb{R}}_{+}}v{\partial}_t\left(1-v^{cd}\rho_{e}(\sigma+\phi^{cd})\right){\partial}_t\sigma dx
-\int_{{\mathbb{R}}_{+}}{\partial}_t\left(v^{cd}{\partial}_x\left(\frac{{\partial}_x\phi^{cd}}{v}\right)\right){\partial}_t\sigma
dx.
\end{split}$$ We turn our attention first to ${\mathcal{J}}_l$ $(1\leq l\leq4)$ which can not be directly controlled. Since ${\sigma}(0,t)={\sigma}(+\infty,t)=0$, by integration by parts and using the cancellation, we find $$\label{CJ24}
\begin{split}
{\mathcal{J}}_2+{\mathcal{J}}_4=-\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v}{\partial}_t{\partial}_x\sigma{\partial}_t{\partial}_x\sigma dx-\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}}{v}{\partial}_t{\partial}_x\sigma{\partial}_t\sigma dx,
\end{split}$$ and $$\label{CJ13}
\begin{split}
{\mathcal{J}}_1+{\mathcal{J}}_3=\int_{{\mathbb{R}}_{+}}\frac{{\partial}_tv^{cd}}{v}{\partial}^2_x\sigma{\partial}_t\sigma dx
+\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v^2}{\partial}_tv{\partial}_x\sigma{\partial}_x{\partial}_t\sigma dx
-\int_{{\mathbb{R}}_{+}}\frac{{\partial}_tv^{cd}}{v^2}{\partial}_xv{\partial}_x\sigma
{\partial}_t\sigma dx.
\end{split}$$ On the other hand, similar to , one has $$\label{taylor2}
1-v^{cd}\rho_{e}(\sigma+\phi^{cd})=-v^{cd}\rho_{e}'(\phi^{cd})\sigma
\underbrace{-v^{cd}\int_{\phi^{cd}}^{\phi}\rho_{e}''(\varrho)(\phi-\varrho)d\varrho}_{{\mathcal{J}}_0},$$ and moreover $$\label{ptR2}
{\partial}_t{\mathcal{J}}_0\thicksim {\partial}_t{\sigma}{\sigma}+{\partial}_tv^{cd}{\sigma}+{\partial}_tv^{cd}{\sigma}^2.$$ Substituting , , and into and applying and , we deduce $$\label{ptsi.es2}
\begin{split}
\int_{{\mathbb{R}}_{+}}&\frac{v^{cd}}{v}|{\partial}_t{\partial}_x\sigma|^2 dx-\int_{{\mathbb{R}}_{+}}vv^{cd}\rho_{e}'(\phi^{cd})|{\partial}_t\sigma|^2dx\\
\leq& \left|\int_{{\mathbb{R}}_{+}}\frac{{\partial}_xv^{cd}}{v}{\partial}_t{\partial}_x\sigma{\partial}_t\sigma dx\right|+\left|\int_{{\mathbb{R}}_{+}}\frac{{\partial}_tv^{cd}}{v}{\partial}^2_x\sigma{\partial}_t\sigma dx\right|+\left|\int_{{\mathbb{R}}_{+}}\frac{v^{cd}}{v^2}{\partial}_tv{\partial}_x\sigma{\partial}_x{\partial}_t\sigma dx\right|
\\&+\left|\int_{{\mathbb{R}}_{+}}\frac{{\partial}_tv^{cd}}{v^2}{\partial}_xv{\partial}_x\sigma
{\partial}_t\sigma dx\right|+C\int_{{\mathbb{R}}_{+}}|{\partial}_x\psi{\partial}_t\sigma|dx+C\int_{{\mathbb{R}}_{+}}|{\partial}_xu{\sigma}{\partial}_t\sigma|dx
\\&+C\int_{{\mathbb{R}}_{+}}|{\partial}_tv^{cd}{\sigma}{\partial}_t\sigma|dx+C\int_{{\mathbb{R}}_{+}}|{\partial}_t{\sigma}{\sigma}{\partial}_t\sigma|dx+C\int_{{\mathbb{R}}_{+}}|{\partial}_tv^{cd}{\sigma}^2{\partial}_t\sigma|dx
\\&+C\int_{{\mathbb{R}}_{+}}|{\partial}_t{\partial}_x^2v^{cd}{\partial}_t\sigma|dx+C\int_{{\mathbb{R}}_{+}}|{\partial}_tv^{cd}{\partial}^2_xv^{cd}{\partial}_t\sigma|dx
+C\int_{{\mathbb{R}}_{+}}|{\partial}_tv^{cd}{\partial}_xv^{cd}{\partial}_xv{\partial}_t\sigma|dx
\\&+C\int_{{\mathbb{R}}_{+}}|{\partial}_xv^{cd}{\partial}_x^2u{\partial}_t\sigma|dx,
\end{split}$$ which yields , according to Cauchy-Schwarz’s inequality, and Lemma \[es.tap\]. This completes the proof of .
[**Acknowledgements:**]{} The first author was supported by grants from the National Natural Science Foundation of China \#11471142 and \#11271160. The second and third authors were supported by the National Natural Science Foundation of China \#11331005, the Program for Changjiang Scholars and Innovative Research Team in University \#IRT13066, the Scientific Research Funds of Huaqiao University (Grant No.15BS201), and the Special Fund Basic Scientific Research of Central Colleges \#CCNU12C01001. The first and second authors would like to thank Professor Renjun Duan for many fruitful discussions on the topic of the paper.
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[^1]: Department of Mathematics, Jinan University, Guangzhou 510632, P.R. China. Email: tsqliu@jnu.edu.cn
[^2]: School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, P.R. China. Email: yinhaiyan2000@aliyun.com
[^3]: Corresponding author. School of Mathematics, South China University of Technology, Guangzhou 510641, P.R. China. Email: cjzhu@mail.ccnu.edu.cn
|
---
abstract: 'We have developed a new technique to measure viscoelasticity in soft materials such as polymer solutions, by monitoring thermal fluctuations of embedded probe particles using laser interferometry in a microscope. Interferometry allows us to obtain power spectra of fluctuating beads from 0.1 Hz to 20 kHz, and with sub-nanometer spatial resolution. Using linear response theory, we determined the frequency-dependent loss and storage shear moduli up to frequencies on the order of a kHz. Our technique measures local values of the viscoelastic response, without actively straining the system, and is especially suited to soft biopolymer networks. We studied semiflexible F-actin solutions and, as a control, flexible polyacrylamide (PAAm) gels, the latter close to their gelation threshold. With small particles, we could probe the transition from macroscopic viscoelasticity to more complex microscopic dynamics. In the macroscopic limit we find shear moduli at 0.1 Hz of $G''=0.11\pm0.03$ Pa and $0.17\pm0.07$ Pa for 1 and 2 mg/ml actin solutions, close to the onset of the elastic plateau, and scaling behavior consistent with $G^*(\omega)\sim\omega^{3/4}$ at higher frequencies. For polyacrylamide we measured plateau moduli of 2.0, 24, 100 and 280 Pa for crosslinked gels of 2, 2.5, 3 and 5% concentration (weight/volume) respectively, in agreement to within a factor of two with values obtained from conventional rheology. We also found evidence for scaling of $G^*(\omega)\sim\omega^{1/2}$, consistent with the predictions of the Rouse model for flexible polymers.'
author:
- |
B. Schnurr, F. Gittes, F.C. MacKintosh, and C.F. Schmidt\
Department of Physics & Biophysics Research Division,\
University of Michigan, Ann Arbor, MI 48109-1120
date: 'July 28, 1997'
title: |
Determining Microscopic Viscoelasticity\
in Flexible and Semiflexible Polymer Networks\
from Thermal Fluctuations
---
Introduction {#introduction .unnumbered}
------------
The usefulness of synthetic polymeric materials, and the functions of biopolymers, are all largely based on complex mechanical properties that derive from a hierarchical structure. The mechanical response of polymer solutions or gels displays characteristics of both fluids (viscosity) and of solids (elasticity), depending on the rate of change of applied stress. This response is conventionally described by frequency-dependent storage and loss shear moduli which are commonly measured by active, mechanically-imposed oscillatory strain in macroscopic samples, in contrast to stationary-flow geometries used for measurements in fluids$^{1-5}$.
Such methods have also been used in the past to study reconstituted biopolymer networks [*in vitro*]{}. A general distinction between synthetic polymers and biopolymers is that the former are typically flexible, since their monomer size is small, whereas many biopolymers are formed from large protein monomers and thus are much less flexible. The viscoelastic properties of flexible and semiflexible polymer systems are quite different. At a given volume fraction of polymer, the shear modulus (or stiffness) of a semiflexible polymer network can be several orders of magnitude larger than that of a flexible one. This may be one reason why biological evolution has favored semiflexible polymer networks for mechanical stability. F-actin is one of the primary components of the [*cytoskeleton*]{} of plant and animal cells, and is largely responsible for the viscoelastic response of cells$^{6,7}$. The $\sim$17 $\mu$m persistence length of actin filaments$^{8,9}$ is about three orders of magnitude larger than their diameter. This tremendous aspect ratio makes them ideal model semiflexible polymers. They are rather rigid on the scale of cytoskeletal networks of typical cells, which have characteristic mesh sizes on the order of microns. In reconstituted F-actin networks, individual filaments can be hundreds of microns long, so that solutions of less than 0.1% polymer (volume fraction) are still strongly entangled.
Macroscopic rheological measurements in reconstituted F-actin solutions have been reported in the literature$^{10-19}$. These experiments have shown that the regime of linear viscoelastic response is small,$^{16}$ of order a few percent, and that this range decreases with increasing F-actin concentration. For larger strains, strain hardening is observed, followed by apparent shear thinning, which may be due to filament breakage. This is to be expected for semiflexible polymer networks, since the amplitudes of thermal undulations are small, and the full extension of individual filaments is thus reached for small strains,$^{20}$ making the response nonlinear. This makes measurements with commercial rheometers problematic, although some custom intruments can deal with strains substantially smaller than 1%.$^{1}$ Active macroscopic measurements on actin have given controversial results$^{16}$ and nonlinearity may be one reason for this.
In attempts to approach [*in vivo*]{} conditions, viscoelasticity has also been studied on a microscopic, cellular scale. Early work included the manipulation of magnetic particles in gelatin $^{21}$ and in living cells$^{22}$. Recent experimental techniques for manipulating and tracking sub-micrometer particles have revived the interest in [*microrheology*]{}. Several studies have used direct manipulation of embedded magnetic beads in solutions of F-actin$^{23-27}$. These experiments have been limited by the spatial and temporal resolution of video microscopy.
=
We introduce here a microscopic method of measuring viscoelastic properties within micrometer-sized sample volumes. We study entangled, but non-crosslinked networks of semiflexible F-actin, as well as low volume fraction gels of crosslinked polyacrylamide (PAAm), a flexible polymer. We observe the thermal fluctuations of micrometer-sized particles embedded in soft gels (Fig. 1). Laser interferometry in a light microscope provides high resolution (less than 1 nm) and bandwidth (from 0.1 Hz to 20 kHz). Using dispersion relations from linear response theory, the frequency-dependent loss and storage shear moduli can be determined from the fluctuation power spectra. The technique is in principle less invasive than active methods, in that no strain at all is imposed on the material.
Diffusing wave spectroscopy (DWS) has been used by others to observe thermal fluctuations of ensembles of particles, and viscoelastic properties in polymers and colloids have thereby been deduced$^{28,29}$. This method measures average viscoelastic properties, as opposed to our local measurements; we further compare this conceptually related method with ours in the Discussion.
Our method measures shear moduli over a larger frequency range than accessible to video based microrheology (while DWS and some other macroscopic methods can reach higher frequencies$^{1,30}$). Besides providing a new way of passively measuring shear moduli in both synthetic and biological polymer systems, our technique can be sensitive to dynamics and material parameters that are inaccessible to macroscopic methods: spatial inhomogeneities in networks can be studied on a micrometer scale. At low frequencies, our method appears to be sensitive to dynamics, such as non-shear deformations of the network with respect to the solvent, that are not seen in macroscopic mechanical rheology. For probe sizes comparable to the mesh size of the polymer network, deviations from continuum elasticity become apparent. With decreasing probe size, the transition from collective network dynamics to single-filament dynamics can be traced to explore the microscopic basis for macroscopic properties. This is particularly relevant for biopolymer networks, with mesh sizes as large as microns. More importantly for biology, microrheology will permit the characterization of small samples, such as living cells.
Theory and Data Analysis Methods$^{31}$ {#theory-and-data-analysis-methods31 .unnumbered}
---------------------------------------
We model our experiments as embedding a spherical bead, of radius $R$, in a linear viscoelastic medium of density $\rho$ with a macroscopic complex shear modulus $G^*$ and Poisson ratio $\nu$.$^{32}$ For motions of micron-sized probe particles, inertial effects are negligible at frequencies $\omega$ where the inertial decay length (i.e., the inverse magnitude $1/|\Gamma|$ of the complex propagation constant$^{1}$ $\Gamma=\left(\rho\omega^2/2G^*\right)^{1/2}$) is larger in magnitude than $R$. At 1 kHz, $1/|\Gamma|$ is at least tens of microns in all our samples. Correspondingly, inertial effects can be expected to be relevant only for frequencies of order 1 MHz. In addition, the Reynolds number for thermal motions on this scale is small: $\sim10^{-3}$. In an incompressible medium, bead motion is then determined entirely by the frequency-dependent complex shear modulus $G^*(\omega)=G'(\omega)+iG''(\omega)$, where $G'$ and $G''$ are the storage and loss moduli.
In extremely low-frequency motions of a gel, the network deforms so slowly that stress in the solvent relaxes completely, and the network can undergo not only shear but compressional deformation, i.e. it behaves as an elastic continuum with shear modulus $G\equiv
G^*(\omega=0)$ and Poisson ratio $\nu<1/2$. Stress $\sigma_{ij}$ is related to strain $u_{ij}$ by$^{32}$ $$\sigma_{ij} = 2G \left[ u_{ij}+\nu\delta_{ij}
\Sigma u_{kk} / (1\!-\!2\nu) \right].
$$ The equation of elastic equilibrium, $\partial\sigma_{ij}/\partial x_i
=0$, can be solved exactly for $u_{ij}$ with no-slip boundary conditions on a rigid spherical surface,$^{33-35}$ yielding an effective compliance (reciprocal spring constant) for sphere displacement
$$\alpha(\omega=0)=\frac{1}{6\pi GR}\left[1+\frac{(\nu-1/2)}{2(\nu-1)}\right].
$$
The static compliance $\alpha$ in the limit $\omega\rightarrow 0$ is significant because it is related by equipartition of energy to the total mean square fluctuations of the probe particle in the gel: $$\langle x^2\rangle=kT\alpha(\omega=0).
$$ Eqs. (3) and (2) together provide an estimate of the static modulus $G$ from the variance $\langle x^2\rangle$ of fluctuations: a priori, $\nu$ is unknown in Eq. (2) but over the relevant range, $0<\nu\le 1/2$, $\nu$ does not strongly affect the result; one can put $\alpha\approx 1/6\pi
GR$ with at most 25% error. However, $\langle x^2\rangle$ is often dominated by low-frequency instrumental drifts. We will instead use a method (to be described below) of estimating $G^*(\omega=0)$ that is less sensitive to such noise.
For frequencies above the $\omega\rightarrow 0$ limit, viscous stresses develop in the gel, and we must reconsider Eq. (2). The solvent and the polymer network are coupled through viscous drag, which becomes stronger with increasing frequency. Eventually, solvent and network are expected to move as one at scales large compared with the network mesh size $\xi$. In this case the polymer network is strongly coupled to the incompressible solvent and so behaves like an incompressible network, which can be described by a Poisson ratio $\nu=1/2$. The crossover frequency, above which incompressible behavior is observed, can be estimated as follows. The viscous force per volume exerted by the solvent on the network$^{36,37}$ is $\sim\eta v/\xi^2$, where $v$ is the velocity of the solvent relative to the network. The local elastic force per volume (exerted by the rest of the network) is $G \nabla^2 u \sim
u/R^2$ at the bead surface, where $u$ is the network displacement field. Viscous coupling will dominate above a crossover frequency $$\omega_c\simeq\frac{G}{\eta}\,\frac{\xi^2}{R^2}.
$$ Eq. (4) is an order-of-magnitude estimate of $\omega_c$ that corresponds to a few Hertz in our F-actin solutions ($G\simeq 1~{\rm Pa}$, $\xi\simeq 0.1R$, $\eta=1~{\rm cP}$). Above $\omega_c$, provided that $\xi$ is small compared with $R$, the bead moves as an inclusion in an incompressible continuum viscoelastic medium. The stress-strain relations in Eq. (1) still hold, but with $\nu=1/2$. The spatial equation of motion is still $\partial\sigma_{ij}/\partial x_i =0$ (at a given $\omega$) and its solution is Eq. (2) with $\nu=1/2$: $$x_{\omega}=\alpha^*(\omega)f_{\omega}=\frac{1}{6\pi G^*(\omega) R}f_{\omega}.
$$ $\alpha^*(\omega)$ and $G^*(\omega)$ are complex because this formula describes sinusoidal sphere displacements $x(t)=\alpha^*(\omega)f(t)$ caused by a force $f(t)\propto e^{-i\omega t}$. This is a generalization of the well-known Stokes formula $f=6\pi\eta r \dot x$, since $\eta=iG^*(\omega)/\omega$ in a purely viscous fluid.
We can rigorously justify Eq. (5) directly, as follows. It applies to an incompressible medium without inertia. The dynamics of the medium are completely determined by its stress-strain relation (Eq. (1)), which becomes $\sigma_{ij} = 2G^*(\omega) u_{ij}$. This is mathematically the same law that describes a simple viscous fluid, but with the (complex and frequency-dependent) quantity $iG^*(\omega)/\omega$ replacing the viscosity $\eta$. Together with the equation of motion, $\partial
\sigma_{ij}/\partial x_i = 0$, the spatial equations to be solved are also mathematically identical. The unique solution to these equations at a frequency $\omega$, that satisfies no-slip boundary conditions on the surface of the moving sphere, is precisely the Stokes law, with the (complex and frequency-dependent) $G^*(\omega)$ replacing the quantity $-i\omega\eta$. This has the simple and remarkable consequence that the well-known Stokes force generalizes to give the correct compliance, involving the complex $G^*(\omega)$.
For $\omega$ between zero and $\omega_c$, the solvent and network may undergo significant relative motion. As a result of this partial “draining” of the network, Eq. (5) will not be valid; the situation must then be described by a “two-fluid” model$^{36,37}$. Force response and fluctuations may become dominated by long-wavelength modes that relax over distances large compared with $\xi(G/\eta\omega)^{1/2}$. This low-frequency draining allows for relative motion of network and solvent, involving not only shear modes but also compressional modes in the network. As the spatial extent of the modes becomes larger than the bead size, the bead motion and its power spectrum are expected to become independent of bead size.
The compliance $\alpha^*(\omega)=\alpha'(\omega)+i\alpha''(\omega)$ is the complex response function for bead displacement. Thus, for frequencies above $\omega_c$, Eq. (5) and the fluctuation-dissipation theorem$^{38}$ relate $\langle x^2_\omega\rangle$, the power spectral density (PSD) of thermal motion, to the imaginary part $\alpha''(\omega)$: $$\langle x^2_\omega\rangle={4kT\alpha''(\omega)\over\omega}.
$$ Provided that $\alpha''(\omega)$ is known over a large enough frequency range, one can recover the real compliance $\alpha'(\omega)$ from the Kramers-Kronig relations$^{38}$ by evaluating a dispersion integral $$\begin{aligned}
\alpha'(\omega)&=&
\frac{2}{\pi}P\int_0^\infty d\zeta\,\frac{\zeta\alpha''(\zeta)}{\zeta^2-
\omega^2}
\\
&=&\frac{2}{\pi}\int_0^\infty dt\,\cos\omega t\int_0^\infty
d\zeta\,\alpha''(\zeta)\sin(\zeta t).
$$ This allows us to explicitly calculate (by Eqs. (5) and (6)) the storage and loss moduli $G'(\omega)$ and $G''(\omega)$. $P$ in Eq. (7) denotes a principal-value integral, meaning the $\epsilon\rightarrow 0$ limit of the sum of two $\zeta$-integrals: from $0$ to $(\omega-\epsilon)$, and from $(\omega+\epsilon)$ to $\infty$. However, as written in Eq. (8), this is equivalent to successive sine and cosine transforms of $\alpha''(\omega)$, which are conveniently performed for the discrete data points of a long time series. Standard Fourier routines$^{39}$ can be used to evaluate the discrete PSD and then to perform the sine and cosine transforms. Either the PSD or the output of Eq. (8) can be smoothed (by averaging within bins of equal logarithmic spacing) without strongly affecting the numerical results. As a control, we have also calculated Eq. (7) directly in the logarithmic domain, finding similar results.
When the behavior of the network plus solvent is well described as a single-component medium (so that Eq. (5) holds), we can take the complex reciprocal of $\alpha^*(\omega)$ to obtain the complex $G^*(\omega)$: $$\begin{aligned}
G'(\omega)&=&\frac{1}{6\pi
R}\;\frac{\alpha'(\omega)}{\alpha'(\omega)^2+\alpha''(\omega)^2}\;,
\\
G''(\omega)&=&\frac{1}{6\pi
R}\;\frac{-\alpha''(\omega)}{\alpha'(\omega)^2+\alpha''(\omega)^2}\;.
$$ Qualitatively, we expect the shear modulus $G^*(\omega)$ to exhibit a characteristic form,$^{4}$ involving up to three distinct dynamical regimes. If the polymer network is entangled but not crosslinked, as in F-actin solutions, viscous flow will occur over times longer than the reptation time $\tau_r$, with $G'(\omega)$ going to zero. The reptation time is hours or days, for actin filaments tens of microns long,$^{40}$ and is thus not accessible to our experiments. For frequencies above $1/\tau_r$, a rubber-like plateau appears, with a frequency-independent elastic response $G'$, as in a crosslinked gel. Above the high-frequency end of the plateau, the moduli are expected to increase with a characteristic power of frequency, $G'$ and $G''\sim\omega^z$, reflecting the increasingly limited relaxations of dynamic modes within a mesh of the network$^{4}$. Power law behavior is expected for the shear moduli of any polymer network above the characteristic mesh relaxation time and below the molecular high frequency cut-off, since there is no other characteristic length or time scale in this regime. In this scaling regime, from Eqs. (5) and (6), the PSD is then expected to follow $$\langle x_\omega^2\rangle\sim\omega^{-(1+z)}.
$$
Materials and Experimental Methods {#materials-and-experimental-methods .unnumbered}
----------------------------------
Actin was purified ($\sim$95% purity) from chicken skeletal muscle, following standard recipes$^{41}$. Its concentration was determined both by staining (BioRad) and by UV absorption at 290 nm (specific absorption: 0.65 cm$^2$/mg). Monomeric actin (G-actin) was rapidly frozen in liquid nitrogen and stored at $-85^\circ$C. Samples were prepared by mixing G-actin with a small number of silica beads (Bangs Laboratories, except for the 0.5 $\mu$m diameter beads, which were kindly provided by E. Matijevic). After adding concentrated polymerization buffer (F-buffer: 2 mM hydroxyethyl-piperazineethanesulfonic acid (HEPES) (pH 7.2), 2 mM MgCl$_2$, 50 mM KCl, 1 mM ethylenebis(oxyethylenenitrilo)tetraacetic acid (EGTA), 1 mM adenosinetriphosphate (ATP)) the mixture was immediately transferred into a sample chamber made from a microscope slide, a cover slip and double-stick tape (with inner dimensions 15mm$\times$3mm$\times$70$\mu$m). In the sealed sample chamber the actin polymerized at room temperature for at least one hour under slow rotation. The samples were stored at 4$^\circ$C and examined within one day (and in one case remeasured the following day, as a control).
Polyacrylamide (PAAm) gels were prepared according to a standard gel electrophoresis recipe,$^{42}$ with concentrations of 2, 2.5, 3 and 5% (weight/volume) and a relative concentration of 3% bis-acrylamide as a crosslinker. Solutions were thoroughly degassed under vacuum, beads added and polymerization initiated with tetramethylethylenediamine (TEMED) and ammonium persulfate (APS). As with actin, polymerizing solutions were then transferred into sample chambers (15mm$\times$6mm$\times$140$\mu$m) and slowly rotated at room temperature for at least one hour before starting experiments.
=
=
Our microscope is a custom-built inverted instrument (optics from Carl Zeiss, Inc.), constructed on an optical rail system and mounted on a vibration isolated optical bench (Fig. 2). To detect the thermal motion of beads imbedded in the gel we used an interferometer$^{43,44}$ (see Fig. 3) with near-infrared laser illumination (1064 nm NdVO$_4$, 3.4 W (cw) max. power, Topaz 106c, Spectra Physics).
A linearly polarized laser beam is split in two beams by the Wollaston prism below the objective, which produces two diffraction limited overlapping foci in the specimen plane. A refractive particle located asymmetrically within the two foci will cause slightly elliptical polarization after recombination of the two beams by the upper Wollaston prism. A quarter-wave plate renders the light close to circularly polarized, whose two perpendicular linear components are then detected by two photodiodes. Deviations from circularity are calibrated to measure particle displacements. The normalized difference between the two signals is calculated by custom-built analog electronics (Fig. 3), amplified and anti-alias filtered slightly below the Nyquist frequency. This analog signal is then digitized and recorded using an A/D interface (MIO 16X, National Instruments) and data acquisition software written in Labview (National Instruments).
Although this is not done here, the focused laser beam can act, at high enough laser power, as an optical trap, exerting forces on the particle in the focus$^{44,45}$. Here, we want to measure unperturbed thermal fluctuations, and thus need to minimize the trapping force. For that purpose, a 1.5 mm diameter pinhole was inserted at a position conjugate to the back focal plane of the objective, broadening the laser focus and increasing the detector range. The laser power was typically 0.6 mW in the specimen, low enough to make optical forces negligible.
Before recording fluctuations of an individual bead, it was centered in the detector range using a piezo-actuated translation stage. For all experiments reported here, the data acquisition rates were 50 or 60 kHz (anti-alias filtered at 25 kHz). Time series data were recorded for at least 17.5 seconds ($2^{20}$ data points) to obtain power spectra ranging from 0.1 Hz to 25 kHz. Each spectrum was evaluated by Fast Fourier Transform (FFT) after applying a Welch window. Power spectra were smoothed by averaging within bins of geometrically increasing width (with factor 1.1).
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The linear range of the detector is about 200 nm for 0.5 $\mu$m beads and increases with bead size. Each observed bead was immediately calibrated following the recording of thermal motion, by moving the sample on the piezo-actuated stage with a triangular signal wave form, producing constant velocity displacements (between 150 nm and 1 $\mu$m peak-to-peak) at a frequency of 0.1 Hz. The driving voltage was digitally synthesized to correct for the non-linearity and hysteresis of the piezo actuators. Driver voltage (not shown) and detector response (Fig. 4 inset) were recorded for about 40 seconds and analyzed by plotting detector response versus bead displacement. Slopes were estimated by linear regression from an averaged response-displacement curve for each individual bead (Fig. 4A and B). The result is a sensitivity factor which is used to convert detector signals into actual displacements.
=
While centering the bead in the detector range, we found that the sensitivity does not vary strongly in the perpendicular direction (Y), out to Y-offsets comparable to the linear range.The sensitivity also did not vary significantly with axial position in a range for which the bead appeared focused in the video image ($\sim 0.6~\mu$m). Detector sensitivities as a function of bead diameter are plotted in Fig. 5. They increase roughly with the third power of bead radius as long as the bead is smaller than the laser focus, since the scattering amplitude is expected to scale with the volume of the scatterer. For larger beads (in the limit of geometrical optics), the sensitivity decreases roughly as $R^{-1}$, the only remaining length scale. In other words, the same relative displacement $\Delta x/R$ should give the same signal $\Delta V$, independent of bead size: $\Delta V\propto \Delta x/R$.
Investigating the origin of scatter in the detector sensitivity, we found that variations in beam alignment dominated bead polydispersity for the actin experiments. Bead polydispersity was checked directly with transmission electron microscopy (TEM; data not shown) and found to be about 10%, with a non-normal distribution. The images showed both larger beads, which appear to have formed by merging nuclei, and very small beads that may be contamination. The scatter in the displacement-response slopes was about $\pm$10% (coefficient of variation) for actin. When no correlation was evident between calibration factors and the amplitudes of the PSD at around 100 Hz (which should scale with bead radius $R$) we averaged the factor over all beads observed in a sample.
For PAAm the scatter in sensitivities for a given bead size appeared significantly larger ($\pm$30-60% coefficient of variation), and response factors were indeed correlated with the PSD, indicating that at least part of the scatter was caused by bead size polydispersity. In these cases, spectra were calibrated by the factors determined for individual beads, which decreased the variance in the spectra between beads in the same sample. Since the response factor scales roughly with the third power of bead diameter for the smaller beads (see Fig. 5), a 10% coefficient of variance can explain a 30-40% scatter in the spectra. The polymer network itself and possible surface adsorption is not expected to influence apparent bead size and thus sensitivity substantially. Furthermore, the index of refraction changes little with polymer volume fraction in PAAm (measured to be about a 1.2% increase in relative index of refraction at 10% polymer w/v). Finally, surface chemistry that could produce a dense layer of polymer directly on the bead surface is not expected to produce more than a monolayer, and is thus limited to a thickness on the order of tens of [Å]{}.
Results {#results .unnumbered}
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### Actin solutions {#actin-solutions .unnumbered}
Fluctuation time series were recorded for silica beads of 0.5, 0.9, 2.1, and 5.0 $\mu$m diameter, embedded in actin gels of 1 and 2 mg/ml concentrations. The fluctuation signal is well above the noise, as shown by comparison to spectra of a bead immobilized on a surface, and of the water background without bead (Fig. 6). These controls display the effects of low-frequency instrumental noise due to thermal drifts in the microscope and the laser beam path, acoustic vibrations, and beam-pointing fluctuations. At high frequencies, noise depends on laser intensity since detection shot noise and preamplifier noise are dominant there. We emphasize that the amplitude of this noise is small, and remains about two orders of magnitude below the signal. Note that all spectra are (analog) anti-alias filtered at 25 kHz, causing the “tails” at about 20 kHz.
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The power spectra of bead fluctuations in actin solutions were reproducible in shape and amplitude between different beads in the same sample (Fig. 7A), as well as between samples (Fig. 7B). Observed coefficients of variance (of the PSD at 100 Hz in Fig. 7A) within a sample (N=6) were 22% (0.5 $\mu$m), 24% (0.9 $\mu$m), 8.7% (2.1 $\mu$m) and 15% (5.0 $\mu$m bead diameter). Fig. 7B shows two spectra from samples in different experiments but from the same protein preparation, and one made with actin from a different laboratory (sample provided by Paul Janmey). We find that beads in different samples do not scatter more than different beads within one sample.
Slow changes have been reported to occur in actin solutions over many hours after polymerization, possibly due to the depletion of ATP in the buffer, or to the annealing of filaments. To address such effects, we remeasured samples the day following their preparation, without seeing significant differences in the spectra (Fig. 7C). Observed beads were typically located at heights between 10 and 50 $\mu$m above the sample chamber surface. Variations in the spectra were not correlated with distance from the boundaries (data not shown).
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=
Power spectral densities (PSDs) of thermal motions of 0.9 and 5.0 $\mu$m silica beads in actin solutions at different concentrations are shown in Fig. 7D. The shape of 1 and 2 mg/ml spectra is similar and the PSD amplitude of the less concentrated sample is smaller, as expected.
A phase transition to a nematic phase has been reported for actin at concentrations of about 2 mg/ml$^{40,46}$. However, we see no qualitative difference in the appearance or in the behavior of our actin gels at 1 and 2 mg/ml. There were no signs of inhomogeneities under polarization microscopy. Furthermore, we do not see any greatly increased scatter in our measured values of $G^*$, which would be expected in a highly anisotropic medium, given our uniaxial detection technique.
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Fig. 8A shows averaged and smoothed spectra obtained from 2 mg/ml actin solutions with beads ranging from 0.5 to 5.0 $\mu$m in diameter. The variance in the smoothed spectra (values at 100 Hz) as a function of bead diameter is plotted in the inset. Part of the scatter presumably reflects local inhomogeneities in the networks. Smaller beads, which probe more locally, should show more scatter. For 0.5 $\mu$m beads, however, some variation is also due to drift out of the linear range of the detector, and to bead polydispersity, as discussed in Materials and Experimental Methods. Detector sensitivity depends on bead size, as we have shown in Fig. 5. The high frequency noise level follows this dependence, with the smallest (0.5 $\mu$m) and largest (5.0 $\mu$m) beads showing the largest relative noise levels.
The largest beads are about a factor of 10 to 20 larger than the average mesh size (0.25 $\mu$m for actin at 2 mg/ml$^{47}$) and a continuum elastic model should apply. Approximate power law behavior, with a slope of about $-1.75$, is evident over about three decades in frequency for the 5 $\mu$m beads, as indicated in Fig. 8. All spectra show a slight downturn at about 3 kHz, an effect which we cannot yet explain. At the high frequency end, the data have not reached the noise bottom evident in the spectrum of a bead fixed to the substrate surface (Fig. 6).
Eq. (5) predicts that spectra should scale with the bead radius $R$ for high frequencies. This is roughly observed as shown in Fig. 8B, but a systematic deviation from $R$-scaling is evident. Small beads show power spectral densities of higher amplitude than expected. We believe that the increased fluctuations are not due to diffusion through the network or constraint release due to filament reptation; in video recordings of up to two hours we did not observe any long-range diffusive motion. Furthermore, we can exclude that the beads themselves bind to F-actin (with potential bunching of filaments as a result), because we observed practically unimpeded diffusion of silica beads (0.2 $\mu$m) with diameters below the mesh size.
We instead attribute the lack of scaling to the steric exclusion of filaments near the bead. This model is elaborated quantitatively in a separate publication$^{34}$ but given qualitatively in the following. A large bead of radius $R$ will affect the distribution of filaments to an exclusion depth $\ell\sim\ell_p$, where $\ell_p$ is the filament persistence length. Since the Stokes flow field around a large bead extends beyond the exclusion depth to a distance on the order of $R$, large beads will mainly see the unperturbed network according to Eq. (5). Small beads ($R\ll \ell_p$) will exclude filaments to a depth of the order of $R$, and thus see an effective compliance larger than that described by Eq. (5). Once in this limit, the compliance should again scale as $R^{-1}$. This is consistent with our observations: the spectra for the smaller beads (0.5 and 0.9 $\mu$m) do indeed superimpose after scaling (Fig. 8B).
At low frequencies the slopes of the spectra decrease, and the PSDs converge for all beads (Fig. 8A). The onset of the slope change, however, [*does*]{} depend on bead size. Therefore, for the actin samples it can [*not*]{} be the onset of the elastic plateau as is the case for the PAAm data shown below. Instead, we interpret the change in slope as the transition, predicted by Eq. (4), from a regime of pure shear fluctuations to long-wavelength draining modes, for which the spectra should become bead-size independent$^{34}$. Such a transition is not seen in PAAm because the smaller mesh size inhibits draining. The estimate for the crossover frequency for our F-actin solutions—on the order of 10 Hz—is also consistent with the data.
Fig. 9A-D show the storage and loss shear moduli $G'(\omega)$ and $G''(\omega)$ obtained for various bead sizes in 2 mg/ml actin solutions by evaluating the Kramers-Kronig integral, Eq. (7). The continuum model should apply to the largest beads (Fig. 9D) which are about 20 times larger than the mesh size. There, scaling is observed directly in both $G'$ and $G''$ in the range from about 5 to 100 Hz. Pure scaling corresponds theoretically (via Eqs. (5) and (7)) to a complex modulus $G^*(\omega)\propto (i\omega)^{z}$, so that their ratio is $G''/G'=\tan
\pi z/2$$^{4}$. This ratio, averaged between 10 and 100 Hz for the largest beads (Fig. 9D), allows a precise estimate of $z=0.75\pm0.02$.
=
Eqs. (5) and (6) imply scaling of the power spectrum, via Eq. (11), as $\langle x^2_\omega\rangle\propto\omega^{-1.75}$, consistent with the apparent slope in (Fig. 8A). The scaling exponent $z$ of about 3/4 in the macroscopic behavior is distinct from the predictions of Rouse-like scaling with exponent 1/2 observed for flexible polymers (see PAAm below). We are unaware of any existing model that predicts $z=3/4$ for the scaling of the complex, macroscopic shear modulus.
We also note that no plateau is (yet) visible in $G'(\omega)$. To compare with the literature, we quote values at the lowest frequencies we observed. $G'$ at 0.1 Hz was $0.11\pm0.03$ Pa for 1 mg/ml and $0.17\pm0.07$ Pa for 2 mg/ml (actin). Ruddies [*et al.*]{}$^{15}$ measured a value of $G'\approx 0.3~Pa$ near the observed onset of the plateau in 0.3 mg/ml actin.
For smaller beads, the apparent $G'(\omega)$ and $G''(\omega)$, calculated under the assumptions of a single fluid behavior, show deviations from macroscopic shear elastic behavior (Fig. 9ABC), and a pure scaling regime is not reached in the data. This confirms that the assumption of an incompressible network is not valid over the whole frequency-range of the spectra for the smaller beads.
### Polyacrylamide gels {#polyacrylamide-gels .unnumbered}
Polyacrylamide gels, even at volume fractions close to the gelation threshold, are so rigid that we approach the limits of our technique. Given the noise floor illustrated in Fig. 6, our technique is limited at present to static shear moduli of up to a few hundred Pascal. A volume fraction of 2% (with relative crosslinker concentration of 3%) was the lowest concentration that resulted in reproducibly solidified gels. Fig. 10 shows PSDs of 0.9 $\mu$m beads embedded in 2, 2.5, 3, and 5% (volume fraction) PAAm gels, linearly averaged over several beads respectively, and log-binned as described in Materials and Experimental Methods. Absence of long-range diffusion of the beads confirms that all gels were crosslinked into solids. Since the mesh size in the PAAm gels (about 5 nm in the 2% gels$^{48}$) is in all cases much smaller than the bead size, a continuum elastic model is expected to apply throughout.
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PSDs for beads in the softest gels (2%) show a power law slope of about $-1.5$ near 100 Hz, which is distinctly less steep than in F-actin. We again observe the slight but unexpected downturn in the spectra above about 3 kHz. The flattening of the spectra below a few Hertz reflects the elastic plateau (compare Fig. 11A). This plateau extends higher for stiffer gels. For concentrations above 2.5%, instrumental noise begins to dominate the low-frequency part of the spectra (and approaches the fixed-bead spectrum, see Fig. 6). Raising the PAAm concentration higher above the gelation threshold leads to dramatically increased gel rigidity, making the observation of fluctuation signals above the noise difficult. After calibrating the spectra with the sensitivity factors determined for each bead (see Fig. 5), the scatter in the spectra typically remained large, unlike for actin. The inset in Fig. 13 shows PSDs at 1 Hz for different bead sizes in 2.5% gels, indicating this large scatter.
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The 2 and 2.5% gels are not much above the gelation threshold and the gel rigidity there changes very rapidly with concentration. Therefore, the observed scatter is presumably partly due to local gel inhomogeneities. Bead polydispersity, as discussed in Materials and Experimental Methods, is a further contribution. As for actin, we find no correlation with distance from the substrate surface. The calculated frequency-dependent storage moduli for the data in Fig. 10 are shown in Fig. 11A. The transformation Eq. (7) has largely eliminated unphysical low-frequency noise in these spectra, allowing the plateau modulus $G'(\omega\rightarrow0)$ to be easily estimated. For 2, 2.5, 3, and 5% gels the plateau values of $G'$ are approximately 2.0, 24, 100, and 280 Pa.
The effective scaling of $G'$ with concentration is 1.8 (for concentrations between 3 and 5%), and steeper for lower concentrations. Our 2.5% gels thus lie in a regime of very rapidly increasing stiffness indeed (this is consistent with Ref. 49). For larger plateau moduli the upturn shifts, as expected, to higher frequencies. The pure scaling regime beyond the plateau is not reached for any of the samples, although its existence is suggested by the slope of $-1.5$ in the PSDs. The corresponding scaling of $G'(\omega)\propto\omega^{1/2}$ is consistent with the Rouse model$^{4}$.
Fig. 11B shows the results of control experiments performed with conventional cone-and-plate rheology (courtesy of M. Osterfield, J. Shah, P. Janmey). Samples were prepared following the same recipes as for the microrheological experiments. Plateau values agree within a factor of 2-3. The data show qualitatively the same steep dependence on concentration as the microrheological results. Elastic plateaus are clearly visible for 2.5 and 3% gels whereas for the 2% gels instrumental limits are reached, as seen from the large scatter of the data.
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Fig. 12A shows averaged spectra for different bead sizes (between 0.5 and 2.1 $\mu$m) in 2.5% PAAm gels; in Fig. 12B these power spectra are multiplied by the respective bead diameters. The $R^{-1}$-dependence in Eq. (5) predicts that the rescaled curves have the same shape and magnitude. This is approximately true for the central frequencies in the spectra, but the high rigidity of the PAAm gels causes non-displacement noise to be significant both at high frequencies (electronic detection noise) and low frequencies (thermal drifts, laser beam-pointing and mode instabilities, etc.). The parts of the spectra that are dominated by noise show no scaling. Due to the bead-size dependence of the detector sensitivity, the low-frequency noise extends higher for smaller beads.
The storage moduli $G'(\omega)$ for three different bead sizes 2.5% gels are shown in Fig. 13(inset). In contrast to what was found for actin, the elastic plateau is clearly visible for all samples, and agrees for different bead sizes in the same concentration to within about a factor of 2. The elastic plateau extends to about the same frequency for all bead sizes. This is in contrast to the low-frequency slope changes in actin, which we interpret as finite-mesh-size effects, supported by a consistent bead-size dependence. With mesh sizes as small as in PAAm, draining effects should not play a role, down to the lowest frequencies we measure. However, the trend in the data is to show apparently smaller moduli for smaller beads. We attribute this artifact to a remaining sensitivity to noise in the integration procedure used to calculate $G'$. The detector sensitivity decreases dramatically for smaller beads, thus including more noise in the integral, leading to the underestimate of $G'$. The plateau moduli estimated by the value of $G'$ at 1 Hz (Fig. 13) are 36 Pa (2.1 $\mu$m), 24 Pa (0.9 $\mu$m), and 17 Pa (0.5 $\mu$m bead diameter).
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Discussion {#discussion .unnumbered}
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We have introduced the direct use of the Kramers-Kronig dispersion integral Eq. (7) to obtain storage and loss moduli from power spectra of thermal motions. Other methods have been used to obtain shear moduli from the thermal mean-square particle displacement as a function of time: Mason and Weitz give an approximate method based on real-valued Laplace transforms$^{28}$; Mason and Wirtz describe a somewhat different approximation$^{29,50}$. A limitation of the former method is that a specific functional form was postulated, fitted to a numerical Laplace transform of particle displacement data, and then analytically continued to the Fourier domain. Possible systematic errors in such a scheme are not known. Such transformation methods have previously been shown to be sensitive to noise and to behavior at the frequency extremes$^{51}$. In contrast with Ref. 28, we have applied the transforms described in Eqs. (5) and (8) directly to our measured spectra. Both of these transform methods are sensitive to the frequency extremes, but we believe that the direct transformations we have employed are better controlled.
In order to demonstrate the dependence of our Kramers-Kronig transformation on the limits of the integration interval, we show in Fig. 14 the transformation of a model power spectrum that decreases as a power of frequency, above a corner frequency that we arbitrarily place at 0.1 Hz. We choose the power law to be $\omega^{-1.75}$, similar to the power law observed in our fluctuation power spectra for actin solutions. This model spectrum corresponds, theoretically, to a complex modulus $G^*$ that increases as $G^* \propto
\omega^{0.75}$ above the corner frequency, and a plateau in the storage modulus $G'$ below the corner frequency. We show in Fig. 14 the results of a transformation carried out with various values of the upper and lower cutoffs. The effect of cutoffs is minimal in the central portion of the transforms.
=
The shear elastic storage and loss moduli ($G'$ and $G''$) we found for actin solutions are consistent with other experiments$^{26,27}$ in the frequency regime where they can be compared. Other reported techniques have been limited in frequency to a maximum of about 10 Hz. Discrepancies persist in the literature,$^{16}$ and values up to 300 Pa have been measured for F-actin at a concentration of 2 mg/ml. We did not observe changes in viscoelastic behavior in F-actin over periods of at least 24 hours after polymerization. This suggests that the local properties of the solutions probed by the micron-sized beads are not affected by slow network changes. On the other hand, macroscopic techniques which have observed such changes may not have probed the linear response regime.
Actin microrheology by others$^{26}$ could be interpreted as consistent with $\omega^{3/4}$ scaling, although the authors there suggest $\omega^{1/2}$ scaling; the frequencies observed were close to the onset of the plateau. Macrorheological work$^{15,19}$ has also reported $G^*\sim\omega^{1/2}$. Here too, the frequencies studied may have been in the transition between plateau and scaling regime. New results from multiple light scattering experiments (D. Weitz, personal communication) are consistent with a scaling exponent close to $3/4$ at even higher frequencies than we are able to measure. This scaling behavior in semiflexible actin solutions may be a universal property of this type of network, but is so far unexplained. In particular, there is no reason to expect Rouse or Zimm scaling in these systems$^{4}$. Bead dynamics in actin networks consistent with the power spectra we observe have also been reported in the limit of very dilute gels (specifically, for $R\simeq\xi$)$^{25}$; the authors suggest a model based on single filament dynamics. We believe that at the (higher) concentrations we used, for which the mesh size is substantially smaller than bead diameters, our continuum elastic approach is correct.
For solutions of semiflexible polymers, where effective entanglement lengths may be much larger than the mesh size,$^{52,53}$ crosslinking is expected to have a very strong effect on the gel rigidity. In cells, the actin cortex is extensively crosslinked while the average filament length is short. Therefore, a rigidity relevant for cell biology has to be measured with a crosslinked actin gel, potentially lessening the discrepancies between reported measurements.
In both F-actin solutions and PAAm gels, one might expect to be able to resolve local inhomogeneities in viscoelastic parameters with increasingly smaller beads. We did see considerable scatter in the fluctuation power spectra, but this spread did not increase for smaller beads, as one might have expected. Therefore the inhomogeneities in the systems we studied may be smaller than effects of bead polydispersity and other instrumental errors. More extensive statistical characterization of the probe particles and improvement of the instrument will reduce the noise, as well as increase the available frequency range. These impovements will allow closer examination of sample inhomogeneities.
Our PAAm gels were very close to the gelation threshold and therefore expected to be inhomogeneous. For gels of 2, 2.5, 3, and 5% we indeed find a very steep increase in $G'$ (Fig. 15). This qualitative behavior and the absolute values are consistent with our control experiments with cone-and-plate rheology and the data match literature values in the 3-5% range$^{49}$. Macroscopic methods tend to get inaccurate for low shear moduli on the order of 1 Pa, whereas our microscopic method underestimates shear moduli above 100 Pa (stiff gels) due to noise in the bead position detection.
=
In PAAm, the scaling of $G^*(\omega)\propto\omega^{1/2}$ is reflected by an apparent power law slope of $-1.5$ in the spectra, consistent with the Rouse model$^{4}$.
A current limitation of our method, so far unexplained, is a steepening of the power spectrum that we consistently observe above a few kHz. We believe this is an artifact, because it occurs even for beads in pure water. We have ruled out electronic filtering as a cause. This effect translates, in the Kramers-Kronig transformation of Eq. (7), to a sharp plunge in the calculated storage modulus at high frequency, which is of course unphysical. Again, this illustrates the subtle nature of the frequency sensitivity of the Kramers-Kronig and related transformations. Although the downturn in our spectra occurs in the vicinity of 10 kHz, its effects are apparent in $G'(\omega)$ approximately one decade below this.
Conclusions {#conclusions .unnumbered}
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We have used thermal fluctuations of micron-sized probes to measure local properties of polymer solutions, by a passive and thus non-invasive method, particularly appropriate for labile biopolymers. Laser interferometry provides dynamic measurements of probe displacements with high resolution and up to frequencies not previously explored for F-actin solutions. With probes that are large compared to the mesh size of the network, macroscopic properties such as the complex shear modulus $G^*(\omega)$ can be estimated from the power spectrum of fluctuations by using dispersion relations from linear response theory. This analysis is made possible by our large detection bandwidth. By sampling micron-sized volumes we can in principle map out spatial inhomogeneities in soft polymer systems.
Probes of sizes on the order of the mesh can be used to observe deviations from macroscopic behavior, for example the transition to microscopic filament dynamics. Observing this breakdown of continuum elasticity may be of fundamental importance for understanding the origins of elasticity in polymer systems, and has been the subject of very recent theoretical attention$^{54-56}$. Furthermore, we have shown that our technique (like other microrheological methods using probe particles in polymer solutions) is predicted to be limited at low frequencies—at least in measuring [*macroscopic*]{} shear moduli. This is because of expected draining dynamics of network-plus-polymer, which is not apparent in macroscopic rheometric methods. On the other hand, inertial effects of both probe and solvent, typically a problem in sensitive macroscopic instruments,$^{1}$ occur only at MHz frequencies in microscopic experiments.
On the micron length scale, a bead in a network should behave somewhat like an organelle suspended in the cytoskeleton of a biological cell, and we expect that the method will be transferable to measurements in living cells, where biological processes are expected to strongly modify viscoelastic parameters. Our technique is currently limited by broadband (mainly acoustic) noise to rather soft materials, up to shear moduli of a few hundred Pascal. This limitation will be extended by technical improvements. Meanwhile, much remains to be done in developing a theoretical understanding of the semiflexible polymer systems.
Acknowledgements {#acknowledgements .unnumbered}
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This work was supported in part by the Whitaker Foundation, the National Science Foundation (Grant Nos. BIR 95-12699 and DMR 92-57544), and by the donors of the Petroleum Research Fund, administered by the ACS. We thank M. Osterfield, J. Shah, and P. Janmey for providing macroscopic rheological data for controls, and P. Janmey for a control batch of actin. We acknowledge generous technical support from the Rowland Institute for Science, particularly by W. Hill. We thank G. Tank and J. Langmore for help with the electron microscopy, and P. Olmsted, P. Janmey, J. Käs, A.C. Maggs, and D. Weitz for helpful discussions. FCM also wishes to thank the Aspen Center for Physics.
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|
---
abstract: 'The acceleration-dependent system with noncommuting coordinates, proposed by Lukierski, Stichel and Zakrzewski \[Ann. Phys. 260, 224 (1997)\] is derived as the non-relativistic limit of Mathisson’s classical electron \[Acta Physica Polonica 6, 218 (1937)\], further discussed by Weyssenhoff and Raabe \[Acta Physica Polonica 9, 7 (1947)\]. The two-parameter centrally extended Galilean symmetry of the model is recovered using elementary methods. The relation to Schrödinger’s Zitternde Elektron is indicated.'
author:
- |
P. A. Horváthy\
Laboratoire de Mathématiques et de Physique Théorique\
Université de Tours\
Parc de Grandmont\
F-37 200 TOURS (France)
title: 'Mathisson’ spinning electron : noncommutative mechanics & exotic Galilean symmetry, 66 years ago\'
---
Acta Physica Polonica [**34**]{}, 2611 (2003) \[`hep-th/0303099`\].
Introduction
============
Non-commutative (quantum) mechanics, where the position coordinates satisfy $$\big\{X_{1}, X_{2}\big\}=\theta,
\label{NCpos}$$ has been at the center of recent interest [@NCQM]. In the plane and in the non-relativistic context, such theories are closely related to the “exotic” Galilean symmetry associated with the two-fold central extension of the planar Galilei group [@exotic]. A model which provides a physical realization of this symmetry has been presented by Lukierski, Stichel and Zakrzewski [@LSZ] who considered the acceleration-dependent Lagrangian $$L=
\frac{m\dot{{{\vec x}}}^2}{2}
+
\frac{\kappa}{2}\,\dot{{{\vec x}}}\times\ddot{{{\vec x}}}.
\label{LSZlag}$$
My aim here is to point out that the model of Lukierski et al. can actually be derived from that published by Mathisson in ‘37 [@Mathisson], and further discussed by Weyssenhoff and Raabe [@WeyRaa]. Not surprisingly, their theory shows interesting analogies also with Schrödinger’s [*Zitternde Elektron*]{} [@Schr].
This Note is dedicated to the memory of these outstanding physicists who, with an extreme courage, tried to continue their scientific activity under those terrible years of World War II.
The Mathisson electron {#Mathisson}
======================
Two years before the outbreak of World War II, Mathisson [@Mathisson] proposed to describe a classical electron with the relativistic equations $$\begin{array}{ll}
m\dot{u}^\alpha+\displaystyle\frac{1}{c^2}S^{\alpha\sigma}
\ddot{u}_{\sigma}=f^\alpha
\\[8pt]
\dot{S}^{\alpha\beta}-
\displaystyle\frac{1}{c^2}S^{\alpha\sigma}
\dot{u}_{\sigma}u^\beta
+
\displaystyle\frac{1}{c^2}S^{\beta\sigma}\dot{u}_{\sigma}u^\alpha
=0
\end{array}
\label{RMeq}$$ where $m$ is the mass, $u^\alpha$ the four-velocity, $f^\alpha$ the force; the dot means differentiation w.r.t. proper time. The antisymmetric tensor $S^{\alpha\beta}$ represents the spin of the electron and is assumed to satisfy the orthogonality condition $$S^{\alpha\beta}u_{\beta}=0.
\label{spinconstr}$$ In the rest frame, the spatial components of $S^{\alpha\beta}$ form therefore a three-vector $\vS$.
In the non-relativistic limit, $\vS$ becomes a constant of the motion. In the absence of external force, the motion is \[apart of free motion along the direction of $\vS$\], in the plane perpendicular to $\vS$ and satisfies the third-order equation $$m\ddot{x}_{i}=-\kappa\epsilon_{ij}\dddot{x_{j}},
\label{Mateq}$$ where the new constant $\kappa$ has been defined by the Jackiw-Nair Ansatz [@JaNa] $$s=\kappa c^2,
\label{JNAnsatz}$$ $s=\vert\vS\vert$ being the length of the spin vector. Eqn. (\[Mateq\]) is precisely the equation of motion put forward by of Lukierski et al. [@LSZ].
From now on we drop the coordinate parallel to $\vS$ and focus our attention to motion in the plane.
Conserved Quantities
====================
The equations of motion (\[Mateq\]) are associated with the Lagrangian (\[LSZlag\]). Then Lukierski et al. derive the conserved quantities associated to the space-time symmetries applying the higher-order version of Noether’s theorem. Let us now reproduce their results using elementary methods.
$\bullet$ An obvious first integral of (\[Mateq\]) is the momentum, $$P_{i}=m\dot{x}_{i}+\kappa\epsilon_{ij}\ddot{x}_{j}.
\label{momentum}$$ Eq. (\[Mateq\]) is in fact $\dot{P}_{i}=0$.
$\bullet$ Multiplying (\[Mateq\]) by the velocity, $\dot{{{\vec x}}}$, yields a total time derivative, where we recognize the conserved [*energy*]{}, $$H=\frac{m\dot{{{\vec x}}}\strut^2}{2}+\kappa\,\dot{{{\vec x}}}\times\ddot{{{\vec x}}}.
\label{energy}$$
$\bullet$ Similarly, taking the vector product of (\[Mateq\]) with ${{\vec x}}$ yields the conserved [*angular momentum*]{}, $$J=m{{\vec x}}\times\dot{{{\vec x}}}+\frac{\kappa}{2}\dot{{{\vec x}}}\strut^2
-\kappa{{\vec x}}\cdot\ddot{{{\vec x}}}.
\label{angmom}$$
$\bullet$ A Galilean boost ${{\vec x}}\to{{\vec x}}+\vb t$ shifts the momentum as $\vP\to\vP+m\vb$. A rest frame where the momentum vanishes can be found, providing us with the conserved [*boost*]{} vector $$K_{i}=mx_{i}-t\big(m\dot{x}_{i}+\kappa\epsilon_{ij}\ddot{x}_{j}\big)
+\kappa\epsilon_{ij}\dot{x}_{j}.
\label{CM}$$
Somewhat surprisingly, one more conserved quantity can be found.
$\bullet$ the vector product of (\[Mateq\]) with the acceleration, $\ddot{{{\vec x}}}$, yields the square of the acceleration, $$I=\frac{\kappa^3}{2m^2}\big(\ddot{{{\vec x}}}\big)^2,
\label{intangmom}$$ where a constant factor has been included for later convenience.
$\bullet$ curiously, multiplying (\[Mateq\]) by $\dddot{{{\vec x}}}$ yields once again the same quantity, namely $(m/2)\big(\ddot{{{\vec x}}}\big)^2=(m/\kappa)^3I$.
The construction of this new quantity reminds one to that of angular momentum and of energy. Its precise origin will be clarified below.
Let us observe that, owing to the conservation of $I$, $\ddot{{{\vec x}}}=0$ can be consistently required. Then the conserved quantities found above reduce to those of an “elementary exotic particle” studied in [@DH].
Zitterbewegung and center-of-mass decomposition {#NRmotion}
===============================================
The equation of motion (\[Mateq\]) is integrated at once. Putting indeed $$Q_{i}=
-\big(\frac{\kappa}{m}\big)^2\epsilon_{ij}\ddot{x}_{j},
\label{intcoord}$$ eqn. (\[Mateq\]) becomes $$\dot{Q}_{i}=\frac{m}{\kappa}\epsilon_{ij}Q_{j},
\label{Qeq}$$ showing that the acceleration rotates uniformly with angular velocity $m/\kappa$. Putting $Q=Q_{1}~+~iQ_{2}$, $Q(t)=Q_{0}e^{-i(m/\kappa)t}$. This is plainly consistent with the conservation of the magnitude of the acceleration, Eq. (\[intangmom\]). Then $$X_{i}=x_{i}+\epsilon_{ij}Q_{j}
\label{CMcoord}$$ moves freely, $$\ddot{X}_{i}=0.
\label{CMmot}$$ In conclusion, the motion has been separated into the free motion of the center of mass coordinate $\vX$, combined with the “Zitterbewegung” \[uniform rotation\] of the internal coordinate $\vQ$.
A key feature of Mathisson’s electron is that the internal variable $\vQ$ measures in fact the extent of how much the momentum, $\vP$, differs from \[$m$-times\] the velocity, $\dot{{{\vec x}}}$, $$\vQ=\frac{\kappa}{m^2}(m\dot{{{\vec x}}}-\vP).
\label{velmom}$$
Re-writing the conserved quantities in terms of the new coordinates confirms the above interpretation. In fact, $$\begin{array}{ccc}
\vP&&=m\dot{\vX}\hfill
\\[3pt]
H&=H_{CM}+H_{int}\hfill&=
\displaystyle\frac{m\dot{\vX}^2}{2}-
\displaystyle\frac{m^3}{2\kappa^2}\vQ^2\hfill
\\[6pt]
J&=J_{CM}+J_{int}\hfill&=
m\vX\times\dot{\vX}+\displaystyle\frac{\kappa}{2}\dot{\vX}^2+
\displaystyle\frac{m^2}{2\kappa}\vQ^2\hfill
\\[8pt]
K_{i}&&=m(X_{i}-\dot{X}_{i}t)+\kappa\epsilon_{ij}X_{j}\hfill
\\[6pt]
I&&=\displaystyle\frac{m^2}{2\kappa}\vQ^2.\hfill
\end{array}
\label{decompCC}$$
Mathisson’s electron is hence a composite system. Note that in (\[decompCC\]) the center of mass behaves precisely as an elementary exotic particle [@DH]; the internal coordinate only contributes to the energy and the angular momentum. In fact, $H_{int}=-\displaystyle\frac{m}{\kappa}I$ and $J_{int}=I$. The new conserved quantity found in (\[intangmom\]) is hence the internal angular momentum and also the internal energy \[which are linked in a $2$-dimensional phase space\].
Let us now observe that the equations of motion (\[Qeq\]-\[CMmot\]) are consistent with the Poisson structure associated with the symplectic form $$\Omega=\Omega_{CM}+\Omega_{int}=
dP_{i}\wedge dX_{i}
+\frac{\kappa}{2m^2}\epsilon_{ij}dP_{i}\wedge dP_{j}
+\frac{m^2}{\kappa}\epsilon_{ij}
dQ_{i}\wedge dQ_{j}.
\label{sympstr}$$ The $6$ dimensional phase space is hence the direct sum of the four-dimensional “exotic” phase space of the center of mass with coordinates $\vX$ and $\vP$, with the two-dimensional internal phase space of the $\vQ$, endowed with a canonical symplectic structure.
The Poisson structure can be used to calculate the algebraic structure of the symmetries. Consistently with Lukierski et al. [@LSZ], we find that $\vP, H, J, \vK$, supplemented with the central charges $m$ and $\kappa$, realize the “exotic” \[two-fold centrally extended\] planar Galilei group. The structure relations of this latter only differ from those of the usual Galilei group in that the Poisson bracket of the boost components yields the “exotic” central charge, $$\big\{K_{1},K_{2}\big\}=\kappa.
\label{exoCR}$$ Similarly, the center-of-mass coordinates have a nonvanishing Poisson bracket, $$\big\{X_{1},X_{2}\big\}=\frac{\kappa}{m^2},
\qquad
\big\{Q_{1},Q_{2}\big\}=-\frac{\kappa}{m^2}.
\label{NCCR}$$ Both the center-of-mass and the internal coordinates are hence noncommuting, cf. (\[NCpos\]) with $\theta=(\kappa/m^2)$ \[while the original coordinates $x_{i}$ commute\]. This is similar to what happens in the Landau problem where the guiding center coordinates are noncommuting, with $\theta=1/eB$.
The additional conserved quantity $I$ in (\[intangmom\]) is actually associated with the [*internal symmetries*]{} of the system. The translations and boosts form indeed an invariant subgroup $K$ of the Galilei group. The quotient $G/K$, which consists of rotations and time translations, is hence a group that can be made to act separately on the center-of-mass and the internal space. We can, e. g., rotate the internal coordinate $\vQ$ alone and leave the center-of-mass coordinate $\vX$ fixed. This is plainly a symmetry, and the associated conserved quantity is the internal angular momentum $J_{int}=I$. (A physical rotation moves both the external and internal coordinates, yielding the total angular momentum in (\[decompCC\])). The internal energy arises in a similar way. In conclusion, the non-relativistic limit of the Mathisson electron admits the direct product of the “exotic” Galilei group with the internal rotations and time translations, ${{\mathop{\rm SO}}({2})}\times {{\bf R}}$, as symmetry. Here the action of the Galilei group is transitive on the submanifolds $I={\mathop{\rm const}\nolimits}$ i. e., $\vQ^2={\mathop{\rm const}\nolimits}$.
The same statement is valid for any composite nonrelativistic system, i. e. one upon which the Galilei group acts by symmetries but not transitively [@SSD].
Relation to Schrödinger’s Zitternde Elektron {#Zitter}
============================================
The results of Section \[NRmotion\] remind those Schrödinger derived in his original paper on Zitterbewegung [@Schr]. Schrödinger starts in fact with the Dirac Hamiltonian $$H=c\valpha\cdot\vP+m^2c^2\beta
\label{Dham}$$ where $\valpha$ and $\beta$ denote the usual Dirac matrices. In the Heisenberg picture, the operators satisfy $$\frac{d\vP}{dt}=0,
\qquad
\frac{dH}{dt}=0
\qquad
\frac{d{{\vec x}}}{dt}=c\valpha.
\label{Heq}$$ The last equation can be rewritten as $
-i\frac{d\veta}{dt}=2H\veta,
$ ($\hbar=1$), where $
\veta=\valpha-cH^{-1}\vP.
$ This can be integrated as $\veta(t)=e^{2iHt}\veta_{0}=\veta_{0}e^{-2iHt}$, where $\veta_{0}$ is a constant operator. Hence $$\begin{aligned}
\frac{d{{\vec x}}}{dt}=c^2H^{-1}\vP+c\veta_{0}e^{-2iHt},\end{aligned}$$ which can again be integrated to yield $${{\vec x}}(t)=
\big\{\vX_{0}+c^2H^{-1}\vP t\big\}
+{{\scriptstyle{\frac{1}{2}}}}ic\vec{\eta}_{0}H^{-1}e^{-2iHt}
\label{zitter}$$ where $\vX_{0}$ is a constant operator. The structure is clearly the same as in (\[CMcoord\]), with the operator $$\vX(t)=\vX_{0}+c^2H^{-1}\vP t$$ representing the freely moving center-of-mass, and the second term describing the internal Zitterbewegung. The precise relation is more subtle, though. Intuitively, dropping the third component and working in the plane, putting $s=1/2$ and $s/c^2\simeq \kappa$ \[which would require the spin to diverge as $c\to\infty$ rather then remain a constant\], setting $c\valpha\simeq \dot{{{\vec x}}}$ and replacing $H\simeq mc^2$, would transform (\[zitter\]) formally into (\[CMcoord\]). In fact, $$\begin{aligned}
\vX(t)\simeq \vX_{0}+\frac{\vP}{m}t,
\qquad
mc\veta\simeq\frac{m^2}{\kappa}\vQ,
\qquad
e^{-i2Ht}=e^{-i(H/s)t}\simeq e^{-i(m/\kappa)t}.\end{aligned}$$ Note that $$mc\veta=mc\valpha-mc^2H^{-1}\vP\simeq m\dot{{{\vec x}}}-\vP.$$ consistently with (\[velmom\]).
A distinctive feature of Schrödinger’s Zitternde Elektron is that the center-of-mass coordinates satisfy the nontrivial commutation relation $$\big[X_{i},X_{j}]=-ic^2E^{-2}\epsilon_{ijk}S_{k},$$ where $E=c\sqrt{\vP^2+m^2c^2}$ and $\vS=-(i/4)\valpha\times\valpha$ is the spin operator. If we assume that the spin is polarized in the third direction, $S_{3}=-s$, and we consider the non-relativistic limit $E\simeq mc^2+\vP^2/2m$ together with the Ansatz (\[JNAnsatz\]), we find for the planar components $$\big[X_{1},X_{2}]\simeq i\frac{s}{c^2m^2}=i\frac{\kappa}{m^2}$$ cf. (\[NCpos\]) with $\theta=\kappa/m^2$. Let us remark that our procedure here is in fact the quantum version of the subtle non-relativistic limit proposed by Jackiw and Nair [@JaNa].
The relativistic description of Weyssenhoff and Raabe
=====================================================
Mathisson’s classical electron was further elaborated by Weyssenhoff and Raabe in a paper published after the War [@WeyRaa]. They posit the equations $$\begin{array}{ll}
\dot{p}^\alpha=0,\qquad
p^\alpha=mu^\alpha+
\displaystyle\frac{1}{c^2}S^{\alpha\beta}\dot{u}_{\beta},
\\[8pt]
\dot{S}^{\alpha\beta}=p^\alpha u^{\beta}-p^\beta u^{\alpha},
\\[6pt]
S^{\alpha\sigma}u_{\sigma}=0,
\end{array}
\label{WeRaeq}$$ where $
m=-\frac{1}{c^2}u_{\beta}p^\beta.
$ Eliminating $p^\alpha$ yields the relativistic Mathisson equations (\[RMeq\]) once again. Eqns. (\[WeRaeq\]) imply that $m$ is constant of the motion, $\dot{m}=0$, identified as the rest-mass of the particle. $S_{\alpha\beta}S^{\alpha\beta}=s^2$ is also a constant of the motion. They also observe that, owing to $\dot{p}^\alpha=0$, the quantity $M$ defined by $p_{\alpha}p^{\alpha}=M^2c^2$ is another constant of the motion. It is worth noting that the position satisfies again a third-order equation analogous to (\[Mateq\]), namely $$m\ddot{x}_{\alpha}=-\frac{1}{c^2}S_{\alpha\sigma}\dddot{x}_{\sigma}.
\label{WR3}$$
Then Weyssenhoff and Raabe proceed to integrate the free relativistic equations of motion. In a suitable inertial frame (called the proper system) the spatial components, $P_{i}$, of the vector $p^\alpha$ can be made to vanish, so that its time component is $Mc$. In this frame $\vS$ is constant. The mass is $m=M/\sqrt{1-(\vv/c)^2}$ where $v_{i}=u_{i}\sqrt{1-(\vv/c)^2}$ denotes the three-velocity. Hence the time component of the four-velocity is also constant so that the four-acceleration is proportional to the three-acceleration, ${\vec a}=d^2{{\vec x}}/dt^2$. Transforming from proper time to $t$, $\vP=0$ reduces finally to $$M\vv+\frac{1}{c^2}\vS\times{\vec a}=0.$$ The particle moves hence along a circle in the plane perpendicular to $\vS$, with uniform angular velocity $$\frac{mc^2}{s}\Big(1-\frac{\vv^2}{c^2}\Big).$$ In a general Lorentz frame, the motion is a superposition of such a motion with a uniform translation.
Our clue is to observe that in the non-relativistic limit these formul[æ]{} reduce, with the Jackiw-Nair Ansatz $s=\kappa c^2$ cf. (\[JNAnsatz\]), to those we derived in Section \[NRmotion\].
It is worth mentionning that the equations of Weyssenhoff and Raabe have again and again re-emerged in the course of the years. Consider, for example, (\[WeRaeq\]) in[*five*]{} dimensions and for $s={{\scriptstyle{\frac{1}{2}}}}$. Multiplication of $p^\beta$ with $S_{\alpha\beta}$ allows us to express the five-vector $\dot{u}_{\alpha}$ as $$\dot{u}_{\alpha}=\frac{4}{c^2}S_{\alpha\sigma}p^\sigma$$ which, together with the remaining relations in (\[WeRaeq\]) and the constraint $u_{\alpha}u^{\alpha}=1$, are precisely the equations proposed by Barut and Zanghi [@Barut] as a “Kaluza-Klein” description of a classical Dirac electron.
Conclusion
==========
In this Note we have shown that the non-relativistic limit of Mathisson’s classical spinning electron yields the acceleration-dependent model of Lukierski et al. [@LSZ]. This latter has non-commuting coordinates and realizes the “exotic” Galilean symmetry.
Our results confirm once again the relation between the relativistic spin and the non-relativistic “exotic” structure, advocated by Jackiw and Nair [@JaNa]. Their rule (\[JNAnsatz\]) is, however, a rather strange one, since it requires the spin to diverge as $c\to\infty$ so that $s/c^2$ remains finite. For this reason, the use of a Dirac equation valid for the fixed velue $s={{\scriptstyle{\frac{1}{2}}}}$ \[as in Section \[Zitter\] above\] is clearly illegitimate, and should be replaced by some anyon equation, valid for any real spin $s$ [@aneq].
Another intriguing feature of this procedure is the following. While the relativistic model is associated with an irreducible representation of the Poincaré group, its dequantized & non-relativistic limit, namely the model of Lukierski et al., only carries a reducible representation of the Galilei group : irreducibility is lost in the procedure.
A final remark concerns the spin constraint (\[spinconstr\]) which appears to lie at the very root of the Zitterbewegung. Trading it for $$S^{\alpha\beta}p_{\beta}=0
\label{Pspinconstr}$$ would in fact eliminate the Zitterbewegung altogether and lead to models of the type discussed in [@Dixon].
I am indebted to Professor J. Lukierski for sending me copies of those old Acta Physica Polonica papers, and also to Professor A. Staruszkiewicz who provided me with some biographical data.
[99]{} It is impossible to provide a complete list of references. See, e. g., V. P. Nair and A. P. Polychronakos, [*Quantum mechanics on the noncommutative plane and sphere*]{}. [*Phys. Lett*]{}. [**B 505**]{}, 267 (2001); J. Gamboa, M. Loewe, F. Méndez, and J. C. Rojas, [*The Landau problem in noncommutative Quantum Mechanics*]{}; S. Bellucci, A. Nersessian, and C. Sochichiu, [*Two phases of the noncommutative quantum mechanics*]{}. [*Phys. Lett.*]{} [**B522**]{}, 345 (2001), etc.
J.-M. Lévy-Leblond, in [*Group Theory and Applications*]{} (Loebl Ed.), [**II**]{}, Acad. Press, New York, p. 222 (1972); A. Ballesteros, N. Gadella, and M. del Olmo, [*Moyal quantization of $2+1$ dimensional Galilean systems*]{}. [*Journ. Math. Phys.*]{} [**33**]{}, 3379 (1992); Y. Brihaye, C. Gonera, S. Giller and P. Kosiński, [*Galilean invariance in $2+1$ dimensions.*]{} `hep-th/9503046` (unpublished); D. R. Grigore, [*Transitive symplectic manifolds in $1+2$ dimensions*]{}. [*Journ. Math. Phys.*]{} [**37**]{}, 240 (1996); [*The projective unitary irreducible representations of the Galilei group in $1+2$ dimensions*]{}. [*ibid*]{}. [**37**]{}, 460 (1996).
J. Lukierski, P. C. Stichel, W. J. Zakrzewski, [*Galilean-invariant $(2+1)$-dimensional models with a Chern-Simons-like term and $d=2$ noncommutative geometry*]{}. [*Annals of Physics*]{} (N. Y.) [**260**]{}, 224 (1997). The model is further discussed in P. A. Horváthy and M. S. Plyushchay, [*Non-relativistic anyons, exotic Galilean symmetry and the non-commutative plane*]{}. [*JHEP*]{} [**06**]{} (2002) 033; J. Lukierski, P. C. Stichel, W. J. Zakrzewski, [*Noncommutative planar particle dynamics with gauge interaction*]{}. [*Annals of Physics*]{} (N. Y.) (in press) \[`hep-th/0207149`\].
M. Mathisson, [*Das Zitternde Elektron und seine Dynamik*]{}. [*Acta Physica Polonica*]{} [**6**]{}, 218-227 (1937). Myron Mathisson (1897-1940), of Jewish origin, taught mathematical physics at Warsaw University as a [*Privatdozent*]{}. He also worked in Kraków, benefitting of a kind of “private scholarship” created for him by Weyssenhoff. Then he spent one year in Kazan, in the Soviet Union. In 1939 he escaped to Britain, where he died. He was remembered by Dirac in the Obituary reproduced below, published in [*Nature*]{} [**146**]{}, 613 (1940) : “The death of Dr. Myron Mathisson on September 13 at the early age of fourty-three has cut short an interesting line of research. Mathisson had been engaged for many years in studying the general dynamical laws governing the motion of a particle, with possibly a spin or a moment, in a gravitational or electromagnetic field, and had developed a powerful method of his own for passing from field equations to particle equations. The subject is of particular interest at the present time, as it has now become clear that quantum mechanics cannot solve the difficulties that arise in connexion with the interaction of point particles with fields, and a deeper classical analysis of the problem is needed. It is much to be regretted that Mathisson’s death has occured before the relations between his method and those of other workers on the subject have been completely elucidated.
Mathisson carried out his work at the Universities of Warsaw and Kazan and at an institute which he started in Cracow, and, since the spring of 1939, at Cambridge.”
\[Sources: A short history of Theoretical Physics at Hoza 69 …and personal communication of Prof. A. Staruszkiewicz.\]
J. Weyssenhoff and A. Raabe, [*Relativistic dynamics of spin fluids and spin particles*]{}. [*Acta Physica Polonica*]{} [**9**]{}, 7-18 (1947). Let us also record the footnote written by Weyssenhoff. “Presented at a meeting of the Cracow Section of the Polish Physical Society on February 28, 1945. \[…\] Most of the results were subject of a lecture at a secret meeting of physicists at Prof. Pieńkowski’s home in Warsaw, October 1942.
Mr. Raabe was a highly gifted young physicist with whom I outlined in all its main features the contents of this paper in 1940/41 in Lwów. We tried to pursue our work in 1942 in Cracow, but unfortunately in June 1942 Mr. Raabe fell victim of a man-hunt in the streets of Cracow; he died four months later in the German concentration camp Oświȩcim \[Auschwitz\].”
Jan Weyssenhoff (1889-1972) came from a prominent Baltic-German aristocratic family, which remained Catholic and become Polish in the XVIIth century. He was a gentleman in the old sense of the word, who used his personal fortune and his wealthy friends to help other colleagues. His father was a succesful writer. His mother came from a very wealthy Jewish banking family which owned, among other things, the Warsaw-Vienna railway.
Weyssenhoff studied in Kraków and in Zürich, where he also met Einstein, who refers to him in his work on Brownian Motion \[available in Dover Publications\]. He was also interested in the Hall effect and wrote his Ph. D. on the theory of paramagnetism. He returned to his country in 1919. He got involved in the study of relativistic spinning particles and fluids in 1937. Between 1939 and 1941 he worked at the Polytechnical University in Lwów, occupied by the Soviet army and attached to Ukrain. In 1942 he returned to Kraków, and was followed by Raabe, who lived in his flat and whom he helped also to get documents, e. g., a “Kennkarte”.
Professor Weyssenhoff also organized secret seminars on physics in his home. Unlike his young collaborator, he survived to the war and continued his scientific work until his death in Kraków, in 1972.
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|
---
abstract: 'A stationary Josephson effect in a weak-link between misorientated nonunitary triplet superconductors is investigated theoretically. The non-self-consistent quasiclassical Eilenberger equation for this system has been solved analytically. As an application of this analytical calculation, the current-phase diagrams are plotted for the junction between two nonunitary bipolar $f-$wave superconducting banks. A spontaneous current parallel to the interface between superconductors has been observed. Also, the effect of misorientation between crystals on the Josephson and spontaneous currents is studied. Such experimental investigations of the current-phase diagrams can be used to test the pairing symmetry in the above-mentioned superconductors.'
address: |
$^1$ Institute for Advanced Studies in Basic Sciences, Zanjan, 45195-1159, Iran\
$^2$ B.Verkin Institute for Low Temperature Physics Engineering of National Academy of Sciences of Ukraine, 47, Lenin ave , 61103, Kharkov, Ukraine\
$^3$ Department of Physics, Faculty of Science, University of Shahrekord, Shahrekord, P.O.Box 115, Iran
author:
- 'G. Rashedi$^{1,3}$ and Yu. A. Kolesnichenko$^{2}$'
title: 'Stationary Josephson effect in a weak-link between nonunitary triplet superconductors'
---
Introduction
============
In recent years, the triplet superconductivity has become one of the modern subjects for researchers in the field of superconductivity [@Ueda; @Maeno; @Mackenzie]. Particularly, the nonunitary spin triplet state in which Cooper pairs may carry a finite averaged intrinsic spin momentum has attracted much attention in the last decade [@Tou; @Machida]. A triplet state in the momentum space $\mathbf{k}$ can be described by the order parameter ${\hat{\Delta}}(\mathbf{k})=i(\mathbf{d}(\mathbf{k})\cdot {\hat{
\sigma}})\hat{\sigma}_{y}$ in a 2$\times $2 matrix form in which $\hat{\sigma}_{j}$ are 2$\times $2 Pauli matrices $\left(
{\hat{\sigma}=}\left(
\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\right) \right)
$. The three dimensional complex vector $\mathbf{d}(\mathbf{k)}$ (gap vector) describes the triplet pairing state. In the nonunitary state, the product ${\hat{\Delta}}(\mathbf{k}){\hat{\Delta}}(\mathbf{k})^{\dagger
}=\mathbf{d}(\mathbf{k})\cdot \mathbf{d}^{\ast
}(\mathbf{k})+i(\mathbf{d}(\mathbf{k})\times \mathbf{d}^{\ast
}(\mathbf{k}))\cdot {\hat{\sigma}}$ is not a multiple of the unit matrix. Thus in a non-unitary state the time reversal symmetry is necessarily broken spontaneously and a spontaneous moment $\mathbf{m}(\mathbf{k})=i\mathbf{d}(\mathbf{k})\times
\mathbf{d}^{\ast }(\mathbf{k})$ appears at each point $\mathbf{k}$ of the momentum space. In this case the macroscopically averaged moment $<\mathbf{m}(\mathbf{k})>$ integrated on the Fermi surface does not vanish. The value $\mathbf{m}(\mathbf{k})$ is related to the net spin average by $Tr[{\hat{\Delta}}(\mathbf{k})^{\dag
}{\hat{\sigma}_{j}}{\hat{\Delta}}(\mathbf{k})]$. It is clear that the total spin average over the Fermi surface can be nonzero. As an application, the nonunitary bipolar state of $f-$wave pairing symmetry has been considered for the $B-$phase of superconductivity in the $UPt_{3}$ compound which has been created at low temperatures $T$ and small values of the magnetic field $H$ [@Machida; @Ohmi].
In the present paper, the ballistic Josephson weak link via an interface between two superconducting bulks with different orientations of the crystallographic axes is investigated. This type of weak link structure can be used for the demonstration of the pairing symmetry in the superconducting phase [@Stefanakis]. Consequently, we generalize the formalism of paper [@Mahmoodi] for the weak link between triplet superconducting bulks with a nonunitary order parameter. In the paper [@Mahmoodi], the Josephson effect in the point contact between unitary $f-$wave triplet superconductors has been studied. Also, the effect of misorientation on the charge transport has been investigated and a spontaneous current tangential to the interface between the $f-$wave superconductors, has been observed.
In this paper the nonunitary bipolar $f-$wave model of the order parameter is considered. It is shown that the current-phase diagrams are totally different from the current-phase diagrams of the junction between the unitary triplet ( axial and planar) $f-$wave superconductors [@Mahmoodi] . Roughly speaking, these different characters can be used to distinguish between nonunitary bipolar $f-$wave superconductivity and the other types of superconductivity. In the weak-link structure between the nonunitary $f-$wave superconductors, the spontaneous current parallel to the interface has been observed as a fingerprint for unconventional superconductivity and spontaneous time reversal symmetry breaking. The effect of misorientation on the spontaneous and Josephson currents is investigated. It is possible to find the value of the phase difference in which the Josephson current is zero but the spontaneous current, which is produced by the interface and is tangential to the interface, is present. In some configurations and at the zero phase difference, the Josephson current is not generally zero but has a finite value. This finite value corresponds to a spontaneous phase difference which is related to the misorientation between the gap vectors$\mathbf{d}$.
The arrangement of the rest of this paper is as follows. In Sec.\[section2\] we describe the configuration that we have investigated. For a non-self-consistent model of the order parameter, the quasiclassical Eilenberger equations [@Eilenberger] are solved and suitable Green functions have been obtained analytically. In Sec.\[section3\] the formulas obtained for the Green functions have been used for the calculation of the current densities at the interface. An analysis of numerical results will be presented in Sec.\[section4\] together with some conclusions in Sec.\[section5\].
Formalism and Basic Equations {#section2}
=============================
We consider a model of a flat interface $y=0$ between two misorientated nonunitary $f-$wave superconducting half-spaces (Fig.1) as a ballistic Josephson junction. In the quasiclassical ballistic approach, in order to calculate the current, we use “transport-like” quasiclassical Eilenberger equations [@Eilenberger] for the energy integrated Green functions $\breve{g}\left(
\mathbf{\hat{v}}_{F},\mathbf{r},\varepsilon _{m}\right) $ $$\mathbf{v}_{F}\cdot \nabla \breve{g}+\left[ \varepsilon
_{m}\breve{\sigma} _{3}+i\breve{\Delta},\breve{g}\right] =0,
\label{Eilenberger}$$ and the normalization condition $\breve{g}\breve{g}=\breve{1}$, where $\varepsilon _{m}=\pi T(2m+1)$ are discrete Matsubara energies $m=0,1,2,...,$ $T$ is the temperature, $\mathbf{v}_{F}$ is the Fermi velocity and $\breve{\sigma}_{3}=\hat{\sigma}_{3}\otimes \hat{I}$ in which $\hat{\sigma} _{j}\left( j=1,2,3\right) $ are Pauli matrices.
![Scheme of a flat interface between two superconducting bulks which are misorientated as much as $\protect\alpha $.[]{data-label="fig1"}](fig1.eps){width="0.9\columnwidth"}
The Matsubara propagator $\breve{g}$ can be written in the form: $$\breve{g}=\left(
\begin{array}{cc}
g_{1}+\mathbf{g}_{1}\cdot \mathbf{\hat{\sigma}} & \left(
g_{2}+\mathbf{g} _{2}\cdot \hat{\sigma}\right) i\hat{\sigma}_{2} \\
i\hat{\sigma}_{2}\left( g_{3}+\mathbf{g}_{3}\cdot
\hat{\sigma}\right) & g_{4}-\hat{\sigma}_{2}\mathbf{g}_{4}\cdot
\hat{\sigma}\hat{\sigma}_{2}
\end{array}
\right),\label{Green's function}$$ where the matrix structure of the off-diagonal self energy $\breve{\Delta}$ in the Nambu space is $$\breve{\Delta}=\left(
\begin{array}{cc}
0 & \mathbf{d}\cdot \hat{\sigma}i\hat{\sigma}_{2} \\
i\hat{\sigma}_{2}\mathbf{{d^{\ast }}\cdot \hat{\sigma}} & 0
\end{array}
\right).\label{order parameter}$$ The nonunitary states, for which $\mathbf{d\times d} ^{\ast }\neq
0,$ are investigated. Fundamentally, the gap vector (order parameter) $\mathbf{d}$ has to be determined numerically from the self-consistency equation [@Ueda], while in the present paper, we use a non-self-consistent model for the gap vector which is much more suitable for analytical calculations [@Kulik]. Solutions to Eq. (\[Eilenberger\]) must satisfy the conditions for the Green functions and the gap vector $\mathbf{d}$ in the bulks of the superconductors far from the interface as follow: $$\breve{g}=\frac{1}{\Omega _{n}}\left(
\begin{array}{cc}
\varepsilon _{m}(1-\mathbf{A}_{n}\cdot \mathbf{\hat{\sigma}}) &
\left[i\mathbf{d}_{n}-\mathbf{d}_{n}\times \mathbf{A}_{n}\right]
\cdot
\hat{\sigma}i\hat{\sigma}_{2} \\
i\hat{\sigma}_{2}\left[ i\mathbf{d}_{n}^{\ast
}+\mathbf{d}_{n}^{\ast }\times \mathbf{A}_{n}\right] \cdot
\hat{\sigma} & -\varepsilon \hat{\sigma}_{2}(1+\mathbf{A}_{n}\cdot
\hat{\sigma})\hat{\sigma}_{2}
\end{array}
\right) \label{Bulk solution}$$ where $$\hspace{-0.4cm}\mathbf{A}_{n}=\frac{i\mathbf{d}_{n}\times
\mathbf{d}_{n}^{\ast }}{\varepsilon _{m}^{2}+\mathbf{d}_{n}\cdot
\mathbf{d}_{n}^{\ast }+\sqrt{(\varepsilon
_{m}^{2}+\mathbf{d}_{n}\cdot \mathbf{d}_{n}^{\ast
})^{2}+(\mathbf{d}_{n}\times \mathbf{d}_{n}^{\ast })^{2}}}$$ and $$\hspace{-0.55cm}\Omega _{n}=\sqrt{\frac{2[(\varepsilon
_{m}^{2}+\mathbf{d} _{n}\cdot \mathbf{d}_{n}^{\ast
})^{2}+(\mathbf{d}_{n}\times \mathbf{d} _{n}^{\ast
})^{2}]}{\varepsilon _{m}^{2}+\mathbf{d}_{n}\cdot \mathbf{d}
_{n}^{\ast }+\sqrt{(\varepsilon _{m}^{2}+\mathbf{d}_{n}\cdot
\mathbf{d} _{n}^{\ast })^{2}+(\mathbf{d}_{n}\times
\mathbf{d}_{n}^{\ast })^{2}}}}$$ $$\mathbf{d}\left( \pm \infty \right) =\mathbf{d}_{2,1}\left(
T,\mathbf{\hat{v}}_{F}\right) \exp \left( \mp \frac{i\phi
}{2}\right) \label{Bulk order parameter}$$ where $\phi $ is the external phase difference between the order parameters of the bulks and $n=1,2$ label the left and right half spaces respectively. It is clear that poles of the Green function in the energy space are in $$\Omega _{n}=0.$$ Consequently, $$(-E^{2}+\mathbf{d}_{n}\cdot \mathbf{d}_{n}^{\ast
})^{2}+(\mathbf{d}_{n}\times \mathbf{d}_{n}^{\ast })^{2}=0$$ and $$E=\pm \sqrt{\mathbf{d}_{n}\cdot \mathbf{d}_{n}^{\ast }\pm
i\mathbf{d} _{n}\mathbf{\times d}_{n}^{\ast }}$$ in which $E$ is the energy value of the poles. The Eq. (\[Eilenberger\]) has to be supplemented by the continuity conditions at the interface between superconductors. For all quasiparticle trajectories, the Green functions satisfy the boundary conditions both in the right and left bulks as well as at the interface. The system of equations (\[Eilenberger\]) and the self-consistency equation for the gap vector $\mathbf{d}$ [@Ueda] can be solved only numerically.
![Component of the current normal to the interface (Josephson current) versus the phase difference $\protect\phi $ for the junction between nonunitary bipolar $f-$wave bulks , $T/T_{c}=0.15$, geometry (i) and different misorientations. Currents are given in units of $j_{0}=\frac{\protect\pi
}{2}eN(0)v_{F}\Delta _{0}(0).$[]{data-label="fig2"}](fig2.eps){width="0.9\columnwidth"}
For unconventional superconductors such solution requires the information about the interaction between the electrons in the Cooper pairs and the nature of unconventional superconductivity in novel compounds which in most cases are unknown. Also, it has been shown that the absolute value of a self-consistent order parameter is suppressed near the interface and at the distances of the order of the coherence length, while its dependence on the direction in the momentum space almost remains unaltered [@Barash]. This suppression of the order parameter changes the amplitude value of the current, but does not influence the current-phase dependence drastically. For example, it has been verified in Refs.[@Coury] for the junction between unconventional $d$-wave superconductors, in Ref.[@Barash] for the case of unitary “$f$-wave” superconductors and in Ref.[@Viljas] for pinholes in $^{3}He,$ that there is good qualitative agreement between self-consistent and non-self-consistent results for not very large angles of misorientation. It has also been observed that the results of the non-self-consistent model in [@Yip] are similar to experiment [@Backhaus]. Consequently, despite the fact that this solution cannot be applied directly to a quantitative analysis of a real experiment, only a qualitative comparison of calculated and experimental current-phase relations is possible. In our calculations, a simple model of the constant order parameter up to the interface is considered and the pair breaking and scattering on the interface are ignored. We believe that under these strong assumptions our results describe the real situation qualitatively. In the framework of such a model, the analytical expressions for the current can be obtained for a certain form of the order parameter.
Analytical results {#section3}
==================
The solution of Eq. (\[Eilenberger\]) allows us to calculate the current densities. The expression for the current is: $$\mathbf{j}\left( \mathbf{r}\right) =2i\pi eTN\left( 0\right)
\sum_{m}\left\langle \mathbf{v}_{F}g_{1}\left(
\mathbf{\hat{v}}_{F},\mathbf{r},\varepsilon _{m}\right)
\right\rangle
\label{charge-current}$$ where $\left\langle ...\right\rangle $ stands for averaging over the directions of an electron momentum on the Fermi surface ${\mathbf{\hat{v}}} _{F},$ and $N\left( 0\right) $ is the electron density of states at the Fermi level of energy.
![Component of the current normal to the interface (Josephson current) versus the phase difference $\protect\phi $ for the junction between nonunitary bipolar $f-$wave bulks , $T/T_{c}=0.15$, geometry (ii) and different misorientations.[]{data-label="fig3"}](fig3.eps){width="0.9\columnwidth"}
We assume that the order parameter is constant in space and in each half-space it equals its value (\[Bulk order parameter\]) far from the interface in the left or right bulks. For such a model, the current-phase dependence of a Josephson junction can be calculated analytically. It enables us to analyze the main features of current-phase dependence for any model of the nonunitary order parameter. The Eilenberger equations (\[Eilenberger\]) for Green functions $\breve{g}$, which are supplemented by the condition of continuity of solutions across the interface, $y=0$, and the boundary conditions at the bulks, are solved for a non-self-consistent model of the order parameter analytically. In the ballistic case the system of equations for functions $g_{i}$ and $\mathbf{g}_{i}$ can be decomposed into independent blocks of equations. The set of equations which enables us to find the Green function $g_{1}$ is: $$\begin{aligned}
v_{F}\hat{\mathbf{k}}\nabla g_{1} &=&i\left( \mathbf{d}\cdot
\mathbf{g}_{3}-\mathbf{d}^{\ast }\cdot \mathbf{g}_{2}\right) ;
\label{a} \\
v_{F}\hat{\mathbf{k}}\nabla \mathbf{g}_{-} &=&-2\left(
\mathbf{d\times
g} _{3}+\mathbf{d}^{\ast }\mathbf{\times g}_{2}\right) ; \label{b} \\
v_{F}\hat{\mathbf{k}}\nabla \mathbf{g}_{2} &=&-2\varepsilon
_{m}\mathbf{g}_{2}+2ig_{1}\mathbf{d}+\mathbf{d}\times
\mathbf{g}_{-}; \label{d} \\ v_{F}\hat{\mathbf{k}}\nabla
\mathbf{g}_{3} &=&2\varepsilon _{m}\mathbf{g}
_{3}-2ig_{1}\mathbf{d}^{\ast }+\mathbf{d}^{\ast }\times
\mathbf{g}_{-}; \label{c}\end{aligned}$$ where $\mathbf{g}_{-}=\mathbf{g}_{1}-\mathbf{g}_{4}.$ The Eqs. (\[a\])-(\[d\]) can be solved by integrating over the ballistic trajectories of electrons in the right and left half-spaces. The general solution satisfying the boundary conditions (\[Bulk solution\]) at infinity is $$g_{1}^{\left( n\right) }=\frac{\varepsilon _{m}}{\Omega
_{n}}+a_{n}e^{-2s\Omega _{n}t}; \label{e}$$ $$\mathbf{g}_{-}^{\left( n\right) }=-2\frac{\varepsilon _{m}}{\Omega
_{n}}\mathbf{A}_{n}+\mathbf{C}_{n}e^{-2s\Omega _{n}t}; \label{f}$$ $$\hspace{-0.3cm}\mathbf{g}_{2}^{\left( n\right)
}=\frac{i\mathbf{d}_{n}-\mathbf{d}_{n}\times
\mathbf{A}_{n}}{\Omega _{n}}-\frac{2ia_{n}\mathbf{d}
_{n}+\mathbf{d}_{n}\times \mathbf{C}_{n}}{2s\eta \Omega
_{n}-2\varepsilon _{m}}e^{-2s\Omega _{n}t}; \label{g}$$ $$\hspace{-0.3cm}\mathbf{g}_{3}^{\left( n\right)
}=\frac{i\mathbf{d}_{n}^{\ast }+\mathbf{d}_{n}^{\ast }\times
\mathbf{A}_{n}}{\Omega _{n}}+\frac{2ia_{n}\mathbf{d}_{n}^{\ast
}-\mathbf{d}_{n}^{\ast }\times \mathbf{C}_{n}}{2s\eta \Omega
_{n}+2\varepsilon _{m}}e^{-2s\Omega _{n}t}; \label{h}$$ where $t$ is the time of flight along the trajectory, $sgn\left(
t\right) =sgn\left( y\right) =s$ and $\eta =sgn\left( v_{y}\right)
.$ By matching the solutions (\[e\]-\[h\]) at the interface $\left( y=0,t=0\right) $, we find constants $a_{n}$ and $\mathbf{C}_{n}.$ Indices $n=1,2$ label the left and right half-spaces respectively. The function $g_{1}\left( 0\right)
=g_{1}^{\left( 1\right) }\left( -0\right) =g_{1}^{\left( 2\right)
}\left( +0\right)$ which is a diagonal term of the Green matrix and determines the current density at the interface, $y=0$, is as follows: $$g_{1}\left( 0\right) =\frac{\eta (\mathbf{d}_{2}\cdot
\mathbf{d}_{2}(\eta \Omega _{1}+\varepsilon
)^{2}-\mathbf{d}_{1}\cdot \mathbf{d}_{1}(\eta \Omega
_{2}-\varepsilon )^{2}+B)}{[\mathbf{d}_{2}(\eta \Omega
_{1}+\varepsilon )+\mathbf{d}_{1}(\eta \Omega _{2}-\varepsilon
)]^{2}} \label{charge-term}$$ where $B=i\mathbf{d}_{1}\times \mathbf{d}_{2}\cdot
(\mathbf{A}_{1}\mathbf{+A}_{2})(\eta \Omega _{2}-\varepsilon
)(\eta \Omega _{1}+\varepsilon ).$ We consider a rotation $\breve{R}$ only in the right superconductor (see Fig.1), i.e., $\mathbf{d}_{2}(\hat{\mathbf{k}})=\breve{R}\mathbf{d}_{1}(
\breve{R}^{-1}\hat{\mathbf{k}});$ $\hat{\mathbf{k}}$ is the unit vector in the momentum space.
![The $x-$component of the current tangential to the interface versus the phase difference $\protect\phi $ for the junction between nonunitary bipolar $f-$wave superconducting bulks, $T/T_{c}=0.15$, geometry (i) and the different misorientations.[]{data-label="fig4"}](fig4.eps){width="0.9\columnwidth"}
The crystallographic $c$-axis in the left half-space is selected parallel to the partition between the superconductors (along the $z$-axis in Fig.1). To illustrate the results obtained by computing the formula (\[charge-term\]), we plot the current-phase diagrams for two different geometries. These geometries correspond to the different orientations of the crystals in the right and left sides of the interface (Fig.1):(i) The basal $ab$-plane in the right side has been rotated around the $c$-axis by $\alpha $; $\hat{\mathbf{c}}_{1}\Vert \hat{\mathbf{c}}_{2}$. (ii) The $c$-axis in the right side has been rotated around the $b$-axis by $\alpha $; $\hat{\mathbf{b}}_{1}\Vert
\hat{\mathbf{b}}_{2}$.Further calculations require a certain model of the gap vector (order parameter) $\mathbf{d}$.
Analysis of numerical results {#section4}
=============================
In the present paper, the nonunitary $f-$wave gap vector in the $B-$phase (low temperature $T$ and low field $H$) of superconductivity in $UPt_{3}$ compound has been considered. This nonunitary bipolar state which explains the weak spin-orbit coupling in $UPt_{3}$ is [@Machida]: $$\mathbf{d}(T,\mathbf{v}_{F})=\Delta
_{0}(T)k_{z}(\hat{\mathbf{x}}\left( k_{x}^{2}-k_{y}^{2}\right)
+\hat{\mathbf{y}}2ik_{x}k_{y}). \label{Model}$$ The coordinate axes $\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}$ are selected along the crystallographic axes $\hat{\mathbf{a}},\hat{\mathbf{b}},\hat{\mathbf{c}}$ in the left side of Fig.\[fig1\]. The function $\Delta _{0}=$$\Delta
_{0}\left( T\right) $ describes the dependence of the gap vector on the temperature $T$ (our numerical calculations are done at the low value of temperature $T/T_{c}=0.1$). Using this model of the order parameter (\[Model\]) and solution to the Eilenberger equations (\[charge-term\]), we have calculated the current density at the interface numerically. These numerical results are listed below:1) The nonunitary property of Green’s matrix diagonal term consists of two parts. The explicit part which is in the $B$ mathematical expression in Eq.(\[charge-term\]) and the implicit part in the $\Omega _{1,2}$ and $\mathbf{d}_{1,2}$ terms. These $\Omega _{1,2}$ and $\mathbf{d}_{1,2}$ terms are different from their unitary counterparts. In the mathematical expression for $\Omega _{1,2}$ the nonunitary mathematical terms $\mathbf{A}_{1,2}$ are presented. The explicit part will be present only in the presence of misorientation between gap vectors,$B=i\mathbf{d}_{1}\times \mathbf{d}_{2}\cdot
(\mathbf{A}_{1}\mathbf{+A}_{2})(\eta \Omega _{2}-\varepsilon
)(\eta \Omega _{1}+\varepsilon )$, but the implicit part will be present always. So, in the absence of misorientation $(\mathbf{d}_{1}\mathbf{\Vert d}_{2})$, although the implicit part of nonunitary exists the explicit part is absent.
![Current tangential to the interface versus the phase difference $\protect\phi $ for the junction between nonunitary bipolar $f-$wave superconducting bulks, $T/T_{c}=0.15$, geometry (ii) and the different misorientations ($x$component) .[]{data-label="fig5"}](fig5.eps){width="0.9\columnwidth"}
This means that in the absence of misorientation, current-phase diagrams for planar unitary and nonunitary bipolar systems are the same but the maximum values are slightly different.2) A component of current parallel to the interface $j_{z}$ for geometry (i) is zero similar to the unitary case [@Mahmoodi] while the other parallel component $j_{x}$ has a finite value (see Fig.\[fig4\]). This latter case is a difference between unitary and nonunitary cases. Because in the junction between unitary $f-$wave superconducting bulks all parallel components of the current ($j_{x}$ and $j_{z}$) for geometry (i) are absent [@Mahmoodi].3) In Figs.\[fig2\],\[fig3\], the Josephson current $j_{y}$ is plotted for a certain nonunitary model of $f-$wave and different geometries. Figs.\[fig2\],\[fig3\] are plotted for the geometry (i) and geometry (ii) respectively. They are completely unusual and totally different from their unitary counterparts which have been obtained in [@Mahmoodi].4) In Fig.\[fig2\] for geometry (i), it is observed that by increasing the misorientation, some small oscillations appear in the current-phase diagrams as a result of the non-unitary property of the order parameter, . Also, the Josphson current at the zero external phase difference $\phi =0$ is not zero but has a finite value. The Josephson current will be zero at the some finite values of the phase difference.5) In Fig.\[fig3\] for geometry (ii), it is observed that by increasing the misorientation the new zeros in current-phase diagrams appear and the maximum value of the current will be changed non-monotonically.
![Current tangential to the interface versus the phase difference $\protect\phi $ for the junction between nonunitary bipolar $f-$wave superconducting bulks, $T/T_{c}=0.15$, geometry (ii) and the different misorientations ($z-$component).[]{data-label="fig6"}](fig6.eps){width="0.9\columnwidth"}
In spite of the Fig\[fig2\] for geometry (i), the Josephson currents at the phase differences $\phi =0$, $\phi =\pi ,$ and $\phi =2\pi $ are zero exactly.6) The current-phase diagram for geometry (i) and $x-$component (Fig.\[fig4\]) is totally unusual. By increasing the misorientation, the maximum value of the current increases. The components of current parallel to the interface for geometry (ii) are plotted in Fig.\[fig5\] and Fig\[fig6\]. All the terms at the phase differences $\phi
=0$, $\phi =\pi,$ and $\phi =2\pi $ are zero. The maximum value of the current-pase diagrams is not a monotonic function of the misorientation.
Conclusions {#section5}
===========
Thus, we have theoretically studied the supercurrents in the ballistic Josephson junction in the model of an ideal transparent interface between two misoriented $UPt_{3}$ crystals with nonunitary bipolar $f-$wave superconducting bulks which are subject to a phase difference $\phi $. Our analysis has shown that misorientation between the gap vectors creates a current parallel to the interface and different misorientations between gap vectors influence the spontaneous parallel and normal Josephson currents. These have been shown for the currents in the point contact between two bulks of unitary axial and planar $f$-wave superconductor in [@Mahmoodi] separately. Also, it is shown that the misorientation of the superconductors leads to a spontaneous phase difference that corresponds to the zero Josephson current and to the minimum of the weak link energy in the presence of the finite spontaneous current. This phase difference depends on the misorientation angle. The tangential spontaneous current is not generally equal to zero in the absence of the Josephson current. The difference between unitary planar and nonunitary bipolar states can be used to distinguish between them. This experiment can be used to test the pairing symmetry and recognize the different phases of $UPt_{3}$.
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---
abstract: 'Explicit inversion formulas for a subclass of integral operators with $D$-difference kernels on a finite interval are obtained. A case of the positive operators is treated in greater detail. An application to the inverse problem to recover canonical system from a Weyl function is given.'
author:
- |
A.L. Sakhnovich, A.A. Karelin, J. Seck-Tuoh-Mora,\
G. Perez-Lechuga, M. Gonzalez-Hernandez
title: 'On explicit inversion of a subclass of operators with $D$-difference kernels and Weyl theory of the corresponding canonical systems'
---
[MSC(2000) Primary 34A55, 45Q05; Secondary 47B65, 47G10]{}
[*Keywords: integral operator with difference kernel, operator with $D$-difference kernel, explicit inversion, canonical system, inverse problem, Weyl function.*]{}
Introduction {#intro}
============
Integral operators with difference kernels are important in mathematics and applications and are actively used in the study of numerous homogenious processes. The papers [@GoKr; @Kr] on the inversion of the operators with difference kernels on the semi-axis became classical. Various results and references on the operators with difference kernels on a finite interval or a system of intervals are given in [@SaLdiff; @SaL1]. Interesting explicit results on the inversion of the operators with exponential type difference kernels on a finite interval one can find in [@BGK; @GoKaSc].
Operators with $D$-difference kernels in $L^2_p(0,l)$, which we shall treat, are bounded operators of the form $$\begin{aligned}
\label{0.1}
&&S_l f=Sf=\frac{d}{dx} \int_0^l
s(x,t)f(t)dt, \quad s(x,t)=\big\{ s_{i j}(x,t) \big\}_{i,j=1}^p,
\\ \label{0.2} && s_{ij}(x,t)=s_{ij}(d_ix-d_j t), \quad s_{ij}(x) \in L^2(-d_j l, \, d_i l), \end{aligned}$$ where $D=D^*={\mathrm{diag}}\{d_1, \, d_2, \, \ldots , \, d_p \}>0$ is a fixed $p \times p$ diagonal matrix. The notion of an operator with a $D$-difference kernel is a natural generalization of the operator with a difference kernel, i.e., of the case $D=I_p$, where $I_p$ is the $p \times p$ identity matrix. The class of operators with $D$-difference kernels on a finite interval includes the operators with difference kernels on systems of intervals, which are important, for instance, in elasticity theory, diffraction theory, and the theory of stable processes (see [@Ka] and Chapter 6 in [@SaL1]).
Explicit inversion formulas for an interesting subclass of operators with $D$-difference kernels are obtained in Section \[Inver\] of this paper using the classical results on semiseparable operators. Note also that the inversion of semiseparable matrices and operators is another interesting and actively developed theory, see [@GGK1; @GK84] and bibliography in [@VVGM]. Some further possible applications are connected with the paper [@KarLT].
Operator identities for the operators with $D$-difference kernels are discussed in Section \[OpId\].
The case of positive and boundedly invertible operators with $D$-difference kernels is treated further in Theorem \[TmPos\] of Section \[InvPr\]. As an application, we solve explicitly in terms of Weyl functions an inverse problem for a subclass of canonical systems. Some results from [@SaL2; @SaL3] are developed further in this section too.
We use the standard notations $\BC$ and $\BC_+$ for the complex plane and upper semi-plane, respectively. By $\{{\cal H}_1, \,{\cal H}_2\}$ we denote the class of the bounded linear operators acting from ${\cal H}_1$ into ${\cal H}_2$, and by $\s(\b)$ we denote the spectrum of $\b$.
Inversion of operators with $D$-difference kernels {#Inver}
==================================================
Consider a self-adjoint operator with $D$-difference kernel $$\label{i1}
S=I+\int_0^l k(x,t) \, \cdot \, dt, \quad k(x,t)=\{k_{ij}(x,t)\}_{i,j=1}^p=k_{ij}(d_i x-d_j t),$$ where $I$ is the identity operator, the $p \times p $ matrix function $k(x)$ on the right hand side of the second relation in (\[i1\]) is given by the equalities $$\label{i2}
k(x)=\T_2^*e^{i x \b^*}\T_1 \quad (x>0), \quad k(-x)=k(x)^*,$$ $\T_m$ ($m=1,2$) is an $n \times p$ matrix, and $\b$ is an $n \times n$ matrix for some integer $n>0$. Without loss of generality we assume further that $$\label{i3}
d_1 \geq d_2 \geq \ldots \geq d_p>0.$$
\[defS\] We suppose that equalities (\[i2\]) hold on $(0, d_1l)$, and so, according to (\[i3\]), each entry $k_{ij}(x)$ is determined by (\[i2\]) on the interval, which contains $(-d_j l, d_il)$, i.e., the operator $S$ of the form (\[i1\]) is determined by (\[i2\]).
Introduce the operator $$\label{i4}
E \in \{L^2_p(0,l), \, L^2(D)\} \quad L^2(D):=L^2(0, d_1l)\oplus L^2(0, d_2l)\oplus \ldots \oplus L^2(0, d_pl)$$ by the equality $(Ef)_j(z)=f_j(z/d_j)$. We shall denote also by $E$ the corresponding operator from $\{L^2_p(0,l), \, L^2_p(0,d_1l)\}$ with the natural embedding of $L^2(D)$ into $L^2_p(0,d_1l)$: $$\label{i5}
(Ef)_j(z)=f_j(z/d_j) \quad (0<z<d_j l), \quad (Ef)_j(z)=0 \quad (d_jl<z<d_1 l).$$ By (\[i1\]) and (\[i5\]) it is easy to see that $$\begin{aligned}
\label{i6}&&
S=E^{-1}\Big(I+\int_0^a \wt k(y, z) \, \cdot \, dz\Big) E, \quad a:=d_1l, \\
&&
\label{i7}
\wt k(y,z)=\{\wt k_{ij}(y, z)\}_{i,j=1}^p, \quad \wt k_{ij}(y, z)=0
\quad {\mathrm{if}}\, \, z>d_j l \, \,
{\mathrm{or}}\, \, y> d_i l, \\
&&
\label{i8}
\wt k_{ij}(y,z)=\frac{1}{d_j}k_{ij}(y- z) \quad {\mathrm{if}}\, \, 0<z<d_j l \, \,
{\mathrm{and}}\, \, 0<y<d_il.\end{aligned}$$ According to (\[i2\]), (\[i7\]), and (\[i8\]), the operator $$\label{i9}
\wt S=I+\int_0^a \wt k(y, z) \, \cdot \, dz$$ is not an operator with a difference kernel but it is a semiseparable operator. Recall [@GGK1] that the integral operator $\wt S$ of the form (\[i9\]) is called semiseparable, when $\wt k$ admits representation $$\label{2v3}
\wt k(y, z)=F_1(y)G_1(z) \quad {\mathrm{for}} \, \,
y> z, \quad \wt k(y, z)=F_2(y)G_2( z)
\quad {\mathrm{for}} \, \, y< z,$$ where $F_1$ and $F_2$ are $p \times n$ matrix functions and $G_1$ and $G_2$ are $n \times p$ matrix functions for some $n>0$. It is assumed that the entries of $F_1$, $F_2$, $G_1$, and $G_2$ are square integrable. When the operator $\wt S$ is invertible and its kernel $\wt k$ is given by (\[2v3\]), the kernel of the operator $\wt T=\wt S^{-1}$ is expressed in terms of the $2 n \times 2 n$ solution $U$ of the differential equation $$\label{2v4}
\Big(\frac{d}{d y}U\Big)(y)=\wt J \wt H(y)U(y), \quad y \geq 0, \quad U(0)=I_{2n},$$ where $$\begin{aligned}
&& \label{2v5}
\wt J \wt H(y):=B(y)C(y), \quad
\wt J=\big( \wt J^*\big)^{-1}=
\left[ \begin{array}{lr} 0 &
-I_p \\ I_p & 0
\end{array} \right].
\\ && \label{2v5'} B(y)=\left[\begin{array}{c}
-G_1(y) \\ G_2(y)
\end{array}
\right] , \quad
C(y)=\left[\begin{array}{lr}
F_1(y) & F_2(y)
\end{array}
\right].\end{aligned}$$ Namely, we have (see, for instance, [@GGK1]) $$\label{2v6}
\wt T=\wt S^{-1}=I+\int_0^a \wt T(y,z) \, \cdot \, dz,$$ $$\label{2v7}
\wt T(y,z)=\left\{\begin{array}{l}
C(y)U(y)\big(I_{2 n}-P^{\times}\big)U(z)^{-1}B(z), \quad y>z, \\
-C(y)U(y)P^{\times}U(z)^{-1}B(z), \quad y<z. \end{array} \right.$$ Here $P^{\times}$ is given in terms of the $n \times n$ blocks $U_{21}(a)$ and $U_{22}(a)$ of $U(a)$: $$\label{2v8}
P^{\times}=\left[\begin{array}{lr}
0 & 0 \\ U_{22}(a)^{-1}U_{21}(a) & I_{n}
\end{array}
\right],$$ and the invertibility of $U_{22}(a)$ is a necessary and sufficient condition for the invertibility of $\wt S$.
When the semiseparable operator $\wt S$ is not invertible, its kernel subspace is given by the equality ([@GGK1], p. 157): $$\label{do1}
\ker \wt S =\{h(y): \, h(y)=C(y)U(y)\left[ \begin{array}{c} 0 \\
g
\end{array} \right], \, U_{22}(a)g=0\}.$$
Rewrite $D$ in the form $$\begin{aligned}
&&\nonumber
D={\mathrm{diag}}
\{\wt d_1I_{p_1}, \ldots, \wt d_k I_{p_k} \}, \quad p_1+ \ldots +p_k=p, \\ &&
\label{i10}
\wt d_{j_1} > \wt d_{j_2}> 0 \quad ( j_1<j_2 \leq k),\end{aligned}$$ and put $$\label{i11}
\wt d_{k+1}=0, \quad P_{k+1}=I_p, \quad
P_j= {\mathrm{diag}}
\{I_{p_1}, \ldots, I_{p_{j-1}}, \, 0, \ldots, 0 \} \, \, (2 \leq j \leq k).$$ Then, in view of (\[i2\]), (\[i7\]), (\[i8\]), and (\[2v5’\]) we have $$\label{i12}
B(y)=e^{-y {\cal A}}
\left[\begin{array}{c}
- \T_1 \\ \T_2
\end{array}
\right] D^{-1}P_j, \quad
C(y)=P_j \left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]e^{y {\cal A}},$$ for $$\label{i13}
\wt d_{j}l<y<\wt d_{j-1} l \quad (2 \leq j \leq k+1),
\quad
{\cal A}:= i\left[\begin{array}{lr}
\b^* &0 \\ 0 & \b
\end{array}
\right].$$
\[Ker\] By (\[i5\]), (\[do1\]) and (\[i12\]), it is immediate that $$\label{do2}
\ker \, \wt S \in {\mathrm{Im}}E.$$ where [[Im]{}]{} means image. The integral parts of $S$ and $\wt S$ are compact operators. Hence, if $\wt S$ is not invertible, then $\ker \wt S\not=0$, and according to (\[do2\]) the subspace $E^{-1}\ker \wt S$ is well defined. In view of (\[i6\]) and (\[i9\]), we have $S E^{-1}\ker \wt S=0$, i.e., $S$ is not invertible too. It follows from (\[i6\]), (\[2v6\]), and (\[2v7\]) that if $\wt S$ is invertible, then $S$ is invertible. In other words, $S$ and $\wt S$ are simultaneousöy invertible.
Next, introduce notations $$\label{i14}
{\cal A}_j^{\times}= {\cal A}+Y_j,
\quad
Y_j=
\left[\begin{array}{c}
- \T_1 \\ \T_2
\end{array}
\right] D^{-1}P_j\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right].$$ For $2 \leq j \leq k+1$, put $$\label{i15}
U(y)=e^{-y {\cal A}}e^{(y-\wt d_jl) {\cal A}_j^{\times}}e^{\wt d_jl {\cal A}}U(\wt d_jl)
\quad (\wt d_{j}l\leq y \leq \wt d_{j-1} l), \quad U(0)=I_{2n},$$ Now, we are prepared to formulate the inversion theorem.
\[S1\] Let $S$ be an operator with the $D$-difference kernel, which has the form (\[i1\]), where $k$ is given by (\[i2\]) and $D$ satisfies (\[i10\]). Let also $\det U_{22}(a)\not=0$ for $U$ given by (\[i15\]). Then $S$ is invertible and its inverse is given by the formula $S^{-1}=E^{-1} \wt TE$, where $E$ is defined by (\[i5\]) and $\wt T$ is given by (\[2v6\])-(\[2v8\]). The matrix functions $B$ and $C$ in (\[2v7\]) are given by (\[i12\]) and the $\wt J$-unitary matrix function $U$ in (\[2v7\]) has the form (\[i15\]).
. To prove the theorem we need to show that $U$ of the form (\[i15\]) satisfies (\[2v4\]). Then by the properties of the semiseparable operators we shall obtain that $\wt S$ given by (\[i9\]) is invertible and that $\wt T=\wt S^{-1}$ is given by (\[2v6\])-(\[2v8\]), (\[i15\]). The formula $S^{-1}=E^{-1}\wt TE$ will be immediate from (\[i6\]).
By formulas (\[i12\]) and (\[i14\]) it is easy to see that $U$ of the form (\[i15\]) satisfies equation $$\begin{aligned}
&& \nonumber
\Big(\frac{d}{d y}U\Big)(y)=
e^{-y {\cal A}}\Big( {\cal A}_j^{\times}
-{\cal A}
\Big)
e^{y {\cal A}}e^{-y {\cal A}}
e^{(y-\wt d_jl) {\cal A}_j^{\times}}e^{\wt d_jl {\cal A}}U(\wt d_jl)
\\ && \label{i16}
=e^{-y {\cal A}}Y_je^{y {\cal A}}U(y)=B(y)C(y)U(y)\end{aligned}$$ for $0 \leq y \leq a$. Hence, by (\[2v5\]) $U$ satisfies (\[2v4\]).
Finally, let us prove that $U$ is $\wt J$-unitary, i.e., $U(y)^*\wt J U(y)=\wt J$. Indeed, according (\[i13\]) we have $$\label{i18}
{\cal A}^*=-\wt J{\cal A}\wt J^*.$$ As we noted in (\[i16\]), the equality $B(y)C(y)=e^{-y {\cal A}}Y_je^{y {\cal A}}$ is true. Thus, taking into account (\[2v5\]), (\[i14\]), and (\[i18\]) we obtain $$\begin{aligned}
&& \nonumber
\wt H(y)=\wt J^* e^{-y {\cal A}}Y_je^{y {\cal A}}=e^{y {\cal A}^*}\wt J^*Y_je^{y {\cal A}}\\ && \label{i19}
=e^{y {\cal A}^*}\left[\begin{array}{c}
\T_2 \\ \T_1
\end{array}
\right] D^{-1}P_j\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]e^{y {\cal A}} \geq 0.\end{aligned}$$ It follows from (\[i19\]) that $\wt H^*=\wt H$. Therefore, formulas (\[2v4\]) and (\[2v5\]) imply $\frac{d}{dy}\Big(U(y)^*\wt JU(y)\Big)=0$. Moreover, from $\frac{d}{dy}\Big(U(y)^*\wt JU(y)\Big)=0$ and $U(0)=I_{2n}$ we get $U(y)^*\wt J U(y)=\wt J$.
\[T\] If $S$ is invertible, then from Theorem \[S1\] we derive $$\label{i20}
T=S^{-1}=I+\int_0^l \{T_{ij}(x,t)\}_{i,j=1}^p \, \cdot \, dt,$$ where for $d_ix>d_jt$ and $e_i=\left[\begin{array}{lr}
\overbrace{0 \quad \ldots \quad 0}^{i-1} \quad 1 \quad 0 \quad \ldots \quad 0
\end{array}
\right]$ we have $$T_{ij}(x,t)=
e_i
\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]e^{d_ix {\cal A}}
U(d_ix)\big(I_{2 n}-P^{\times}\big)U(d_jt)^{-1}
e^{-d_j t {\cal A}}
\left[\begin{array}{c}
- \T_1 \\ \T_2
\end{array}
\right]e_j^*,$$ and for $d_i x<d_j t$ we have $$T_{ij}(x,t)=
-e_i
\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]e^{d_ix {\cal A}}
U(d_ix)P^{\times}U(d_jt)^{-1}
e^{-d_j t {\cal A}}
\left[\begin{array}{c}
- \T_1 \\ \T_2
\end{array}
\right]e_j^*.$$
Operator identities for operators with $D$-difference kernels {#OpId}
=============================================================
According to [@SaL1] (Ch. 6) a bounded in $L^2_p(0,l)$ operator $S$ with $D$-difference kernel, that is, an operator of the form (\[0.1\]), (\[0.2\]) satisfies the operator identity $$\label{1.1}
AS-SA^*=i \Pi J\Pi^*,$$ where $A_l=A \in \{L^2_p(0,l), \, L^2_p(0,l)\}$, $\Pi_l=\Pi=[\Phi_1 \quad \Phi_2]$, $\Phi_k \in
\{\BC^p, \, L^2_p(0,l)\}$, the index “$l$” is often omitted in our notations, and $$\label{1.2}
A=i D \int_0^x \, \cdot \, dt, \quad \Phi_1 g =D s(x,0)g, \quad \Phi_2 g \equiv g.$$ It is said that $A$, $S$, and $\Pi$, which satisfy (\[1.1\]), form an $S$-node. Further we assume that $A$ and $\Phi_2$ have the form (\[1.2\]). Operator identities play an important role in the study of structured operators [@SaL1; @SaL20; @SaL3].
Let us show that not only the operator with the $D$-difference kernel satisfies (\[1.1\]) but the inverse statement is also true, i.e., (\[1.1\]) implies that $S$ is an operator with a $D$-difference kernel (see also the corresponding statement in Example 1.2, p. 104 [@SaL3]). Quite similar to the proof of Theorem 1.3 ([@SaL1], p. 11), where the case $D=I_p$ was treated, one can prove the following theorem
\[TmId\] Suppose a bounded operator $T \in \{L^2_p(0,l), \, L^2_p(0,l)\}$ satisfies the operator identity $$\label{p1}
TA-A^*T=i \int_0^l Q(x,t) \, \cdot \, dt,$$ $$\label{p1'}
Q(x,t)=Q_1(x)Q_2(t),$$ where $Q$, $Q_1$, and $Q_2$ are $p \times p$, $p \times \wh p$, and $\wh p \times p$ $(\wh p >0)$ matrix-functions, respectively. Then $T$ has the form $$\label{p2}
Tf=\frac{d}{dx} \int_0^l
\frac{\p}{\p t}\Upsilon(x,t)f(t)dt,$$ where $\Upsilon(x,t)=\{\Upsilon_{ij}(x,t)\}_{i,j=1}^p$ is absolutely continuous in $t$, and $$\label{p3}
\Upsilon_{ij}(x,t):=(2d_i d_j)^{-1}\int_{d_i x+d_j t}^{f_{min}}
Q\Big(\frac{u+d_i x-d_j t}{2d_i}, \frac{u-d_i x+d_j t}{2d_j}\Big)du,$$ $$\label{p4} f_{min}:=\min\big(d_i(2l-x)+d_jt, \, d_i x+d_j(2l-t)\big).$$
In fact, Theorem \[TmId\] is true for a much wider class of functions $Q$ than the one given by (\[p1’\]). Similar to Theorem 2.2 in [@SaL1], the next theorem is immediate from Theorem \[TmId\] and equality $\wh U A \wh U =A^*$ ($(\wh U f)(x)=\ov{f(l-x)}$).
\[TmId2\] Suppose $S \in \{L^2_p(0,l), \, L^2_p(0,l)\}$ satisfies the operator identity $AS-SA^*=i \int_0^l\big(\Phi_1(x)+\wh \Phi_1(t)\big) \, \cdot \, dt$, where $\Phi_1(x)$ and $\wh \Phi_1(t)$ are $p \times p$ matrix functions with the entries from $L^2(0,l)$. Then $S$ is an operator with a $D$-difference kernel, i.e., the operator of the form (\[0.1\]), (\[0.2\]), and $s(u,0)=D^{-1}\Phi_1(u)$, $s(0,u)=-D^{-1}\wh \Phi_1(u)$. Moreover, when (\[1.1\]) holds, that is, $\wh \Phi_1(t)=\Phi_1(t)^*$ we have $$\label{p12} s(x,t)=-D^{-1}s(t,x)^*D, \quad S=S^*.$$
Positive operators $S$ and an inverse problem for canonical system {#InvPr}
==================================================================
Operators with $D$-difference kernels are essential for the construction of solutions of an inverse problem for an important subclass of canonical systems [@SaA1; @SaL2; @SaL3]. Canonical system is a system of the form $$\label{1.3}
\frac{d}{d x}w(x,\lambda )=i\lambda JH(x)w(x,\lambda ),
\quad
H(x) \geq 0, \quad J=
\left[ \begin{array}{lr} 0 &
I_p \\ I_p & 0
\end{array} \right],$$ where the Hamiltonian $H$ is a $m \times m$ ($m=2p$) locally summable matrix function. A Weyl function of the canonical system on the semi-axis $x \geq 0$ is a $p \times p$ matrix function $\vp(\la)$, which is analytic in $\BC_+$ and satisfies the condition [@SaL3] $$\label{1.6}
\int_0^\infty
\left[ \begin{array}{lr}
I_p &i \varphi (\la)^* \end{array} \right]
w(x, \la)^*H(x)w(x, \la)
\left[ \begin{array}{c}
I_p \\ -i \varphi (\la) \end{array} \right]
dx < \infty, \quad \la \in \BC_+.$$ The corresponding inverse problem is the problem to recover $H$ or, equivalently, canonical system from the Weyl function. In the case of rational Weyl matrix functions several inverse problems were solved explicitly using a GBDT version of the Bäcklund-Darboux transformation [@FKS1; @GKS1; @GKS6; @MST; @SaA2]. (See [@D; @GeT; @Gu; @MS; @SaA2; @ZM] and references therein for various versions of the Bäcklund-Darboux transformation and commutation methods.) However, taking into account that the positivity of operators $S$ and the application of the inversion formulas for semiseparable operators is of independent interest, we shall use a general scheme [@SaL0; @SaL3] and its modification [@SaA1] for the inverse problem treated in this section. As a result of the application of the general scheme to rational matrix functions, semiseparable operators appear. Inverse problems for self-adjoint and skew-self-adjoint Dirac-type systems were studied using semiseparable operators in [@AGLKS] and [@FKS], respectively.
Consider rational Herglotz $p \times p$ matrix functions $\vp$. The statement below is immediate from Theorem 5.2 [@GKS6].
\[RkW\] If $\vp$ is a rational matrix function such that $$\label{fW}
\lim_{\la \to \infty} \vp(\la)=\frac{i}{2}D, \quad \Im \vp( \la) \geq 0 \quad (\la \in \BC_+),$$ then $\vp$ admits a representation (i.e., realization in terms of control theory) $$\label{p6}
\vp(\la)=\frac{i}{2}D+\T_1^*(\b - \la I_n)^{-1}\T_2,$$ where $\T_1$ and $\T_2$ are $n \times p$ matrix functions, $n$ is some positive integer number, and $n \times n$ matrix $\b$ satisfies the matrix identity $$\label{p6'}
\b^*-\b =i\big(\T_2 -\T_1\big) D^{-1} \big(\T_2 -\T_1\big)^*.$$
Dirac systems and Weyl matrix functions $\wt \vp$, which have the form $\wt \vp =2D^{-\frac{1}{2}}\vp D^{-\frac{1}{2}}$, were studied in [@GKS1]. The next proposition follows from the Step 1 of the proof of Theorem 4.3 [@GKS1] (see also [@GKS6]).
\[NevRepr\] Let relations (\[p6\]) and (\[p6’\]) hold. Then $\Im \vp(\la)>0$ $(\la \in \BC_+)$ and $\vp$ admits Herglotz representation $$\label{p7}
\varphi ( \lambda )= \nu + \int_{- \infty }^{ \infty }
\Big( \frac{1}{z- \lambda } - \frac{z}{1+z^{2}}\Big)d \tau (z) \quad (\nu= \nu^*),$$ where $$\label{p8}
\tau ( z )= \int_{0}^{ z }\rho(t)d t + \sum_{ z_{k}< z } \nu_{k},$$ numbers $z_1<z_2< \ldots $ are the real eigenvalues of $\b$, $$\label{p9}
\nu_{k}={ \mathrm {res}}_{z=z_{k}} \T_{2}^{*}(zI_{n}- \beta )^{-1}
\T_{2} \geq 0,$$ and $\rho$ is $p \times p$ rational matrix function: $$\label{p10}
\rho( t)=\frac{1}{2 \pi}\zeta( t)^* D\zeta( t) \geq 0, \hspace{1em}
\zeta( t):=
I_{p}-i D^{-1} \big(\T_2 -\T_1\big)^*
( t I_{n}- \beta )^{-1}
\T_{2}.$$
It is easy to see from (\[1.2\]) that $$\label{p11}
(I-zA)^{-1}\Phi_2=e^{izxD}, \quad \Phi_2^*(I-zA^*)^{-1}f=\int_0^le^{-izxD}f(x)dx \quad (f \in L^2_p(0,l)).$$ By (\[p8\])-(\[p11\]) the right-hand side of the equality $$\label{p13}
S:=\int_{-\infty}^{\infty}(I-zA)^{-1}\Phi_2d \tau(z)\Phi_2^*(I-zA^*)^{-1}$$ weakly converges, and so the equality defines an operator $S$. Moreover, it is easy to see that the inequalities $$\label{p13'}
c(f,f)_{L^2}>(Sf,f)_{L^2}>0$$ hold for some fixed $c>0$ and arbitrary $f \not=0$. (Here $(\cdot , \cdot)_{L^2}$ denotes the scalar product in $L^2_p(0,l)$.) Thus, $S$ is a bounded and positive operator. We shall show that operators $S$ belong to a subclass of operators of the form (\[i1\]), (\[i2\]).
\[TmPos\] Let the matrix identity (\[p6’\]) hold. Then the operator $S$ given by (\[i1\]) and (\[i2\]) is positive and boundedly invertible.
. The theorem is obtained by proving that $S$ of the form (\[i1\]), (\[i2\]) admits representation (\[p13\]).
First, consider $S$ given by (\[p13\]). It can be calculated directly (see also Section 1.1 in [@SaL20]) that this operator $S$ satisfies the operator identity (\[1.1\]), where $\Pi=[\Phi_1 \quad
\Phi_2]$ and $$\label{p14}
\Phi_1=i\left(\nu -\int_{-\infty}^{\infty}\Big(A(I-zA)^{-1}+\frac{z}{1+z^2}I\Big)\Phi_2d \tau(z)
\right).$$ Here the operator $\Phi_1$ is an operator of multiplication by the matrix function, which we denote by $\Phi_1(x)$. From the identity (\[1.1\]) and Theorem \[TmId2\] it follows that $S$ is an operator with a $D$-difference kernel $s(x,t)=\{ s_{ij}(d_ix-d_j t)\}_{i,j=1}^p$ and $s(x,0)=D^{-1}\Phi_1(x)$. Introduce $S=S_l$ and $\Phi_1=\Phi_{1,l}$ by (\[p13\]) and (\[p14\]), respectively, for all $0<l<\infty$. Then the kernel $s(x)$ of the integral operators $S_l$ is determined on $\BR$ by the equalities $$\label{p15}
s_{ij}(x)=d_i^{-1}\big(\Phi_1\big)_{ij}(x/d_i) \quad (x>0), \quad s_{ij}(-x)=-\frac{d_j}{d_i}\ov{s_{ji}(x)}.$$ For $\vp$ satisfying (\[p7\]), according to Statement 3 in [@SaA1], after the corresponding change of notations we get $$\label{1.7}
\varphi (\la)=\la \int_0^{\infty}s(x,0)^*e^{i\la x D}dx D^2=\la \int_0^{\infty}e^{i\la x }s(x)^*dx D.$$ Note that in view of formula (\[p14\]) and Proposition \[NevRepr\] we can present $s$ as a sum $s(x)=s_1(x)+s_2(x)$, where the entries of $s_1$ are bounded and the entries of $s_2$ belong $L^2(0,\infty)$. Finally, we apply Fourier transform to derive from (\[1.7\]) the equality $$\label{1.7'}
e^{-\eta x}s(x)^*
=\frac{1}{2 \pi}
{\mathrm{
l.i.m.}}_{a \to \infty} \int_{- a}^{a}e^{-i \xi
x} \lambda^{-1} \vp(\lambda ) D^{-1}d \xi \quad (\lambda= \xi +i \eta , \quad \eta>0),$$ the limit l.i.m. being the limit in $L^2(0,l)$ ($0<l<\infty$). Using (\[p6\]) and (\[1.7’\]), we obtain $$\label{p16}
e^{-\eta x}s(x)^*
=\frac{1}{2 \pi}
{\mathrm{
l.i.m.}}_{a \to \infty} \int_{\G_a}e^{-i \xi
x} \lambda^{-1} \vp(\lambda ) D^{-1}d \xi \quad (\lambda= \xi +i \eta , \quad \eta>0),$$ where $\G_a$ is a clockwise oriented contour: $$\G_a=[-a, \, a]\cup\{\xi: \, |\xi|=a, \, \Im \xi<0\}.$$ It is easy to see that $$\label{p16'}
\frac{1}{2 \pi}
{\mathrm{
l.i.m.}}_{a \to \infty} \int_{\G_a}e^{-i \xi
x} \lambda^{-1} d \xi =-ie^{-\eta x}.$$ According to (\[p6’\]) we have $\s(\b)\subset \ov{\BC_-}$, where $\s$ is spectrum. Similar to [@FKS] we turn to zero $\ve$ in the equality $\la^{-1}(\b_{\ve}-\la I_n)^{-1}=\b_{\ve}^{-1}\big(\la^{-1}I_n+(\b_{\ve}-\la I_n)^{-1}\big)$, where $\det \b_{\ve}\not=0$, $\|\b -\b_{\ve}\|<\ve$, and thus obtain $$\label{p17}
\frac{1}{2 \pi}
{\mathrm{
l.i.m.}}_{a \to \infty} \int_{\G_a}e^{-i \xi
x} \lambda^{-1} (\b-\la I_n)^{-1}d \xi =e^{-\eta x}\int_0^x\exp(-iu\b)du.$$ Here we take into account that, when the spectrum of some matrix ${\cal{K}}$ is situated inside the anti-clockwise oriented contour $\G$ we have $$\frac{1}{2\pi i}\int_{\G}e^{- i \la x}(\la I_n - {\cal{K}})^{-1}d\la=\exp(-ix{\cal K}).$$ By (\[p6\]) and (\[p16\])-(\[p17\]) we get $$\label{p18}
s(x)
=\frac{1}{2 }I_p+D^{-1}\T_2^*\int_0^x\exp(iu\b^*)du\T_1 \quad (x>0).$$ It follows from (\[p15\]) that $s(x)=-D^{-1}s(-x)^*D$ ($x<0$), and so according to (\[p18\]) $s(x)$ is continuously differentiable for $x \not=0$. As the functions $s_{ij}(x)$ are continuous at $x=0$ for $i\not=j$, and $s_{ii}(+0)-s_{ii}(-0)=1$, formulas (\[0.1\]) and (\[0.2\]) imply (\[i1\]), where $k(x)=D\Big(\frac{d}{dx}s\Big)(x)$. Therefore we have $$\label{p19}
k(x)=
\T_2^*\exp(ix\b^*)\T_1 \quad (x>0), \quad k(x)=k(-x)^*.$$
Now, note that equalities (\[i2\]) and (\[p19\]) coincide. In other words, the operator $S$, which is considered in the theorem, admits representation (\[p13\]). Hence, by (\[p13’\]) this operator is bounded and positive, and so in view of (\[i1\]) and (\[i2\]) it is also boundedly invertible.
The matrix function $\tau$ of the form (\[p8\])-(\[p10\]) and the $S$-node given by (\[1.2\]), (\[p13\]), and (\[p14\]) satisfy conditions of Theorem 2.4 [@SaL3], p. 57. Therefore $\vp(\la)$ given by (\[p7\]) can be presented as a linear-fractional transformation $$\label{p20}
\varphi (\la ) =i\big( {\cal W}_{11}(\la )R_1(\la )+ {\cal W}_{12}(\la )R_2(\la )\big)
\big( {\cal W}_{21}(\la )R_1(\la )+ {\cal W}_{22}(\la )R_2(\la )\big)^{-1},$$ where ${\cal W}_{ij}(\la )$ are $p \times p$ blocks of the matrix function ${\cal W}$, $$\label{p21}
{\cal W}(\la ):=W(l, \ov \la)^*, \quad W(l, \la)=I_{2p}+i\la J\Pi^*S^{-1}(I -\la A)^{-1}\Pi,$$ and $R_1(\la)=R_1(l,\la)$, $R_2(\la)=R_2(l,\la)$ is a pair of $p \times p$ matrix functions, which are meromorphic in $\BC_+$ and have property-$J$, that is, $$\label{p22}
R_1(\la )^*R_1(\la )+R_2(\la )^*R_2(\la )>0,\quad \left[\begin{array}{lr}
R_1(\la )^* & R_2(\la )^* \end{array}\right]\, J\,
\left[\begin{array}{c} R_1(\la ) \\ R_2(\la ) \end{array}\right]\geq 0.$$ It is easy to see from (\[p21\]) that $\lim_{l \to +0}W(l, \la)=I_{2p}$ and thus we put $W(0,\la)=I_{2p}$. Now, by Theorem 2.1 from [@SaL3], p.54 the matrix function $W$ satisfies for $x \geq 0$ the equation $$\label{p23}
W(x, \la)=I_{2p}+i\la J \int_0^x\big(dB_1(r))W(r, \la), \quad B_1(r):=\Pi_r^*S_r^{-1}\Pi_r,$$ where $S_r \in \{L^2_p(0,r), \, L^2_p(0,r)\}$, $\Pi_r \in \{\BC^{2p}, \, L^2_p(0,r)\}$. As the operators $S_{r}$ ($0<r \leq l <\infty$) are invertible, the operators $S_l$ admit triangular factorisation (see [@GoKrb], p. 184). It follows that $B_1$ is differentiable, and we rewrite (\[p23\]) as the canonical system $$\begin{aligned}
\label{p24}&&
\frac{d}{d x}W(x, \la)=i\la J H(x)W(x, \la),
\\ &&\label{p24'}
H(x):=\frac{d}{dx}\Big(\Pi_x^*S_x^{-1}\Pi_x\Big).\end{aligned}$$ Moreover, in view of Remark \[T\] the kernel $T_r(x,t)$ of the integral operator $S_r^{-1}$ is continuous with respect to $x,t,r$ excluding the lines $d_i x = d_j t$. Therefore, for $d_i x \not= d_j r$ ($1\leq i,j \leq p$) similar to the continuous kernels ([@GoKrb], p.186) we have $$\label{p31}
k(x,r)+T_r(x,r)+\int_0^rk(x,u)T_r(u,r)du=0, \quad x \leq r \leq l.$$ Introduce an upper triangular operator $$\label{p32}
V_+=I+\int_x^lT_r(x,r) \, \cdot \, dr \in \{L^2_p(0, \, l)\}.$$ According to (\[p31\]) and (\[p32\]) the operator $S_lV_+$ is a lower triangular operator. Hence, the operator $V_+^*S_lV_+$ is a lower triangular operator. On the other hand $V_+^*S_lV_+$ is selfadjoint, and so the integral part of $V_+^*S_lV_+$ equals zero, i.e., $V_+^*S_lV_+=I$ or equivalently $$\label{p33}
S_l^{-1}=V_+V_+^*, \quad V_{+,l}^*=V_+^*=I+\int_0^xT_x(x,r) \, \cdot \, dr.$$ In the second equality above we used formula (\[p32\]) and relation $T_x(r,x)^*=T_x(x,r)$ ($x \geq r$).
\[aInvPr\] Let $\vp$ be a rational function, which satisfies (\[fW\]). Then $\vp$ is a Weyl function of the canonical system (\[p24\]), where the Hamiltonian $H$ has the form $$\label{p34}
H(x)=\g(x)^*\g(x), \quad \g(x)=\Big(V_+^*[\Phi_1 \quad \Phi_2]\Big)(x) \quad (x\leq l<\infty),$$ and the operator $V_+^*$ is given by (\[p33\]) and is applied columnwise to the matrix functions $\Phi_1(x)=\{d_is_{ij}(d_i x)\}_{i,j=1}^p$ and $\Phi_2 \equiv I_p$. The matrix function $s(x)$ is given by (\[p18\]) and the matrix function $T_x(x,r)$ in (\[p33\]) is given in Remark \[T\].
. It follows from (\[p24\]) that $$\begin{aligned}
&& \label{p25}
\frac{d}{dx}\Big(W(x, \ov \la)^*JW(x, \la)\Big)=0, \\
&& \label{p26}
\frac{d}{dx}\Big(W(x, \la)^*JW(x, \la)\Big)
=i( \la - \ov \la)W(x, \la)^*H(x)W(x, \la).\end{aligned}$$ In view of (\[p26\]) we obtain $$\label{p27}
\int_0^l
W(x, \la)^*H(x)W(x, \la)
dx = i(\ov \la - \la)^{-1}\Big(W(l, \la)^*JW(l, \la)-J\Big).$$ Note also that according to (\[p25\]) the equality $W(l, \ov \la)^*JW(l, \la)=J$ holds, or equivalently $$\label{p28}
W(l, \ov \la)^*=JW(l, \la)^{-1}J.$$
By Proposition \[RkW\] $\vp$ admits representation (\[p6\]) and identity (\[p6’\]) is valid. So, by Proposition \[NevRepr\] $\vp$ admits Herglotz representation, where the matrix function $\tau(t)$ has the form (\[p8\])-(\[p10\]). Hence, as it was shown above, the representation (\[p20\]) of $\vp$, where ${\cal W}$ is expressed via the matrizant $W(l, \la)$ and the pair $R_1$, $R_2$ satisfies (\[p22\]), is also true. Using (\[p28\]), we rewrite (\[p20\]) in the form $$\label{p29}
\left[\begin{array}{c}
I_p \\ -i\vp(\la)
\end{array}
\right]=W(l, \la)^{-1}J\left[\begin{array}{c}
R_1(\la) \\ R_2(\la)
\end{array}
\right]\big( {\cal W}_{21}(\la )R_1(\la )+ {\cal W}_{22}(\la )R_2(\la )\big)^{-1}.$$ Taking into account (\[p22\]), (\[p27\]), and (\[p29\]) we derive $$\begin{aligned}
&& \nonumber
\int_0^l
\left[ \begin{array}{lr}
I_p &i \varphi (\la)^* \end{array} \right]
W(x, \la)^*H(x)W(x, \la)
\left[ \begin{array}{c}
I_p \\ -i \varphi (\la) \end{array} \right]
dx \leq i( \la - \ov \la)^{-1}\\ \label{p30} &&
\times \left[ \begin{array}{lr}
I_p &i \varphi (\la)^* \end{array} \right]
J
\left[ \begin{array}{c}
I_p \\ -i \varphi (\la) \end{array} \right],
\quad \la \in \BC_+.\end{aligned}$$ As the right-hand side in the inequality (\[p30\]) does not depend on $l$ we can substitute $\infty$ instead of the limit $l$ of integration in the left-hand side. Hence $\vp$ is a Weyl function of the constructed system.
According to the second relation in (\[p33\]) we obtain $(V_{+,l}^*f)(x)=(V_{+,x}^*\wt f)(x)$ for $x \leq l$, where $\wt f$ is the restriction of $f$ on the interval $[0,x]$. Therefore, relations (\[p24’\]) and (\[p33\]) imply (\[p34\]).
\[appl\] Let the conditions of Theorem \[aInvPr\] hold and let $\det \b \not=0$. Then we have $$\label{p35}
\g(x)=\Big(V_+^*[\frac{1}{2}D+i\T_2^*(\b^*)^{-1}\T_1 \qquad I_p]\Big)(x)-i[\g_0(x) \quad 0],$$ where the $s$-th row of $\g_0$ $(p \geq s \geq 1)$ is given by the equality $$\begin{aligned}
\nonumber &&
e_s\g_0(x)=e_s \Big(\T_2^* e^{id_sx {\b^*}}+
\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]e^{d_sx {\cal A}}
U(d_s x)
\\ \label{p36} && \times
\big(P^{\times}U(d_1x)^{-1}- U(d_sx)^{-1}+ I_{2 n}-P^{\times}\big)\left[\begin{array}{c}
I_p \\ 0
\end{array}
\right]\Big)(\b^*)^{-1}\T_1,\end{aligned}$$ $U$ in (\[p36\]) is defined by (\[i15\]) after substitution $l=x$, and $P^{\times}$ is defined by (\[2v8\]) after substitution $a=d_1l=d_1x$.
. By (\[0.2\]), (\[1.2\]), and (\[p18\]) the equality $$\label{p37}
e_s[\Phi_1(x) \quad \Phi_2]=e_s[\frac{1}{2}D+i\T_2^*(\b^*)^{-1}\T_1 -i \T_2^* e^{id_sx {\b^*}}
(\b^*)^{-1}\T_1 \qquad I_p]$$ is true. Using (\[p37\]) and the second equlity in (\[p34\]) we obtain (\[p35\]), where $$\label{p38}
\g_0(x)=\Big(V_+^*\{e_s\T_2^* e^{id_sx {\b^*}}
(\b^*)^{-1}\T_1\}_{s=1}^p\Big)(x).$$ From (\[i13\]) it follows that $$\label{p39}
\T_2^* e^{id_sx {\b^*}}=[\T_2^* \quad \T_1^*]e^{d_sx {\cal A}}\left[\begin{array}{c}
I_p \\ 0
\end{array}
\right].$$ According to the representation of $V_+^*$ in (\[p33\]), Remark \[T\], formula (\[p39\]) and second relation in (\[i14\]) we get $$\label{p40}
V_+^*\{e_s\T_2^* e^{id_sx {\b^*}}
\}_{s=1}^p=\{e_s\T_2^* e^{id_sx {\b^*}}
\}_{s=1}^p
+\{{\cal F}_s(x)
{\cal G}_s(x)
\}_{s=1}^p,$$ where $$\label{p41}
{\cal F}_s(x)=e_s\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]e^{d_sx {\cal A}}
U(d_s x),$$ $$\begin{aligned}
&& \nonumber
{\cal G}_s(x)=
\Big((I_{2 n}-P^{\times})\int_0^{d_s x}U(z)^{-1}e^{-z{\cal A}}Y(z)e^{z {\cal A}}dz
\\ \label{p42} &&
-P^{\times}\int_{d_s x}^{d_1 x}U(z)^{-1}e^{-z{\cal A}}Y(z)e^{z {\cal A}}dz\Big)\left[\begin{array}{c}
I_p \\ 0
\end{array}
\right],\end{aligned}$$ $$Y(z)=\sum_{j: \, d_j >\wt d_m}\frac{1}{d_j}\left[\begin{array}{c}
- \T_1 \\ \T_2
\end{array}
\right] e_j^*e_j\left[\begin{array}{lr}
\T_2^* & \T_1^*
\end{array}
\right]=Y_m \quad {\mathrm{for}} \quad \wt d_m x\leq z \leq \wt d_{m-1}x.$$ Taking into account (\[i16\]) rewrite (\[p42\]) in the form $$\label{p43}
{\cal G}_s(x)=
\Big((I_{2 n}-P^{\times})\big(I_{2 n}-U(d_s x)^{-1}\big)
+P^{\times}\big(U(d_1x)^{-1}-U(d_s x)^{-1}\big)\Big)\left[\begin{array}{c}
I_p \\ 0
\end{array}
\right].$$ Finally, formulas (\[p38\]), (\[p40\]), (\[p41\]), and (\[p43\]) imply (\[p36\]).
In view of Corollary \[appl\], to recover $\g$ and Hamiltonian $H$ we need only to calculate the action of $V_+^*$ on constant vectors.
The matrix function $\g(x)$, which is recovered in Theorem \[aInvPr\], satisfies the equality $$\label{p44}
\g(x)J\g(x)^*\equiv D.$$ Indeed, by (\[1.1\]), the first equality in (\[p33\]), and the second equality in (\[p34\]) we have $$\label{p45}
V_+^*A(V_+^*)^{-1}-V_+^{-1}A^*V_+=i\g(x)J\int_0^l \g(t)^*\, \cdot \, dt.$$ As $V_+^*A(V_+^*)^{-1}$ is a lower triangular operator and $V_+^{-1}A^*V_+$ is an upper triangular operator, we derive $$\label{p46}
V_+^*A(V_+^*)^{-1}=i\g(x)J\int_0^x \g(t)^*\, \cdot \, dt.$$ Rewrite (\[p46\]) in terms of the kernels of the corresponding integral operators and put $t=x$ to get (\[p44\]).
As it is stated in the proposition below, equality (\[p44\]) means that we recover canonical systems from the subclass of systems with linear similar matrix functions $JH(x)$, though (differently from [@SaL3], p. 104) the kernel of $S^{-1}$ is not necessarily continuous.
\[sim\]Let the conditions of Theorem \[aInvPr\] hold. Then $JH(x)$ is similar to the matrix $JH_0$, where $$\label{p47}
H_0:=\left[\begin{array}{lr}
D & 0 \\ 0 & 0
\end{array}
\right]$$
. Fix $x \geq 0$ and denote by $X$ a $p \times 2p$ matrix such that it has rank $p$ and satisfies the equality $XJ\g(x)^*=0$. As the maximal $J$-nonnegative subspaces are $p$-dimensional, it easily follows from $\g(x)J\g(x)^*>0$ and $XJ\g(x)^*=0$ that $XJX^*<0$. Then, we have $$\label{p48}
\wt X J \wt X^*=-I_p, \quad \wt X J \g(x)^*=0 \quad {\mathrm{for}} \quad
\wt X:=(-XJX^*)^{-\frac{1}{2}}X.$$ Now, put $$\label{p49}
L:=\left[\begin{array}{c}
D^{-\frac{1}{2}}\g(x) \\ \wt X
\end{array}
\right].$$ By (\[p44\]), (\[p48\]), and (\[p49\]) the equality $$\label{p50}
L^{-1}=[J\g(x)^*D^{-\frac{1}{2}} \quad -J\wt X^*]$$ is true. According to (\[p47\]), (\[p49\]), and (\[p50\]) we get $L^{-1}H_0L=J\g(x)^*\g(x)$. In view of (\[p34\]) the last equality yields $L^{-1}H_0L=JH(x)$.
[**Acknowledgement.**]{} The work of A.L. Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant no. Y330, and his visit to Mexico was supported by the PIFI grant P/CA-9 2007-14-17. A.L. Sakhnovich is grateful to the Autonomous University of Hidalgo for its hospitality.
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*A.L. Sakhnovich,\
Fakultät für Mathematik, Universität Wien,\
Nordbergstrasse 15, A-1090 Wien, Austria\
e-mail: [al$_-$sakhnov@yahoo.com ]{}*
*A.A. Karelin,\
Universidad Autonoma del Estado de Hidalgo, Instituto de Ciencias Basicas e Ingenieria Centro de investigaci—n Avanzada en Ingenier’a Industrial Carretera Pachuca-Tulancingo, Km. 4,5 Ciudad Universitaria, C.P. 42180, Pachuca, Hidalgo, Mexico\
karelin@uaeh.edu.mx*
*J. Seck-Tuoh-Mora,\
Universidad Autonoma del Estado de Hidalgo, Instituto de Ciencias Basicas e Ingenieria Centro de investigaci—n Avanzada en Ingenier’a Industrial Carretera Pachuca-Tulancingo, Km. 4,5 Ciudad Universitaria, C.P. 42180, Pachuca, Hidalgo, Mexico\
jseck@uaeh.edu.mx*
*G. Perez-Lechuga,\
Universidad Autonoma del Estado de Hidalgo, Instituto de Ciencias Basicas e Ingenieria Centro de investigaci—n Avanzada en Ingenier’a Industrial Carretera Pachuca-Tulancingo, Km. 4,5 Ciudad Universitaria, C.P. 42180, Pachuca, Hidalgo, Mexico\
glechuga2004@hotmail.com*
*M. Gonzalez-Hernandez,\
Universidad Autonoma del Estado de Hidalgo, Instituto de Ciencias Basicas e Ingenieria Centro de investigaci—n Avanzada en Ingenier’a Industrial Carretera Pachuca-Tulancingo, Km. 4,5 Ciudad Universitaria, C.P. 42180, Pachuca, Hidalgo, Mexico\
mghdez@uaeh.edu.mx*
|
---
abstract: 'This paper is concerned with longtime dynamics of semilinear Lamé systems $$\partial^2_t u - \mu \Delta u - (\lambda + \mu) \nabla {\rm div} u + \alpha \partial_t u + f(u) = b,$$ defined in bounded domains of $\mathbb{R}^3$ with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to a critical forcing $f(u)$. Writing $\lambda + \mu$ as a positive parameter $\varepsilon$, we discuss some physical aspects of the limit case $\varepsilon \to 0$. Then, we show the upper-semicontinuity of attractors with respect to the parameter when $\varepsilon \to 0$. To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before.'
author:
- |
**Lito Edinson Bocanegra-Rodr[í]{}guez\
Institute of Mathematical and Computer Sciences, University of São Paulo, 13566-560 São Carlos, SP, Brazil\
**To Fu Ma\
Department of Mathematics, University of Brasília, 70910-900 Brasília, DF, Brazil\
**Paulo Nicanor Seminario-Huertas [^1]\
Academic Department of Mathematics, National University of Callao, Bellavista 07011, Callao, Peru\
Department of Mathematics, University of Brasília, 70910-900 Brasília, DF, Brazil\
**Marcio Antonio Jorge Silva\
Department of Mathematics, State University of Londrina, 86057-970 Londrina, PR, Brazil********
title: '**Longtime dynamics of a semilinear Lamé system**'
---
[**Keywords:**]{} System of elasticity, global attractor, gradient system, upper-semicontinuity.
[**MSC:**]{} 35B41, 74H40, 74B05.
Introduction {#sec-introduction}
============
The Lamé system is a classical model for isotropic elasticity. In three dimensions, it is given by $$\begin{aligned}
\left\lbrace
\begin{array}{ll}
\partial^2_{t} u - \mu \Delta u - (\lambda+\mu) \nabla {\rm div} u = 0 & \text{in} \,\; \Omega\times\mathbb{R}^{+}, \smallskip \\
u=0 & \text{on} \,\; \partial \Omega\times\mathbb{R}^{+}, \smallskip \\
u(0)=u_0, \,\; \partial_t u(0)=u_1 & \text{in} \; \Omega,
\end{array}
\right.
\label{1}\end{aligned}$$ where $\Omega$ is a bounded domain of $\mathbb{R}^3$ with smooth boundary $\partial \Omega$, representing the elastic body in its rest configuration. Here, the vector $u=(u_1,u_2,u_3)$ denotes displacements and $\lambda,\mu$ are Lamé’s constants with $\mu > 0$. In this model, the stress tensor is given by $$\label{ten}
\sigma(u)_{ij}=\lambda {\rm div} u \, \delta_{ij} + \mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right).$$ We refer the reader to [@Achenbach; @Ciarlet; @Love; @Teodorescu] for modeling aspects and [@Cerv2001; @KK; @Pujol] for some applications of vector waves. Later, we discuss the physical justification of taking limit $\lambda + \mu \to 0$.
We note that the energy functional corresponding to the linear system is given by $$E_{\ell}(t) = \frac{1}{2}\int_{\Omega}\left( |\partial_t u|^2 + \mu |\nabla u |^2 + (\lambda + \mu) |{\rm div}u|^2 \right) \, dx,$$ which is conservative since we have formally $\frac{d}{dt}E_{\ell}(t)=0$. This motivated several papers on such systems where the main feature is finding suitable damping and controllers in order to get stabilization and controllability, respectively. Let us recall some related results. The exponential stabilization of Lamé systems, defined in exterior domains of $\mathbb{R}^3$ with Dirichlet boundary, was studied by Yamamoto [@Yama]. Uniform stabilization by nonlinear boundary feedback was studied by Horn [@Horn]. Polynomial stabilization with interior localized damping was studied by Astaburuaga and Charão [@Asta]. By adding viscoelastic dissipation of memory type, Bchatnia and Guesmia [@BG] established the so-called general stability. More recently, Benaissa and Gaouar [@Benaissa] studied strong stability of Lamé systems with fractional order boundary damping. With respect to controllability, we refer the reader to, for instance, [@Alabau; @BL; @Lagnese; @Lions; @Liu].
Our objective in the present article is different and goes further than considering stabilization. We are concerned with longtime dynamics of Lamé systems under nonlinear forces. Here, the above linear system (\[1\]) becomes $$\begin{aligned}
\left\lbrace
\begin{array}{ll}
\partial^2_{t} u - \mu \Delta u - (\lambda+\mu) \nabla {\rm div} u + \alpha \partial_t u + f(u) = b & \text{in} \,\; \Omega\times\mathbb{R}^{+}, \smallskip \\
u=0 & \text{on} \,\; \partial \Omega\times\mathbb{R}^{+}, \smallskip \\
u(0)=u_0, \,\; \partial_t u(0)=u_1 & \text{in} \; \Omega,
\end{array}
\right.
\label{problem}\end{aligned}$$ where $\alpha \partial_t u$ ($\alpha >0$) represents a frictional dissipation, $f(u)$ stands for a nonlinear structural forcing, and $b=b(x)$ represents some external force. As far as we know, the long-time dynamics of semilinear Lamé systems has not been studied before. We present two main results. Firstly, we establish the existence of global attractors with finite fractal-dimensional. Secondly, by taking $\lambda + \mu = \varepsilon >0$, we study the upper semicontinuity of attractors with respect to $\varepsilon \to 0$.
In what follows we summarize the main contributions of the paper.
$(i)$ Our first result establishes existence of global attractors for dynamics of problem (\[problem\]) under nonlinear forces with critical growth $|f_i(u)| \approx |u|^p + |u_i|^3$, $p<3$, $i=1,2,3$. Under careful energy estimates, we show that the system is gradient and quasi-stable in the sense of [@chueshov-book; @Von]. Then we conclude that the attractors are smooth and have finite fractal dimension. See Theorem \[3.16\].
$(ii)$ In Section \[sec-phys\], we discuss the physical meaning of the limit case $\lambda + \mu \to 0$ in real world applications. This arises mainly in Seismology.
$(iii)$ Finally, setting $\varepsilon = \lambda + \mu \to 0$, we consider the $\varepsilon$-problem $$\partial^2_{t} u - \mu \Delta u - \varepsilon \nabla {\rm div} u + \alpha \partial_t u + f(u) = b,$$ depending on a parameter $\varepsilon \ge 0$. In Theorem \[singular\] we show that the weak solutions of $\varepsilon$-problem converges to the vectorial wave equation with $\varepsilon=0$. Then we provide all necessary analysis to prove that corresponding family of attractors $\mathcal{A}_{\varepsilon}$ is upper semicontinuous with respect to $\varepsilon \to 0$. This is given in a suitable phase space. See Theorem \[upper\].
Preliminaries {#sec-well-posed}
=============
Physical aspects of $\lambda + \mu \to 0$ {#sec-phys}
-----------------------------------------
From the Hooke law and from the constitutive law (\[ten\]) referring to elastic bodies, one derives the equation $$\label{ex}
\rho\partial^2_{t} u-\mu \Delta u-(\lambda+\mu)\nabla{\rm div}u=\rho \mathcal{F},$$ which may represent the displacement of vector particles for an elastic, isotropic and homogeneous body subject to external forces $\mathcal{F}$.
In Poisson [@Po1829], Timoshenko [@Ti1953], Hudson [@Hu1980], among others, it has been shown that equation provides information about different [*body waves*]{}. In a scalar sense ($P$-waves), where the notation ${\rm div}u$ stands for fractions of volume changes from the strain tensor, it explains the behavior of compression and rarefaction in the interior of the body. From the mathematical point of view, it can be given by the identity $$\partial^2_{t}({\rm div}u)-\alpha^2\Delta ({\rm div}u)={\rm div}\mathcal{F},$$ where $\alpha=\sqrt{\frac{\lambda+2\mu}{\rho}}$ represents speeds of wave propagation.
On the other hand, by considering the case $\nabla \times u$, one obtains the behavior of vector waves ($S$-waves) that model small rotations of lineal elements from shear forces acting within the body. In this way, the following equation arises $$\partial^2_{t} (\nabla \times u)-\beta^2\Delta (\nabla \times u)=\nabla \times\mathcal{F},$$ where $\beta=\sqrt{\frac{\mu}{\rho}}$ means the speeds of $S$-wave propagation.
The analysis of the dynamics for has shown great applications in the effect of seismic waves on various materials (e.g. harzburgite, garnet, pyroxenite, amphibolites, granite, gas sands, quartz, etc), where the propagation of the $P$-waves represents the change of volume in the interior of the body under compression and dilatation in the wave direction, see Figure \[fig-w\](b), whereas the $S$-waves are cross displacements that produce vibrations in a perpendicular direction (normal to the traveling wave), see Figure \[fig-w\](c).
![In (a) we have the elastic body in a rest position. In (b) we have the effect of $P$-waves propagation on the material, where small contraction and dilation are produced in the same direction of the wave propagation. In (c) we exemplify the effect of transversal $S$-waves on the material, which are generated from the shear forces and are effective in normal directions with respect to the direction of the wave propagation.[]{data-label="fig-w"}](waves.pdf)
A general existing scenario is when earthquakes generate shear waves, say $S$-waves, that are more effective than compression waves, say $P$-waves, and therefore the most damage on the body displacements is due to the “stronger” vibrations caused by $S$-waves. On the other hand, $P$-waves commonly propagate at a higher speed in relation to $S$-waves, reaching their highest speed, namely, the highest value for $ \beta $, near the basis of the body. Thus, from this viewpoint, it is worth mentioning that the approximation $\lambda \to -\mu$ symbolizes the approaching of the velocities with respect to $S$-waves in relation to $P$-waves.
For instance, when one considers the approach of $\lambda$ to $-\mu$ on sedimentary rocks, one has atypical cases concerning bulk modulus or Poisson’s ratio. This is the case when one considers e.g. $\lambda<-\frac{2\mu}{3}$ which is the case where we have negative bulk modulus or when $\lambda\sim-\mu$ which is the case where the Poisson’s ratio is not defined, being $\pm \infty$ in the left or right approximation, respectively. These results seem to contradict the physical notion that we have regarding the study of thermodynamics on this type of materials, but several studies show that the compressibility of the material is closely related to the constant $\lambda$ instead of approximations coming from the bulk modulus or the Poisson’s ratio, see e.g. Goodway [@AVO].
Other examples of such approximations are considered as follows. Indeed, in Moore et al. [@Moore] the authors reveal the possibility of considering negative incremental bulk modulus on open cell foams on porous media. Also, Lakes and Wojciechowski [@LW] show the possibility of taking negative Poisson’s ratio and bulk modulus for the same type of materials, which proves its structural stability. These are examples that show us the existence of materials (e.g., gas sands [@AVO] and open cell foams [@Moore; @LW]) that, under certain circumstances, allow us to consider the limit situation of $\lambda$ to negative values. Thus, it makes sense to consider for example $\lambda \to -\mu$.
Moreover, in Ji et al. [@Ji] the authors show that for quartz materials under a confining pressure of $600$ MPa and a temperature around $650 \, ^0$C, the transmission between High–Low Quartz demonstrates a significant decreasing in the speed of $P$-wave propagation $\left(\alpha=\sqrt{\frac{\lambda+2\mu}{\rho}}\right)$ in relation to the perturbation of the speed of $S$-wave propagation $\left(\beta=\sqrt{\frac{\mu}{\rho}}\right)$. Therefore, to consider the approximation $$\frac{\alpha}{\beta} \to 1 \quad \mbox{ wich means } \quad \lambda\to-\mu$$ in the dynamic of seismic waves, it is equivalent to study the state of transition between High–Low Quartz in materials (say rocks) containing quartz (as for example granite, diorite, and felsic gneiss) and its behavior with respect to the wave speeds of propagation for transverse and compressible waves in the material, under proper conditions of temperature and pressure.
Assumptions
-----------
The following assumptions shall be considered throughout this paper for the functions defined on a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary $\partial\Omega$.
1. The damping coefficient $\alpha$ and the Lamé coefficients $\lambda, \mu$ fulfill $$\label{2}
\alpha, \mu>0 \ \ \mbox{ and } \ \ \lambda\in\mathbb{R} \ \ \mbox{with} \ \ \mu +\lambda \geq 0.$$
2. The external vector force $b$ satisfies $$\label{hip-h}
b \in (L^2(\Omega))^3.$$
3. The nonlinear vector field $f=(f_1,f_2,f_3)$ is assumed to satisfy: there exist a vector field $g=(g_1,g_2,g_3) \in (C^1(\mathbb{R}^3))^3$, and functions $G \in C^2(\mathbb{R}^3)$ and $h_i \in C^2(\mathbb{R})$, $i=1,2,3$, such that
$$f_i(u_1,u_2,u_3)=g_i(u_1,u_2,u_3) + h_i(u_i), \quad i=1,2,3,$$ $$f_i(0)=g_i(0)=h_i(0)=0, \quad i=1,2,3,$$ $$g=(g_1, g_2, g_3)=\nabla G.$$ In addition, there exist constants $M, m_f \geq 0$ such that $$\begin{aligned}
f(u) \cdot u -G(u) - \sum_{i=1}^3 \int_0^{u_i} h_i(s)ds \geq - M |u|^2 - m_f, \,\,\forall u \in \mathbb{R}^3, \label{9.1}\\
G(u) +\sum_{i=1}^3 \int_0^{u_i} h_i(s)ds \geq -M |u|^2 -m_f, \,\,\forall u \in \mathbb{R}^3, \label{4.1}
\end{aligned}$$ with $$\label{4.2}
0 \leq M < \frac{\mu \lambda_1}{2},$$ where $\lambda_1>0$ denotes the first eigenvalue of the Laplacian operator $-\Delta$. Moreover, with respect to functions $g_i$ and $h_i$, $i=1,2,3$, we additionally assume:
- $g$ fulfills the subcritical growth restriction: there exist $1 \leq p < 3 $ and $M_g >0$ such that, for $i=1,2,3$, $$\label{4.3}
|\nabla g_i(u)| \leq M_g (1+ |u_1|^{p-1}+|u_2|^{p-1} + |u_3|^{p-1}), \ \ \forall \, u=(u_1,u_2,u_3) \in \mathbb{R}^3.$$
- For each $i=1,2,3$, $h_i$ fulfills the critical growth restriction: there exists a constant $c_h>0$ such that $$\begin{aligned}
\label{aa}
|h'_i(x)| \leq c_h(1+|x|^2),\quad \forall \ x \in \mathbb{R}, \ i=1,2,3.
\end{aligned}$$
Functional setting
------------------
We denote the inner product in $L^2(\Omega)$ by $\left\langle u , v\right\rangle = \int_{\Omega} uv dx$ for $u,v \in L^2(\Omega)$. For the sake of simplicity, we use the same notation to the inner product in $(L^2(\Omega))^3$, that is, given $u=(u_1,u_2,u_3), v=(v_1,v_2,v_3) \in (L^2(\Omega))^3$, $$\begin{aligned}
\left\langle u , v\right\rangle := \sum_{i=1}^3 \left\langle u_i,v_i \right\rangle. \end{aligned}$$ Similarly, $\left\langle \nabla \cdot, \nabla \cdot \right\rangle $ stands for the inner product in $H_0^1(\Omega)$ as well as the inner product in $(H_0^1(\Omega))^3$. Thus, given $u=(u_1,u_2,u_3), v=(v_1,v_2,v_3) \in (L^2(\Omega))^3$, $$\begin{aligned}
\left\langle \nabla u , \nabla v\right\rangle := \sum_{i=1}^3 \left\langle \nabla u_i, \nabla v_i \right\rangle. \end{aligned}$$ In addition, for $p>0$, we denote the norms in the spaces $L^p(\Omega)$ and $(L^p(\Omega))^3$ by $|\cdot|_p$ and $\|\cdot \|_p$, respectively, that is, $$\begin{aligned}
&|u|_p:=\left(\int_{\Omega} |u|^pdx \right)^{\frac{1}{p}}, \;\; u \in L^p(\Omega),\\
&\|u\|_p^p:=\sum_{i=1}^3 |u_i|_p^p, \;\; u=(u_1,u_2,u_3) \in (L^p(\Omega))^3 .\end{aligned}$$ In particular, for $p=2$, one reads $$\|u\|_2^2=\left\langle u, u\right\rangle \ \mbox{ for } \ u \in (L^2(\Omega))^3 \ \ \mbox{and} \ \
|u|_2^2=\left\langle u, u\right\rangle \ \mbox{ for } \ u \in L^2(\Omega).$$ The elasticity operator $\mathcal{E}$, with domain $D(\mathcal{E}):= (H^2(\Omega) \cap H_0^1(\Omega))^3 $, is given by $$\label{operator}
{\mathcal{E}} u=-\mu \Delta u - (\lambda +\mu)\nabla(\nabla \cdot u).$$ We consider the Hilbert space $\left( (H_0^1(\Omega))^3, \left\langle \cdot, \cdot \right\rangle_{e} \right)$, where the inner product $\left\langle \cdot, \cdot \right\rangle_{e} $ is given by $$\left\langle v,w \right\rangle_{e} = \mu \left\langle \nabla v,\nabla w \right\rangle+ (\lambda+\mu) \left\langle \text{div}u, \text{div}w \right\rangle.$$
\[lemma0.1\] Under the above notations, it is easy to verify that the norms $\| \cdot \|_{e}^2 :=\sqrt{\left\langle \cdot, \cdot \right\rangle_{e}}$ and $\| \nabla \cdot \|_2^2 := \sqrt{\left\langle \nabla \cdot, \nabla \cdot \right\rangle}$ are equivalent in $(H_0^1(\Omega))^3$. More precisely, one has $$\begin{aligned}
\mu\|\nabla u \|_2^2 \leq \| u \|_{e}^2 \leq a_0 \| \nabla u \|_2^2, \ \ \forall \, u=(u_1,u_2,u_3)\in(H_0^1(\Omega))^3,
\end{aligned}$$ where $ a_0=\max\{\mu, 3(\lambda+\mu)\}$.
Additionally, if $u \in D(\mathcal{E}) $ and $v \in (H_0^1(\Omega))^3$, then it is easy to verify that $$\begin{aligned}
\label{9.2}
\left\langle \mathcal{E} u ,v \right\rangle = \left\langle u, v \right\rangle_{e}.\end{aligned}$$ From , Remark \[lemma0.1\] and the compact embedding of $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$, one sees that $\mathcal{E}$ is a positive self-adjoint operator. We denote the fractional power associated to $\mathcal{E}$ by $\mathcal{E}^r$ with domain $X^r := D({\mathcal{E}}^r)$, which is endowed with the natural inner product $\left\langle \cdot, \cdot \right\rangle_{r} := \left\langle {\mathcal{E}}^r \cdot, {\mathcal{E}}^r \cdot \right\rangle$. In particular, $$\begin{aligned}
X^0&=((L_2(\Omega))^3; \left\langle \cdot, \cdot \right\rangle),\\
X^{1/2}&=\left( (H_0^1(\Omega))^3; \left\langle {\mathcal{E}}^{1/2} \cdot, {\mathcal{E}}^{1/2} \cdot \right\rangle \right),\\
X^1&=(D({\mathcal{E}}); \left\langle \mathcal{E} \cdot, \mathcal{E} \cdot \right\rangle).\end{aligned}$$
From Riesz’s Theorem along with density arguments and continuity, we have $$\left\langle u, v \right\rangle_{1/2}=\left\langle u, v \right\rangle_{e}, \quad \forall \ u, v \in (H_0^1(\Omega))^3.$$
Finally, we define the (Hilbert) weak phase space $\mathcal{H}:= X^{1/2}\times X^0$ with the usual inner product and induced norm $\|\cdot \|_\mathcal{H}$; and the (Hilbert) strong phase space $
\mathcal{H}^1:=X^1 \times X^{1/2} .
$
Well-posedness and energy estimates
-----------------------------------
Under the above assumptions and notations, we are able to state the Hadamard well-posedness of $(\ref{problem})$. We start by denoting $$\label{eq}
U=\left[\begin{array}{c}
u \\
\partial_t u
\end{array}\right], \ \ \mathbb{E}=\left[\begin{array}{cc}
0 & -I \\
\mathcal{E} & \alpha
\end{array}\right], \ \ \mathbb{F}=\left[\begin{array}{cc}
0 & 0 \\
f(\cdot) & 0
\end{array}\right], \ \ \mathbb{B}=\left[\begin{array}{c}
0 \\
b(x)
\end{array}\right].$$ Then, problem (\[problem\]) is equivalent to the Cauchy problem
$$\label{abs-system}
\partial_t U+\mathbb{E}U+\mathbb{F}U=\mathbb{B}, \ \ U(0)=\left[\begin{array}{c}
u_0 \\
u_1
\end{array}\right],$$
where $\mathbb{E}:D(\mathbb{E})\subset \mathcal{H}\to \mathcal{H}$ with domain $$D(\mathbb{E})=\{ (u,v) \in \mathcal{H} \ | \ \mathcal{E}u+\alpha v \in X^0, \, v \in X^{1/2}\}=\mathcal{H}^1.$$
\[Theorem3.4\] Let us assume that $(\ref{2})$-$(\ref{aa})$ hold. Then,
- For $(u_0, u_1) \in \mathcal{H}$, system $(\ref{abs-system})$ possesses a unique mild solution $$\begin{aligned}
U \in C(\mathbb{R}^+;\mathcal{H}).
\end{aligned}$$
- For $(u_0, u_1) \in \mathcal{H}^1$, system $(\ref{abs-system})$ possesses a unique regular solution $$\begin{aligned}
U \in C(\mathbb{R}^+;\mathcal{H}^1).
\end{aligned}$$
- For any $T>0$ and any bounded set $B \subset \mathcal{H}$, there exists a constant $C_{BT}>0$ such that for any two solutions $z^i = (u^i,\partial_{t} u^i)$ of with initial data $z_0^i \in B$, $i=1,2$, we have $$\begin{aligned}
\label{cc}
\|z^1(t)- z^2(t)\|_{\mathcal{H}}^2 \leq C_{BT} \|z_0^1-z_0^2\|_{\mathcal{H}}^2.
\end{aligned}$$
It is easy to check that operator $\mathbb{E}$ set in is a maximal monotone operator and also, under assumption (A3), $\mathbb{F}$ is a locally Lipschitz on $\mathcal{H}$. Therefore, applying the classical theory of linear semigroups, see e.g. [@arrieta; @hale; @pazy], items $(i)$-$(ii)$ are concluded. The continuous dependence $(iii)$ is also obtained by using standard computations in the difference of solutions.
In what follows we give some useful inequalities involving the energy functional. The total energy functional associated with problem (\[problem\]) is given by $$\label{16}
E(t)=\frac{1}{2}\| (u, \partial_t u)\|_{\mathcal{H}}^2 + \int_{\Omega}G(u)dx + \sum_{i=1}^3 \int_{\Omega} \int_0^{u_i} h_i(s)ds dx - \left\langle b(x), u\right\rangle.$$
Under the hypotheses $(\ref{2})$-$(\ref{aa})$, we have:
- the energy $E(t)$ is non-increasing with $E(t)\leq E(0)$ for all $t\geq0;$
- there exist positive constants $K_1, K_2$ and $K_3$ such that $$\label{18}
K_2 \| (u, \partial_t u)\|_{\mathcal{H}}^2 - K_3 \leq E(t)\leq K_1 \| (u, \partial_t u)\|_{\mathcal{H}}^4 + K_3, \ \ \forall \, t\geq0.$$
$(i)$ Taking the multiplier $u_t$ in problem , then a straightforward computation leads us to $$\begin{aligned}
\label{17}
E'(t) &=
-\alpha \|\partial_t u\|_2^2 \leq 0 , \quad \forall \ t >0,
\end{aligned}$$ from where it readily follows the stated in item $(i)$.
$(ii)$ From conditions $(\ref{2})$-$(\ref{aa})$ and Young’s inequality with $\epsilon>0$, the expression $$\begin{aligned}
I =\int_{\Omega}G(u)dx + \sum_{i=1}^3 \int_{\Omega} \int_0^{u_i} h_i(s)ds dx - \left\langle b(x), u\right\rangle
\end{aligned}$$ can be estimated from below and above as follows $$\begin{aligned}
I & \geq \, - \ m_f |\Omega|- \frac{\epsilon}{4}\|b\|_2^2 - \left(\frac{M}{\lambda_1\mu} + \frac{1}{\lambda_1\mu \epsilon}\right) \|(u,\partial_t u)\|_{\mathcal{H}}^2,\\
I & \leq C_{f}|\Omega|+ \frac{1}{2}\|b\|_2^2 + \frac{C_{g}}{\mu^{\frac{p+1}{2}}}\| (u,\partial_t u)\|_{\mathcal{H}}^{p+1} \\
& \ \ \ \ + \frac{C_{h}}{\mu^2}\| (u,\partial_t u)\|_{\mathcal{H}}^{4} +\frac{1}{2\sqrt{\lambda_1}\mu} \|(u, \partial_t u)\|_{\mathcal{H}}^2,\end{aligned}$$ where the positive generic constants depend on their index and some embedding with $H_0^1(\Omega)$, for example $C_{h}$ depends on the constant $c_h$ in (\[aa\]) and the compact embedding $H_0^1(\Omega) {\hookrightarrow} L^4(\Omega)$. From this and the definition of $E(t)$ in , we infer $$\begin{aligned}
E(t) &\leq C_{f}|\Omega| +\frac{1}{2}\|b\|_2^2 + \frac{1}{2\sqrt{\lambda_1}\mu}+ \frac{C_{g}}{\mu^{\frac{p+1}{2}}} \\
& \ \ \ +\left(\frac{1}{2}+\frac{C_{g}}{\mu^{\frac{p+1}{2}}} + \frac{C_{h}}{\mu^2}\right) \|(u,\partial_t u)\|_{\mathcal{H}}^4,\\
E(t) &\geq -m_f |\Omega|- \frac{\epsilon}{4}\|b\|_2^2 + \left(\frac{1}{2} - \frac{M}{\lambda_1\mu} - \frac{1}{\lambda_1 \mu \epsilon}\right) \|(u,\partial_t u)\|_{\mathcal{H}}^2.
\end{aligned}$$ Therefore, from a proper choice of $\epsilon>0$ and using condition (\[4.2\]), one can conclude the existence of positive constants $K_1,K_2$ and $K_3$ satisfying .
\[remark1\] We emphasizes that above constants $K_1, K_2$ and $K_3$ in $(\ref{18})$ do not depend on the parameter $\lambda$.
Long-time dynamics {#sec-long-time-dyn}
==================
From Theorem \[Theorem3.4\], one can define a dynamical system $(\mathcal{H}, S(t))$ associated with problem $(\ref{problem})$, where the evolution operator $S(t)$ corresponds to a non-linear $C_0$-semigroup (locally Lipschitz) on $\mathcal{H}$.
Our main goal in this section is to prove that $(\mathcal{H}, S(t))$ possesses a finite dimensional global attractor $\mathcal{A}$ as well as to reach its qualitative properties such as characterization and regularity. To this end, we first recall some concepts in the theory of dynamical systems, by following e.g. the references [@chueshov-book; @Von].
Some elements of dynamical systems
----------------------------------
For the sake of completeness, we recall some basic facts on dynamical systems.
- A [*global attractor*]{} for a dynamical system $(\mathcal{H}, S(t))$ is a compact set $\mathcal{A} \subset \mathcal{H}$ which is fully invariant and uniformly attracting, it means, for any bounded subset $B \subset \mathcal{H}$ $$S(t)\mathcal{A}=\mathcal{A} \text{ and } \lim_{t\to \infty } d_\mathcal{H} (S(t)B,\mathcal{A}) =0.$$
- The [*fractal dimension*]{} of a compact set $B \subset \mathcal{H}$ is defined as $$\dim_{f} B = \limsup_{\epsilon \to 0} \frac{ln N_\epsilon (B)}{ln(1/\epsilon)},$$ where $N_\epsilon (B)$ is the minimal number of closed balls of radius $2 \epsilon$ necessary to cover $B$.
- The set of [*stationary points*]{} $\mathcal{N}$ of a dynamical system $(\mathcal{H}, S(t))$ is defined as $$\begin{aligned}
\mathcal{N} = \left\{ V \in \mathcal{H} \ |\ S(t)V=V, \ \ \forall \, t >0 \right\}.
\end{aligned}$$
- A dynamical system $(\mathcal{H},S(t))$ is called [*gradient*]{} if there exists a strict Lyapunov functional $\Psi$, that is, for any $z \in \mathcal{H}$, $\Psi(S(t)z)$ is decreasing with respect $t \geq 0$ and $\Psi$ is constant on the set of stationary points $\mathcal{N}$.
- Given a set $B \subset \mathcal{H}$, its [*unstable manifold*]{} $W^u(B)$ is the set of points $z \in \mathcal{H}$ that belongs to some complete trajectory $\{y(t) \}_{t \in \mathbb{R}}$ and satisfies $$y(0) =z \text{ and } \limsup_{t \to -\infty} \text{dist}(y(t),B)=0.$$
- [*Quasi-stability.*]{} Let $X,Y$ be reflexive Banach spaces with compact embedding $X \overset{c}{\hookrightarrow} Y$ and $\mathcal{H}=X \times Y$. Let us suppose $(\mathcal{H},S(t))$ is given by $$S(t)z =(u(t), \partial_t u(t)) , \,\, z=(u_0,u_1) \in \mathcal{H},$$ where $$u \in C(\mathbb{R}^{+} ; X ) \cap C^1 (\mathbb{R}^{+} ; Y),$$ Then, $(\mathcal{H},S(t))$ is called [*quasi-stable*]{} on a set $B \subset \mathcal{H}$ if there exists a compact semi-norm $\eta_X$ on $X$ and non-negative scalar functions $a_1(t)$ and $a_3(t)$ locally bounded in $\mathbb{R}^{+}$ and $a_2(t) \in L^1(\mathbb{R}^{+})$ with $\lim_{t \to \infty} a_2(t)=0$ such that $$\begin{aligned}
\|S(t)z^1 -S(t)z^2\|_\mathcal{H}^2 \leq a_1(t)\|z^1 -z^2\|_\mathcal{H}^2,
\end{aligned}$$ and $$\begin{aligned}
\|S(t)z^1 -S(t)z^2\|_\mathcal{H}^2 \leq a_2(t)\|z^1 -z^2\|_\mathcal{H}^2 + a_3(t)\sup_{0 \leq s \leq t} \left[ \eta_X (u^1(s) -u^2(s) ) \right]^2,
\end{aligned}$$ for any $z^1, z^2 \in B$.
\[corollary\] Let $(\mathcal{H},S(t))$ be a gradient asymptotically smooth dynamical system. Additionally, if its Lyapunov function $\Psi(x)$ is bounded from above on any bounded subset of $\mathcal{H}$, the set $\Psi_R = \left\{x \in \mathcal{H} : \Psi(x) \leq R\right\}$ is bounded for every $R$ and the set $\mathcal{N}$ of stationary points of $(\mathcal{H},S(t))$ is bounded, then $(\mathcal{H},S(t))$ possesses a compact global attractor characterized by $ \mathcal{A}=W^u(\mathcal{N})$.
\[corollary1\] Let us assume that the dynamical system $(\mathcal{H},S(t))$ is quasi-stable on every bounded forward invariant set $B\subset \mathcal{H}.$ Then, $(\mathcal{H},S(t))$ is asymptotically smooth.
\[corollary2\] Let $(\mathcal{H},S(t))$ a quasi-stable dynamical system. If $(\mathcal{H},S(t))$ possesses a compact global attractor $\mathcal{A}$ and is quasi-stable on $\mathcal{A}$, hen the attractor $\mathcal{A}$ has a finite fractal dimension $\dim_{f} \mathcal{A}<\infty.$
Main result and proofs
----------------------
We are now in condition to state and prove the main result concerning global attractors associated with problem $(\ref{problem})$. It reads as follows.
\[3.16\] Under the assumptions $(\ref{2})$-$(\ref{aa})$, we have:
- The dynamical system $(\mathcal{H}, S(t))$ corresponding to problem has a unique global attractor $\mathcal{A}$ with finite fractal dimension $\dim_{f} \mathcal{A}<\infty$, and is characterized by the unstable manifold $ \mathcal{A}=W^u(\mathcal{N})$ emanating from the set of stationary points $\mathcal{N}$ of $(\mathcal{H}, S(t))$.
- Moreover, if $h_i=0$, $i=1,2,3$, then $\mathcal{A}$ is bounded in the strong phase space $\mathcal{H}^1$. In particular, any full trajectory $\{(u(t),\partial_t u(t)), t \in \mathbb{R}\}$ that belongs to $\mathcal{A}$ has the following regularity properties $$\label{reg-1}
\partial_t u \in L^{\infty}(\mathbb{R};(H_0^1(\Omega))^3) \cap C(\mathbb{R};(L^2(\Omega))^3),\,\,\, \partial^2_{t} u \in L^{\infty}(\mathbb{R};(L^2(\Omega))^3),$$ and there exists $R>0$ such that $$\label{reg-2}
\|(\partial_{t} u(t),\partial^2_{t} u(t))\|_\mathcal{H}^2 \leq R^2,$$ where $R$ does not depend on $\lambda$.
The proof of Theorem \[3.16\] will be concluded at the end of this section as a consequence of some technical results provided in the sequel.
### Gradient property
\[Lemma3.9\] Under the assumptions of Theorem $ \ref{3.16}, $ let us define the functional $$\begin{array}{rcl}
\Psi: \, \mathcal{H} & \rightarrow & \mathbb{R} \\
z & \mapsto & \Psi(z):=\Psi(u, v)
\end{array}$$ given by $$\Psi(u, v)=\frac{1}{2}\| (u, v)\|_{\mathcal{H}}^2 + \int_{\Omega}G(u)dx + \sum_{i=1}^3 \int_{\Omega} \int_0^{u_i} h_i(s)ds dx - \left\langle b(x), u\right\rangle.$$ Then:
1. $\Psi$ is a strict Lyapunov functional;
2. $\Psi(z) \rightarrow \infty $ if and only if $\|z\|_{\mathcal{H}} \rightarrow \infty$;
3. $\mathcal{N}$ is bounded on $\mathcal{H}$
As a consequence, the dynamical system $(\mathcal{H}, S(t))$ associated with problem is a gradient system.
Let fix $z_0 \in \mathcal{H}$ and recall that $\mathcal{N}$ is the set of stationary points of $(\mathcal{H}, S(t))$. Also, from one sees that $\Psi(u(t),\partial_{t}u(t))=E(u(t),\partial_{t}u(t)):=E(t)$. Then, we infer:
- From (\[17\]), it is clear that $\Psi(S(t)z_0)$ is decreasing with respect to time and from (\[18\]), $\Psi(z)=\Psi(S(0)z) \rightarrow \infty $ if and only if $||z||_{\mathcal{H}} \rightarrow \infty$.
- Let us consider the stationary problem: $$\begin{aligned}
\label{bb}
\left\lbrace
\begin{array}{lcr}
\mathcal{E} u+ f(u) = b(x) & \text{in}& \Omega,\\
u=0 & \text{on}&\partial \Omega.
\end{array}
\right.
\end{aligned}$$ Thus, a simple computation shows that $\mathcal{N}$ is given by $$\mathcal{N}=\left\{(u,0) \in \mathcal{H} \ |\ u \text{ is the solution of } (\ref{bb}) \right\}.$$ In addition, from (\[17\]) it is easy to prove that $\Psi$ is constant on $\mathcal{N}$. Finally, multiplying (\[bb\]) by $u$, integrating on $\Omega$ and using (\[9.1\]) and (\[4.1\]), we obtain that for any $\epsilon >0$ $$\begin{aligned}
\label{eq23}
\left( 1-\frac{2M}{\lambda_1\mu} -\frac{1}{4 \lambda_1 \mu \epsilon}\right) \|u\|_{e}^2 &\leq 2m_f|\Omega|+ \epsilon \|b\|_2^2,
\end{aligned}$$ from where (along with (\[4.2\])) we conclude that $\mathcal{N}$ is bounded on $\mathcal{H}$, for $\epsilon>0$ properly chosen.
Therefore, the items 1 - 3 are proved.
### Quasi-stability property
\[Theorem8\] Under the assumptions of Theorem $ \ref{3.16},$ let us consider a bounded subset $B\subset \mathcal{H}$ and two weak solutions $\tilde{z}=(v,\partial_t v)$ and $ z=(u, \partial_t u)$ of problem $(\ref{problem})$ with initial data $\tilde{z}(0)=(v_0,v_1)$, $z(0)=(u_0,u_1)\in B$. Then, $$\label{est-est}
\|\tilde{z}(t)-z(t)\|_{\mathcal{H}}^2 \leq a_2(t)\|\tilde{z}(0)-z(0)\|_{\mathcal{H}}^2+ c(t)\sup_{0\leq s\leq t} \|v-u\|_{p_0}^2,$$ where $p_0=\max\{4, \frac{6}{4-p}\}< 6$, $b\in L^1(\mathbb{R}^+)$ with $\displaystyle\lim_{t \to \infty} a_2(t)=0$ and $c(t)$ is a locally bounded function.
The estimate is one of the main cores of the present article. Its proof is quite technical and long, and for this reason we are going to proceed in several steps as follows.
[*Step 1. Setting the difference problem and functionals.*]{} Let us denote $w=u-v$. Then, a simple computation shows that $w$ is a solution (in the weak and strong sense) of the following problem $$\begin{aligned}
\label{29}
\left\lbrace
\begin{array}{lcr}
\partial^2_{t} w+ \mathcal{E} w + \alpha \partial_t w +f(u)-f(v) =0 &\text{on}& \Omega\times\mathbb{R}^{+}, \smallskip \\
w=0 &\text{in}& \partial \Omega\times\mathbb{R}^{+}, \smallskip \\
w(x,0)=u_0(x)-v_0(x), \ x\in \Omega, && \smallskip \\
\partial_t w(x,0)=u_1(x)-v_1(x) , \ x\in\Omega.&&
\end{array}
\right.\end{aligned}$$ The energy associated with system (\[29\]) is given by $$\label{Xi}
\Xi(t):= \frac{1}{2}\| (w, \partial_t w)\|_{\mathcal{H}}^2= \frac{1}{2} \|\tilde{z}(t)-z(t)\|_{\mathcal{H}}^2, \ \ t\geq0.$$ We also set the functional $$\begin{aligned}
\chi(t) &= \left\langle w, \partial_{t} w \right\rangle,\end{aligned}$$ and the perturbed energy functional $$\begin{aligned}
\Upsilon(t) = \epsilon_1 \Xi(t) + \epsilon_2\chi(t) ,\end{aligned}$$ where the constants $\epsilon_1, \epsilon_2 >0$ will be chosen later.
[*Step 2. Equivalence.*]{} There exist constants $C_1, C_2 >0$ such that $$\label{26}
C_2\Xi(t) \leq \Upsilon(t) \leq C_1 \Xi(t).$$ Indeed, the inequalities in follow by taking $K'=\max\{\frac{C^2}{\mu},1\}$, $\epsilon_1 > \epsilon_2K' $, $C_2=\epsilon_1 - \epsilon_2K'$ and $C_1=\epsilon_1 + \epsilon_2K' $.
[*Step 3. Estimate for $\Xi'$.*]{} Given $\xi>0$, there exists a constant $C(\xi,B)>0$, which depends on $\xi$ and $B$, such that $$\begin{aligned}
\label{Xi'}
\Xi'(t) \leq -{\alpha} \|\partial_t w\|_2^2+C(\xi,B)\|w\|_{\frac{6}{4-p}}^2+\xi\|\partial_t w\|_2^2 + I,\end{aligned}$$ where we set $$\label{I}
I := \displaystyle\sum_{i=1}^3 \left\langle h_i(v_i)-h_i(u_i), \partial_t w_i \right\rangle.$$ In fact, we first observe that deriving $\Xi(t)$ and using (\[29\]), we get $$\begin{aligned}
\Xi'(t) =& -\alpha \|\partial_t w\|_2^2 -\left\langle g(u)-g(v), \partial_t w \right\rangle+I .\end{aligned}$$ Since $$\begin{aligned}
|\left\langle g(u)-g(v), \partial_t w \right\rangle| \leq \sum_{i=1}^{3} \int_{\Omega} M_g \left\lbrace 1 + \sum_{i=1}^3 |v_i|^{p-1} + \sum_{i=1}^3 |u_i|^{p-1} \right\rbrace|w| |\partial_t w_i|dx,\end{aligned}$$ then applying Hölder’s inequality, we obtain $$\begin{aligned}
|\left\langle g(u)-g(v), \partial_t w \right\rangle| \leq \sum_{i=1}^{3} \tilde{C}_f\|w\|_{\frac{6}{4-p}} \|\partial_t w_i\|_2 , \end{aligned}$$ where $$\begin{aligned}
\tilde{C}_f = M_f \left\lbrace |\Omega|^{\frac{p-1}{6}} + \sum_{i=1}^3\|v_i\|_{6}^{p-1} + \sum_{i=1}^3 \|u_i\|_6^{p-1} \right \rbrace \leq C(B)<\infty,\end{aligned}$$ is a constant depending on $B$. Therefore, the estimate follows from Young’s inequality with $\xi>0$.
[*Step 4. Estimate for $\chi'$.*]{} There exists a constant $C(B)>0$ depending on $B$ such that $$\begin{aligned}
\label{chi'}
\chi'(t) \leq -\Xi(t) -\frac{1}{2}\|w\|_e^2 +\frac{\alpha}{2}\| w\|_2^2 + C(B)\|w\|_{4}^2 + \frac{3+\alpha}{2}\|\partial_t w\|_2^2.\end{aligned}$$ Indeed, multiplying (\[29\])$_1$ by $w$ and integrating on $\Omega$, we obtain $$\begin{aligned}
\chi' (t)
&=-\Xi(t) -\frac{1}{2}\|w\|_e^2 + \frac{\alpha}{2}\| w\|_2^2 + \frac{\alpha+3}{2}\|\partial_t w \|_2^2 \\
&\hspace{3cm}- \left\langle g(u)-g(v),w\right\rangle +\sum_{i=1}^3 \left\langle h_i(u_i)-h_i(v_i) ,w_i \right\rangle.\end{aligned}$$ Now, noting that $$\begin{aligned}
\left\langle g(v)-g(u),w \right\rangle
\leq 3 \left\{|\Omega|^{p-1} + \sum_{i=1}^3 \|u_i\|_{p+1}^{p-1} + \sum_{i=1}^3 \|v_i\|_{p+1}^{p-1} \right\} \|w\|_{p+1}^2 \leq \tilde{C}_B \|w\|_{p+1}^2,\end{aligned}$$ and $$\begin{aligned}
\sum_{i=1}^3 \left\langle h_i(u_i)-h_i(v_i) ,w_i \right\rangle
\leq \sum_{i=1}^3( |\Omega|^2 +\|v_i\|_4^2 + \|u_i\|_4^2)|w_i|_4^2 \leq C_B \|w\|_4^2,\end{aligned}$$ where the constants $\tilde{C}_B,C_B>0$ depend only on $B$, then the estimate follows.
[*Step 5. Estimate for $\Upsilon$.*]{} There exists a constant $C_3>0$ depending on $B$ such that $$\begin{aligned}
\label{23}
\Upsilon(t)\leq e^{-\frac{\epsilon_2 t}{C_1}}\Upsilon(0) +C_3 \int_0^t e^{-\frac{\epsilon_2 }{C_1}(t-s)}\|w(s)\|_{p_0}^2ds + \epsilon_1 e^{-\frac{\epsilon_2t }{C_1}}J , \end{aligned}$$ where $C_1>0$ comes from and we set $$\begin{aligned}
\label{Jota}
J:=\int_0^t e^{\frac{\epsilon_2s }{C_1}}Ids =\sum_{i=1}^3 \int_0^t e^{\frac{\epsilon_2s }{C_1}} \left\langle h_i(v_i(x,s))-h_i(u_i(x,s)), \partial_t w_i(x,s) \right\rangle ds.\end{aligned}$$ First, we note that from and , one has $$\begin{aligned}
\Upsilon'(t)
\leq & -\epsilon_2\Xi(t)+\frac{\alpha\epsilon_2}{2}\|w\|_2^2 + \epsilon_2C(B) \|w\|_{4}^2 +\epsilon_1C(\xi,B)\|w\|_{\frac{6}{4-p}}^2 \\
& \ + \epsilon_1 I + \left( \frac{3\epsilon_2 + \alpha \epsilon_2}{2} +\epsilon_1\xi-\alpha \epsilon_1\right) \|\partial_t w\|_2^2.\end{aligned}$$ We now choose $\epsilon_1,\epsilon_2,\xi>0$ small enough such that $$\begin{aligned}
\epsilon_2K' < \epsilon_1 \quad \mbox{ and } \quad
\frac{3\epsilon_2 + \alpha\epsilon_2}{2} +\epsilon_1\xi <\alpha \epsilon_1.\end{aligned}$$ It is worth mentioning that $\epsilon_1, \epsilon_2, \xi >0$ do not depend on $\lambda$. Thus, from this choice, setting $p_0=\max\{\frac{6}{4-p},4\}$ and using , there exists a constant $C_3=C(B)>0$ such that $$\begin{aligned}
\Upsilon'(t) & \leq -\frac{\epsilon_2}{C_1}\Upsilon(t) + C_3 \|w\|_{p_0}^2 + \epsilon_1 I,\end{aligned}$$ from where it follows the estimate with $J$ given in .
Since the choices for $\epsilon_1, \epsilon_2$ do not depend on $\lambda$, then $C_3>0$ is a constant that does not depend on $\lambda$ as well.
[*Step 6. Estimate for $J$.*]{} There exist constants $\gamma_0>0$ and $C_4>0$ depending on $B$ such that $$\label{est-J}
J \leq C_4 e^{\gamma_0 t} \sup_{0<s<t} \|w\|_4^2 + C_4 \int_0^t (\|\partial_t u(s)\|_2 + \|\partial_t v (s)\|_2) e^{\gamma_0 s} \Upsilon(s)ds.$$ Firstly, in view of assumption and following verbatim the same arguments as in [@Ma Lemma 4.9], for any constant $\gamma>0$ and each $i=1,2,3,$ there exists a constant $K_i'>0$ such that $$\begin{aligned}
\int_0^t e^{\gamma s} &\left\langle h_i(v_i(s))-h_i(u_i(s)), \partial_t w_i(s) \right\rangle ds \notag\\
& \leq K_i' e^{\gamma t} \sup_{0<s<t} \|w_i\|_4^2 + K_i' \int_0^t \left(| u_i'(s) |_2 + | v_i'(s) |_2\right) e^{\gamma s} |\nabla w_i(s)|_2^2ds.\label{22}\end{aligned}$$ Therefore, from and it prompt follows $$\begin{aligned}
J \leq \sum_{i=1}^3K'_i \sup_{0<s<t} \|w\|_4^2 + \max\{K'_i\} \int_0^t e^{\gamma_0 s} (\| \partial_t u \|_2 + \|\partial_t v \|_2) \|\nabla w(s)\|_2^2ds,\end{aligned}$$ for $\gamma_0=\frac{\epsilon_2}{C_1}>0.$ Additionally, taking $C_4= \max\{K'_1+K'_2+K'_3, \frac{2\max\{K'_i\}}{\mu C_2}\}>0$ and noting that $$\begin{aligned}
\|\nabla w(s)\|_2^2 \leq \frac{1}{\mu}\| w\|_{e}^2 \leq \frac{2}{\mu} \Xi(s) \leq \frac{2}{\mu C_2} \Upsilon(s),
\end{aligned}$$ then estimate follows as desired.
We emphasize that constants $\gamma_0$ and $C_4$ do not depend on $\lambda$.
[*Step 7. Conclusion of the proof.*]{} We are finally in position to complete the proof of . Indeed, from and , there exists a constant $C_5>0$ depending on $B,$ but independently of $\lambda$, such that $$\begin{aligned}
e^{\gamma_0 t}\Upsilon(t) \leq & \, C_5\Upsilon(0) +C_5 e^{\gamma_0 t} \sup_{0<s<t} \|w\|_{p_0}^2 \\
& \ +C_5 \int_0^t (\|\partial_t u(s)\|_2 + \|\partial_t v (s)\|_2) e^{\gamma_0 s} \Upsilon(s)ds,\end{aligned}$$ and applying Gronwall’s inequality, one gets $$\begin{aligned}
\Upsilon(t)\leq C_5\left\lbrace e^{-\gamma_0 t} \Upsilon(0) + \sup_{0<s<t} \|w\|_{p_0}^2 \right\rbrace
e^{\left(C_5 e^{-\gamma_0 t}\int_0^t (\|\partial_t u(s)\|_2 + \|\partial_t v (s)\|_2) e^{\gamma_0 s}ds\right)}. \label{31}\end{aligned}$$ Now, from and (\[17\]), and also in view of Remark \[remark1\], we have $$\int_0^t \|\partial_t u(s)\|_2^2ds = -\frac{1}{\alpha}\int_0^t E'(s)ds \leq \frac{2|E(0)|}{ \alpha} \leq Q,$$ where $Q >0$ is a constant depending on $B$ and $f$, but independent of $\lambda.$ The same computation holds true for $\int_0^t \|\partial_t v(s)\|_2^2ds$. Thus, using Hölder and Young’s inequalities, we obtain $$\begin{aligned}
e^{-\gamma_0 t}\int_0^t (\|\partial_t u(s)\|_2 + \|\partial_t v (s)\|_2) e^{\gamma_0 s}ds \leq 2\sqrt{Q}\sqrt{t}\leq \epsilon t + \frac{2Q}{\epsilon},\end{aligned}$$ for any $t>0$ and $\epsilon>0$. Replacing the latter estimate in (\[31\]), we arrive at $$\begin{aligned}
\Upsilon(t) &\leq C_5e^{(\epsilon C_5 t + \frac{2C_5Q}{\epsilon})}\left\lbrace e^{-\gamma_0 t}\Upsilon(0) + \sup_{0<s<t} \|w\|_{p_0}^2 \right\rbrace .\end{aligned}$$ Taking $\epsilon = \frac{\gamma_0}{2C_4}$ and using (\[26\]), we have $$\label{last}
\Xi(t) \leq \frac{C_1C_4e^{\frac{Q}{\gamma_0}}}{C_2} e^{\frac{-\gamma_0}{2} t}\Xi(0) + \frac{C_4e^{\frac{Q}{\gamma_0}}}{C_2} e^{\frac{\gamma_0}{2}t} \sup_{0<s<t} \|w\|_{p_0}^2.$$ Finally, regarding the definition of $\Xi(t)$, $t\geq0$, in and setting $$\label{functions}
a_2(t):=\frac{C_1C_4e^{\frac{Q}{\gamma_0}}}{C_2} e^{\frac{-\gamma_0}{2} t} \ \ \mbox{ and } \ \ a_3(t):=\frac{2C_4e^{\frac{Q}{\gamma_0}}}{C_2} e^{\frac{\gamma_0}{2}t},$$ then leads to as desired.
The proof of Proposition \[Theorem8\] is therefore concluded.
\[cor-qs\] Under the assumptions of Theorem $ \ref{3.16},$ the dynamical system $(\mathcal{H}, S(t))$ associated with problem is quasi-stable on any bounded set $B\subset\mathcal{H}$.
It is a direct consequence of Theorem \[Theorem3.4\] - $(iii)$ and Proposition \[Theorem8\] by noting the semi-norm given by $n_{H^{1}_{0}}(v-u)=\|v-u\|_{p_0}$ is compact.
### Conclusion of the proof of Theorem $ \ref{3.16} $
- From Proposition \[corollary1\] and Corollary \[cor-qs\], the dynamical system $(\mathcal{H}, S(t))$ related to problem is asymptotically smooth. Therefore, using Lemma \[Lemma3.9\] and Propositions \[corollary\] and \[corollary2\], the conclusion of Theorem $ \ref{3.16} $ - $ (i)$ is complete.
- In case $h_i=0$, $i=1,2,3$, then going back to , one sees that $I=0$ and, consequently, from one gets $J=0$. Thus, reduces to $$\begin{aligned}
\Upsilon(t)\leq e^{-\frac{\epsilon_2 t}{C_1}}\Upsilon(0) +\frac{C_3 C_1}{\epsilon_2} \sup_{0 \leq s \leq t} \|w(s)\|_{p_0'}^2 \big(1-e^{-\frac{\epsilon_2 }{C_1}t}\big) ,
\end{aligned}$$ $p_0'=\max\{\frac{6}{4-p},p+1\}$. In this way, one reaches (respec. ) with $$a_3(t):= \frac{C_3 C_1}{\epsilon_2} \big(1-e^{-\frac{\epsilon_2 }{C_1}t}\big),$$ instead of $a_3(t)$ given in . Thus, $c_{\infty}=\sup _{t \in \mathbb{R}^{+}} a_3(t)<\infty$, and from [@Von Theorem 7.9.8], the regularity properties - are ensured, that is, the conclusion of Theorem $ \ref{3.16} $ - $ (ii)$ is complete.
Therefore, the proof of Theorem $ \ref{3.16} $ is ended.
Upper semicontinuity {#sec-upper-sem}
====================
Along this section $\varepsilon$ denotes a real number in $[0, 1]$ and assume $\lambda + \mu = \varepsilon$. Thus, problem (\[problem\]) can be rewritten as follows $$\begin{aligned}
\left\lbrace
\begin{array}{ll}
\partial^2_{t} u - \mu \Delta u - \varepsilon \nabla {\rm div} u + \alpha \partial_t u + f(u) = b & \text{in} \,\; \Omega\times\mathbb{R}^{+}, \smallskip \\
u=0 & \text{on} \,\; \partial \Omega\times\mathbb{R}^{+}, \smallskip \\
u(0)=u_0, \,\; \partial_t u(0)=u_1 & \text{in} \; \Omega,
\end{array}
\right.
\label{problem-sing}\end{aligned}$$ In this way, instead of operator , we write the $\varepsilon$-operator $$\begin{aligned}
\mathcal{E}_{\varepsilon}u:= -\mu \Delta u - \varepsilon \nabla \text{div}(u), \ \ \mbox{ for } \ \ u=(u_1,u_2,u_3).\end{aligned}$$ Hereafter, we denote by $P_\varepsilon$ the $\varepsilon$-problem and, in view of Theorem \[3.16\], we also denote by $\mathcal{A}_\varepsilon$ the compact finite dimensional global attractor of its associated dynamical system. The energy corresponding to $P_\varepsilon$ is still given by (\[16\]) and denoted here as $E_\varepsilon(t)$.
Using the same notation as in Section \[sec-well-posed\], we define the inner-product $$\left\langle v,w \right\rangle_{\varepsilon} = \mu \left\langle \nabla v,\nabla w \right\rangle+ \varepsilon \left\langle \text{div}u, \text{div}w \right\rangle.$$ Then, the norm $\|\cdot \|_\varepsilon = \sqrt{\left\langle v,w \right\rangle_{\varepsilon}}$ satisfies that $$\begin{aligned}
\label{up}
\mu\|\nabla \cdot \|_2^2 \leq \| \cdot \|_{\varepsilon}^2 \leq \max\{\mu, 3\} \| \nabla \cdot \|_2^2.\end{aligned}$$ Additionally, let us denote by $$\mathcal{H}_\varepsilon = ((H_0^1(\Omega))^3, \|\cdot\|_\varepsilon) \times ((L^2(\Omega))^3, \|\cdot \|_2)$$ the space of weak solutions associated to $P_\varepsilon$, and $$\mathcal{H}_0^1 = (D(-\Delta), \|\mu \Delta \cdot\|_2) \times ((H_0^1(\Omega))^3, \|\mu \nabla \cdot\|_2)$$ the space of strong solutions associated to $P_0$.
Analogously, we denote by $(\mathcal{H}_\varepsilon,S_\varepsilon(t))$ the dynamical system associated with $P_\varepsilon$, and by $\mathcal{N}_\varepsilon$, its corresponding set of stationary solutions. The existence of a global attractor $\mathcal{A}_\varepsilon$ as well as its properties are ensured by Theorem \[3.16\].
In this section, our main goal is to study the upper semicontinuity of attractors $\mathcal{A}_\varepsilon$ with respect to the parameter $\varepsilon \to 0$. More precisely, our main results are presented in Theorems \[singular\] and \[upper\]. To this end, we need to prove the following two properties: the existence of an absorbing set which does not depends on $\varepsilon$ and the convergence in some sense of the solutions of $P_\varepsilon$ when $\varepsilon \to 0$.
For the existence of an absorbing set we need the following result which is a direct consequence of [@Von Remark 7.5.8].
\[Remark3.11\] Under the conditions $(\ref{2})$-$(\ref{aa})$, the following inequality holds true for the attractor $\mathcal{A}$ in Theorem $ \ref{3.16} $ and $\Psi$ given in Lemma $ \ref{Lemma3.9} $: $$\sup\{\Psi(u,\partial_{t} u): (u,\partial_{t} u) \in \mathcal{A} \} \leq \sup\{\Psi(u,0): (u,0) \in \mathcal{N} \}.$$
With respect to the convergence of solutions, it is important to note that the phase space $\mathcal{H}_\varepsilon$ changes when $\varepsilon \to 0$. So, the convergence of the solutions of $P_\varepsilon$ is *singular* in the same sense proposed in [@singular].
\[Lemma4.1\] Under the conditions $(\ref{2})$-$(\ref{aa})$, there exists a bounded absorbing set $\mathcal{B}$ for $(\mathcal{H}_\varepsilon, S_\varepsilon(t))$, that does not depend on $\varepsilon$.
Denoting by $\Psi_\varepsilon$ the Lyapunov functional defined on $\mathcal{H}_\varepsilon$, then from (\[18\]), Remarks \[remark1\] and Theorem \[Remark3.11\], $$\begin{aligned}
\sup_{z \in \mathcal{A}_\varepsilon} \|z\|_{\mathcal{H}_\varepsilon}^2 &\leq \frac{\sup_{z \in \mathcal{A}_\varepsilon}\Psi_\varepsilon(z) + K_3}{K_2}\\
&\leq \frac{\sup_{z \in \mathcal{N}_\varepsilon}\Psi_{\varepsilon}(z) + K_3}{K_2}\\
&\leq \frac{ K_1\sup_{z \in \mathcal{N}_\varepsilon}\|z\|_{\mathcal{H}_\varepsilon}^4 + 2K_3}{K_2}.
\end{aligned}$$ Thus, from (\[eq23\]) there exists a constant $R_1$ which does not depend on $\varepsilon$ such that $$\begin{aligned}
\sup_{z \in \mathcal{A}_\varepsilon} \|z\|_{\mathcal{H}_\varepsilon}^2 \leq R_1^2, \,\,\ \forall \varepsilon \in [0,1] .
\end{aligned}$$ Let us define $\mathcal{B}= \left\{z \in \mathcal{H}_0 : \|z\|_{\mathcal{H}_0}^2 \leq R_1+1 \right\}$, then from (\[up\]) $A_\varepsilon \subset \mathcal{B}$, for all $\varepsilon$.
\[lemma15\] Let $B$ be a bounded subset in $\mathcal{H}_0$ and $\{z_\varepsilon =(u_\varepsilon, v_\varepsilon)\}_\varepsilon \subset B$ a family of initial data related to each $P_\varepsilon$ with solutions $\{S_\varepsilon(t) z_\varepsilon\}_\varepsilon$. Then there exists a constant $\hat{C}$ that does not depend on $t,\varepsilon$ such that $$E_\varepsilon(t) \leq \hat{C} \quad \mbox{ and } \quad
\|S_\varepsilon(t) z_\varepsilon\|_{\mathcal{H}_\varepsilon} \leq \hat{C}, \quad \forall \ \varepsilon,t >0.$$
From (\[18\]), (\[up\]), Remark \[remark1\] and the fact that for each $\varepsilon$, $E_\varepsilon$ is decreasing, we have $$\begin{aligned}
K_2\|S_\varepsilon(t) z_\varepsilon\|_{\mathcal{H}_\varepsilon} - K_3 & \leq E_\varepsilon(t) \leq E_\varepsilon(0) \\
&\leq K_1(\|u_\varepsilon \|_\varepsilon^2 + \|v_\varepsilon \|_2^2)^4 + K_3 \\
& \leq K_1\hat{K}(B,\mu)+K_3
\end{aligned}$$ where $\hat{K}(B,\mu)$ is a constant which depends only on $B$ and $\mu$, and $K_1, K_2$ and $ K_3$ do not depend on $t$ and $\varepsilon$.
Now we are in position to state and prove the main results of this section.
\[singular\] Under the assumptions of $(\ref{2})$-$(\ref{aa})$. Given a sequence $\{\varepsilon_n\} $ of positive numbers, let $(u_n(t), \partial_t u_n (t))$ be the weak solution to $P_{\varepsilon_n}$ with initial data $(v_0,v_1) \in \mathcal{H}_0$. Then if $\varepsilon_n \to 0$ when $n \to \infty$, there exist a weak solution $(u(t),\partial_t u(t))$ of $P_0$ with the same initial data, such that for any $T>0$: $$\begin{aligned}
u_n &\overset{*}{\rightharpoonup} u \text{ in } L^{\infty}(0,T; (H_0^1(\Omega))^3),\\
\partial_t u_n &\overset{*}{\rightharpoonup} \partial_t u \text{ in } L^{\infty}(0,T; (L^2(\Omega))^3).
\end{aligned}$$
Using Lemma \[lemma15\] for $B=\{(v_0,v_1)\}$ and equation (\[up\]), for some constant $K$, $$\begin{aligned}
\label{36}
\| (u_n(t), \partial_t u_n(t)) \|_{\mathcal{H}_0} \leq K.
\end{aligned}$$ Then, we have for any $T>0$, $$\begin{aligned}
u_n &\overset{*}{\rightharpoonup} u \text{ in } L^{\infty}(0,T; (H_0^1(\Omega))^3),\\
\partial_t u_n &\overset{*}{\rightharpoonup} \partial_t u \text{ in } L^{\infty}(0,T; (L^2(\Omega))^3).
\end{aligned}$$ Fixing $n$ and multiplying $P_{\varepsilon_n}$ by a function $\phi \in (H_0^1(\Omega))^3$, we get $$\begin{aligned}
\label{33}
\frac{d}{dt}\left\langle \partial_{t} u_n,\phi\right\rangle+\mu \left\langle \nabla u_n,\nabla \phi\right\rangle
&+ \varepsilon_n \left\langle \text{div}u_n, \text{div}\phi\right\rangle \\
&+ \alpha \left\langle \partial_t u_n, \phi\right\rangle + \left\langle f (u_n), \phi\right\rangle = \left\langle b,\phi\right\rangle. \notag
\end{aligned}$$ It is clear that $$\begin{aligned}
\left\langle \nabla u_n,\nabla \phi\right\rangle &\underset{n \to \infty}{\longrightarrow} \left\langle \nabla u,\nabla \phi\right\rangle,\\
\left\langle \partial_t u_n, \phi\right\rangle &\underset{n \to \infty}{\longrightarrow} \left\langle \partial_t u, \phi\right\rangle, \\
\varepsilon_n \left\langle \text{div}u_n, \text{div}\phi\right\rangle &\underset{n \to \infty}{\longrightarrow} 0 \,\,\,\text{ from } (\ref{36}).
\end{aligned}$$ Additionally, we have $$\begin{aligned}
\left\langle f(u_n)-f(u),\phi\right\rangle &\leq \sum_{i=1}^3 \int_{\Omega} |f_i(u_n)-f_i(u)||\phi_i|\\
&\leq \sum_{i=1}^3 \int_{\Omega} M_g(1 + \sum_{j=1}^3|u_n^j |^{p-1} + |u^j |^{p-1})|u_n-u| |\phi_i|.
\end{aligned}$$ Then proceeding analogously to (\[23\])-(\[Jota\]) and using Simon’s compactness theorem [@simon] we have that $$\begin{aligned}
\|u_n-u\|_2, \|u_n-u\|_{\frac{6}{4-p}} \to 0,
\end{aligned}$$ which implies $$\begin{aligned}
\left\langle f(u_n),\phi\right\rangle &\underset{n \to \infty}{\longrightarrow} \left\langle f(u),\phi\right\rangle.
\end{aligned}$$ Therefore (\[33\]) converges to $$\begin{aligned}
\label{34}
\frac{d}{dt}\left\langle \partial_{t} u,\phi\right\rangle+\mu \left\langle \nabla u,\nabla \phi\right\rangle
+ \alpha \left\langle \partial_t u, \phi\right\rangle + \left\langle f (u), \phi\right\rangle = \left\langle b,\phi\right\rangle,
\end{aligned}$$ which means that $(u,\partial_t u)$ is a weak solution of $P_0$ and $u(0)=u_0$.
Finally we multiply equations (\[33\]) and (\[34\]) by a test function $\psi \in H^1([0,T])$ such that $\psi(0)=1$, $\psi(T)=0$ and integrating on $[0,T]$, we obtain for all $\phi \in (H_0^1(\Omega))^3$, $$\begin{aligned}
&\int_0^T\frac{d}{dt}\left\langle \partial_{t} u_n,\phi\right\rangle \psi dt+\mu\int_0^T \left\langle \nabla u_n,\nabla \phi\right\rangle \psi dt + \varepsilon_n \lambda_0 \int_0^T \left\langle \text{div}u_n, \text{div}\phi\right\rangle \psi dt \\
&\hspace{3.2cm}\alpha \int_0^T\left\langle \partial_t u_n, \phi\right\rangle \psi dt + \int_0^T\left\langle f (u_n), \phi\right\rangle \psi dt= \int_0^T\left\langle b,\phi\right\rangle \psi dt,\\
&\int_0^T\frac{d}{dt}\left\langle \partial_{t} u_,\phi\right\rangle \psi dt+\mu \int_0^T\left\langle \nabla u_,\nabla \phi\right\rangle \psi dt + \alpha \int_0^T\left\langle \partial_t u, \phi\right\rangle \psi dt + \int_0^T\left\langle f (u), \phi\right\rangle \psi dt \\
&\hspace{9.3cm}= \int_0^T\left\langle b,\phi\right\rangle \psi dt.
\end{aligned}$$ Solving the integrals and taking $n \to \infty$, $$\begin{aligned}
&-\left\langle v_1,\phi\right\rangle - \int_0^T\left\langle \partial_{t} u,\phi\right\rangle \frac{d}{dt}\psi dt +\mu\int_0^T \left\langle \nabla u,\nabla \phi\right\rangle \psi dt + \alpha \int_0^T\left\langle \partial_t u, \phi\right\rangle \psi dt\\
&\hspace{7.0cm}+\int_0^T\left\langle f (u), \phi\right\rangle \psi dt = \int_0^T\left\langle b,\phi\right\rangle \psi dt,\\
&-\left\langle \partial_{t} u(0),\phi\right\rangle - \int_0^T\left\langle \partial_{t} u,\phi\right\rangle \frac{d}{dt}\psi dt +\mu \int_0^T\left\langle \nabla u_,\nabla \phi\right\rangle \psi dt + \alpha \int_0^T\left\langle \partial_t u, \phi\right\rangle \psi dt \\
&\hspace{7.0cm} +\int_0^T\left\langle f (u), \phi\right\rangle \psi dt = \int_0^T\left\langle b,\phi\right\rangle \psi dt.
\end{aligned}$$
Therefore, $\partial_{t} u(0)=v_1$, which ends the proof.
From Theorem \[singular\] and its proof, it is worth making two comments as follows:
- the limit in the previous theorem is singular in the sense that $\mathcal{H}_{\varepsilon}$ is not the same for $\varepsilon$ varying the range $[0, 1]$;
- the space $\mathcal{H}_{\varepsilon}$ for weak solutions is defined by means of $(H_0^1(\Omega))^3 \times (L^2)^3$, where $H_0^1(\Omega)$ is provided with the norm $\|\cdot\|_{\varepsilon}$. Therefore, it makes sense to consider $(v_0,v_1)$ as initial data to any $P_\varepsilon$.
\[upper\] Under the assumptions $(\ref{2})$-$(\ref{aa})$, the family of attractors $\{\mathcal{A}_\varepsilon\}$ is upper semicontinuous with restect $\varepsilon \to 0$. More precisely, $$\lim_{\varepsilon \to 0 } d_{\mathcal{H}_0} (\hat{\i}_\varepsilon (\mathcal{A}_\varepsilon) , \mathcal{A}_0)=0,$$ where $d_{\mathcal{H}_0}$ denotes Hausdorff semi-distance and $\hat{\i}_\varepsilon: \mathcal{H}_\varepsilon \rightarrow \mathcal{H}_0$ is the identity map.
The proof is done by contradiction arguments and follows similar lines as presented e.g. in [@hale1988; @geredeli; @singular].
Let us assume, for some $\epsilon>0$, that
$$\sup_{y \in \mathcal{A}_\varepsilon} \inf_{z \in \mathcal{A}_0} \|\hat{\i}_{\varepsilon}(y)-z \|_{\mathcal{H}_0} \geq \epsilon.$$ Since for any $\varepsilon$, $\mathcal{A}_{\varepsilon}$ is compact, there exists a sequence $\{y_n^0 \}_n$ such that $y_n^0 \in \mathcal{A}_{\varepsilon_n}$ and $$\inf_{z \in \mathcal{A}_0}\|\hat{\i}_{\varepsilon_n}(y_n^0)-z \|_{\mathcal{H}_0} \geq \epsilon.$$ Let $y_n(t)=(u_n(t), \partial_t u_n(t) )$ be a full trajectory in $\mathcal{A}_{\varepsilon_n}$ such that $y_n(0)=y_n^0$. From Lemma \[Lemma4.1\], $$\begin{aligned}
\label{35}
\|y_n(t)\|_{\mathcal{H}_0} \leq R_1 +1.
\end{aligned}$$ Also, from Theorem \[3.16\], for each $n \in \mathbb{N}$, there exists $R_2^{\varepsilon_n}> 0$ such that $$\|\partial_t y_n(t)\|_{\mathcal{H}_{0}} \leq \| \partial_t y_n(t)\|_{\mathcal{H}_{\varepsilon_n}} \leq R_2^{\varepsilon_n} .$$ Additionally, from (\[up\]), we obtain the existence of $R_2>0$, that does not depend on $\varepsilon_n$ for all $n$, such that $$\|\partial_t y_n(t)\|_{\mathcal{H}_{0}} \leq \|\partial_t y_n(t)\|_{\mathcal{H}_{\varepsilon_n}} \leq R_2, \quad \forall \ t, n.$$ In this way, one sees that $$\mathcal{E}_{\varepsilon_n} u =- \alpha \partial_t u - f(u) -\partial_{tt} u +h \in (L^2(\Omega))^3.$$ Thus, multiplying this identity by $\mathcal{E}_{\varepsilon_n} u$, integrating and using Hölder’s inequality, there exists $R_3 > 0$, not depending on $\varepsilon_n$, such that $$\|\mathcal{E}_{ \varepsilon_n} u (t)\|_2 \leq R_3, \quad \forall \ t, n,$$ from where it follows that $$\begin{aligned}
(y_n) &\text{ is bounded on } L^\infty(\mathbb{R},\mathcal{H}_0 ), \\
(\partial_t y_n) &\text{ is bounded on } L^\infty(\mathbb{R},\mathcal{H}_0 ).
\end{aligned}$$ Using Simon’s Theorem of compactness for the spaces $\mathcal{H}_0^1 \overset{c}{\hookrightarrow} \mathcal{H}_0 \hookrightarrow \mathcal{H}_0$, we have that for any $T>0$, there exists a subsequence $\{y_{n_l}\}$ and $y \in C([-T,T], \mathcal{H}_0 )$ such that $$\begin{aligned}
\lim_{l \to \infty}\sup_{t \in [-T,T]} \|y_{n_l}(t)-y(t) \|_{\mathcal{H}_0} =0.
\end{aligned}$$ In particular, $$\begin{aligned}
\lim_{l \to \infty} \|\hat{\i}_{\varepsilon_{n_l} }( y_{n_l}^0)-y(0) \|_{\mathcal{H}_0} =0.
\end{aligned}$$ In order to get the desired contradiction, it remains to prove $y(0) \in \mathcal{A}_0$. In fact, since $\{ y_{n_l}^0\}_l$ is bounded on $\mathcal{H}_0$, we can process as in the proof of Theorem \[singular\], and prove that $y$ is a solution of $P_0$ for time varying $t \in [-T,T]$ with initial data $y(0)$. Since $T>0$ is arbitrary and (\[35\]) holds true, then $y(t)$ is a bounded full trajectory of $P_0$. This implies that $y(0)\in \mathcal{A}_0$. The proof Theorem \[upper\] is complete.
Acknowledgment {#acknowledgment .unnumbered}
==============
L. E. Bocanegra-Rodríguez was supported by CAPES, finance code 001 (Ph.D. Scholarship). M. A. Jorge Silva was partially supported by Fundação Araucária grant 066/2019 and CNPq grant 301116/2019-9. T. F. Ma was partially supported by CNPq grant 312529/2018-0 and FAPESP grant 2019/11824-0. P. N. Seminario-Huertas was partially supported by INCTMat-CNPq and CAPES-PNPD.
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[**Email addresses:**]{}
L. E. Bocanegra-Rodr[í]{}guez: [lito@icmc.usp.br]{}
M. A. Jorge Silva: [marcioajs@uel.br]{}
T. F. Ma: [matofu@mat.unb.br]{}
P. N. Seminario-Huertas: [pseminar@icmc.usp.br]{}
[^1]: Corresponding author.
|
---
abstract: 'We provide a general formalism to calculate the infrared correlators of multiple interacting scalar fields in the de Sitter space by means of the stochastic approach. These scalar fields are treated as test fields and hence our result is applicable to the models such as the curvaton scenario where the fields that yield initially isocurvature modes do not contribute to the cosmic energy density during inflationary expansion. The stochastic formalism combined with the argument of conformal invariance of the correlators reflecting the de Sitter isometries allows us to fix the form and amplitude of the three-point functions completely and partially for the four-point functions in terms of calculable quantities. It turns out that naive scaling argument employed in the previous literature does not necessarily hold and we derive the necessary and sufficient condition for the correlator to obey the naive scaling. We also find that correlation functions can in principle exhibit more complicated structure than argued in the literature.'
---
RESCEU-43/12
.5in
[**Consequences of a stochastic approach to the conformal invariance of inflationary correlators** ]{} .45in
[Hayato Motohashi$^{1,2}$, Teruaki Suyama$^2$ and Jun’ichi Yokoyama$^{2,3}$ ]{}
.45in [ *$^1$ Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan* ]{}\
[*$^2$ Research Center for the Early Universe (RESCEU), Graduate School of Science,\
The University of Tokyo, Tokyo 113-0033, Japan* ]{}\
[*$^3$ Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),\
The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan* ]{}
.4in
Introduction
============
De Sitter spacetime is as fundamental as Minkowski spacetime. It describes the accelerated expansion of the Universe sourced by the cosmological constant, or the effective vacuum energy mimicked by some scalar field. In addition to the observation that present Universe is undergoing accelerated expansion, it has become a standard paradigm that the small inhomogeneity, [*i.e.*]{}, tiny deviation from the Friedmann-Lemaître-Robertson-Walker Universe, is also thought to be explained by conversion of the quantum fluctuations of light scalar fields generated during de Sitter expansion in the early Universe [@Liddle-Lyth]. Because of this, it is important to study statistical properties of the scalar field in de Sitter space.
The simplest scenario for the generation of the curvature perturbation is to assume that inflaton which causes inflation simultaneously generates observed amplitude of the primordial curvature perturbation $\sim 10^{-5}$ [@Liddle-Lyth]. Despite its simplicity and that its predictions are consistent with observations [@Komatsu:2010fb], this scenario is not a prediction of (unknown) fundamental theory but rather an assumption. Actually, there are also many alternatives to this scenario most of which introduce other scalar fields which are originally isocurvature modes and convert to the curvature perturbation after inflation. The curvaton [@Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct] and the modulated reheating models [@Dvali:2003em; @Kofman:2003nx] are the representative ones that belong to this category. The scalar fields in this category are generically negligible during inflation, that is, they do not affect the dynamics of the inflationary expansion. Because of this, the correlation functions among such scalar fields become invariant under the de Sitter isometries. Note that this is not necessarily true for the case of the inflaton fluctuation since the inflaton can affect the expansion of the Universe.
The purpose of this paper is to study generic statistical properties of the correlation functions of scalar fields which enjoy full de Sitter invariance. In particular, we are concerned with infrared limit of the correlators. This is equivalent to large distance limit or to late time limit in de Sitter space since any two points that are close originally is eventually stretched to arbitrary large distance by accelerated expansion.
Some literature [@Antoniadis:1996dj; @Antoniadis:2011ib; @Creminelli:2011mw; @Kehagias:2012pd] already discuss the generic shape of the correlators consistent with de Sitter invariance. In these papers, the scalar fields appearing in the correlators are (implicitly) assumed to scale like $(-\eta)^{\Delta_a}$ for $-\eta \ll H^{-1}$ (late time), where $\eta$ is the conformal time of the de Sitter metric given by $$ds^2=\frac{-d\eta^2+d{\vec x}^2}{H^2 \eta^2},$$ and $\Delta_a$ is a constant specific to the scalar field $\phi_a$ and cannot be constrained only by the de Sitter invariance. For instance, $\langle \phi_a (\eta_1,{\vec x_1}) \phi_b (\eta_2,{\vec x_2}) \rangle \sim {(-\eta_1)}^{\Delta_a} {(-\eta_2)}^{\Delta_b}$ etc., where $\Delta_a$ neither depends on the number of fields nor the type of fields appearing in the correlator under consideration but just depends on $\phi_a$. This time dependence of the correlators then allows us to (partially) fix their dependence on the spatial coordinates by requiring that the correlators are invariant under the de Sitter isometric transformations. However, it is not clear how wide class of models satisfy the above scaling.
In order to evaluate the correlation functions for the de Sitter invariant state without introducing the scaling assumption a priori, we make use of the stochastic formalism. This formalism was introduced and developed in [@Starobinsky:1982ee; @Starobinsky:1984fx; @Starobinsky:1986fx; @Sasaki:1987gy; @Starobinsky:1994bd]. It has been since then employed for various inflationary models to study the infrared behavior of the scalar fields ([*e.g.*]{},[@Nakao:1988yi; @Nambu:1988je; @Mollerach:1990zf; @Starobinsky:1994bd; @Martin:2005ir; @Hattori:2005ac; @Finelli:2010sh; @Lorenz:2010vf; @Martin:2011ib; @Kawasaki:2012bk]). The formalism has been also used to determine the distribution of the initial value of the scalar field that may become important in the late Universe. This is important, for example, to provide the initial value of the quintessence field which causes the accelerated expansion of the present Universe [@Martin:2004ba; @Ringeval:2010hf].
The stochastic formalism solves the infrared dynamics by treating the long wavelength modes as the classical statistical variables that are sourced by stochastic noises coming from the short wavelength quantum modes. Since the dynamics can be solved, we can fix the form of the correlation functions unlike in the literature where they are constrained only by the argument of the de Sitter invariance. The stochastic formalism is especially useful when the nonperturbative effect is crucial to obtain the correct correlation functions as in the case of massless self-interacting scalar field [@Starobinsky:1994bd]. This is simply because infrared dynamics is solved without invoking perturbative expansion, that is, nonperturbative effect is automatically taken into account in the stochastic formalism.
Despite the many existing applications of the stochastic formalism to particular inflationary models, we do not find any paper that discusses general consequences of the stochastic approach on the infrared properties of the correlators of the multiple interacting scalar fields, which motivated us to address this issue. This article is the report of the calculations of the two-, three- and four-point functions of the multiple scalar fields derived by the use of the stochastic formalism. As will be demonstrated explicitly, in addition to the scaling index of the correlators, their amplitudes, which are completely unconstrained within the framework of the symmetry argument, can also be expressed in terms of quantities that are reasonably calculable in the stochastic approach. Thus we can completely fix the (infrared) correlation functions. We will find that the naive universal scaling $\phi_a \sim (-\eta)^{\Delta_a}$ in the correlators does not always hold and provide the necessary condition for this scaling to hold by explicitly constructing three-point and four-point functions.
Basics of the stochastic formalism
==================================
Our purpose is to study superhorizon evolution of (weakly) interacting multiple fields in de Sitter space and to calculate the resulting correlation functions in the stochastic formalism. Before doing this, in this section, let us briefly review the basic points of this formalism. For more details, see, for example, [@Starobinsky:1982ee; @Starobinsky:1984fx; @Starobinsky:1986fx; @Sasaki:1987gy; @Starobinsky:1994bd].
The first step of the stochastic formalism is to split the field (operator) into the long wavelength part ${\pmb \phi}_L$ and the short wavelength part; $${\pmb \phi} (t,{\vec x})={\pmb \phi}_L (t,{\vec x})+\int \frac{d^3 k}{{(2\pi)}^{\frac{3}{2}}} \theta (k-\epsilon a(t) H) \left( {\pmb a}_{\vec k} {\pmb \phi}_{\vec k}(t)e^{-i {\vec k} \cdot {\vec x}}+h.c. \right),$$ where $\theta (x)$ is a step function and we have used a bold letter ${\pmb \phi} =(\phi_1,\phi_2,\cdots,\phi_N)$ to make it refer to the multiple scalar fields. The second term on the right-hand side represents the contribution from short wavelength modes whose wavenumbers are greater than $\epsilon a(t) H$. Since we are assuming that the background spacetime is de Sitter one ($a(t) = e^{Ht}$), any mode that originally belongs to the short wavelength part eventually enters ${\pmb \phi}_L$. Although the evolution of the short wavelength part depends on the nature of the interactions, in this paper, we assume that any interaction is negligible for the short wavelength part and ${\pmb \phi}_{\vec k}$ obeys the massless Klein-Gordon equation in de Sitter space whose solution is given by $$\phi_{a,{\vec k}}=\frac{H}{\sqrt{2k}} \left( \eta-\frac{i}{k} \right) e^{-ik\eta},$$ where the subscript $a$ of $\phi$ runs from $1$ to $N$ and $\eta =-\frac{1}{aH}$ is the conformal time. Then we can treat ${\pmb a}_{\vec k}$ and ${\pmb a}^\dagger_{\vec k}$ as the standard annihilation and creation operators.
The second step is to consider the evolution equation for the long wavelength part ${\pmb \phi}_L$. Applying the slow-roll approximation and neglecting the higher-order terms in the short-wave modes and mode mixing terms, it reads $${\dot \phi_a}(t,{\vec x}) = -\frac{1}{3H}V_a({\pmb \phi})+f_a(t,{\vec x}), \label{eom-phi}$$ where $V_a\equiv \partial V/\partial \phi_a$, and we have abbreviated the subscript $L$ for the coarse-grained field. This equation may be regarded as a classical Langevin equation with a stochastic noise term, $f_a$, which is given by time derivative of the short wavelength part and represents modes of $k=\epsilon a(t) H$ that join ${\pmb \phi}_L$ at time $t$. Straightforward calculation shows that $f_a$ is random Gaussian whose two-point function is given by $$\langle f_a (t_1,{\vec x_1}) f_b (t_2,{\vec x_2}) \rangle =\delta_{ab} \frac{H^2}{4\pi^2} \delta (t_1-t_2) j_0 (\epsilon a(t) H |{\vec x_1}-{\vec x_2}|),$$ where $j_0(x) \equiv \frac{\sin x}{x}$ is the unnormalized sinc function.
Then we can easily show that the one-point probability density $\rho_1 ({\pmb \phi}({\vec x}),t)\equiv \rho (\phi_1({\vec x}),\phi_2({\vec x}),\cdots,t)$ obeys the Focker-Planck equation: $$\frac{\partial}{\partial t} \rho_1 ({\pmb \phi},t) =\frac{\partial}{\partial \phi_a} \left( \frac{V_a}{3H} \rho_1 ({\pmb \phi},t) \right)+\frac{H^3}{8\pi^2} \delta_{ab} \frac{\partial^2}{\partial \phi_a \partial \phi_b} \rho_1 ({\pmb \phi},t).$$ Introducing the dimensionless potential by $v({\pmb \phi}) \equiv 4\pi^2 V({\pmb \phi})/(3H^4)$, a general solution of the above equation can be written as $$\rho_1 ({\pmb \phi},t)=e^{-v({\pmb \phi})} \sum_n a_n \Phi_n ({\pmb \phi}) e^{-\Lambda_n (t-t_0)}, \label{sol-rho1}$$ where $\Phi_n ({\pmb \phi})$ is an eigenfunction of $N$-dimensional Schrödinger equation: $$\sum_a \left( -\frac{1}{2} \frac{\partial^2}{\partial \phi_a \partial \phi_a}+\frac{1}{2} (v_a v_a -v_{aa}) \right) \Phi_n ({\pmb \phi})=\frac{4\pi^2 \Lambda_n}{H^3} \Phi_n ({\pmb \phi}). \label{Schrodinger}$$ To be definite, we only consider the case the eigenfunctions are normalizable and satisfy $$\int d{\pmb \phi} ~\Phi_m ({\pmb \phi})\Phi_n ({\pmb \phi})=\delta_{mn}.$$ The left-hand side of Eq. (\[Schrodinger\]) can be written as $$\sum_a \frac{1}{2} \left( -\frac{\partial}{\partial \phi_a}+v_a \right) \left( \frac{\partial}{\partial \phi_a}+v_a \right) \Phi_n ({\pmb \phi}).$$ By multiplying $\Phi_n ({\pmb \phi})$ to the above expression from left and integrating it by parts, the left-hand side of becomes the integral of $\left[\left( \frac{\partial}{\partial \phi_a}+v_a \right) \Phi_n ({\pmb \phi})\right]^2$. Therefore, $\Lambda_n \ge 0$. In particular, the eigenfunction $\Phi_0$ having the minimum eigenvalue ([*i.e.*]{}, $\Lambda_0=0$) is given by $$\Phi_0 ({\pmb \phi})={\cal N} e^{-v({\pmb \phi})},~~~~~{\cal N}={\left( \int d{\pmb \phi}~e^{-2v({\pmb \phi})} \right)}^{-\frac{1}{2}}.$$ At sufficiently late time, all the modes having the positive $\Lambda_n$ decays in Eq. (\[sol-rho1\]) and $\rho_1$ becomes independent of time; $$\rho_1 ({\pmb \phi},t) \to \rho_{\rm eq} ({\pmb \phi})={\cal N}^2 e^{-2 v({\pmb \phi})}. \label{de-S-inv}$$ Here we used $a_0={\cal N}$ which holds from the normalization $\int d{\pmb \phi} \rho_1({\pmb \phi},t) =1$. This is a distribution function for an equilibrium state achieved in the de Sitter space.
It is possible that the integral appearing in the definition of ${\cal N}$ diverges. In such a case, Eq. (\[sol-rho1\]) does not possess any static solution and hence there is no equilibrium state. This happens, for example, for massless free scalar field.
Correlation functions
=====================
The formalism explained above enables us to evaluate correlation functions between multiple fields. In the following, we will derive the infrared behaviors of the two-point, three-point and four-point functions separately by means of the stochastic formalism. Derivation of the expression of the two-point functions mostly follows the one developed in [@Starobinsky:1994bd], which also showed their de Sitter invariance constructed from the equilibrium distribution function.
Two-point functions
-------------------
The (de Sitter invariant) spatial correlators (on superhorizon scales) at equal time can be written as $$\langle \phi_a (t,{\vec x_1}) \phi_b (t,{\vec x_2}) \rangle=\int d{\pmb \phi}^1
d{\pmb \phi}^2~\phi^1_a \phi^2_b ~\rho_2({\pmb \phi}^1,{\pmb \phi}^2,t), \label{two-point0}$$ where $\rho_2({\pmb \phi}^1,{\pmb \phi}^2,t)$ is the probability density of finding ${\pmb \phi}^i$ at ${\vec x_i}$ ($i=1,2$). It can be shown that $\rho_2$ obeys the following equation, $$\begin{aligned}
\frac{\partial \rho_2}{\partial t}=\sum_{i=1}^2 \left[ \frac{\partial}{\partial \phi_a^i} \left( \frac{V_a ({\pmb \phi}^i)}{3H} \rho_3 \right)+\frac{H^3}{8\pi^2} \frac{\partial^2 \rho_3}{\partial \phi_a^i \partial \phi_a^i} \right]+\frac{H^3}{4\pi^2} \frac{\partial^2 \rho_3}{\partial \phi_a^1 \partial \phi_a^2} j_0 (\epsilon a(t) H|{\vec x_1}-{\vec x_2}|). \label{evo-rho2}\end{aligned}$$ At early times when the points ${\vec x_1}$ and ${\vec x_2}$ are deeply inside the same Hubble patch, $j_0$ on the right-hand side becomes unity. For such a case, it can be shown that $$\rho_2 ({\pmb \phi}^1,{\pmb \phi}^2,t)= \delta ({\pmb \phi}^2-{\pmb \phi}^1) \rho_{\rm eq}({\pmb \phi}^1), \label{two-eq}$$ constitutes a static solution. This is very reasonable since setting $j_0=1$ means that ${\pmb \phi_1}$ and ${\pmb \phi_2}$ are fully correlated, [*i.e.*]{}, ${\pmb \phi_1}={\pmb \phi_2}$ and the appearance of $\rho_{\rm eq}$ reflects Eq. (\[de-S-inv\]) that guarantees the de Sitter invariance [@Starobinsky:1994bd]. Therefore, Eq. (\[two-eq\]) can be used as the initial condition for Eq. (\[evo-rho2\]).
It is hard to find the analytic solution of Eq. (\[evo-rho2\]) with the initial condition (\[two-eq\]). Here we make an approximation that $$j_0 (\epsilon a(t) Hr)\simeq \theta (1-\epsilon a(t) Hr).$$ This drastically simplifies the equation without losing any essential point of the stochastic formalism. This approximation allows us to write down the solution of Eq. (\[evo-rho2\]) after ${\vec x_1}$ and ${\vec x_2}$ are separated by super-horizon distance; $$\rho_2 ({\pmb \phi}^1,{\pmb \phi}^2,t)=
\int d{\pmb \phi}_r \Pi ({\pmb \phi}^1,t;{\pmb \phi}_r,t_r)
\Pi ({\pmb \phi}^2,t;{\pmb \phi}_r,t_r) \rho_{\rm eq} ({\pmb \phi}_r),$$ where $t_r$ is a solution of $\epsilon a(t_r) H|{\vec x_1}-{\vec x_2}|=1$ and represents the time when ${\pmb \phi}$ at ${\vec x_1}$ and ${\pmb \phi}$ at ${\vec x_2}$ get uncorrelated. Here $\Pi ({\pmb \phi}^1,t_1;{\pmb \phi}^2,t_2)$ is the transition probability from ${\pmb \phi}={\pmb \phi}^2$ at $t=t_2$ to ${\pmb \phi}={\pmb \phi}^1$ at $t=t_1$. Its expression in terms of the eigenfunctions is given by [@Starobinsky:1994bd] $$\Pi ({\pmb \phi}^1,t_1;{\pmb \phi}^2,t_2)=e^{-v({\pmb \phi}^1)+v({\pmb \phi}^2)} \sum_{n=0}^\infty \Phi_n ({\pmb \phi}^1) \Phi_n ({\pmb \phi}^2) e^{-\Lambda_n (t_1-t_2)}. \label{sol-Pi}$$ This expression of $\rho_2$ appeals to our intuition, that is, $\rho_2$ is given by the product of the probability of ${\pmb \phi}$ going to ${\pmb \phi_1}$ from ${\pmb \phi_r}$ by the stochastic process described by Eq. (\[eom-phi\]) and that of ${\pmb \phi}$ going to ${\pmb \phi_2}$ from ${\pmb \phi_r}$ with a weight $\rho_{\rm eq}({\pmb \phi_r})$. Using this picture for $\rho_2$, we have $$\begin{aligned}
\langle \phi_a (t,{\vec x_1}) \phi_b (t,{\vec x_2}) \rangle = \int d{\pmb \phi}^1 d{\pmb \phi}^2 ~\phi^1_a \phi^2_b \int d{\pmb \phi}_r \Pi ({\pmb \phi}^1,t;{\pmb \phi}_r,t_r) \Pi ({\pmb \phi}^2,t;{\pmb \phi}_r,t_r) \rho_{\rm eq} ({\pmb \phi}_r). \label{two-point}\end{aligned}$$ Substituting Eq. (\[sol-Pi\]) into Eq. (\[two-point\]), we find $$\langle \phi_a (t,{\vec x_1}) \phi_b (t,{\vec x_2}) \rangle ={\cal N}^2 \sum_{n=0}^\infty A^{(n)}_{a} A^{(n)}_{b} {(HR_{12})}^{-\frac{2 \Lambda_n}{H}} \exp \left( -\frac{2\Lambda_n}{H} \ln \epsilon \right), \label{two-point-2}$$ where $R_{12}=a(t)|{\vec x_1}-{\vec x_2}|$ is the physical distance between ${\vec x_1}$ and ${\vec x_2}$ and $A^{(n)}_a$ is defined by $$A^{(n)}_a = \int d {\pmb \phi}~\phi_a e^{-v({\pmb \phi})} \Phi_n ({\pmb \phi}). \label{def-A}$$ Equation (\[two-point-2\]) is our expression for the two-point functions. To minimize the effect of $\epsilon$, we choose $\epsilon$ so that it satisfies $\exp \left( -\frac{2\Lambda_n}{H} \ln \epsilon \right) \sim 1$ for the dominant mode contributing to the two-point functions, as suggested in [@Starobinsky:1994bd]. Now let us consider correlation function (\[two-point-2\]) on sufficiently late time or (equivalently) large scales in which case $a(t) |{\vec x_1}-{\vec x_2}|/H^{-1}$ is quite large. Then the leading contribution to the correlator is from a state $\Phi_{\bar n}({\pmb \phi})$ labeled by an integer ${\bar n}$ having minimum $\Lambda_{\bar n}$ (apart from the ground state $n=0$ which has $\Lambda_n=0$) with nonvanishing $A^{({\bar n})}_{a} A^{({\bar n})}_{b}$. Generally, ${\bar n}$ depends on the choice of fields $(\phi_a,~\phi_b)$ and can vary for different set of fields of the correlators. In particular, it may happen that ${\bar n}$ for $A^{({\bar n})}_{a} A^{({\bar n})}_{a}$ which we denote by $n_a$ is different from $n_b$ and $\Lambda_{n_a} \neq \Lambda_{n_b}$. In such a case, ${\bar n}$ for $A^{({\bar n})}_{a} A^{({\bar n})}_{b}$ which we denote by $n_{ab}$ may or may not exist. If $n_{ab}$ does not exist, it means no correlation between $\phi_a$ and $\phi_b$. Alternatively, our result can also allow a possibility that $n_{ab}$ exists, in which case $\phi_a$ and $\phi_b$ are correlated. By a simple consideration, we find that such $n_{ab}$ is either equal to or larger than $n_a$ or $n_b$ whichever is greater. In short, two different fields having different ${\bar n}$ ([*i.e.*]{}, $n_a \neq n_b$) can in principle have correlation between them, which is consistent with our findings for the simple example demonstrated in the Introduction.
On the contrary, if some of the fields are interacting so that the minimum integers for the nonvanishing of $A^{(n)}_a$ for such fields are all the same, then the correlator exhibits a universal behavior in the sense that the scaling index for any field takes the same value and is completely given by $\Lambda_{\bar n}$, eigenvalue of the $N$-dimensional Schrödinger equation. In this case, correlators at late time scales as $$\langle \phi_a (t,{\vec x_1}) \phi_b (t,{\vec x_2}) \rangle \sim {|{\vec x_1}-{\vec x_2}|}^{-\frac{2\Lambda_{\bar n}}{H}}.$$
Three-point functions
---------------------
What we want to evaluate is the spatial three-point functions evaluated at equal time $t$; $$\langle \phi_a (t,{\vec x_1}) \phi_b (t,{\vec x_2}) \phi_c (t,{\vec x_3}) \rangle=\int d{\pmb \phi}^1 d{\pmb \phi}^2 d{\pmb \phi}^3~\phi^1_a \phi^2_b \phi^3_c ~\rho_3 ({\pmb \phi}^1,{\pmb \phi}^2,{\pmb \phi}^3,t), \label{three-point}$$ where $\rho_3({\pmb \phi}^1,{\pmb \phi}^2,{\pmb \phi}^3,t)$ is the probability density of finding ${\pmb \phi}^i$ at ${\vec x_i}$ ($i=1,2,3$). It can be shown that $\rho_3$ obeys the following equation, $$\begin{aligned}
\frac{\partial \rho_3}{\partial t}=\sum_{i=1}^3 \left[ \frac{\partial}{\partial \phi_a^i} \left( \frac{V_a ({\pmb \phi}^i)}{3H} \rho_3 \right)+\frac{H^3}{8\pi^2} \frac{\partial^2 \rho_3}{\partial \phi_a^i \partial \phi_a^i} \right]+\frac{H^3}{4\pi^2} \sum_{i<j} \frac{\partial^2 \rho_3}{\partial \phi_a^i \partial \phi_a^j} j_0 (\epsilon a(t) H|{\vec x_i}-{\vec x_j}|). \label{evo-rho3}\end{aligned}$$ At early times, all the three points are inside the Hubble radius, [*i.e.*]{}, $a(t)|{\vec x_i}-{\vec x_j}|H \ll 1$ for any $1 \le i,j \le 3$, and the fields are maximally correlated each other. During this epoch, the evolution equation for $\rho_3$ can therefore be well approximated by Eq. (\[evo-rho3\]) with all the $j_0$ being replaced by unity. We can verify that this equation allows the following solution, $$\rho_3 ({\pmb \phi}^1,{\pmb \phi}^2,{\pmb \phi}^3,t)=\delta ({\pmb \phi}^3-{\pmb \phi}^2) \delta ({\pmb \phi}^2-{\pmb \phi}^1) \rho_{\rm eq}({\pmb \phi}^1),$$ whose physical meaning is obvious from the reasoning we made earlier. This solution is independent of time and can be used as a de Sitter invariant initial condition of Eq. (\[evo-rho3\]). Then the problem is to solve Eq. (\[evo-rho3\]) with such an initial condition until sufficiently late time when all the points are separated by super-horizon length and any correlation between the fields at different points is turned off. Although this is a well defined mathematical problem and we can in principle solve Eq. (\[evo-rho3\]) and perform the integrals appearing in Eq. (\[three-point\]) to get the three-point functions, we find it difficult in practice to solve Eq. (\[evo-rho3\]) which is highly involved partial differential equation. Fortunately, as long as we are only concerned with sufficiently late time behavior of the three-point functions, which is actually the present case, there is a way to derive an analytic expression without directly solving Eq. (\[evo-rho3\]) as we will demonstrate below. The point is to utilize the de Sitter isometries which allows us to find the three-point functions at general points by implementing the coordinate transformation that preserves the de Sitter metric from some extreme configuration of the points where analytic evaluation of the three-point functions (to a very good approximation) is feasible.
As is well known, there are 10 isometries for the metric of the de Sitter spacetime. Among 10 isometries, translations and rotations for the spatial coordinate constitute 6 isometries. We also have the dilatation isometry which amounts to multiply both time and spatial coordinates by the same constant factor. The remaining 3 isometries are complex mixing whose infinitesimal form is given by $$\eta'=\eta-2\eta ({\vec b} \cdot {\vec x}),~~~~~{\vec x}'={\vec x}+{\vec b}(-\eta^2+x^2)-2 ({\vec b} \cdot {\vec x}) {\vec x},$$ where $x^2 \equiv{\vec x}^2$ and ${\vec b}$ is an infinitesimal constant vector. On super-horizon scales, or on sufficiently late time, in which $\eta$ is much smaller than ${\vec x}$, the finite version of the above transformation can be written as $$\eta'=\frac{\eta}{1+2{\vec b}\cdot {\vec x}+b^2 x^2},~~~~~{\vec x}'=\frac{{\vec x}+x^2 {\vec b}}{1+2{\vec b}\cdot {\vec x}+b^2 x^2}. \label{transform}$$ The transformation of the spatial coordinate does not involve time and becomes exactly what is known as the special conformal transformation. The special conformal transformation combined with the dilatation, rotation and translation transformations constitute the conformal transformation [@francesco]. We will come back to this point later when we utilize the conformal symmetry to fix the correlators.
Now let us consider the three-point function in the squeezed limit with different time coordinates for the different points; $$\langle \phi_a (t_1,{\vec y_1}) \phi_b (t_2,{\vec y_2}) \phi_b (t_3,{\vec y_3}) \rangle. \label{3pt-squeezed}$$ Although what we are interested in is the equal time correlators ($t_1=t_2=t_3=t$), for the moment, we let them to be independent due to the reason which will become clear later. To be definite, we take ${\vec y_3}={\vec 0}$ and $|{\vec y_1}| \gg |{\vec y_2}|$ (squeezed limit), which is always possible without a loss of generality. For convenience, we define $R\equiv |{\vec y_1}|$ and $r\equiv |{\vec y_2}|$. All the time coordinates $t_1,t_2$ and $t_3$ are assumed to be very large so that any two different points are eventually separated by super-horizon size distance. Instead of directly solving Eq. (\[evo-rho3\]), the following physical consideration enables us to evaluate Eq. (\[3pt-squeezed\]). By definition, ${\vec y_2}$ and ${\vec y_3}$(this is actually ${\vec 0}$) are close together compared to ${\vec y_1}$. Therefore, a field at ${\vec y_1}$ first gets uncorrelated and starts to evolve independently when the physical distance between ${\vec y_1}$ and other two points becomes equal to ${(\epsilon H)}^{-1}$. Strictly speaking, the epoch when $a H |{\vec y_1}-{\vec y_2}|=1$ occurs is different from that when $a H |{\vec y_1}-{\vec y_3}|=1$ is satisfied. But the difference between these little affects the final result in the squeezed limit and we can safely take them as being equal. At this time, ${\vec y_2}$ and ${\vec y_3}$ are still deeply inside the Hubble radius and fields at those two points take the same value. As the Universe expands, the physical distance between ${\vec y_2}$ and ${\vec y_3}$ then becomes equal to ${(\epsilon H)}^{-1}$. After this time, all the fields at different points evolve separately. This picture, which is a good approximation when there is a huge hierarchy among the lengths of the sides of the triangle, enables us to write the three-point function in the following form, $$\begin{aligned}
\langle \phi_a (t_1,{\vec y_1}) \phi_b (t_2,{\vec y_2}) \phi_c (t_3,{\vec y_3}) \rangle=&&\int d{\pmb \phi}^1 d{\pmb \phi}^2 d{\pmb \phi}^3d{\pmb \phi}_Rd{\pmb \phi}_r~\phi^1_a \phi^2_b \phi^3_c ~\Pi ({\pmb \phi}^1,t_1;{\pmb \phi}_R,t_R) \Pi ({\pmb \phi}^2,t_2;{\pmb \phi}_r,t_r) \nonumber \\
&&\times \Pi ({\pmb \phi}^3,t_3;{\pmb \phi}_r,t_r) \Pi ({\pmb \phi}_r,t_r;{\pmb \phi}_R,t_R) \rho_{\rm eq}({\pmb \phi}_R). \end{aligned}$$ Substituting Eq. (\[sol-Pi\]), the above expression reduces to $$\begin{aligned}
\langle \phi_a (t_1,{\vec y_1}) \phi_b (t_2,{\vec y_2}) \phi_b (t_3,{\vec y_3}) \rangle&=&{\cal N}^2 \sum_{\ell,m,n} A_a^{(\ell)}A_b^{(m)}A_c^{(n)} B_{\ell m n} \nonumber \\
&&\times e^{-\Lambda_\ell(t_1-t_R)}e^{-\Lambda_m(t_2-t_r)}
e^{-\Lambda_n (t_3-t_r)} e^{-\Lambda_\ell(t_r-t_R)}, \label{squeezed-3pt}\end{aligned}$$ where $B_{\ell m n}$ is defined by $$B_{\ell m n} \equiv \int d{\pmb \phi}~e^{v({\pmb \phi})} \Phi_\ell ({\pmb \phi})\Phi_m ({\pmb \phi})\Phi_n ({\pmb \phi}), \label{def-B}$$ and is totally symmetric under the permutation of the indices.
Now let us consider the late time behavior of Eq. (\[squeezed-3pt\]). Since all the eigenvalues satisfy $\Lambda_n \ge 0$, the leading contribution comes from terms with a particular set of $({\bar \ell},{\bar m},{\bar n})$ having nonvanishing $A_a^{({\bar \ell})}A_b^{({\bar m})}A_c^{({\bar n})} B_{{\bar \ell} {\bar m} {\bar n}}$ with the minimum decay rate $e^{-\Lambda_{\bar \ell} t_1}e^{-\Lambda_{\bar m} t_2}e^{-\Lambda_{\bar n} t_3}$. This means that each integer of $({\bar \ell},{\bar m},{\bar n})$ is determined by the lowest value of the eigenvalues with the condition that $A_a^{({\bar \ell})}A_b^{({\bar m})}A_c^{({\bar n})} B_{{\bar \ell} {\bar m} {\bar n}}$ does not vanish. Note that this condition does not necessarily fix $({\bar \ell},{\bar m},{\bar n})$ uniquely apart from the trivial permutations and, depending on the interactions among the scalar fields, it is possible that there are more than one set of $({\bar \ell},{\bar m},{\bar n})$. Generally speaking, all the numbers can be different from each other, can be partially equal or completely coincide and concrete values of $({\bar \ell},{\bar m},{\bar n})$ needs specification of the underlying model. This may be understood by considering the simplest case, [*i.e.*]{}, single field case in which $B_{111}$ and $B_{112}$ vanish and the lowest contributions are either $B_{122} A^{(1)}A^{(2)}A^{(2)}+{\rm perms.}$ (when $\Lambda_3 > 2\Lambda_2-\Lambda_1$) or $B_{113} A^{(1)}A^{(1)}A^{(3)}+{\rm perms.}$ (when $\Lambda_3 < 2\Lambda_2-\Lambda_1$). If the model yields $\Lambda_3 = 2 \Lambda_2-\Lambda_1$ by chance, both two contributions decay in time at the same rate and none of the two terms can be neglected even at sufficiently late time. We will come back to the single field case later.
Using the equations for $t_r$ and $t_R$ given by $$t_r=-\frac{1}{H} \ln \left( \epsilon R H \right),~~~~~t_R=-\frac{1}{H} \ln \left( \epsilon r H \right),$$ we find that the three-point functions for sufficiently late time become $$\begin{aligned}
\langle \phi_a (t_1,{\vec y_1}) \phi_b (t_2,{\vec y_2}) \phi_b (t_3,{\vec y_3}) \rangle \approx &&
{\cal N}^2 \sum_{\rm min} B_{{\bar \ell} {\bar m} {\bar n}}
A_a^{({\bar \ell})}A_b^{({\bar m})}A_c^{({\bar n})}
e^{\Lambda_{\bar \ell}t_1+\Lambda_{\bar m}t_2+\Lambda_{\bar n} t_3} \nonumber \\
&&\times (HR)^{-\frac{2\Lambda_{\bar \ell}}{H}}
(Hr)^{-\frac{\Lambda_{\bar m}+\Lambda_{\bar n}-\Lambda_{\bar \ell}}{H}}, \label{squee}\end{aligned}$$ where the summation indicated by “min” is done for all the possible sets of $({\bar \ell},{\bar m},{\bar n})$ satisfying the condition mentioned above. Here, we fixed $\epsilon$ in the same way as the case of two-point functions. This is the late time three-point functions in the squeezed limit.
This form allows us to write the expressions for the general shape of the triangle formed by ${\vec x_1},~{\vec x_2}$ and ${\vec x_3}$ by using that the left-hand side of Eq. (\[squee\]) is invariant under the transformation that preserves de Sitter isometry. To understand this, notice that we can always move one of the point, say ${\vec x_1}$, to a vector having very long length by performing the transformation (\[transform\]). Indeed, if we choose ${\vec b}$ as $-\frac{\vec x_1}{x_1^2}+{\vec \xi}$, then we find that the transformed point is ${\vec y_1}={\vec \xi}/\xi^2$, whose distance from the origin can be arbitrary large in the limit $\xi \ll 1$. By this transformation, time coordinates, which have the same value in the original frame, take different values in the new frame. Thus, equal time correlator for arbitrary configuration of points is related to the squeezed correlator with different time coordinates for different points by isometry-preserving transformation. This is the reason why only the information in the squeezed limit is enough to obtain the three-point functions for any configuration of points.
The de Sitter isometries for the spatial coordinates at late time $|\eta| \to 0$ become the conformal transformation which serves a base for the conformal field theory [@Antoniadis:2011ib]. In the language of the conformal field theory [@francesco], focusing on any one particular term in Eq. (\[squee\]), the field $\phi_a$ can be interpreted as a primary field of a conformal weight $-\Lambda_a/H$. Since it is well established how to obtain the general expression of the three-point functions of the conformal fields out of the squeezed limit [@francesco], we do not expand the detailed discussion here and we only give the final result for the three-point functions which is given by $$\begin{aligned}
\langle \phi_a (t,{\vec x_1}) \phi_b (t,{\vec x_2}) \phi_c (t,{\vec x_3}) \rangle &\approx&
{\cal N}^2\sum_{\rm min} B_{{\bar \ell} {\bar m} {\bar n}} A_a^{({\bar \ell})}A_b^{({\bar m})}A_c^{({\bar n})}
{\left( H R_{12} \right)}^{-\frac{\Lambda_{\bar \ell}+\Lambda_{\bar m}-\Lambda_{\bar n}}{H}} \nonumber \\
&&\times
{\left( H R_{23} \right)}^{-\frac{\Lambda_{\bar m}+\Lambda_{\bar n}-\Lambda_{\bar \ell}}{H}}
{\left( H R_{31} \right)}^{-\frac{\Lambda_{\bar n}+\Lambda_{\bar \ell}-\Lambda_{\bar m}}{H}}, \label{3pt-final}\end{aligned}$$ where $R_{ij} \equiv a(t) |{\vec x_i}-{\vec x_j}|$ represents the physical distance between ${\vec x_i}$ and ${\vec x_j}$. This equation is one of our primary result. This shows that all the information regarding the three-point functions can be obtained once we know the eigenfunctions and eigenvalues of the $N$ dimensional Schrödinger equation (\[Schrodinger\]). As a consistency check, we can verify that squeezed limit of Eq. (\[3pt-final\]) ($R_{12}=R_{13}=R,~R_{23}=r$) gives back Eq. (\[squee\]). Also, we can implement the similar derivation of the three-point functions for the equilateral case in which $|{\vec y_1}-{\vec y_2}|=|{\vec y_2}-{\vec y_3}|=|{\vec y_3}-{\vec y_1}|$. This case also allows the evaluation of the three-point function without resorting to the direct computation of Eq. (\[evo-rho3\]). It can be verified that the result coincides with the equilateral case of Eq. (\[3pt-final\]).
Now there are several points to be remarked. First, as mentioned earlier, $({\bar \ell},{\bar m},{\bar n})$ is given by the condition that it is a set of integers as small as possible with nonvanishing $A_a^{({\bar \ell})}A_b^{({\bar m})}A_c^{({\bar n})} B_{{\bar \ell} {\bar m} {\bar n}}$. In principle, this integer set can vary for different choice of fields $(a,b,c)$. This suggests that only a knowledge of correlators of the three product of the same field ([*i.e.*]{}, $\langle \phi_a \phi_a \phi_a \rangle$ etc.) is not enough to know the scaling behavior of the three-point function of the three different fields. Furthermore, if at least one of the field appearing in the correlator is different from others, it can happen that the correlator cannot be given by a single term with a power-law form. Instead the correlator becomes a sum of up to six terms each of which exhibits the different power-law behavior. Notice that all of those terms are not necessarily nonvanishing and it is also possible that only some of them remain nonzero.
Secondly, the result (\[3pt-final\]) is obtained without the use of the perturbative expansion in terms of the strength of the interactions among fields. In the standard approach, the three-point functions (and higher order functions as well) are calculated by using the so-called in-in formalism which usually uses perturbative expansion and truncation of the calculations at some order to yield an analytic expression. Although this approach is completely justified as long as the higher-order terms contribute much less to the final result, it is known that some particular model (for example, massless scalar field with a quartic self-interaction) requires nonperturbative treatment to obtain the reliable correlators. In more general terms, the perturbative approach fails when the system does possess the de Sitter invariant state only if the interactions are present. In such a case, the standard perturbative approach needs some care, if not impossible, to get the correct result. On the other hand, our result (\[3pt-final\]) is derived by the stochastic approach. As is well known, the stochastic approach includes the nonperturbative effects coming from the long wavelength modes. Except for some simple models, numerical computations are required to solve Eq. (\[Schrodinger\]) in order to obtain the eigenvalues $\Lambda_n$. But whichever computation method is used, the obtained $\Lambda_n$ contains the nonperturbative effect. This is also true for the amplitude of the correlator, [*i.e.*]{}, $A_a^{({\bar \ell})}A_b^{({\bar m})}A_c^{({\bar n})} B_{{\bar \ell} {\bar m} {\bar n}}$. The point is that everything is reasonably calculable in the stochastic approach while it is hard to take into account the higher-order or nonperturbative effects in the standard perturbative expansion of the in-in formalism. Therefore, our result will be quite useful when the nonperturbative effect is crucial to get the correct correlators.
Four-point functions
--------------------
Contrary to the case of the three-point function where consideration in the squeezed limit is enough to get the correlator for the general triangle, this is not the case for the four-point function. The reason behind this is that the transformation specified by the vector ${\vec b}$ is not sufficient to convert any quadrangle to the one having hierarchy among all the sides of the quadrangle. Thus we need to solve the evolution equation for the probability distribution function for four variables even to obtain the late time behaviour of the correlator. Yet, it would be interesting to see to what extent we can restrict the form of the four-point function in analytic way.
[ ![Double squeezed quadrangle for which $R_{14} \simeq R_{12} \gg R_{23} \simeq R_{24} \gg R_{34}$, where $R_{ij}$ denotes the physical distance between ${\vec y}_i$ and ${\vec y}_j$.[]{data-label="fig1"}](fig1.eps "fig:") ]{}
Let us first consider the double squeezed quadrangle for which $R_{14} \simeq R_{12} \gg R_{23} \gg R_{34}$, where $R_{ij}$ denotes the distance between ${\vec y}_i$ and ${\vec y}_j$ (see Fig. \[fig1\]). For this quadrangle, using the similar reasoning we made in the case of the three-point function, the four-point function can be written as $$\begin{aligned}
\left\langle \prod_{i=1}^4 \phi_{a_i} (t_i,{\vec y_i}) \right\rangle =&&{\cal N}^2 \sum_{m,n,p,q,s}B_{mns} B_{spq}A^{(m)}_{a_1} A^{(n)}_{a_2} A^{(p)}_{a_3} A^{(q)}_{a_4} e^{-\Lambda_m (t_1-t_X)-\Lambda_n (t_2-t_Y)}\nonumber \\
&&\times e^{-\Lambda_p (t_3-t_Z)-\Lambda_q (t_4-t_Z)-\Lambda_s (t_Z-t_Y)-\Lambda_m (t_Y-t_X)},\end{aligned}$$ where $X =R_{14},~Y=R_{24}$ and $Z=R_{34}$ ($X \gg Y \gg Z$). Interestingly, the four-point function in this limit is written solely in terms of the quantities characterizing the three-point functions. At sufficiently late time, this reduces to $$\begin{aligned}
\left\langle \prod_{i=1}^4 \phi_{a_i} (t_i,{\vec y_i}) \right\rangle &&={\cal N}^2 \sum_{\rm min}B_{{\bar m}{\bar n}{\bar s}} B_{{\bar s}{\bar p}{\bar q}}A^{({\bar m})}_{a_1} A^{({\bar n})}_{a_2} A^{({\bar p})}_{a_3} A^{({\bar q})}_{a_4} e^{-\Lambda_{\bar m} t_1-\Lambda_{\bar n} t_2}\nonumber \\
&&\times e^{-\Lambda_{\bar p} t_3-\Lambda_{\bar q} t_4} {\left( HX \right)}^{-\frac{2\Lambda_{\bar n}}{H}}
{\left( HY \right)}^{-\frac{\Lambda_{\bar m}+\Lambda_{\bar s}-\Lambda_{\bar n}}{H}}
{\left( HZ \right)}^{-\frac{\Lambda_{\bar p}+\Lambda_{\bar q}-\Lambda_{\bar s}}{H}}, \label{4pt-doublesqueeze}\end{aligned}$$ where the meaning of the “min” in the summation is the same as the case for the three-point function. Now, the expression of the right-hand side of the above equation must be the double squeezed limit of the correlator for the general quadrangle. Using again the procedure of restricting the form of the four-point function in the conformal field theory [@francesco], we find that the four-point function (\[4pt-doublesqueeze\]) for the general quadrangle becomes $$\begin{aligned}
\left\langle \prod_{i=1}^4 \phi_{a_i} (t,{\vec x_i}) \right\rangle &&=
{\cal N}^2 \sum_{\rm min}f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}} \left( \frac{R_{12}R_{34}}{R_{13}R_{24}},~\frac{R_{12}R_{34}}{R_{23}R_{14}} \right) \prod_{i<j} {\left( HR_{ij} \right)}^{\frac{\Delta}{3}-\Delta_i-\Delta_j}, \label{4pt-general}\end{aligned}$$ where $\Delta_1=\frac{\Lambda_{\bar m}}{H},~\Delta_2=\frac{\Lambda_{\bar n}}{H}$ etc. and $\Delta=\Delta_1+\Delta_2+\Delta_3+\Delta_4$. The function $f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}$ depending on the two arguments that are invariant under the de Sitter isometric transformation (only for late time or on superhorizon scales) is completely arbitrary function at the level of the symmetry argument. Some more nontrivial information is needed to (even partially) fix the form of it.
Although we cannot find analytic form of $f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}$ for the whole domain of the arguments, we can derive its asymptotic behavior or a value at specific point for some limiting cases of the arguments by considering the corresponding squeezed shape of the quadrangle. For instance, comparison between Eqs. (\[4pt-doublesqueeze\]) and (\[4pt-general\]) leads to the following asymptotic form of $f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}} (z,z)$ when $z \ll 1$ as $$f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}(z,z) \simeq B_{{\bar m}{\bar n}{\bar s}} B_{{\bar s}{\bar p}{\bar q}}A^{({\bar m})}_{a_1} A^{({\bar n})}_{a_2} A^{({\bar p})}_{a_3} A^{({\bar q})}_{a_4} z^{\frac{\Lambda_{\bar s}}{H}-\frac{\Lambda_{\bar m}+\Lambda_{\bar n}+\Lambda_{\bar p}+\Lambda_{\bar q}}{3H}},~~~~~{\rm for}~~ z\ll 1. \label{trib}$$ In a similar way, consideration of the other quadrangles with different shapes shown in Fig. \[fig2\] leads to the following expression; $$\begin{aligned}
&&f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}(z,1) \simeq B_{{\bar m}{\bar n}{\bar s}} B_{{\bar s}{\bar p}{\bar q}}A^{({\bar m})}_{a_1} A^{({\bar n})}_{a_2} A^{({\bar p})}_{a_3} A^{({\bar q})}_{a_4} z^{-\frac{\Lambda_{\bar s}}{H}+\frac{\Lambda_{\bar m}+\Lambda_{\bar n}+\Lambda_{\bar p}+\Lambda_{\bar q}}{3H}},~~~~~{\rm for}~~ z\gg 1, \label{case1}\\
&&f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}(1,z) \simeq B_{{\bar m}{\bar q}{\bar s}} B_{{\bar s}{\bar n}{\bar q}}A^{({\bar m})}_{a_1} A^{({\bar n})}_{a_2} A^{({\bar p})}_{a_3} A^{({\bar q})}_{a_4} z^{-\frac{\Lambda_{\bar s}}{H}+\frac{\Lambda_{\bar m}+\Lambda_{\bar n}+\Lambda_{\bar p}+\Lambda_{\bar q}}{3H}},~~~~~{\rm for}~~ z\gg 1, \label{case2}\\
&&f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}(1,1) \simeq T_{{\bar m}{\bar n}{\bar p}{\bar q}}A^{({\bar m})}_{a_1} A^{({\bar n})}_{a_2} A^{({\bar p})}_{a_3} A^{({\bar q})}_{a_4}, \label{case3}\end{aligned}$$ In Eq. (\[case3\]), we have introduced $T_{{\bar m}{\bar n}{\bar p}{\bar q}}$ defined by $$T_{{\bar m}{\bar n}{\bar p}{\bar q}}=\int d{\pmb \phi}~e^{2v({\pmb \phi})} \Phi_m ({\pmb \phi})\Phi_n ({\pmb \phi}) \Phi_p ({\pmb \phi})\Phi_q ({\pmb \phi}),$$ which is a new quantity to characterize the four-point function. These expressions for the several limiting cases indicate that four-point function having the double squeezed quadrangle is determined by $B_{mns}$, [*i.e.*]{}, related to (square of) the three-point function while the correlator having the single squeezed one is determined by $T_{mnpq}$ which has nothing to do with the three-point function. Finally, let us remark that Eqs. – cover all the possible limiting cases which we can obtain. One can show that we cannot make such a quadrangle which provides $f_{a_1a_2a_3a_4}^{{\bar m}{\bar n}{\bar p}{\bar q}}(z,w)$ with $z=1,~w\ll 1$, or $z\ll 1,~w=1$ or $z,w\gg 1$.
[ ![Left quadrangle having $R_{12} \approx R_{34} \gg R_{13} \approx R_{24}$ yields Eq. (\[case1\]). Middle quadrangle having $R_{12} \approx R_{34} \gg R_{14} \approx R_{23}$ yields Eq. (\[case2\]). Right quadrangle having $R_{12} \approx R_{13} \approx R_{14} \gg R_{23} = R_{34} = R_{24}$ yields Eq. (\[case3\]).[]{data-label="fig2"}](fig2.eps "fig:") ]{}
Discussion and summary
======================
In this paper, we have developed a general formalism to calculate three- and four-point functions among test light scalar fields in de Sitter space in the framework of the stochastic formalism. Stochastic approach treats the coarse grained fields as classical variables that are affected by the random Gaussian noises reflecting the effect that the short wavelength modes enter the long wavelength modes due to the accelerated expansion. We investigated behaviors of those correlators at sufficiently late time when the fields at different spatial points evolve independently and derived general formulae of the correlators. In the stochastic approach, one generally needs to solve the Focker-Planck equation to evaluate the time evolution of the correlator since its solution (probability density) appears in the definition of the correlator. This program (with a reasonable approximation) works for the two-point functions and the result is given by Eq. (\[two-point-2\]). This result clearly shows that the amplitude of the two-point functions is determined by $A^{(n)}_a$ defined by Eq. (\[def-A\]) which represents the expectation value of $\phi_a$ for the $n$-th state $e^{-v({\pmb \phi})} \Phi_n ({\pmb \phi})$ corresponding to the eigenvalue $\Lambda_n$ which determines the scaling index. It is important to notice that by construction the stochastic formalism incorporates the dynamics of the long wavelength modes without expanding the basic equations in terms of the amplitude of the coarse grained fields or the strength of the interactions. Therefore, once $A^{(n)}_a$ and $\Lambda_n$ and hence the two-point functions are obtained, they are the results that do not rely on the perturbative expansion and automatically involve the nonperturbative effects. Except for some simple cases, Eq. (\[Schrodinger\]) does not allow the analytic solution and numerical computation is generally required to obtain the concrete values of $A^{(n)}_a$ and $\Lambda_n$. However, solving Eq. (\[Schrodinger\]) is a mathematically well-defined problem and, in principle, it can be solved (especially for the case with a small number of fields) without any principal difficulty. This plausible feature of the stochastic formalism is in striking contrast to the standard calculation of the correlation function that usually uses perturbative expansion in terms of the strength of the interactions and truncates the expansion at some order. In most cases, the truncation is done at the lowest order that yields nonvanishing contribution and extension to the higher order entails very messy expression that requires careful consideration to obtain physically meaningful results.
Direct manipulation of the evolution equation for the probability density becomes very difficult for the case of the three-point functions. Therefore, in this paper, we adopted another approach which works as long as we are only interested in the late time behavior of the correlators. The basic idea is to utilize the de Sitter isometries and to consider only the squeezed case in which one of the three spatial points is separated far way than the others. This limiting case allows us to build up the analytic expressions of the correlators at late time. We then converted the correlators to the ones for the arbitrary configuration of points by using the fact that the value of the correlator itself remains the same under the isometric transformation for the de Sitter invariant state. Among the ten de Sitter isometries, the three reduces to the special conformal transformation for the spatial transformation. It is this transformation that enables us to transform any triangle into the arbitrarily squeezed one and to obtain the analytic form of the (only late time) three-point function for any configuration of points which is given by Eq. (\[3pt-final\]).
Now it would be interesting to consider the consequences of Eq. (\[3pt-final\]) focusing on the single field case for simplicity. Even in this case, we find nontrivial and interesting properties of the three-point function. Noting that we can always make $A^{(0)}$ be zero by suitably redefining the field, the leading term that remains at late time is given by $B_{111} {(A^{(1)})}^3$ provided neither $B_{111}$ nor $A^{(1)}$ vanishes. Since this combination $(1,1,1)$ provides the lowest decaying rate of the correlator, this is the only leading term that survives at late time. Therefore, Eq. (\[3pt-final\]) in this case becomes $$\langle \phi (t,{\vec x_1}) \phi (t,{\vec x_2}) \phi (t,{\vec x_3}) \rangle = {\cal N}^2
B_{111} {(A^{(1)})}^3 {\left(HR_{12} \right)}^{-\frac{\Lambda_1}{H}}{\left(HR_{23} \right)}^{-\frac{\Lambda_1}{H}}
{\left(HR_{31} \right)}^{-\frac{\Lambda_1}{H}}. \label{3pt-single}$$ In this case, two-point function at late time becomes $$\langle \phi (t,{\vec x_1}) \phi (t,{\vec x_2}) \rangle ={\cal N}^2
{(A^{(1)})}^2 {\left(HR_{12} \right)}^{-\frac{2\Lambda_1}{H}}. \label{2pt-single}$$ Scaling behavior of Eqs. (\[3pt-single\]) and (\[2pt-single\]) coincide with the one given in [@Antoniadis:1996dj; @Antoniadis:2011ib; @Creminelli:2011mw; @Kehagias:2012pd]. In these references, the scaling behavior was derived by combining the de Sitter isometries like we have done in this paper and the assumption that $\phi$ (not at the level of the correlators) at sufficiently late time scales as $\phi \sim {(-\eta)}^\Delta$ and this scaling directly enters the scaling of the correlator, for instance, $\langle \phi(\eta_1) \phi (\eta_2) \rangle \sim {(-\eta_1)}^\Delta {(-\eta_2)}^\Delta$. On the other hand, as we have shown, the stochastic formalism can provide a necessary and sufficient condition in order for the above naive assumption to hold, which is given by $A^{(1)} B_{111} \neq 0$. In addition to this, the stochastic formalism also gives amplitudes of the correlators in terms of the calculable quantities.
What happens if $B_{111}$ accidentally vanishes, which is possible for some models? In this case, the leading contribution to the three-point function comes from a term $B_{112} {(A^{(1)})}^2 A^{(2)}$ unless it vanishes [^1]. Then, the two-point function is still given by Eq. (\[2pt-single\]), but the three-point function becomes $$\langle \phi (t,{\vec x_1}) \phi (t,{\vec x_2}) \phi (t,{\vec x_3}) \rangle = {\cal N}^2
B_{112} {(A^{(1)})}^2 A^{(2)}
\bigg[ {\left(HR_{12} \right)}^{-\frac{2\Lambda_1-\Lambda_2}{H}}
{\left(HR_{23} \right)}^{-\frac{\Lambda_2}{H}} {\left(HR_{31} \right)}^{-\frac{\Lambda_2}{H}}+2~{\rm perms.} \bigg].
\label{b112}$$ This is very different from Eq. (\[3pt-single\]) on two counts; it does not obey the single power-law, and it is not given by a single term but by three terms that are mutually related by permutations. As far as we know, this type of three-point functions has been overlooked in literature. If $B_{112} {(A^{(1)})}^2 A^{(2)}$ vanishes too, then we need to consider $B_{122}A^{(1)}{(A^{(2)})}^2 $ or $B_{113}A^{(1)}{(A^{(2)})}^2$, whichever yields the lower decaying rate. If both of these have the same decaying rate, we must keep both of them in the correlator. Obviously, any of these leads to the multi-scaling expressions of the three-point function like Eq. (\[b112\]). These examples suggest that knowledge of the scaling behavior of the two-point function is not necessarily enough to know the scaling behavior of the three-point function. In principle, three-point function can exhibit more complicated structure than the native expectation.
The single field with $B_{111}\neq 0$ gives the following form of the four-point function, $$\left\langle \prod_{i=1} \phi (t,{\vec x_i}) \right\rangle =
{\cal N}^2 f^{1111} \left( \frac{R_{12}R_{34}}{R_{13}R_{24}},~\frac{R_{12}R_{34}}{R_{23}R_{14}} \right) \prod_{i<j} {\left( HR_{ij} \right)}^{-\frac{2\Lambda_1}{3H}}.$$ This expression is the same as the one given in [@Creminelli:2011mw]. The function $f^{1111}$ for some limiting cases are given in Eqs. (\[trib\])-(\[case3\]). For instance, we find $$f^{1111}(z,z) =B_{111}^2 {(A^{(1)})}^4 z^{-\frac{\Lambda_1}{3H}},~~~~~{\rm for}~z\ll 1. \label{single-4pt}$$ Generally speaking, $B_{111}=0$ does not imply the vanishing of $f^{1111}$ but only changes the form of $f^{1111}$. Assuming that $B_{112}$ does not vanish, $f^{1111}$ corresponding to Eq. (\[single-4pt\]) now becomes $$f^{1111}(z,z) =B_{112}^2 {(A^{(1)})}^4 z^{\frac{\Lambda_2}{H}-\frac{4\Lambda_1}{3H}},~~~~~{\rm for}~z\ll 1,$$ which contains information of the second excited state $\Phi_2$. As it should be, the power of $z$ is higher by $(\Lambda_2-\Lambda_1)/H$ than Eq. (\[single-4pt\]).\
[**Acknowledgments:**]{} We would like to thank A. Riotto and A. Kehagias for explaining the applicability of the results of [@Kehagias:2012pd]. This work was supported by JSPS Research Fellowships for Young Scientists (HM), Grant-in-Aid for JSPS Fellows No. 1008477 (TS), JSPS Grant-in-Aid for Scientific Research No. 23340058 (JY), and the Grant-in-Aid for Scientific Research on Innovative Areas No. 21111006 (JY).
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[^1]: If the potential has a reflection symmetry $V(\phi)=V(-\phi)$, $\Phi_1(\phi)$ for the bound state is an odd function, so that $B_{111}$ in Eq. (\[def-B\]) vanishes. In this case, however, $B_{112}$ should also vanish because of the same symmetry.
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abstract: 'According to the statistical interpretation of quantum theory, quantum computers form a distinguished class of probabilistic machines (PMs) by encoding $n$ qubits in $2n$ pbits (random binary variables). This raises the possibility of a large-scale quantum computing using PMs, especially with neural networks which have the innate capability for probabilistic information processing. Restricting ourselves to a particular model, we construct and numerically examine the performance of neural circuits implementing universal quantum gates. A discussion on the physiological plausibility of proposed coding scheme is also provided.'
address: 'Department of Theoretical Physics, University of Łódź, Pomorska 149/153, 90-236, Łódź, Poland'
author:
- 'P. Gralewicz'
title: Quantum computing in neural networks
---
[2]{}
Introduction
============
Neural networks are naturally evolved systems for information processing. Despite decades of experimental and theoretical research, there is no agreement upon the information encoding employed by these circuits – the problem of what exactly is being communicated *via* seemingly chaotic spike trains is still largely open [@RWRB1997]. Advancement in understanding of this neural language is obstructed by variety of cell types, working conditions and molecular factors to be taken into account [@ChS1992]. Generally accepted schemes, the [*rate code*]{} and the [*phase code*]{}, may turn out to be only the first two in a sequence of progressively more intricate codes, where higher order correlations within cellular complexes are utilized.
Quantum information science, on the other hand, had matured over the last two decades making significant contributions to both information theory and quantum mechanics (). The latter, having historical roots in particle physics, is still often identified with the micro-world. Yet, there is nothing in the mathematical foundations of which could justify that point of view. In fact, apart from that microscopic realizations, quantum theory has found many avatars, from mechanical [@A1984], linguistic [@AC2003], purely geometric [@B2004], to statistical [@Bal70; @P1995; @PeTe98; @CFS2002]. In this article, that last, widely accepted interpretation, is being used to study the feasibility of a hypothesis that spike trains may actually encode for quantum states.
Such hypothesis appears particularly attractive in that the Nature is notorious in repeating itself at various scales, and if quantum computing () proves to be practical, it would be rather surprising if one could not find it implemented at a higher level. From this point of view neural networks are the obvious candidates for such implementations. By examining two neural circuits, designed to perform quantum operations ($1$-qubit rotations, and $2$-qubit gate), we demonstrate the feasibility of our hypothesis within the limits of a simple model. Although quantum registers are realized efficiently with just two neurons per qubit, the major costs are in the processing of information carried by the spike trains. The simulations provided are intended to emphasize the amount of these resources as well as the functionality required for implementation.
We begin with a short review of the formalism which allows for the identification of pairs of spiking neurons with qubits. In Section \[sc:ann\] a reduced model of neural network is described, which in Sec. \[sc:q-gates\] is further used as a basis for construction of quantum gates. The results of simulations, in terms of achieved fidelity and coherence, are promising enough to look toward more realistic implementations. We touch briefly on these issues in the last section.
Manipulation of quantum states embedded in probabilistic space {#sc:povm}
==============================================================
The operational approach to quantum mechanics, through the formalism of positive operator-valued measures (s), allows one to express the states of a quantum system defined in a finite-dimensional Hilbert space $\spH$, in terms of probability distributions. If the dimension is $d:=\dim\spH$, then a generic density matrix $\hat{\rho}$ representing the state has $d^2-1$ degrees of freedom (s). A distribution obtained through particular has length $d^2$, and – due to normalization constraint – the same number of s as the density matrix [@PeTe98]. For $n$-qubit states, this distribution can be associated with joint probability of $2n$ binary random variables[^1].
Let $\hat{\rho}$ be a generic density matrix of a $1$-qubit state, which using summation convention, we write as $$\hat{\rho}
:= \rho^\mu\,\qbs_\mu
= \begin{pmatrix}
\thalf + \rho^3 & \rho^1 + \imu \rho^2 \\
\rho^1 - \imu \rho^2 & \thalf - \rho^3
\end{pmatrix},$$ where $\qbs_0=\idmatrix$, $\qbs_{1,2,3}$ are the Pauli matrices, $\rho^0\equiv \thalf$, and $\rho^1,\rho^2,\rho^3\in[-\thalf,\thalf]$ are the three real coordinates of a Bloch vector. Let $\{\hat{A}^{z}\}$ be a normalized $4$-element positive operator-valued measure $$\label{eq:Anorm1}
\sum_{z} \hat{A}^{z} = \idmatrix,
\qquad z=0\Till 3.$$ Typically, one associates such a with the Pauli basis, that is $$\hat{A}^{z}
:= A^z\null_\mu \qbs^\mu
= \tfrac{1}{2}\idmatrix + A^z\null_i \qbs^i,\qquad
i=1\Till 3.$$ where $A^z\null_0\equiv \tfrac{1}{2}$, and $\qbs^\mu:=\qbs_\mu^\dagger=\qbs_\mu$, is the basis dual with respect to the scalar product $
{\langle{\qbs^\mu},{\qbs_\nu}\rangle}
:=\thalf\operatorname{tr}[\qbs^\mu\qbs_\nu]
=\delta^\mu\null_\nu
$. Although not a strict necessity, it is reasonable to assume the same for all $n$ qubits within a register, and consequently take the entire as a $n$-fold tensor product $$\hat{A}^{z_1...z_n}
:= \hat{A}^{z_1}\otimes\cdots\otimes\hat{A}^{z_n}.$$ This leads to the following distribution $$\hspace{-5mm}
\begin{aligned}
\null&p^{z_1...z_n}
:= {\langle{\hat{A}^{z_1...z_n}},{\hat{\rho}}\rangle}
= 2^{-2n} + A^{z_1}\null_{i_1}\cdots A^{z_n}\null_{i_n}\rho^{i_1...i_n},\\
\null&\qquad \sum_{z_1...z_n} p^{z_1...z_n} = 1.
\end{aligned}$$ Introducing the event basis $\{\ebs_{z}\}$, the transformation can concisely be written as $$\label{eq:rho_p}
\hat{p} = A \hat{\rho},$$ where $$\hspace{-5mm}
\begin{aligned}
\hat{p} &= p^{z_1...z_n}\,\ebs_{z_1}\otimes\cdots\otimes\ebs_{z_1},\\
A &= A^{z_1}\null_{\mu_1}\cdots A^{z_n}\null_{\mu_n}\,
\ebs_{z_1}\otimes\cdots\otimes\ebs_{z_1}\otimes
\qbs^{\mu_1}\otimes\cdots\otimes\qbs^{\mu_n}.
\end{aligned}$$ Conversely, if $\{\hat{A}^z\}$, are linearly independent, then one can invert the relation and take the distribution $\hat{p}$ as an equivalent representation of the quantum state $\hat{\rho}=A^{-1}\hat{p}$.
A unitary transformation $U\in\grU(2^n)$ of the state is a linear operator[^2] $L\in 1\oplus\grSO(2^{2n}-1)$ $$\hat{\rho}
\mapsto U^\dagger\!\hat{\rho}\,U
= L\hat{\rho},$$ with elements $$\label{eq:L}
\begin{aligned}
\null&L^{\mu_1...\mu_n}\null_{\nu_1...\nu_n}\\
\null&\quad
= {\langle{\qbs^{\mu_1}\otimes\cdots\otimes\qbs^{\mu_n}},{U^\dagger\qbs_{\nu_1}\otimes\cdots\otimes\qbs_{\nu_n}U}\rangle}.
\end{aligned}$$ After transformation of the basis $A^{-1}:\{\qbs_\mu\}\to\{\ebs_{z}\}$ one has the same operation acting on probability distribution $$\label{eq:Lp}
\hat{p} \mapsto (A L A^{-1}) \hat{p}.$$ There is, however, an important difference between the linear dynamics of Eq. and Markovian transitions usually considered in association with stochastic evolution: Denote by $\Omega^{2n}$ the space of joint probability distributions of $2n$ pbits. Since the operator $A L A^{-1}$ is by definition invertible, it follows that, in general, it is not a positive one, hence only a subset of $\Omega^{2n}$ will be mapped back into itself. We denote this subset – the (closure of) positive domain of quantum operators – by $$\overline{\Omega^{2n}_+}:=\{\hat{p}\in\Omega^{2n}\mid A L A^{-1}\hat{p}\in\Omega^{2n}\}.$$ This is simply the image of all quantum states under the $A$. The boundary $\Omega^{2n}_0 := \partial\overline{\Omega^{2n}_+}$, which is the image of the Bloch sphere in $\Omega^{2n}$ contains pure states, while its interior $\Omega^{2n}_+ := \overline{\Omega^{2n}_+}\setminus\Omega^{2n}_0$ is the subset of mixed states. All remaining distributions $\Omega^{2n}_- := \Omega^{2n}\setminus \overline{\Omega^{2n}_+}$ are mapped by $A^{-1}$ to the exterior of the Bloch sphere. Therefore, the partitions the set of possible distributions into three disjoint subsets:
--------------------------------------------------- --------------------------
$\Omega^{2n}_0$ – pure quantum states
$\Omega^{2n}_+ = \operatorname{int}\Omega^{2n}_0$ – mixed/decohered states
$\Omega^{2n}_- = \operatorname{ext}\Omega^{2n}_0$ – overcohered states
--------------------------------------------------- --------------------------
To explain the term [*overcohered*]{} used above, let us take a closer look at the limitations imposed by the on distributions in $\overline{\Omega^{2n}_+}$. Positivity of $A$ implies, that the probabilities are bound by $$p^{z_1...z_n}\leq 2{\langle{p}\rangle}=2(A^z\null_0\rho^0)^{n}=2^{1-2n}.$$ Furthermore, if, as we assume, $A$ is non-degenerate, then for any quantum state only one of the elements $\{p^{z_1...z_n}\}$ can either vanish, or reach the maximal value $2{\langle{p}\rangle}$. This means, that there is a non-zero lower bound on the entropy of distributions in $\overline{\Omega^{2n}_+}$, and hence no distribution with certain outcome can represent a quantum state. Moreover, all single-pbit marginals are non-vanishing.
A quantitative characterization of the [*coherence*]{} can be given by the radius of the state’s Bloch vector. The metric $g:\Omega^{2n}\times\Omega^{2n}\to\fR$ induced by the on the distribution space permits to obtain this radius directly for an arbitrary $\hat{p}\in\Omega^{2n}$. Let $\hat{p}_1,\hat{p}_2\in\overline{\Omega^{2n}_+}$, and $\hat{\rho}_1$, $\hat{\rho}_2$ be corresponding quantum states. Then $g$ is given by $$g(\hat{p}_1,\hat{p}_2)
:= \operatorname{tr}[\hat{\rho}_1^\dagger\hat{\rho}_2]
= (A^{-1}\hat{p}_1)^\dagger\cdot(A^{-1}\hat{p}_2).$$ Because this is a bilinear map with coefficients independent of $\hat{p}_1,\hat{p}_2$, one is free to extend its domain onto the entire space $\Omega^{2n}$. Since $A^0\null_\mu=\thalf$, the radius is $$r^2 := \|\vec{\rho}\,\|^2 = 2^{-n} g(\hat{p},\hat{p})-2^{2n}.$$ In particular, for a pure state $$r_\mathrm{pure}^2 = 2^{-n}(1-2^{-n}),$$ and the Bloch radius of any mixed state is always bound by $r<r_\mathrm{pure}$. The ratio $$\label{eq:defR}
R := \frac{r}{r_\mathrm{pure}},$$ can be adopted for a measure of coherence – ranging from $R=0$ for maximally decohered state $\hat{\rho}=2^{-n}\idmatrix$, through $R=1$ for any pure $\hat{\rho}=\hat{\rho}^2$, and beyond $R>1$ for all overcohered ones.
In order to quantify the performance of circuits considered latter in this article, we will also employ another, independent measure by which one can estimate the angular disparity between expected and obtained states. The [*fidelity*]{} or normalized overlap between $\hat{p},\hat{q}\in\Omega^{2n}$ is defined here as $$F := \frac{g(\hat{p},\hat{q})}{\sqrt{g(\hat{p},\hat{p})}\sqrt{g(\hat{q},\hat{q})}}.$$
We choose fidelity as a commonly adopted measure, for the purpose of comparison, notwithstanding direct estimate of the unitary error between the desired pure state $\hat{p}\in\Omega^{2n}_0$ and obtained distribution $\hat{q}\in\Omega^{2n}$, which is readily computable: $$\alpha = \arccos\left(\frac{g(\hat{p},\hat{q})-2^{-n}}{r(\hat{q})\sqrt{2^{n}-1}}\right).$$ The two quantities $\alpha$ and $F$, are nevertheless dependent.
A toy-model neural-network {#sc:ann}
==========================
The information in neural networks is carried by spike trains, which after appropriate discretization can be transformed to binary strings. The model network described in this section is a much simplified version of what usually is considered realistic – the purpose of such reduction is to retain only the essential features. Consistently with discretization of transmitted signals, the model operates in explicitly step-wise manner, instead of continuous-time evolution. Likewise, the delays effected along the inter-neuron paths are also taken to be integers. Let $\grG=(\stV,\stE)$, be a multiply-connected digraph, where $\stV=\{\vbs_i\}$ is the vertex basis of neurons, (we shall also write $\stV_{N}$ to explicit the number $N$ of vertices involved) and $\stE=\stV\otimes\stV\otimes\fN$ is the basis of edges, that is the possible synaptic connections. The actual couplings between $i^\mathrm{th}$ and $j^\mathrm{th}$ neuron are set by the weights $W^i\null_{js}$ where $s\in\fN$ enumerates the delays introduced along multiple edges. For each vertex we define two variables: the binary [*output state*]{} $X^i\in\{0,1\}$, and the [*residual potential*]{} $u^i\in\fR$.
We adopt the discrete [*integrate-and-fire*]{} scheme for the dynamics of this network. In each time step the potential is first updated by accumulating the incoming signals $$u^{i,t-1}\mapsto u^{it}_\star := u^{i,t-1} + W^i\null_{js}X^{j,t-s}$$ where the summation runs over connected vertices ($j$) and edge delays ($s\geq 1$). Subsequent spike generation ($X^{it}=1$) occurs with probability $P(u^{it}_\star)$, where $P:\fR\to[0,1]$, is a ‘noisy’ activation function with firing threshold fixed at $u_\mathrm{thr}=\thalf$. Its actual form used in simulations is given by $$P(u_\star) := \half\left(1+\operatorname{erf}\frac{u_\star - u_\mathrm{thr}}{\sigma}\right),$$ where $\sigma\geq 0$ is a global control parameter characterizing the noise standard deviation (). In particular, in the limit $\sigma\to 0$ the spikes are produced deterministically, as $P$ becomes a step-function. The excited state $u^{it}_\star$ is eventually reduced by release of a spike (refractory potential), and further quenched with a bound, nonlinear map $S$ $$u^{it}_\star \mapsto u^{it} = S(u^{it}_\star- X^{it}).$$ We assume $S$ to have an attractive fixed point at the origin (the resting potential), $
\forall_u:\;\lim_{t\to\infty} S^t(u)=0
$, to be linear in its neighborhood $
S'(0) = 1
$, and having finite, but non-zero asymptotes $
|S(\pm\infty)|<\infty
$. The motivation for introduction of this mapping is twofold: First, the physiological mechanisms of signal transmission imply existence of [*saturations*]{} in both positive and negative direction. The cell can be depolarized or hyperpolarized through synaptic channels only to certain extent, and adding more excitatory or inhibitory connections will not have a significant effect. Second, the reason to have $u=0$ for an attractive fixed point, is to imitate the ‘leaky’ integration scenario, by which in the absence of input the potential returns back to its resting point. In the simulations this function was taken to be a simple, skew-symmetric mapping $$S(u) := \gamma\tanh\frac{u}{\gamma},\qquad \gamma \geq 0.$$ Here, the asymptotes are $S(\pm\infty)=\pm\gamma$, therefore we call $\gamma$ the ‘saturation parameter’. If we assume, that the neuron is left without input and some residual $u$, so that no spikes are generated, then the potential $u$ will decay sub-exponentially in time, as $$u \mapsto S^t(u) \underset{t\gg 1}{\to} \frac{\gamma u}{\sqrt{\gamma^2 + \tfrac{2}{3} t u^2}},$$ where $S^t$ means $t$-fold composition. In the limit $\gamma\to 0$, the residual potential is reset to zero after each cycle, and this situation can be associated with time steps longer than the total refractory time ($\sim 20\,\mathrm{ms}$), within which the cell relaxes to its resting point. If $\gamma>0$, then the probability of a consecutive spike is modified by the residual potential: The cell is within the *relative* refractory period, when the the potassium channels are still open, but the sodium gates are already reverted to their normal state. This mode corresponds to time steps of order $\sim 5\,\mathrm{ms}$. Shorter times are generally unrealistic due to high suppression of spike generation during the *absolute* refractory period, when the sodium channels are closed.
The choice of a specific value of $\gamma$ is therefore indirectly related to the time scale, and consequently to the discretization window of action potentials. If this window is too short, the discretization becomes ambiguous and the model breaks down – this is another reason not to consider high saturation values.
The qualitative behavior of the above model is best understood by analyzing single neuron at the limits of the two control parameters $\sigma$, and $\gamma$. Assume the cell is fed with a stimulus at a constant frequency $\nu_\mathrm{in}\in[0,1]$, and consider at first the noiseless regime $\sigma=0$. If $\gamma=0$, then the only memory of past input values is stored in delayed connections. The cell fires only if the value of the convolution $W_{js}X^{j,t-s}$ exceeds the threshold $u_\mathrm{thr}$. Such neurons acts like a high-pass filter and its firing rate is $\nu_\mathrm{out}=P(\nu_\mathrm{in}\sum_{s,j} W_{js})$. By increasing the noise $\sigma$, the shape of this filtering function changes along with the spiking probability $P$, nevertheless it never becomes close to an ideal multiplier – the response is always nonlinear.
If $\gamma\to\infty$ the cell accumulates and ‘remembers’ the residual value of convolution left over after subtraction of generated spikes. This makes it into a perfect multiplier with spike rate $\nu_\mathrm{out}=\nu_\mathrm{in}\sum_{s,j} W_{js}$. Raising $\sigma$ above zero does not change this average response, but the determinism initially apparent in the spike patterns is gradually being washed away.
In between of these two regimes, lies a surprisingly complex area of fractal-spaced frequency thresholds and output patterns, particularly conspicuous at $\sigma=0$ and $\nu_\mathrm{in}=1$. Presence of these features, found in many nonlinear deterministic systems do not critically depend on the specific shape of the function $S$.
Implementation of universal quantum gates {#sc:q-gates}
=========================================
According to the discussion provided in section \[sc:povm\], one needs $2n$ random binary variables to implement an $n$-qubit register. In our model of the neural network, these variables are identified with discretized spikes registered at $2n$ network sites. The question we set up to address in this section is, whether there are circuits which can implement state-independent rotations of the joint probability distributions, that is – quantum gates.
The set of gates universal for quantum computation [@BBCVMSSSW1995] includes the whole algebra of $1$-qubit rotations, and an arbitrary $2$-qubit entangling gate, typically chosen to be the (controlled-). Although probabilistic encoding of qubits is efficient ([[*i.e.*]{}]{} linear in $n$), manipulation of their $2^{2n}$ degrees of freedom (s), by definition requires exponential amount of resources. From this perspective the construction of circuits described below should appear at least conceptually straightforward: The space of binary functions over the vertices $\stV_{2n}$ is $\stV^{2n}_*=\fZ_{2n}$. We first embed an element $X^t=\{X^{it}\}\in\stV^{2n}_*$ into $\Omega^{2n}$, then apply the gate $G := A L A^{-1}$, and finally project the result back onto $\stV^{2n}_*$. The entire quantum gate transforming one set of spike trains $X^t$ to another $Y^t\in\stV^{2n}_*$, is then a composition $$\Pi\circ G\circ\Pi^{-1}:\; X^t \mapsto Y^t,$$ where $\Pi^{-1}:\stV^{2n}_*\to\Omega^{2n}$, and $\Pi\circ\Pi^{-1}=\operatorname{id}_{\stV^{2n}_*}$. The main problem in this approach is to construct a reliable projection $\Pi$, since any information loss during that operation will affect the quality of entire gate.
Concrete realization, requires also to decide upon particular being used. It is possible to choose this transformation in such a way, that some of the gates will be significantly simplified, for instance acquiring convenient form of permutations. Our choice is dictated by the optimization of the gate, discussed latter in this section. This is given by[^3] $$\label{eq:A}
A^z\null_\mu := \frac{1}{2}\begin{pmatrix}
1 & - \tfrac{1}{\sqrt{3}} & - \tfrac{1}{\sqrt{3}} & - \tfrac{1}{\sqrt{3}} \\
1 &\phantom{-}\tfrac{1}{\sqrt{3}} &\phantom{-}\tfrac{1}{\sqrt{3}} & - \tfrac{1}{\sqrt{3}} \\
1 & - \tfrac{1}{\sqrt{3}} &\phantom{-}\tfrac{1}{\sqrt{3}} &\phantom{-}\tfrac{1}{\sqrt{3}} \\
1 &\phantom{-}\tfrac{1}{\sqrt{3}} & - \tfrac{1}{\sqrt{3}} &\phantom{-}\tfrac{1}{\sqrt{3}}
\end{pmatrix}.$$
Single-qubit gates
------------------
The neural circuit implementing arbitrary $1$-qubit gate is presented in [Fig. \[fig:G1D\]]{}. The projection $\Pi$ which transforms the ‘sparse’ code $\{\Xl{00'},\Xl{01'},\Xl{10'},\Xl{11'}\}\in\Omega^2$ onto a ‘dense’ one $\{\Xl{A'},\Xl{B'}\}\in\stV^2_*$, is a linear mapping implemented with weights $$W_\Pi
= \begin{pmatrix}
0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1
\end{pmatrix}.$$ But its inverse, $\Pi^{-1}$ is nonlinear and we realize this function, in a two-step linear-feedback operation. The first step requires, apart from the input signals, an additional supply of constant ‘current’ of units from the vertex $\vbsl{1}$. The effect of such a coupling to unity on a cell is to alter its firing threshold. The weights of this part, effecting a linear injection from $\{\Xl{A},\Xl{B},1\}$ to $\{\Xl{00},\Xl{01},\Xl{10},\Xl{11}\}$ are $$W_{\Pi^{-1}} = \begin{pmatrix}
- 1 & - 1 &\phantom{-}1 \\
- 1 &\phantom{-}1 &\phantom{-}0 \\
\phantom{-}1 & - 1 &\phantom{-}0 \\
\phantom{-}1 &\phantom{-}1 & - 1
\end{pmatrix}.$$ While the composition $W_\Pi W_{\Pi^{-1}}=\operatorname{id}_{\stV^2_*}$ as required, the reciprocal is not an identity and needs a rectifying feedback sent from the ‘winning’ neuron to its neighbors, in order to bring their residual potentials back to zero. Because of the one-step delay, this signal has to be adjusted to match the attenuation already done by the function $S$. Hence the weight matrix of this rectification step is determined by $$\label{eq:Wrec}
\begin{aligned}
{[}W_\mathrm{rec}{]}^i\null_j
&= -S\big([W_{\Pi^{-1}}W_\Pi]^i\null_j - \delta^i\null_j\big)\\
&= -S(-1)[\idmatrix_{\Omega^2} - W_{\Pi^{-1}}W_\Pi]^i\null_j.
\end{aligned}$$ Note, that for vanishing saturation parameter, this correction also disappears due to $S\equiv 0$.
In the absence of noise ($\sigma=0$), the conversion $\Pi^{-1}$ between dense and sparse coding is completely error-free. As $\sigma$ increases, the imperfections start to appear in the form of either multiple, or ‘void’ spiking in the first hidden layer. Although we found the circuit to behave stably in these conditions, an improvement, in terms of both fidelity and coherence, can be achieved by adding a second, normalizing feedback (not shown explicitly in [Fig. \[fig:G1D\]]{}).
\[fig:G1D\]
Normalizing feedbacks are commonly proposed for explanation of the observed behavior in cortical neurons [@CHM1997; @TLDRN1998]. The main difference between these and our proposal is that while the former are multiplicative, this one acts additively. Its role is to adjust the residual potentials for the difference $$1 - \sum_i X^i,\qquad i=00,01,10,11.$$ Because we do not know which of the four neurons spiked mistakenly, the normalizing signal is sent evenly to all of them. Its strength is determined by the average excess of a signal encountered on a double-spike event: $$\tfrac{1}{4}\sum_j [W_\mathrm{rec}]^i\null_j = -\tfrac{1}{4}S(-1).$$ The deficit, which happens upon lack of a single spike has the same magnitude but opposite sign. Therefore, the weight matrix of this normalization reads $$W_\mathrm{nor}
= -S(-1)\begin{pmatrix}
-\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & 1 \\
-\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & 1 \\
-\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & 1 \\
-\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & -\tfrac{1}{4} & 1
\end{pmatrix},$$ where the last column refers to the unit vertex $\vbsl{1}$.
Making the embedding $\Pi^{-1}$ robust is crucial for achieving correct projection $\Pi$. Multiple, or void events linearly projected *via* $W_\Pi$ typically lead to a significant loss of coherence (although with less impact on fidelity). In order to compensate for these non-exclusive events, the spiking neuron sends a composite inhibitory signal down the hierarchy. This is implemented with double connections: the first transmits inhibitory signal at certain level $\eta$, and is followed by a delayed, excitatory one aimed at bringing the residual potential of target neuron back to its prior value. The full accomplishment of this goal is impossible with nonlinear function $S$ – the value can only be fully restored in the linear limit $\gamma\to\infty$. Given the inhibitory coupling $\eta$, our best estimate of the following excitation strength is $\eta'={\langle{u}\rangle} - S({\langle{u}\rangle}-\eta)$, where ${\langle{u}\rangle}$ is the average residual potential. Because ${\langle{u}\rangle}\approx 0$, we set $\eta'=-S(-\eta)$. The optimal value of $\eta$ was found numerically, by minimizing the variation of fidelity over a range of gates acting on test states (see Results).
Application of a gate $G$ requires no additional node of the network, only manipulation of the weights between the embedding and projecting parts. In simplest case these are directly set to $$W_G = A L A^{-1} = G.$$ We have found however, that within some limits, the mechanism of [*synaptic averaging*]{} may provide improvement of the performance. In real networks, a single synapse contributes only a tiny fraction of the total input signal [@ChS1992]. Multiple connections of similar lengths lead to signal accumulation, different delays – to temporal averaging. In our toy-model, the first case is replaced by single, but strong connections, while for implementation of the latter we directly use several edges having different lengths with proportionally attenuated couplings. In the case of a single-qubit gate, of the several configurations tested, the best results were obtained with just two-step average $\tAvr=2$, hence, the connections were fixed at $W_{G,1}=W_{G,2}=\thalf G$.
#### Results.
In order to reduce statistical uncertainties, all gates were tested on a fixed set of $14$ pure states approximately evenly distributed on the Bloch sphere: $$\hspace{-1mm}
\begin{aligned}
\Big\{&{{|{0}\rangle}}, {{|{1}\rangle}},
\tfrac{1}{\sqrt{2}}\big({{|{0}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{1}\rangle}}\big),\\
\null&\frac{\sqrt{\sqrt{3}\pm 1}{{|{0}\rangle}}+{\mathrm{e}^{\imu(2k+1)\pi/4}}\sqrt{\sqrt{3}\mp 1}{{|{1}\rangle}}}
{\sqrt{2\sqrt{3}}}\Big\},
\end{aligned}
\quad
k \in \fZ_4.$$ While the input nodes were fed with spike trains $\{\Xl{A},\Xl{B}\}$ of joint probability distributions $\hat{p}_\mathrm{in}\in\Omega^2_0$ corresponding to the above states, the output $\{\Xl{A'},\Xl{B'}\}$ was tested for its coherence $R$, and fidelity with the desired distribution $G\hat{p}_\mathrm{in}$. The initial test runs were made for several gates including Hadamard’s, , the antipode (non-unitary), and two rotations: $U_\theta = \exp(\imu\theta\hat{\sigma}_2/2)$, and $U_\phi = \exp(\imu\phi\hat{\sigma}_3/2)$. For the representative, the phase gate $U_\phi$ was selected – the effects of other operations were quantitatively similar, or better. Its representation $G_\phi=A L_\phi A^{-1}$, acting in $\Omega^2$ reads $$\hspace{-7mm}
G_\phi
= \half\begin{pmatrix}
1 + \cos\phi & 1 - \cos\phi &-\sin\phi & \sin\phi \\
1 - \cos\phi & 1 + \cos\phi & \sin\phi &-\sin\phi \\
\sin\phi &-\sin\phi & 1 + \cos\phi & 1 - \cos\phi \\
-\sin\phi & \sin\phi & 1 - \cos\phi & 1 + \cos\phi
\end{pmatrix}.$$ The results presented in [Fig. \[fig:FC-1D\]]{} are averages over $36$ rotation angles evenly spaced across the entire interval $[0,2\pi)$. The best performance was observed for $\phi=0$ (identity) and $\phi=\pi$, while the worst cases were encountered around $\phi\approx \pm \pi/2$ (but not exactly at these angles). For each setting $(\sigma,\gamma)$, the inhibition level $\eta$ was adjusted to minimize the variance of fidelity across the test states and rotation angles ([[*cf.*]{}]{} [Fig. \[fig:FC-1D\]]{}-insets). Note that while this optimization was mainly coincident with maximization of the fidelity itself, the trend in coherence was typically opposite. Had we chosen to optimize for purity of states ($R\to 1$), the figures would look different.
The prominent feature of [Fig. \[fig:FC-1D\]]{}a, is the overcoherence of output states in the limit $\gamma\to 0$. This means these distributions are too sharp to represent quantum states, and any subsequent application of another gate would certainly lead to a loss of accuracy. Interestingly, the average fidelity remains at relatively high level. This suggests a possibility of correcting the distributions by rescaling about the average. On the other hand, the fidelity is significant for small saturations, and becomes comparable with statistical uncertainties only above $\gamma\gtrsim 1$.
The conclusion drawn from [Fig. \[fig:FC-1D\]]{}b is clear: the circuit considered here is designed to work in deterministic regime $\sigma\to 0$. This makes an interesting contrast between stochastic nature of quantum states and the determinism of gates acting on them. As we are going to show, this dichotomy is not limited to the $1$-qubit gate, but persists also in the case of entangling operation .
Finally, we have sought for an estimate of the time needed to complete the quantum rotations with this gate. Apart from the spatial resources, measured in terms of cells and connections being used, time is an important factor contributing to the overall cost of the realization. To assess this property, we have run the circuit while varying the [*signal length*]{} $\tSig$: After an initial transient of $\tGate = 4 + \tAvr$, the network was ran for $\tSig\geq 1$ successive steps, after which the cells were re-set to their initial state ($u^i=0$, $X^i=0$), ensuring that all memory traces stored in residual potentials were erased. This procedure was repeated until satisfactory statistics ($N\tSig\approx 10^4$) was gathered.
\[fig:FC-1D\]
The results provided in [Fig. \[fig:FC-1D\]]{}c evidently show that the real temporal cost is not only the delay $\tGate$, but a significant number of further steps are needed to ‘tune’ this gate to a signal. After approximately $\tSig\approx 30$ events the output quality no longer improves, and consequently one can identify $\tSig$ with the statistics needed for maximal efficiency. Since the latter is a function of saturation $\gamma$ and noise $\sigma$, one expects $\tSig$ to raise monotonically with $\gamma$ and decrease as $\sigma$ increases. In particular, the ideal case $\gamma\to\infty$, $\sigma\to 0$ would also require infinite statistics to achieve the best performance. One therefore finds yet another reason for the low saturation values: The finiteness of signals encoded in spike trains, limits the attainable efficiency of transformations, and high saturation values cannot provide improvement beyond these limitations.
The CNOT gate
-------------
Unlike the single-qubit gates which can, by means of a special choice of the , be transformed to a permutation, the operation does not admit such representation[^4]. With $A$ given by , its operator $G_\mathrm{CNOT}=A L_\mathrm{CNOT} A^{-1}$ has the following structure $$\hspace{-9mm}
G_\mathrm{CNOT}
= \begin{pmatrix}
H_1 - J_1 & H_1^\mathrm{T} + J_2 & H_2 & H_2^\mathrm{T} \\
H_1^\mathrm{T} + J_2 & H_1 - J_1 & H_2^\mathrm{T} & H_2 \\
-H_2 &-H_2^\mathrm{T} & H_1 - J_2 & H_1^\mathrm{T} + J_1 \\
-H_2^\mathrm{T} &-H_2 & H_1^\mathrm{T} + J_1 & H_1 - J_2
\end{pmatrix},$$ where $$\begin{aligned}
H_1 &= \frac{1}{4}\begin{pmatrix}
1 & \tfrac{1}{\sqrt{3}} & 1 & \tfrac{1}{\sqrt{3}}\\
-\tfrac{1}{\sqrt{3}} & 1 &-\tfrac{1}{\sqrt{3}} & 1\\
1 & \tfrac{1}{\sqrt{3}} & 1 & \tfrac{1}{\sqrt{3}}\\
-\tfrac{1}{\sqrt{3}} & 1 &-\tfrac{1}{\sqrt{3}} & 1
\end{pmatrix},\\
H_2 &= \frac{1}{4}\begin{pmatrix}
\tfrac{1}{\sqrt{3}} &-1 &-\tfrac{1}{\sqrt{3}} & 1\\
1 & \tfrac{1}{\sqrt{3}} &-1 &-\tfrac{1}{\sqrt{3}}\\
-\tfrac{1}{\sqrt{3}} & 1 & \tfrac{1}{\sqrt{3}} &-1\\
-1 &-\tfrac{1}{\sqrt{3}} & 1 & \tfrac{1}{\sqrt{3}}
\end{pmatrix},\\
J_1 &= \frac{1}{\sqrt{3}}\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 0 &-1 \\
0 & 0 & 1 & 0 \\
0 &-1 & 0 & 0
\end{pmatrix},\\
J_2 &= \frac{1}{\sqrt{3}}\begin{pmatrix}
0 & 0 & 1 & 0 \\
0 &-1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 &-1
\end{pmatrix}.\end{aligned}$$ As already mentioned, the has been chosen to optimize the gate. Indeed, the linear projection $\Pi:\Omega^4\to\stV^4_*$, $$W_\Pi =
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & \ldots & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & \ldots & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & \ldots & 1 & 1 \\
0 & 1 & 0 & 1 & 0 & \ldots & 0 & 1
\end{pmatrix},$$ when combined with $G_\mathrm{CNOT}$, shows the two pbits $\Xl{A}$ and $\Xl{D}$ are invariant under $G_\mathrm{CNOT}$. These can be directly copied to the output $\Xl{A'}$, $\Xl{D'}$, as shown in [Fig. \[fig:CNOT\]]{}. For that same reason, while implementing the embedding part $\Pi^{-1}$, we pair $\{\Xl{A},\Xl{D}\},\{\Xl{B},\Xl{C}\}$, rather than using the natural order $\{\Xl{A},\Xl{B}\},\{\Xl{C},\Xl{D}\}$. The section $\Pi^{-1}$ is a two-stage procedure: First, with the same construction as in $1$-qubit case we separately embed the two marginals $\{\Xl{A},\Xl{D}\}$ and $\{\Xl{B},\Xl{C}\}$: $$\Pi^{-1}_\mathrm{I}:\stV^4_*\to\Omega^2\times\Omega^2.$$ Next, we combine these into a single map $$\Pi^{-1}_\mathrm{II}:\Omega^2\times\Omega^2\to\Omega^4.$$ Since this is done with a linear mapping, there is again a rectifying feedback $W_\mathrm{rec,II}$ obtainable from Eq. applied to $\Pi_\mathrm{II}$ on $\Omega^4$. In contrast to $\Pi^{-1}_\mathrm{I}$, the second-stage embedding $\Pi^{-1}_\mathrm{II}$ turns out to be unstable against noise, and the normalization feedback is now a necessity. Because of $$\forall_i\; 2^{-4}\sum_j [W_\mathrm{rec,II}]^i\null_j =-\tfrac{9}{16} S(-1),$$ the normalization weights are set to $$W_\mathrm{nor,II} = -9S(-1)(-D_4\oplus 1),$$ where $[D_4]^i\null_j\equiv 2^{-4}$ is the diffusion operator, and ‘$1$’ refers to the unit vertex $\vbsl{1}$.
\[fig:CNOT\]
Thanks to the invariance of two pbits $\Xl{A'}=\Xl{A}$, $\Xl{D'}=\Xl{D}$, the hierarchical projection $\Pi$ have been significantly simplified (in comparison to what is needed for general $2$-qubit gate). The four partial projections from $G_\mathrm{CNOT}$, shown on the right hand side of [Fig. \[fig:CNOT\]]{}, are modulated directly by the marginal $\Pi^{-1}_\mathrm{I}(\Xl{A},\Xl{D})$. The mechanism of this modulation is the same as explained before (inhibition followed by attenuated excitation) and the same parameter $\eta$ is set common on those connections.
Finally, the gate edges were multiplied, in order to use the synaptic averaging mechanism. We found no dramatic improvement while varying the averaging length $\tAvr$, at $\gamma\gtrsim 1$, nevertheless the performance was significantly better for small values of the saturation parameter. At $\gamma=1$ the optimal length was $\tAvr=4$.
#### Results.
The performance was assessed upon a testing set of $28$ pure states, which included both separable and entangled ones: $$\hspace{-9mm}
\begin{aligned}
\big\{
&{{|{00}\rangle}},{{|{01}\rangle}},{{|{10}\rangle}},{{|{11}\rangle}},\\
&\tfrac{1}{\sqrt{2}}\big({{|{00}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{01}\rangle}}\big),\,&
&\tfrac{1}{\sqrt{2}}\big({{|{00}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{10}\rangle}}\big),\\
&\tfrac{1}{\sqrt{2}}\big({{|{00}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{11}\rangle}}\big),\,&
&\tfrac{1}{\sqrt{2}}\big({{|{01}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{10}\rangle}}\big),\\
&\tfrac{1}{\sqrt{2}}\big({{|{01}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{11}\rangle}}\big),\,&
&\tfrac{1}{\sqrt{2}}\big({{|{10}\rangle}}+{\mathrm{e}^{\imu k\pi/2}}{{|{11}\rangle}}\big)
\big\},
\end{aligned}
\quad
k\in\fZ_4.$$ Interestingly, although some of these states are ‘preferred’ in terms of achieved fidelity $F$, there was no correlation between this measure and the entanglement property. This observation should not be surprising, because the mapping of $2$-qubit states into joint probability distributions does not make entangled states distinct. It follows that even imperfect gate implementation should not distinguish these states from separable ones.
The results of simulations are presented in [Fig. \[fig:FC-CNOT\]]{}. Like before, for each setting of the control parameters $(\gamma,\sigma)$, the inhibition level $\eta$ was adjusted to minimize the variance of fidelity across the test states.
Qualitatively, the figures \[fig:FC-CNOT\] are largely similar to what had been obtained for $1$-qubit gates ([Fig. \[fig:FC-1D\]]{}), the major difference is in the range of achieved fidelities and output coherences. Standard deviations of coherence $R$ increased evenly by approximately a factor of $2$, while the fidelity multiplied by about $4-5$. The most dramatic changes are observed in [Fig. \[fig:FC-CNOT\]]{}a: Whereas at low saturation values ($\gamma<1$) the $1$-qubit gate worked relatively well, in the case of a huge overcoherence takes place along with significant fidelity loss.
For a reasonable performance at $\sigma=0$ and $\tSig \gtrsim 30$ one needs $\gamma \gtrsim 1$. In this regime one finds $F\gtrsim 0.97\,(-0.03,+0.02)$, corresponding to the unitary error $\alpha \lesssim 14^\circ\,(-5,+4)$; with noise at $\sigma=0.3$ and $\gamma=1$ the fidelity drops down to $F=0.77\,(-0.11,+0.15)$, or $\alpha=42^\circ\,(-22,+12)$, what is hardly acceptable for a large-scale quantum computation. While comparing these values with the best to-date experimental achievements ($F \sim 0.7-0.8$ with trapped ions [@SKHRGLDBREB2003], $F \sim 0.6-0.8$ with Josephson junctions [@YPANT2003], $F \sim 0.85$ in optical setup [@BPWR2003]) one has to take into account the many simplifications of our toy-model. More realistic simulations, or ultimately – realizations, may not necessarily prove as good as this one, and would probably require additional resources to implement some of the error correcting schemes, nevertheless the principle of quantum computing with neural networks has been demonstrated.
\[fig:FC-CNOT\]
Discussion {#sc:done}
==========
We have studied the potential of an artificial neural network to operate on correlated spike trains assuming the latter to encode quantum states. The model neurons are reduced here down to the essential ingredients of computational capability. Few comments concerning the simplifications made are in order at this point:
First, we have completely neglected the synaptic noise, by assuming the signals to be relayed undisturbed between cells. The justification is that here the few edges of each node represent averages over $10^3-10^4$ real synaptic connections therefore the impact of faulty transmission through a single synapse is greatly reduced. But inclusion of this likely source of errors may still be a significant factor reducing the overall performance.
Second, the time duration of processed signals are assumed to be much shorter than the synaptic plasticity scale. Adaptation is an inherent element of information processing in the brain, but it conflicts with the objective of reliable signal transformations in that there is a trade-off between computing efficiency and adaptive capability. The resolution is provided by separation in time scales between the two processes – transformations act over short signals, typically in response to rapidly varying external stimuli. This is consistent with the optimal signal length which was found here for both $1$- and $2$-qubit gates to be of order $\sim 30$ steps. Assuming the time step is set to $\sim 5\,\mathrm{ms}$ leads to a realistic signal duration of $\sim 150\,\mathrm{ms}$.
Third, the detrimental effect of cellular noise on the performance of quantum gates clearly shows the deterministic regime to be preferable at least for the coding scheme considered here. On one hand, a sharp firing threshold needed for the neurons to act as ‘counters’ which discretize linearly accumulated input signals, corroborates with the theoretical analysis of optimality in terms of information encoding [@BRP2003]. But on the other, the noise itself which blurs this threshold has been shown to be a viable resource acting through the mechanisms of stochastic resonance [@GHJM1998]. This suggests to consider alternative quantum coding schemes, which would make use of the inherent uncertainty in spike generation, provided the relevant conditions are stable enough ([[*e.g.*]{}]{}, noise variance at a constant, moderate level).
It is worthwhile to note at this point, that the quantum states are not absolute entities, and the same set of spike trains may be ‘quantized’ in many different ways depending on the assumed definition of a state. Accordingly, the quantum transformations as well as their implementations will differ. We have discussed here only two coding schemes (referred to as the ‘dense’ and ‘sparse’ spatial code), but it appears plausible, that the real networks may actually alternate (or combine) many different encodings, depending on the nature of the input signal and the functional properties of the circuit. An evident possibility is the *sparse temporal code* based on probability waves, particularly attractive for at least two reasons: First, the brain waves provide the frequency basis necessary for phase discrimination, and there is an experimental indication for independence between rate and phase variables [@HBK2003]. The question is not whether the spiking probability oscillation does have a role, but rather what is the relevant number of modes involved in computation (if more than two then one should consider qudits instead of just qubits). Second, while the ‘dense’ code requires two random binary variables per qubit, by trading spatial for temporal resources, probability waves allow to encode one-qubit per neuron. The drawback is that the mechanisms of short term synaptic plasticity [@MLFS1997; @BP1998; @FD2002] makes the neural circuits operating on this form of a code susceptible to unwanted modifications [@FLBR2001]. From this perspective, the use of sparse spatial coding [@F1987; @BNMLL1990; @OF1996; @ASGA1996; @BS1997; @KBTGA2003], appears to be advantageous, since such spike trains have by definition no temporal correlations, and hence the circuits operating in this fashion are expected to be more stable.
In summary, we have demonstrated the principle of employing quantum coding in artificial neural networks, by providing examples of circuits which realize quantum gates. There is a room for improvement and further investigation with more realism put into the model, alternative circuits, and algorithm implementations. Exploring the possible ways in which neural networks can handle quantum codes, can certainly benefit both the quantum mechanics and neuroscience. On one hand, applications of to neural systems broaden the range of possibilities to be considered when seeking to understand the language of spikes, on the other – macroscopic realizations can provide clues about the microscopic phenomena upon which originated.
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[^1]: This is in close analogy to complex numbers which extend the reals, and at the same time are embeddable in a real vector space of doubled dimension equipped with complex structure.
[^2]: Note, that the embedding allows to consider a wider range of isometries to be implemented, not only the ones corresponding to unitary operations. For instance the $1$-qubit antipode (unfortunately also called the quantum universal-NOT) can only be approximated in unitary QM [@BHW1999; @MBSS2002]. In probabilistic approach one can realize it exactly.
[^3]: From the point of view of state estimation, the optimal POVM is a conformal transformation, which maps the Bloch sphere into a sphere inscribed in the standard simplex of $\fR^{2^{2n}}$. Thanks to the many symmetries of such geometric configuration, some of the rotations are expressible as permutations of the simplex’ vertices and can be implemented with high efficiency. In the case of $A$ given by Eq. , the permutation $(00,10)(01,11)$ corresponds to the $1$-qubit NOT gate. With a different POVM one can bring the Hadamard’s gate H to a permutation, therefore if an algorithm relies on frequent applications of this operation, that could be a preferred choice.
[^4]: Would it be possible, then either the gate could have no entangling capability, or the marginal probabilities of the two qubits be not conserved.
|
---
abstract: 'An alternative approach to two-part ’critical compression’ is presented. Whereas previous results were based on summing a lossless code at reduced precision with a lossy-compressed error or noise term, the present approach uses a similar lossless code at reduced precision to establish absolute bounds which constrain an arbitrary lossy data compression algorithm applied to the original data.'
author:
- John Scoville
title: Bounding Lossy Compression using Lossless Codes at Reduced Precision
---
Introduction
============
Another possible implementation of critical data compression[@critical] compresses the critical bits of a data object losslessly, as before, while simultaneously compressing the entire object using lossy methods, as opposed to lossy coding only an error or residual value. In principle, this results in the coding of redundant information. In practice, however, the lossy coding step is often more effective when, for example, an entire image is compressed rather than just the truncated bits. Encoding an entire data object tends to improve prediction in the lossy coder, while encoding truncated objects often leads to the high spatial frequencies which tend to be lost during lossy coding. Such a redundant lossy coding of the original data often results in the most compact representation available, making this approach desirable for many applications relating to lossy coding. This may not always be the case, for instance, when the desired representation is nearly lossless such a scheme may converge more slowly than one encoding truncated data. On the other hand, however, this scheme is simple generally applicable to any type of lossy coding, whereas greater care must be taken when lossy coding truncated data (in video data, for instance) to avoid introducing high-frequency artifacts and noise to the data. Furthermore, compressing and decompressing data using this approach potentially requires fewer operations than critically compressing data via truncation, as no normalization needs to be performed.
The Algorithm
=============
Once the desired critical bit depth has been selected, the original data’s precision is reduced from the original bit depth of $d$ to the critical bit depth of $n$ and the resulting object is coded losslessly, as before, while the original data is simultaneously coded using lossy compression. As before, this operation may take place on a channel-by-channel basis with arbitrary parameters and with data having undergone an arbitrary set of transformations, e.g. color space rotations.
Upon decompression, the lossless code is used to establish exact upper and lower bounds for lossy compression. The lossless code establishes a lower bound since the truncated values never exceed the original values. Likewise, since $2^{d-n}-1$ is the largest quantity which could have be truncated, adding this to the reduced-precision data produces an upper bound on any value which could be truncated to produce the losslessly coded reduced-precision data.
The decompression scheme is as follows: If the $n$ leading bits of the value predicted by the lossy code match the lossless code, the value predicted by the lossy code is returned as the decompressed datum. If the $n$ leading bits predicted by the lossy code are less than the value coded by the lossless code, the value of the lossless code is returned as the decompressed datum since it is a lower bound on the original value. Otherwise, if the $n$ leading bits predicted by the lossy code are greater than the the value coded by the lossless code, the value coded by the lossless code is increased by $2^{d-n}-1$ and returned as the decompressed datum since this is an upper bound on the value of the original data.
Example
=======
As an example of this approach, we will consider the critical compression of another test image, the 8-bit (per channel) RGB color ’fireworks’ image from ’The New Test Images’, available at imagecompression.info. The image is 3136 by 2152 pixels, which is 20,246,016 bytes of raw data. We will compress the image in YCC space, with the luminance channel being critically compressed at a bit depth of 4 and chrominance data is taken from the lossy compression. Two objects are stored, one is a JPEG2000 compressed version of the original RGB color space image (which uses a YCC color space internally) whose compression ratio is 1000:1 and the other is a PAQ-compressed lossless representation of a luminance (Y) channel whose precision has been reduced from an 8-bit (256 shade) grayscale image to a 4-bit (16 shade) grayscale image. The JPEG2000 representation of the original image may be seen in figure 1, and the lossless representation of a 4-bit luminance channel derived from the original image in figure 2.
{width="4.7in"}\
Figure 1\
{width="4.7in"}\
Figure 2\
During decompression, the luminance value implied by the color of each pixel in the JPEG2000 channel is compared to the 4-bit luminance channel. The largest possible truncation error in the 4-bit luminance channel is 15, so the luminance should be between the value coded by the 4-bit lossless channel and this value increased by fifteen. If the luminance predicted by the lossy JPEG2000 code falls within this range, then the color predicted by JPEG2000 is used for the pixel. If the JPEG2000-predicted luminance falls below this range, then the value coded by the 4-bit lossless channel is used for the luminance, being combined with the chrominance values implied by the JPEG2000-predicted color value before being rotated back into the RGB color space. If the JPEG2000-predicted luminance falls above the allowed range, then the value coded by the 4-bit lossless channel is increased by fifteen before used for the luminance and combined with the chrominance values implied by the JPEG2000-predicted color and being rotated back into the RGB color space. The decompressed image resulting from this coding and decoding scheme may be seen below. Examination of this image reveals that the absolute bounds obtained from lossless brightness values have greatly increased the contrast of the resulting picture, as compared to the JPEG2000 representation.
{width="4.7in"}\
Figure 3\
Disclosures
===========
This research was funded entirely by the author, John Scoville, and the method described is part of a pending patent application.
[10]{}
J. Scoville, *Critical Data Compression*, arXiv:1005.1684 **1**
|
---
abstract: 'We make use of the *Hubble Space Telescope* proper-motion catalogs derived by @bellini2014 to produce the first radial velocity-dispersion profiles $\sigma(R)$ for blue straggler stars (BSSs) in Galactic globular clusters (GCs), as well as the first dynamical estimates for the average mass of the entire BSS population. We show that BSSs typically have lower velocity dispersions than stars with mass equal to the main-sequence turnoff mass, as one would expect for a more massive population of stars. Since GCs are expected to experience some degree of energy equipartition, we use the relation $\sigma \propto M^{-\eta}$, where $\eta$ is related to the degree of energy equipartition, along with our velocity-dispersion profiles to estimate BSS masses. We estimate $\eta$ as a function of cluster relaxation from recent Monte Carlo cluster simulations by @bianchini2016b and then derive an average mass ratio ${M_\mathrm{BSS}}/{M_\mathrm{MSTO}}= 1.50 \pm 0.14$ and an average mass ${M_\mathrm{BSS}}= 1.22 \pm 0.12$ [M$_\odot$]{}from 598 BSSs across 19 GCs. The final error bars include any systematic errors that are random between different clusters, but not any potential biases inherent to our methodology. Our results are in good agreement with the average mass of ${M_\mathrm{BSS}}= 1.22 \pm 0.06$ [M$_\odot$]{}for the 35 BSSs in Galactic GCs in the literature with properties that have allowed individual mass determination.'
author:
- 'A. T. Baldwin$^{1,2}$, L. L. Watkins$^2$, R. P. van der Marel$^2$, P. Bianchini$^3$, A. Bellini$^2$, J. Anderson$^2$'
bibliography:
- 'refs.bib'
title: '*Hubble Space Telescope* proper motion (HSTPROMO) catalogs of Galactic globular clusters$^{\ast}$. IV. Kinematic profiles and average masses of blue straggler stars'
---
Introduction
============
Blue-straggler stars (BSSs) are hydrogen-burning stars that occupy a region of the optical color-magnitude diagram (CMD) brighter and bluer than the main-sequence turnoff (MSTO). They were first discovered in M3 by @sandage1953 and have since been detected in all Galactic globular clusters (GCs). We refer to @cannon2015 for a more detailed review of early BSS research. BSSs appear to extend the main sequence into higher-mass stars, which should have evolved into giants or stellar remnants if they had formed at the same time as the rest of the cluster. BSSs mimic a younger population of stars, but GCs do not contain sufficient gas to support recent or ongoing star formation. In order to explain BSSs, then, there must be some mechanism through which pre-existing main-sequence stars can increase in mass and luminosity. This is primarily expected to occur through mass transfer in evolved binary systems [@sollima2008; @knigge2009; @geller2011; @leigh2013; @gosnell2014] or through stellar collisions in the cluster core [@hurley2005; @geller2013; @chatterjee2013]. Neither model can adequately produce the entire BSS population so the most likely explanation is some combination of the two [@ferraro2009; @dalessandro2013].
The masses of individual BSSs are not, in general, well known and their status as higher-mass stars was initially inferred solely from their position on the CMD. This hypothesis was finally put to the test by @shara1997 who measured the surface gravity of an individual BSS in 47Tuc to derive a mass of $1.7 \pm 0.4$ [M$_\odot$]{}; nearly twice the MSTO mass in the cluster. Individual BSS masses have since been measured for 35 stars via spectroscopic analysis [@demarco2005] and stellar pulsations [@gilliland1998; @fiorentino2014]. These studies seem to suggest that a typical BSS is significantly more massive than stars at the MSTO, but not more than twice as massive, as one would expect for a population of stars formed via mass transfer or mergers of main-sequence stars. Stars significantly more massive than the MSTO will have long since evolved off of the main sequence, along with any BSSs they produced in the past. There are, of course, some exceptions to this trend such as S1082 [@vandenberg2001; @sandquist2003] or WOCS7782 [@geller2009], BSS systems that contain significantly more than twice the MSTO mass and that are likely the result of multiple stellar mergers or multiple dynamical interactions respectively, but these systems are rare and do not represent a typical BSS.
Galactic GCs have all been shown to contain BSSs and they provide a unique environment in which we can study not only BSSs, but also their dynamical interactions with the rest of the cluster. Over time, dynamical friction is expected to cause more massive objects such as BSSs to migrate towards the cluster core. This has been observed as a bimodal distribution in BSS density [e.g. @ferraro1997] consisting of a strong central concentration, followed by a dip at intermediate radii and a subsequent rise at large radii. This distribution arises because relaxation time is a function of radius: there is some critical radius within which BSSs will have had time to migrate to the cluster core, and beyond which BSSs will have remained largely undisturbed. Another crucial result of dynamical interactions is energy equipartition. Frequent two-body interactions will tend to equalize the energy of all stars within a cluster and so more massive populations typically have lower velocity dispersions. As such, we anticipate that BSSs should have a lower velocity dispersion profile than typical main-sequence stars at a given distance from the centre. This was observed to be true for 47Tuc, where the BSS velocity dispersion is related to the dispersion of stars at the turnoff mass by ${\sigma_\mathrm{BSS}}/ {\sigma_\mathrm{MSTO}}\approx \frac{1}{\sqrt{2}}$ [@mclaughlin2006]. This is consistent with a population of stars with twice the turnoff mass in a state of energy equipartition with the rest of the cluster.
Recently, we presented a set of proper-motion catalogues for 22 Galactic GCs, compiled from archival *Hubble Space Telescope* (*HST*) observations taken at multiple epochs [@bellini2014 hereafter ]. Our proper-motion catalogs have several advantages over radial-velocity surveys: since proper motions are measured by determining how far stars have moved from one epoch to another, we are able to measure a large sample of stars from imaging for which it would be prohibitively time consuming to acquire the individual spectra needed for a radial-velocity survey. Further, we are able to observe fainter stars for which reliable spectra may not be available at all. In @watkins2015a [hereafter ], we used the catalogs to study the radial velocity-dispersion and velocity-anisotropy profiles of all the bright stars in 22 Galactic GCs. In @watkins2015b, we compared the bright-star dispersion profiles against literature line-of-sight dispersion profiles to estimate dynamical distances and mass-to-light ratios. These studies are all part of the *Hubble Space Telescope* Proper Motion (HSTPROMO) collaboration [@vandermarel2014].
Here, we present the first large-scale kinematic survey of BSSs in Galactic GCs using the proper-motion catalogs. The paper is laid out as follows: introduces a series of cuts to our catalog, first to produce a sample of stars with a narrow range of masses and second to select the BSS population. In , we divide our BSS population into radial bins, use a maximum-likelihood method to estimate the velocity dispersion of each bin, and fit a dispersion profile to the binned velocity dispersion estimates. In , we estimate the typical mass of BSSs in each cluster and of BSSs as a whole, and make comparisons with previous results. Our results are summarized in .
Cluster Data {#sect:data}
============
Although presented *HST* proper-motion catalogs for 22 Galactic GCs, in this paper we make use of the catalogs for only 19 of these clusters. NGC6535 and NGC7099 (M30) are excluded from our analysis as our catalogs contain relatively few stars for these clusters and we do not detect enough BSSs in either cluster to make a meaningful estimate of their velocity dispersion. We also exclude NGC6715 (M54) due to the risk of contamination from the Sagittarius dwarf spheroidal galaxy. Some characteristic properties are provided in () for the 19 clusters used for this study.
These catalogs measure *relative* proper motions rather than absolute proper motions. This is due to a lack of “fixed” background sources bright enough to be observed through the dense core region of a GC. Consequently, the average velocity of cluster members in any small region of the sky should be zero by design and these catalogs cannot be used to measure the differential rotation or bulk motion of the cluster as a whole. In this paper, we are only concerned with measuring velocity dispersions, which can be calculated from relative proper motions. We refer back to for a more detailed explanation of these issues.
We select our BSSs from the high-quality bright-star samples described in . Here we briefly describe the bright-star samples before explaining our BSS selection procedure.
Bright-Star Catalog {#sect:brightcat}
-------------------
Accurate kinematic estimates require high-quality velocity measurements with reliable uncertainties. Including stars for which the positions are poorly determined (often due to blending with a neighbouring star) or for which the velocity uncertainties have been underestimated tends to artificially increase the velocity distributions. Contaminants – i.e. stars that are not members of the cluster – can also introduce biases.
To avoid such sources of error, we do not use the full catalogs from , but instead the cleaned samples of bright stars from . We refer to Section 2 of for further details regarding these cuts, but briefly summarize them here: To select these samples, we started with a magnitude cut at 1 magnitude below the MSTO to select only bright stars. Next, a series of quality cuts were made on: 1) the number of individual measurements used for the proper-motion estimate; 2) the quality of the proper-motion fits; and 3) the quality of the point-spread-function fits. Finally, velocity outliers and stars with large velocity uncertainties were removed.
In , we were interested only in radial changes of the kinematics and wished to neglect the effects of stellar mass. The magnitude cut was made to restrict the range of stellar mass in each cluster sample. As we will see, this cut is fainter than the faint magnitude limit we will impose on our BSS samples, so it will not interfere with our selection.
Blue-Straggler Selection
------------------------
BSSs are an apparent extension of the main sequence, both brighter and bluer than the MSTO, so we must make cuts in both color and magnitude to separate them from the rest of the bright-star catalog. We first identify the MSTO as follows: we bin all of the stars in our catalog into bins 0.1 mag wide, fit a Gaussian to the color distribution of each bin, and take the bin with the bluest mean to be the MSTO . We then identify the color and color dispersion of the MSTO by selecting all of the stars within 0.1 mag of the MSTO and calculating both the 5$\sigma$-clipped mean color and its standard deviation, which we denote as ${\sigma_\mathrm{c}}$.
--------- ----------- ------------ --------- ---------- ------------------ -- --
Cluster Faint cut Bright cut Red cut Blue cut $N_\mathrm{BSS}$
(mag) (mag) (mag) (mag)
(1) (2) (3) (4) (5) (6)
NGC104 16.9 14.8 0.56 0.28 25
NGC288 18.4 16.4 0.52 0.10 27
NGC362 18.2 16.5 0.50 0.10 40
NGC1851 18.9 17.2 0.51 0.10 24
NGC2808 18.8 16.6 0.67 0.20 58
NGC5139 17.9 15.7 1.090 0.20 73
NGC5904 17.8 17.0 0.51 0.10 16
NGC5927 18.8 16.5 0.97 0.60 65
NGC6266 18.8 16.5 2.15 1.00 19
NGC6341 18.0 16.0 0.46 0.00 37
NGC6362 18.0 16.0 0.58 0.27 21
NGC6388 19.6 17.5 0.91 0.40 58
NGC6397 15.7 14.5 0.64 0.07 10
NGC6441 20.2 18.0 1.03 0.60 25
NGC6624 18.7 16.8 0.80 0.50 12
NGC6656 16.8 15.6 0.85 0.46 34
NGC6681 18.5 17.0 0.58 0.10 14
NGC6752 16.6 15.0 0.52 0.00 16
NGC7078 18.6 17.3 0.52 0.10 21
--------- ----------- ------------ --------- ---------- ------------------ -- --
: Color and magnitude cuts for each cluster.[]{data-label="table:bsscuts"}
**Notes.** Columns: (1) cluster identification (2) faint magnitude cut; (3) manual bright magnitude cut; (4) red color cut; (5) manual blue color cut; (6) number of BSSs used in our analysis. In some cases, the number of BSSs used is lower than the number detected in our catalog since some BSS were too isolated to be sensibly binned for dispersion estimates. For most clusters, the magnitudes given are F814W and colors are defined as F606W-F814W. For NGC5139, the given magnitude is F625W and the color is defined is F435W-F625W. For NGC6266, the given magnitude is F658N and the color is defined as F390W-F658N.
Now that we have characterized the MSTO, we are ready to select BSSs. We first select stars that are at least 0.1 mag brighter than the MSTO. This number is chosen as we have only constrained the MSTO to within a 0.1 mag bin, which is large relative to the photometric uncertainty of our measurements. This cut is sufficient to ensure that only stars brighter than the MSTO appear in our BSS catalog.
Next, we select for stars that are bluer than the MSTO. A binary or multiple-star system of main-sequence stars near the MSTO could mimic a BSS. To account for this, we select only stars that are at least 3${\sigma_\mathrm{c}}$ bluer than the MSTO. A binary or multiple-star system will appear brighter on the CMD, but it will not appear any bluer so this cut should be sufficient to ensure that we select only stars that are truly distinct from the main sequence.
After making these cuts, we often need to make additional cuts in both color and magnitude to remove the horizontal branch from our BSS sample. These extra cuts are made by eye on a cluster-by-cluster basis. All cuts are given in .
As an example, we show the CMD for NGC362 in with our selected BSSs shown as blue diamonds and all other stars plotted as black points. The red diamond marks the adopted location of the MSTO. The black lines mark our BSS selection cuts. Figures \[fig:cmds1\] and \[fig:cmds2\] present the CMDs for the rest of the clusters in our sample.
![CMD for NGC362, illustrating our BSS selection. The black points show stars from the bright-star catalog, the blue diamonds show the selected BSSs, and the red diamond marks the adopted position of the MSTO. The black lines show the cuts made to isolate the BSS population.[]{data-label="fig:cmd_example"}](f1.pdf){width="\linewidth"}
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Results {#sect:results}
=======
Our aim is to compare the radial velocity-dispersion profiles of BSSs with the radial velocity-dispersion profiles of stars with masses on the order of the turnoff mass. In , we calculated velocity-dispersion profiles for the bright stars in each cluster. To each we fit a monotonically-decreasing fourth-order polynomial that was defined to be flat at small radii. These polynomial fits are described in detail and displayed in . The best fits for each cluster are used in this paper as a morphological model for our BSS dispersion profiles, as we will later discuss.
Two-body interactions between stars are known to preferentially equalize the kinetic energy of the two stars. That is to say, kinetic energy is most-often transferred from a higher-energy star to a lower-energy star. Stars within the ancient and densely-populated environment of a GC will have experienced many such interactions during their lifetime and, as a result, we expect them to evolve towards a state where all stars in the cluster have the same kinetic energy. This state is called energy equipartition. The kinetic energy of a population of $N$ stars of mass $M$, average velocity $\bar{v} = 0$, and velocity dispersion $\sigma$ is proportional to $\sum_{i=1}^{N} $ $M\sigma^2$. So, for a system in complete energy equipartition, we would expect $\sigma(M) \propto M^{-0.5}$. In practice, GCs are not found in complete energy equipartition [@trenti2013; @anderson2010], and instead follow the power law, $$\sigma(M) \propto M^{-\eta},
\label{eqn:sigmeta}$$ where $\eta$ is a constant between 0 and 0.5 that depends on the type of system in question and the parameters of the GC as a whole (e.g. core concentration, relaxation time).[^1]
Recall from that we have limited ourselves to a catalog of bright stars, where ‘bright’ is defined as no more than 1 mag below the MSTO. This is advantageous for our present analysis because it represents a narrow range of stellar masses, which allows us to produce standard profiles against which we can compare our BSS profiles.
Previous work [eg. @sabbi2004; @ferraro2006; @dalessandro2008] has shown that the relative radial distributions of BSSs in clusters can be flat, bimodal, or centrally peaked, depending on the dynamical state of the cluster.[^2] In the case of a bimodal radial distribution, it is possible that core stragglers could have different properties from the outer stragglers. However, as our BSS samples are primarily from the cluster cores, we do not consider such differences here.
Blue-Straggler Dispersion Profiles {#sect:bssdisps}
----------------------------------
To estimate BSS kinematic profiles for each cluster, we bin each BSS population in radius and then estimate the velocity dispersion in each bin using the maximum-likelihood method described in Section 3.1 of . Due to the relative rarity of BSSs, the radial distribution of BSS varies significantly between clusters and we are unable to apply a single binning algorithm to all of the clusters within our sample. Each cluster is, therefore, manually binned in an effort to minimize the radial extent of each bin while maximizing the number of stars per bin and the number of available bins. Ideally, we try to make our bins small enough that the dispersion profile should not vary appreciably across the radial extent of the bin but this is not always possible for sparsely populated clusters.
BSSs make up a tiny fraction of each cluster; our catalogs have fewer than 75 BSSs, even for the most-populated clusters in our sample. We reduce the minimum number of stars in each bin from 25 to 7 to increase the spatial resolution of our dispersion profiles. The fractional error on our velocity dispersion measurement is given by $$\frac{\Delta \sigma}{\sigma} = \sqrt{\frac{1}{2N_\mathrm{v}}} ,$$ where $N_\mathrm{v}$ is the number of velocity measurements. In this case, $N_\mathrm{V}$ is twice as large the number of stars in each bin since we have both radial and tangential proper motions for each star. With a bin size of 7, we achieve a fractional error less than 0.2, which is reasonable for our purposes.
It is difficult to constrain the shape of a BSS dispersion profile based on a small number of radial bins. Instead, we use the fact that, as a system moves towards energy equipartition, we expect velocity dispersion to vary as a function of mass, following . We therefore assume for simplicity that the BSSs follow a dispersion profile morphologically similar to stars of the turnoff mass modified by some factor, $$\alpha \equiv \frac{{\sigma_\mathrm{BSS}}}{{\sigma_\mathrm{MSTO}}} = \left( \frac{{M_\mathrm{BSS}}}{{M_\mathrm{MSTO}}}
\right)^{-\eta},
\label{eqn:alpha}$$ where ${\sigma_\mathrm{BSS}}$ and ${M_\mathrm{BSS}}$ are the dispersion and mass of the BSSs, ${\sigma_\mathrm{MSTO}}$ and ${M_\mathrm{MSTO}}$ are the dispersion and mass of the turnoff stars, and $\eta$ represents the degree of energy equipartition experienced by BSSs in the cluster, which for simplicity is assumed here to be independent of radius.
Let us define $f(R)$ to be the polynomial fit to the bright-star dispersion from . Then for any given scale factor $\alpha$, the model dispersion profile is $\sigma(R) = \alpha f(R)$. For a bin $i$ at position ${R_\mathrm{i}}$, the likelihood ${\mathcal{L}_\mathrm{i}}$ of the observed velocity dispersion ${\sigma_\mathrm{i}}\pm {\Delta\sigma_\mathrm{i}}$ given the model prediction is, $$\begin{aligned}
{\mathcal{L}_\mathrm{i}}& = p \left( {\sigma_\mathrm{i}}| \alpha, {R_\mathrm{i}}, {\Delta\sigma_\mathrm{i}}\right) \nonumber \\
& = \frac{1}{\sqrt{2 \pi {\Delta\sigma_\mathrm{i}}^2}}
\exp \left[-\frac{\left( {\sigma_\mathrm{i}}- \alpha f(R_\mathrm{i}) \right)^2}{2 {\Delta\sigma_\mathrm{i}}^2} \right].\end{aligned}$$ The posterior probability ${\mathcal{P}_\mathrm{i}}$ of the model $\alpha$ given the observed properties of bin $i$ is then, $$\begin{aligned}
{\mathcal{P}_\mathrm{i}}& = p \left( \alpha | {\sigma_\mathrm{i}}, {R_\mathrm{i}}, {\Delta\sigma_\mathrm{i}}\right) \nonumber \\
& = p \left( {\sigma_\mathrm{i}}| \alpha, {R_\mathrm{i}}, {\Delta\sigma_\mathrm{i}}\right) p \left( \alpha \right) \nonumber \\
& = {\mathcal{L}_\mathrm{i}}p \left( \alpha \right),\end{aligned}$$ where $p \left( \alpha \right)$ is the prior probability of $\alpha$, which we will assume is constant. We wish to find the value of $\alpha$ that maximises the total posterior for all $N$ bins, however, as we are assuming a flat prior on $\alpha$, we need only maximise the total likelihood $\mathcal{L}$, $$\mathcal{L} = \prod_{i=1}^{N} {\mathcal{L}_\mathrm{i}}.$$
We use <span style="font-variant:small-caps;">emcee</span>, an affine-invariant Markov Chain Monte Carlo (MCMC) sampler [@foreman-mackey2013], to explore the parameter space and sample the region of best fit. We use 250 trial points (walkers) per step, and find 200 steps to be sufficient for our walkers to converge. We take the final position of each of our 250 walkers to represent a family of fits to our data. This method returns an approximately Gaussian distribution of values for $\alpha$; we take the mean to be our estimate for $\alpha$ and the dispersion to be our $1\sigma$ error estimate.
![Velocity dispersion profile for NGC362. The orange points show the binned dispersion profile for all bright stars (with masses of the order of the MSTO mass) from . The black points show the binned dispersion estimates for the BSS sample; these points clearly fall below the bright-star profile, as expected for a population of higher mass stars in a system with some degree of energy equipartition. The orange line shows a polynomial fit to the bright-stars . To the BSS points, we fit a profile with the same shape as the bright-star profile, but scaled by a factor $\alpha = \left( {M_\mathrm{BSS}}/ {M_\mathrm{MSTO}}\right)^{-\eta}$ to estimate the relative mass difference. The blue lines show draws from the MCMC fit; the adopted ‘best’ fit is shown in black.[]{data-label="fig:disp_example"}](f4.pdf){width="0.9\linewidth"}
We show the radial velocity-dispersion profile for NGC362 in . The orange points represent the binned dispersion estimates for all bright stars from and the black points show our binned dispersion estimates for BSSs. In this case, the BSS dispersion profile clearly falls well below the dispersion profile for all bright stars, as we would expect for a more massive population of stars in a system approaching energy equipartition. The orange line shows the best-fit polynomial to the bright stars . We use this to represent the dispersion profile for stars with mass comparable to the turnoff mass and then scale the profile to fit the BSS profile. The blue lines show a family of fits to the data obtained from the MCMC sampling. The adopted ‘best’ estimate is shown in black.
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We show similar velocity-dispersion profiles for all clusters in Figures \[fig:disps1\] and \[fig:disps3\]. As for NGC362, many clusters have BSS populations that exhibit lower dispersions than the other bright stars, indicating that they are indeed more massive. However, for some clusters, the BSS dispersion profiles are similar to, or even slightly higher than, the bright-star dispersion profiles. This could indicate that these clusters are not very relaxed and so the BSSs have not had enough time to come into equipartition.
This is not unexpected. As briefly discussed earlier, the radial distributions of BSSs in GCs can be flat, bimodal, or unimodal, depending on their dynamical histories; and, in fact, these radial-distribution shapes can be used as a dynamical ‘clock’ [@ferraro2012].[^3] Over time, BSSs in a cluster will relax and sink towards the centre via dynamical friction; relaxation times in cluster cores are shorter than in the outer regions, so the centres will tend to relax first. Consequently, dynamically-old clusters are expected to have a centrally-concentrated population of BSSs as all of the BSSs will have had time to sink towards the centre; clusters of dynamically-intermediate age are expected to show a bimodal radial distribution because the central BSSs will have relaxed and moved to the centre, while the outer BSSs will not have had enough time to relax; and dynamically-young clusters are expected to have flat radial distributions because none of the BSSs will have had enough time to relax.
Following the same lines of reasoning, for dynamically-old clusters, we would expect to see a clear separation between the BSS dispersions and the evolved-star dispersions; for the dynamically-intermediate age clusters, we would expect to see a clear separation between the BSSs dispersions and evolved-star dispersions near the centre, but little or no difference in the dispersions in the outer regions; and, for the dynamically-young clusters, we would expect little or no difference in the dispersions across the whole cluster. Though, as our BSS populations are mostly restricted to the central regions of clusters, the dynamically-old and dynamically-intermediate age clusters will likely be largely indistinguishable in this study.
Now let us consider the clusters in our sample for which radial BSS distributions have been measured. NGC5139 ($\omega$ Cen) shows a flat radial distribution [@ferraro2006] and we find that the BSS and evolved-star dispersions are almost identical, so both result are consistent and suggest that the cluster is dynamically-young. NGC104 (47 Tuc) shows a bimodal radial distribution [@ferraro2004] and we see a clear offset in the BSS velocity dispersion profile, so again the results are consistent, but this time suggesting that the cluster is of dynamically-intermediate age. The radial distributions and kinematics are also consistent for NGC6388 (where both the bimodal radial distribution [@dalessandro2008] and the clear offset between the BSS and evolved-star kinematics suggest a dynamically-intermediate age) and NGC362 (where the centrally-concentrated radial distribution [@dalessandro2013] and the offset in the kinematics are consistent with the cluster being dynamically old). Our conclusions for these four clusters are also consistent with the number of relaxation times that we estimate the clusters to have experienced (see ).
However, for both NGC6752 and NGC5904, the two results are at odds: the radial distributions are clearly bimodal [@sabbi2004; @lanzoni2007], suggesting that the clusters are of dynamically-intermediate age, whereas we find that the BSS velocity dispersions suggest that they are dynamically young. NGC6752 is a core-collapsed cluster, so our results may indicate that there are additional dynamical processes at work in its very dense core that have washed out any velocity dispersion differences, although other known core-collapsed clusters, such as NGC6681, do show clear velocity dispersion differences. Further, NGC5904 is not thought to be core-collapsed, so the reasons for the mismatch in this case is unclear.
It is also worth noting that any systematics in our analysis could cause us to overestimate some dispersion profiles and underestimate others; these will be accounted for in our final average, but may explain why the results are inconsistent for NGC6752.
--------- ------------------ ---------- ----------- ---------- ---------------- -- --
Cluster $N_\mathrm{BSS}$ R $\Delta$R $\sigma$ $\Delta\sigma$
(arcsec) (arcsec) (mas/yr) (mas/yr)
(1) (2) (3) (4) (5) (6)
NGC104 8 30.2056 6.2296 0.5777 0.1048
9 49.7092 4.3549 0.3774 0.0647
8 74.4922 12.5776 0.4158 0.0758
NGC288 9 22.0483 8.5347 0.0622 0.0120
9 37.7978 4.4031 0.0533 0.0110
--------- ------------------ ---------- ----------- ---------- ---------------- -- --
: Blue straggler radial dispersion profiles.[]{data-label="table:profiles"}
**Notes:** Columns: (1) cluster ID; (2) number of BSS in the bin; (3) mean radius of the bin; (4) error on the mean radius; (5) binned velocity dispersion; (6) error on the binned velocity dispersion.
(This table is available in its entirety in a machine-readable form. A portion is shown here as an example of form and content.)
We provide our binned BSS velocity-dispersion profiles in .
Estimates of equipartition {#sect:eta}
--------------------------
We have directly measured $\alpha$ and would like to use these measurements to determine the average mass of the BSS population via . To do this, we must estimate $\eta$. A direct measurement would require velocity-dispersion profiles for stars well below the turnoff mass and is beyond the scope of this paper. Instead, we turn to simulations of GCs.
@trenti2013 used a set of N-body simulations to show that GCs do not achieve complete energy equipartition even after several relaxation times, and that even more massive objects, such as compact remnants and BSSs, approach a value of $\eta$ well below 0.5 within the core (their Figure 1). More recently, @bianchini2016b studied a set of 7 Monte Carlo cluster simulations, with varying concentrations, binary fractions, and total cluster masses. Each simulation was analysed at 4, 7 and 11 Gyr, yielding 21 snapshots in total. They showed that the degree of equipartition reached by stars in a simulated cluster depends on stellar mass, such that stars more massive than some threshold mass ${M_\mathrm{eq}}$ are in complete equipartition (where $\eta = 0.5$), while stars less massive than ${M_\mathrm{eq}}$ have values of $\eta$ that vary linearly as a function of stellar mass. That is, $$\eta \left( M \right) = \begin{cases}
\frac{1}{2} \frac{M}{{M_\mathrm{eq}}} & \mbox{if } M \le {M_\mathrm{eq}}, \\
\frac{1}{2} & \mbox{if } M > {M_\mathrm{eq}}.
\end{cases}
\label{eqn:eta}$$ They also showed that the threshold mass ${M_\mathrm{eq}}$ varied from cluster to cluster and was strongly correlated with the number of relaxation times experienced by the cluster ${n_\mathrm{rel}}= {T_\mathrm{age}}/ {T_\mathrm{rc}}$, where ${T_\mathrm{age}}$ is the age of the cluster and ${T_\mathrm{rc}}$ is the core relaxation time, such that, $${M_\mathrm{eq}}= 1.55 + 4.10 \, {n_\mathrm{rel}}^{-0.85},
\label{eqn:Meq}$$ (see panel C of their Figure 6); this correlation is independent of concentration, binary fraction or initial mass. So it is clear that we cannot assume that the clusters have reached full equipartition, and we must consider both the approximate mass of the BSSs and the relaxation of the cluster when determining values of $\eta$.
To begin, we determine the degree of relaxation experienced by our clusters. @vandenberg2013 estimated ages – via isochrone fitting near the MSTO – for 55 Milky Way GCs, 15 of which overlap with our sample, leaving 4 of our clusters without age estimates. We split the clusters with age estimates into two groups based on their ${\rm [Fe/H]}$ metallicities [taken from @harris1996 2010 edition], denoting clusters with ${\rm [Fe/H]}\le -1.5$ as metal poor and clusters with ${\rm [Fe/H]}> -1.5$ as metal rich. Next we take the average ages separately of the metal-poor and metal-rich clusters and assign the appropriate age to the remaining 4 clusters depending on their metallicity (rounded to the nearest 0.25 Gyr to match the precision of the @vandenberg2013 ages). Combining these age estimates with estimates of the core relaxation times [also from @harris1996 2010 edition], we estimate ${n_\mathrm{rel}}= {T_\mathrm{age}}/ {T_\mathrm{rc}}$. Then we use these ${n_\mathrm{rel}}$ estimates, along with , to estimate ${M_\mathrm{eq}}$ values for each cluster.
Now, let us address the issue of stellar mass. The turnoff mass is typically around 0.8 [M$_\odot$]{}for Galactic GCs, and we expect that BSS masses will typically fall somewhere between the turnoff mass and twice the turnoff mass, so most BSSs within our sample should have masses between 0.8 [M$_\odot$]{}and 1.6 [M$_\odot$]{}. For our purposes, we do not need to evaluate $\eta$ as a function of $M$, instead we require an average $\eta$ across this mass range. So we use our ${M_\mathrm{eq}}$ estimates and to estimate $\eta$ at $M=1.2$ [M$_\odot$]{}(the middle of the range of interest) for each cluster, and adopt these as representative $\eta$ values across the putative BSS range.
![Velocity dispersion as a function of stellar mass for single stars (solid circles) and binary stars (open circles) as predicted by one of the Monte Carlo simulations of @bianchini2016b. The dashed orange line corresponds to an exponential fit across the full range of stellar masses; it is clear that a straight-line fit to the whole mass range would be poor. The solid orange line shows the slope of the fit at $M=1.2$ [M$_\odot$]{}, which we use to approximate the fit across the 0.8-1.6 [M$_\odot$]{}range (highlighted by the dotted lines) in which we expect to find BSSs; a straight line is a reasonable fit in this limited mass range, and the slope of this line provides an estimate of $\eta$, the degree of equipartition reached by the simulation.[]{data-label="fig:etasims"}](f7.pdf){width="\linewidth"}
In , we show velocity dispersion as a function of stellar mass for Simulation 1 from @bianchini2016b. The filled circles show the dispersion profile for single stars and the open circles show the dispersion profile for binary systems; these are consistent across the whole mass range. This is important as it allows us to henceforth consider the single and binary populations together. The dashed orange line shows an exponential fit to the dispersion profile from [@bianchini2016b their equation 3]; the instantaneous slope of this line gives the value of $\eta$ for any given stellar mass (). The solid orange line shows the tangent of the exponential fit at $M=1.2$ [M$_\odot$]{}(the middle of the putative BSS range). The dotted lines mark the 0.8-1.6 [M$_\odot$]{}mass range in which we are interested for our BSS study.
Across the whole mass range, the exponential function clearly performs better than a simple straight-line fit. However, in the BSS mass range, we do find that the dispersion profile can be well approximated by a straight line, implying that $\eta$ can be assumed constant in this range. Further, the deviation of the exponential fit from the tangent evaluated at $M=1.2$ [M$_\odot$]{}is small in the BSS mass range, indicating that the value of $\eta$ at $M=1.2$ [M$_\odot$]{}can indeed by used a representative value, as we have done.
Finally, we must consider the uncertainty in the $\eta$ values we have determined. There are a number of sources of uncertainty, including: uncertainty on the ages and relaxation times used to calculate ${n_\mathrm{rel}}$; scatter in ; scatter in . Furthermore, the @bianchini2016b simulations start from a specific set of initial conditions that may not accurately represent the initial conditions of our clusters, and the 11 Gyr evolution of the simulated clusters may also not accurately reproduce the history of our clusters [see also @bianchini2016a]. Finally, the value of $\eta$ at $M=1.2$ [M$_\odot$]{}is intended to be representative of $\eta$ across the range 0.8-1.6 [M$_\odot$]{}, but we do know that $\eta$ does change with stellar mass, so this may be a further source of uncertainty. Also consider that our choice of $0.8-1.6$ [M$_\odot$]{}as an expected BSS mass range was motivated by typical turnoff mass in clusters, but there will also be cluster-to-cluster variations in the turnoff mass. To encompass all of these sources, we adopt a generous systematic uncertainty of $\sfrac{\eta}{3}$ for each cluster.
We provide ${\rm [Fe/H]}$ metallicities, ages, relaxation times, ${n_\mathrm{rel}}$ estimates, ${M_\mathrm{eq}}$ estimates and $\eta$ estimates in .
[cccccccccccc]{} Cluster ID & ${\rm [Fe/H]}$ & ${T_\mathrm{age}}$ & $r_{\rm c}$ & $\log_{10} {T_\mathrm{rc}}$ & $\log_{10} {n_\mathrm{rel}}$ & ${M_\mathrm{eq}}$ & $\eta$ & ${M_\mathrm{MSTO}}$ & $\alpha$ & $f$ & ${M_\mathrm{BSS}}$\
& (dex) & (Gyr) & (arcsec) & (Gyr) & & ([M$_\odot$]{}) & & ([M$_\odot$]{}) & & & ([M$_\odot$]{})\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) & (12)\
\
NGC104 & -0.72 & 11.75 & 21.6 & -1.16 & 2.23 & 1.60 & 0.37 & 0.87 & $0.77 \pm 0.06$ & $2.04_{-0.41}^{+0.64}$ & $1.78_{-0.35}^{+0.56}$\
NGC288 & -1.32 & 11.50 & 81.0 & -0.01 & 1.07 & 2.05 & 0.29 & 0.79 & $0.80 \pm 0.07$ & $2.18_{-0.55}^{+0.88}$ & $1.71_{-0.43}^{+0.69}$\
NGC362 & -1.26 & 10.75 & 10.8 & -1.24 & 2.27 & 1.60 & 0.38 & 0.80 & $0.69 \pm 0.04$ & $2.72_{-0.52}^{+0.91}$ & $2.18_{-0.42}^{+0.73}$\
NGC1851 & -1.18 & 11.00 & 5.4 & -1.57 & 2.61 & 1.57 & 0.38 & 0.83 & $0.85 \pm 0.07$ & $1.53_{-0.27}^{+0.45}$ & $1.27_{-0.22}^{+0.38}$\
NGC2808 & -1.14 & 11.00 & 15.0 & -0.76 & 1.80 & 1.67 & 0.36 & 0.83 & $0.95 \pm 0.05$ & $1.15_{-0.14}^{+0.20}$ & $0.95_{-0.11}^{+0.17}$\
NGC5139 & -1.53 & 12.75 & 142.2 & 0.60 & 0.51 & 3.07 & 0.20 & 0.76 & $1.02 \pm 0.04$ & $0.89_{-0.15}^{+0.23}$ & $0.68_{-0.12}^{+0.17}$\
NGC5904 & -1.29 & 11.50 & 26.4 & -0.72 & 1.78 & 1.68 & 0.36 & 0.80 & $0.99 \pm 0.04$ & $1.03_{-0.23}^{+0.36}$ & $0.82_{-0.18}^{+0.29}$\
NGC5927 & -0.49 & 10.75 & 25.2 & -0.61 & 1.64 & 1.72 & 0.35 & 0.89 & $0.87 \pm 0.04$ & $1.50_{-0.19}^{+0.27}$ & $1.33_{-0.17}^{+0.25}$\
NGC6266 & -1.18 & 11.25 & 13.2 & -1.10 & 2.15 & 1.61 & 0.37 & 0.81 & $1.10 \pm 0.09$ & $0.77_{-0.17}^{+0.22}$ & $0.62_{-0.13}^{+0.18}$\
NGC6341 & -2.31 & 12.75 & 15.6 & -1.04 & 2.15 & 1.61 & 0.37 & 0.76 & $0.78 \pm 0.05$ & $1.96_{-0.31}^{+0.51}$ & $1.49_{-0.23}^{+0.39}$\
NGC6362 & -0.99 & 12.50 & 67.8 & -0.20 & 1.30 & 1.87 & 0.32 & 0.81 & $1.10 \pm 0.10$ & $0.74_{-0.19}^{+0.28}$ & $0.60_{-0.15}^{+0.22}$\
NGC6388 & -0.55 & 11.25 & 7.2 & -1.28 & 2.33 & 1.59 & 0.38 & 0.85 & $0.85 \pm 0.04$ & $1.53_{-0.18}^{+0.27}$ & $1.31_{-0.16}^{+0.22}$\
NGC6397 & -2.02 & 13.00 & 3.0 & -4.06 & 5.17 & 1.55 & 0.39 & 0.76 & $0.95 \pm 0.12$ & $1.12_{-0.27}^{+0.54}$ & $0.84_{-0.20}^{+0.42}$\
NGC6441 & -0.46 & 11.25 & 7.8 & -1.07 & 2.12 & 1.61 & 0.37 & 0.84 & $0.98 \pm 0.08$ & $1.05_{-0.21}^{+0.29}$ & $0.88_{-0.17}^{+0.25}$\
NGC6624 & -0.44 & 11.25 & 3.6 & -2.39 & 3.44 & 1.55 & 0.39 & 0.87 & $0.73 \pm 0.10$ & $2.28_{-0.67}^{+1.26}$ & $1.99_{-0.58}^{+1.09}$\
NGC6656 & -1.70 & 12.50 & 79.8 & -0.47 & 1.57 & 1.74 & 0.34 & 0.77 & $0.80 \pm 0.06$ & $1.94_{-0.37}^{+0.61}$ & $1.49_{-0.28}^{+0.47}$\
NGC6681 & -1.62 & 12.75 & 1.8 & -3.18 & 4.29 & 1.55 & 0.39 & 0.76 & $0.78 \pm 0.08$ & $1.86_{-0.41}^{+0.76}$ & $1.42_{-0.31}^{+0.58}$\
NGC6752 & -1.54 & 12.50 & 10.2 & -2.12 & 3.22 & 1.56 & 0.39 & 0.77 & $1.11 \pm 0.10$ & $0.76_{-0.17}^{+0.22}$ & $0.59_{-0.13}^{+0.17}$\
NGC7078 & -2.37 & 12.75 & 8.4 & -1.16 & 2.27 & 1.60 & 0.38 & 0.76 & $0.95 \pm 0.08$ & $1.15_{-0.23}^{+0.30}$ & $0.87_{-0.17}^{+0.24}$\
Mean & …& …& …& …& …& …& …& …& $0.90 \pm 0.03$ & $1.48 \pm 0.13$ & $1.20 \pm 0.11$\
**Notes.** Columns: (1) NGC identification; (2) ${\rm [Fe/H]}$ metallicity from @harris1996 [2010 edition]; (3) cluster age (in Gyr), mostly taken from @vandenberg2013, others derived from average @vandenberg2013 values and ${\rm [Fe/H]}$ metallicity – see text for details; (4) core radius $r_{\rm c}$ (in arcsec) [@harris1996 2010 edition]; (5) logarithm of the core relaxation time (in Gyr) [@harris1996 2010 edition]; (6) logarithm of the number of relaxation times ${n_\mathrm{rel}}= {T_\mathrm{age}}/{T_\mathrm{rc}}$; (7) equipartition mass ${M_\mathrm{eq}}$ (in [M$_\odot$]{}) above which stars are in complete energy equipartition [see @bianchini2016b for further details]; (8) equipartition parameter $\eta$; (9) MSTO mass ${M_\mathrm{MSTO}}$ (in [M$_\odot$]{}); (10) $\alpha = ({M_\mathrm{BSS}}/{M_\mathrm{MSTO}})^{-\eta}$ estimated from our dispersion profiles; (11) average BSS mass as a multiple of the MSTO mass, $f = {M_\mathrm{BSS}}/{M_\mathrm{MSTO}}$; (12) average BSS mass ${M_\mathrm{BSS}}$ (in [M$_\odot$]{}).
(This table is available in machine-readable form.)
Blue Straggler Mass Fractions {#sect:massfracs}
-----------------------------
Now, we are ready to estimate the mass ratio for each cluster. We begin with the values of $\alpha$ returned by the MCMC sampling in that we will turn into a mass ratio $$f \equiv \frac{{M_\mathrm{BSS}}}{{M_\mathrm{MSTO}}},$$ by solving for $$f = \alpha^{-\frac{1}{\eta}}.
\label{eqn:f}$$
To accurately propagate our uncertainty in $\eta$, we draw 1000 values of $\eta$ from a boxcar distribution with half-width $\sfrac{\eta}{3}$ centered on our best estimate of $\eta$ for each cluster (). We then take the median of all mass fractions returned by this method to be our best estimate of the average BSS mass ratio in the cluster, and the distance between the 16th and 84th percentiles to be the lower and upper error bars, respectively. Mass ratio estimates for each cluster are presented in .
The reduced $\chi^2$ for the sample compared to its unweighted mean is 3.4. That this value is larger than unity indicates that there is more cluster-to-cluster scatter in our inferred BSS masses than can be explained by our random errors. That is, there are likely further sources of systematic uncertainty, at a level comparable to the random errors, for which we have not accounted.
For this reason, it is not appropriate to include the error bars when calculating a sample mean for all the clusters, so we use a simple unweighted average to calculate the mean sample BSS mass ratio. Similarly, to estimate the error on the mean we use $\sigma/\sqrt{N}$, where $\sigma$ is the scatter between measurements for different clusters. This yields a result that is based only on the scatter between the mass ratios for each cluster, and ignores the random error bars in the individual measurements.
Calculated in this way, we find an average mean mass ratio for the sample of $1.50 \pm 0.14$. The error bar on our final mean result is symmetric; this is reasonable as the distribution of BSS masses inferred for different clusters is not strongly asymmetric (unlike the random errors for individual clusters, which do often tend to be strongly asymmetric). We have experimented with other statistics for calculating the sample mean, and find that the results of alternative methods are generally consistent with this result within the error bars.
shows mass-ratio estimates for all clusters in our sample. The orange line represents the average mass ratio across all the clusters, and the dashed lines represent the standard error on the mean.
![Estimates of BSS mass as a multiple of MSTO mass for each cluster. The orange line shows the mean BSS mass ratio for the sample and the orange dashed lines represent the standard error on the mean.[]{data-label="fig:fmasses"}](f8.pdf){width="0.99\linewidth"}
Blue Straggler Masses
---------------------
So far, we have estimated BSS masses as a multiple of the turnoff mass in a cluster. To determine the intrinsic mass of BSSs, we require estimates for the turnoff masses, which we obtain via isochrone fitting using isochrones from the Dartmouth Stellar Evolution Database [@dotter2008]. Interpolating between isochrones is beyond the scope of this paper, so to select representative isochrones for each cluster, we require: ${\rm [Fe/H]}$ metallicities to the nearest 0.5 dex; $\alpha$-element abundances ${\rm [\alpha/Fe]}$ to the nearest 0.2 dex; and ages to the nearest 0.5 Gyr.
As we did in , we use ${\rm [Fe/H]}$ values from @harris1996 [2010 edition] and ages from @vandenberg2013 (see also ), appropriately rounded. In general, GCs have $\alpha$-element abundances ${\rm [\alpha/Fe]}\sim 0.3$ [eg. @carney1996], but they do show a mild correlation with ${\rm [Fe/H]}$ metallicity, such that more metal-poor clusters are more $\alpha$ enhanced [eg. @kirby2008; @johnson2011]. As such, for metal-poor clusters with ${\rm [Fe/H]}\le -1.5$, we assume $\alpha$-element abundance ${\rm [\alpha/Fe]}= 0.4$; for metal-rich clusters with ${\rm [Fe/H]}\geq -1.5$, we assume $\alpha$-element abundance ${\rm [\alpha/Fe]}= 0.2$.
We use these metallicities, $\alpha$-element abundances and ages to extract representative isochrones and then we adjust the isochrone magnitudes for distance and extinction using distances and reddening values from @harris1996 [2010 edition] and extinction coefficients from @sirianni2005. We interpolate along the isochrone to extract mass estimates for our stars based on their apparent magnitudes. These mass estimates may not be reliable for stars that have evolved off the main-sequence. However, this is of no consequence here as we are only interested in the masses of stars near the turnoff for which this method is robust. Finally, we adopt the median mass of all stars within 0.05 mag of the MSTO as the turnoff mass of the cluster for all clusters, except NGC6266 for which we have no isochrone fit due to its unusual combination of filters; instead, for NGC6266 we adopt a value of 0.81 [M$_\odot$]{}, which is the average turnoff mass for all of the other clusters in our sample.
The actual average mass estimate for BSSs within a cluster is then simply the mass ratio derived in multiplied by our estimate of the MSTO mass; uncertainties in the turnoff mass are negligible compared to the uncertainty in the mass ratio. The BSS mass estimates and our turnoff mass estimates are also given in .
As discussed in detail in , we calculate an unweighted mean and the standard error on the mean to estimate an average BSS mass of ${M_\mathrm{BSS}}= 1.22 \pm 0.12$ [M$_\odot$]{}for the whole sample.
Discussion {#sect:discussion}
==========
Our best mass estimate of ${M_\mathrm{BSS}}= 1.22 \pm 0.12$ [M$_\odot$]{}is in very good agreement with the average mass of ${M_\mathrm{BSS}}= 1.22 \pm 0.06$ [M$_\odot$]{}taken from 35 individual BSSs in the literature [@shara1997; @gilliland1998; @demarco2005; @fiorentino2014].[^4]
As we have derived our BSS masses from cluster dynamics, we have measured the average *total* mass of the BSS systems, which includes any possible binary companions. The BSS population of open cluster NGC188 has a binary fraction of 76% [@mathieu2009]; GCs are typically more dense environments than open clusters, which may affect the fraction of BSSs found in binaries, however many common formation channels for BSSs involve binaries in some fashion so the binary fraction is likely to be high in GCs as well. Furthermore, close 3-body interactions involving binary systems tend to eject the *least* massive object, and so, as some of the most massive objects in a cluster, BSSs that exist in a binary system are likely to remain a part of one.
There are three main formation theories for blue stragglers: stable mass-transfer in binary systems, stellar mergers, and stellar collisions; although cluster dynamics are complicated so BSS formation histories seldom follow just one of these channels [eg. @chatterjee2013]. Indeed, @leigh2016 recently showed that binary mass-transfer can be interrupted by a dynamical encounter with another star, particularly in lower-mass clusters. Nevertheless, let us consider each mechanism in turn and consider the resulting BSS.
In the case of stable mass transfer in a binary system, the more-massive star in a binary fills its Roche lobe and transfers mass to its companion [@mccrea1964]. We would expect BSSs formed via mass transfer to retain at least a helium white dwarf companion with a mass on the order of 0.5 [M$_\odot$]{}.[^5] If the mass-transfer formation channel for BSSs is active in GCs, then we might expect our dynamical mass estimates to be somewhat larger than those derived from spectra or pulsations which only consider the BSS itself and not the additional mass of a companion.
BSSs resulting from mergers are expected to form via two alternative pathways: 1) unstable mass transfer in a binary system can lead to the complete merger of the two stars [@chen2009], leaving behind a single BSS; or 2) the Kozai effect [@kozai1962] can cause the inner two stars of a hierarchical triple to merge [@perets2009], leaving behind a BSS with the third star as a binary companion. The second formation mechanism is thought to be significant in open clusters, but not in the more dense environments found in GCs [@perets2009]. So the primary merger channel active in GCs is likely to be unstable mass transfer; in this case, we would expect our dynamical estimates to be consistent with those derived from spectra or pulsations.
Stellar collisions can occur as single-single encounters, binary-single encounters, or binary-binary encounters [eg. @hut1983; @sigurdsson1993], or even triple-single, triple-binary or triple-triple encounters [eg. @leigh2011a]. It is likely that single-single encounters may result in a single BSS, however binary-single, binary-binary and other multiple interactions are likely to leave a BSS that exists as part of a binary or multiple system. As binary or multiple encounters are generally more common than single-single encounters in GCs [eg. @leonard1989; @leigh2011b], collisions are more likely to result in a BSS with a binary companion. Again, in this case, we would expect our dynamical BSS mass estimates to be higher than the literature values for individual BSSs.
Simulations, such as those studied in @chatterjee2013, suggest that collisions and stable-mass transfer are the dominant mechanisms for BSS formation. That our dynamical estimates are in such good agreement with previous studies may imply that the binary fraction of BSSs is lower than expected, and may further imply that stellar mergers resulting from unstable mass transfer play a more significant role in BSS formation in GCs than predicted.
However, we must consider that the individual BSS measurements from previous studies may not represent an unbiased sample of the BSS population: @demarco2005 only included stars with effective temperatures greater than 5750K whereas @gilliland1998 and @fiorentino2014 focused exclusively on pulsating BSSs. By contrast, we have measured the average mass for *all* 598 BSSs detected within our sample of 19 Galactic GCs, which should provide an unbiased representation of the BSS population.
Recently, @xin2015 simulated a population of BSSs formed via mass transfer to mimic the population in NGC7099 (M30). Within the simulation, systems that eventually form BSSs had a total mean binary mass of $1.21 \pm 0.03$ [M$_\odot$]{}. Unfortunately, NGC7099 is one of the clusters for which we have insufficient BSSs in the bright-star catalog to produce a BSS dispersion profile and, thus, estimate a mass. However, we can predict a BSS mass from our data by combining our mean BSS mass fraction (${M_\mathrm{BSS}}/{M_\mathrm{MSTO}}= 1.50 \pm 0.14$) with our turnoff mass estimate for NGC7099 (${M_\mathrm{MSTO}}= 0.76$ [M$_\odot$]{}) to predict an average BSS mass of ${M_\mathrm{BSS}}= 1.14 \pm 0.10$ [M$_\odot$]{}, which is consistent with the @xin2015 prediction to within 1$\sigma$.
Our results should be taken with a few crucial caveats:
- The velocity dispersion profile of BSSs need not be a simply-scaled version of the bright-star profile, since $\eta$ has been shown to vary as a function of radius [see @trenti2013].
- The relationship between $\eta$ and cluster relaxation that arises in the simulations from @bianchini2016b may not be correct for real GCs if they evolved from initial conditions that do not exactly match the simulated clusters.
- All of the clusters in our sample are known to host multiple populations of stars [see e.g. @piotto2015]. Second-generation stars typically make up a sizeable fraction of the cluster as a whole; they tend to be He-enhanced and, hence, have a lower MSTO mass, so our BSS masses may be overestimated.
As a final approach, we can define a model-independent *minimum* mass ratio by assuming that each cluster is in complete energy equipartition with $\eta = 0.5$. Doing so returns a minimum average mass ratio of $f \ge 1.32 \pm 0.08$. This implies with 4$\sigma$ confidence that BSSs, on average, have masses greater than the turnoff mass.
Finally, we note that the final error bars on our mass-ratio and mass estimates include any systematic errors that are random between different clusters, but do not include the possible impact of any potential systematic errors (i.e., a bias) that would shift the mass estimates for different clusters in the same direction. We have discussed various potential sources of systematic error in our data-model comparisons, and have not identified any individual source that we expect to introduce a significant bias, but this does not prove that biases may not exist. As noted, our final estimate agrees with literature estimates based on other methods to within the random error of our measurement. This suggests than any potential systematic biases in our final estimate are no larger than the random error.
Conclusions {#sect:conclusions}
===========
We have produced velocity-dispersion profiles for the BSS populations in 19 Galactic GCs based on the *HST* proper-motion catalogs presented in . From these profiles:
- We found that BSSs typically have lower velocity dispersions than stars at the MSTO, as one would expect for a more massive population of stars in a system with some degree of energy equipartition.
- We derived an average mass ratio of ${M_\mathrm{BSS}}/{M_\mathrm{MSTO}}= 1.50 \pm 0.14$ for all 598 BSSs across all 19 clusters; this corresponds to an average mass of ${M_\mathrm{BSS}}= 1.22 \pm 0.12$ [M$_\odot$]{}.
- We confirmed at the $4\sigma$ level that BSSs are on average more massive than the turnoff mass.
Our dynamical estimates are in very good agreement with previous estimates for BSS masses [@shara1997; @gilliland1998; @demarco2005; @fiorentino2014] based on the properties of individual stars.
We wish to thank Yu Xin for providing us with us data on mass transfer BSS simulations for M30, and Nathan Leigh for interesting discussions about blue stragglers and useful comments on the draft. We also wish to thank the anonymous referee for the useful reports that improved the discussion of our results. Support for this work was provided by grants for *HST* programs AR-12845 (PI: Bellini) and AR-12648 (PI: van der Marel), provided by the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. PB acknowledges support from the International Max Planck Research School in Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD).
This research made use of Astropy[^6], a community-developed core Python package for Astronomy [@astropy2013]. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
This project is part of the HSTPROMO collaboration[^7], a set of HST projects aimed at improving our dynamical understanding of stars, clusters and galaxies in the nearby Universe through measurement and interpretation of proper motions.
[^1]: As we will discuss in , this power-law approximation is generally only valid for limited mass ranges but does not hold globally for GCs [see also @bianchini2016b].
[^2]: Typically, the radial distribution of BSS stars is compared to the radial distribution of some reference population, such as horizontal-branch or red-giant-branch stars.
[^3]: Although recent results from Monte Carlo and N-body models have challenged this claim [@hypki2016].
[^4]: We include the variable BSSs from @demarco2005 in this average, but neglect the BSSs from @fiorentino2014 for which the pulsation mode was ambiguous.
[^5]: This represents the compact remnant of an evolved donor star after mass transfer has ceased [see @gosnell2014].
[^6]: <http://www.astropy.org>
[^7]: <http://www.stsci.edu/~marel/hstpromo.html>
|
---
abstract: |
In this paper we answer two recent questions from [@CKPZ] and [@Harju] about self-shuffling words. An infinite word $w$ is called self-shuffling, if $w=\prod_{i=0}^\infty
U_iV_i=\prod_{i=0}^\infty U_i=\prod_{i=0}^\infty V_i$ for some finite words $U_i$, $V_i$. Harju [@Harju] recently asked whether square-free self-shuffling words exist. We answer this question affirmatively. Besides that, we build an infinite word such that no word in its shift orbit closure is self-shuffling, answering positively a question from [@CKPZ].
**Keywords:** infinite words, shuffling, square-free words, shift orbit closure, self-shuffling words
author:
- |
Mike Müller[^1]\
Institut für Informatik\
Christian-Albrechts-Universit[ä]{}t zu Kiel\
Germany\
mimu@informatik.uni-kiel.de\
- |
Svetlana Puzynina\
LIP, ENS de Lyon\
Université de Lyon, France\
and Sobolev Institute of Mathematics\
Novosibirsk, Russia\
s.puzynina@gmail.com
- |
Michaël Rao\
LIP, CNRS, ENS de Lyon\
Université de Lyon, France\
michael.rao@ens-lyon.fr
date: |
\
Mathematics Subject Classifications: 68R15
title: '**Infinite square-free self-shuffling words**'
---
Introduction
============
A self-shuffling word, a notion which was recently introduced by Charlier et al. [@CKPZ], is an infinite word that can be reproduced by shuffling it with itself. More formally, given infinite words $x, y \in \Sigma^\omega$ over a finite alphabet $\Sigma,$ we define ${\mathscr{S}}(x,y)\subseteq \Sigma^\omega$ to be the collection of all infinite words $z$ for which there exists a factorization $$z=\prod_{i=0}^\infty U_i V_i$$ with each $U_i, V_i \in \Sigma^*$ and with $x=\prod_{i=0}^\infty
U_i$, $y=\prod_{i=0}^\infty V_i$. An infinite word $w\in
\Sigma^\omega$ is *self-shuffling* if $w\in{\mathscr{S}}(w,w)$. Various well-known words, e.g., the Thue-Morse word or the Fibonacci word, were shown to be self-shuffling.
Harju [@Harju] studied shuffles of both finite and infinite square-free words, i.e., words that have no factor of the form $uu$ for some non-empty factor $u$. More results on square-free shuffles were obtained independently by Harju and Müller [@HaMU13], and Currie and Saari [@CuSa14]. However, the question about the existence of an infinite square-free self-shuffling word, posed in [@Harju], remained open. We give a positive answer to this question in Sections 2 and 3.
The *shift orbit closure* $S_w$ of an infinite word $w$ can be defined, e.g., as the set of infinite words whose sets of factors are contained in the set of factors of $w$. In [@CKPZ] it has been proved that each word has a non-self-shuffling word in its shift orbit closure, and the following question has been asked: Does there exist a word for which no element of its shift orbit closure is self-shuffling (Question 7.2)? In Section 4 we provide a positive answer to the question. More generally, we show the existence of a word such that for any three words $x,y,z$ in its shift orbit closure, if $x$ is a shuffle of $y$ and $z$, then the three words are pairwise different. On the other hand, we show that for any infinite word there exist three different words $x,y,z$ in its shift orbit closure such that $x \in \mathscr{S} (y,z)$ (see Proposition \[prop\_xyz\]).
Apart from the usual concepts in combinatorics on words, which can be found for instance in the book of Lothaire [@Lothaire], we make use of the following notations: For every $k \geq 1$, we denote the alphabet $\{0, 1, \ldots, k-1\}$ by $\Sigma_k$. For a word $w = uvz$ we say that $u$ is a *prefix* of $w$, $v$ is a *factor* of $w$, and $z$ is a *suffix* of $w$. We denote these prefix- and suffix relations by $u {\leq_p}w$ and $v {\leq_s}w$, respectively. By $w[i,j]$ we denote the factor of $w$ starting at position $i$ and ending after position $j$. Note that we start numbering the positions with $0$.
A *prefix code* is a set of words with the property that none of its elements is a prefix of another element. Similarly, a *suffix code* is a set of words where no element is a suffix of another one. A *bifix code* is a set that is both a prefix code and a suffix code. A morphism $h$ is *square-free* if for all square-free words $w$, the image $h(w)$ is square-free.
A square-free self-shuffling word on four letters
=================================================
Let $g : \Sigma_4^* \rightarrow \Sigma_4^*$ be the morphism defined as follows: $$\begin{aligned}
g(0) &= 0121, \\
g(1) &= 032, \\
g(2) &= 013, \\
g(3) &= 0302.\end{aligned}$$
We will show that the fixed point $w = g^\omega(0)$ is square-free and self-shuffling in the following. Note that $g$ is not a square-free morphism, that is, it does not preserve square-freeness, as $g(23) = 0130302$ contains the square $3030$.
\[no\_3u1u3\] The word $w = g^\omega(0)$ contains no factor of the form $3u1u3$ for some $u \in \Sigma_4^*$.
We assume that there exists a factor of the form $3u1u3$ in $w$, for some word $u \in \Sigma_4^*$. From the definition of $g$, we observe that $u$ can not be empty. Furthermore, we see that every $3$ in $w$ is preceded by either $0$ or $1$. If $1 {\leq_s}u$, then we had an occurrence of the factor $11$ in $w$, which is not possible by the definition of $g$, hence $0 {\leq_s}u$. Now, every $3$ is followed by either $0$ or $2$ in $w$ and $01$ is followed by either $2$ or $3$. Since both $3u$ and $01u$ are factors of $w$, we must have $2 {\leq_p}u$. This means that the factor $012$ appears at the center of $u1u$, which can only be followed by $1$ in $w$, thus $21 {\leq_p}u$. However, this results in the factor $321$ as a prefix of $3u1u3$, which does not appear in $w$, as seen from the definition of $g$.
The word $w = g^\omega(0)$ is square-free.
We first observe that $\{g(0), g(1), g(2), g(3)\}$ is a bifix code. Furthermore, we can verify that there are no squares $uu$ with $|u| \leq 3$ in $w$. Let us assume now, that the square $uu$ appears in $w$ and that $u$ is the shortest word with this property. If $u = 02u'$, then $u' = u''03$ must hold, since $02$ appears only as a factor of $g(3)$, and thus $uu$ is a suffix of the factor $g(3)u''g(3)u''$ in $w$. As $w = g(w)$, also the shorter square $3g^{-1}(u'')3g^{-1}(u'')$ appears in $w$, a contradiction. The same desubstitution principle also leads to occurrences of shorter squares in $w$ if $u = xu'$ and $x \in \{01, 03, 10, 12, 13, 21, 30, 32\}$.
If $u = 2u'$ then either $03 {\leq_s}u$ or $030 {\leq_s}u$ or $01 {\leq_s}u$, by the definition of $g$. In the last case, that is when $01 {\leq_s}u$, we must have $21 {\leq_p}u$, which is covered by the previous paragraph. If $u' = u''030$, then $uu$ is followed by $2$ in $w$ and we can desubstitute to obtain the shorter square $g^{-1}(u'')3g^{-1}(u'')3$ in $w$. If $u = 2u'$ and $u' = u''03$, and $uu$ is preceded by $03$ or followed by $2$ in $w$, we can desubstitute to $1g^{-1}(u'')1g^{-1}(u'')$ or $g^{-1}(u'')1g^{-1}(u'')1$, respectively. Therefore, assume that $u = 2u''03$ and $uu$ is preceded by $030$ and followed by $02$ in $w$. This however means that we can desubstitute to get an occurrence of the factor $3g^{-1}(u'')1g^{-1}(u'')3$ in $w$, a contradiction to Lemma \[no\_3u1u3\].
We now show that $w = g^\omega(0)$ can be written as $w=\prod_{i=0}^\infty U_iV_i=\prod_{i=0}^\infty
U_i=\prod_{i=0}^\infty V_i$ with $U_i,V_i \in \Sigma_4^*$.
The word $w = g^\omega(0)$ is self-shuffling.
In what follows we use the notation $x=v^{-1}u$ meaning that $u=v
x$ for finite words $x, u, v$. We are going to show that the self-shuffle is given by the following:
$$\begin{aligned}
& U_0=g^2(0),& &\hskip-9pt U_1=0,& &\hskip-10pt \dots,\hskip-8pt & &U_{6i+2}=g^i(0^{-1}g(0)0),&
&\hskip-6pt U_{6i+3}=g^{i}(0^{-1}g(3)0),\\& & & & & &
&U_{6i+4}=g^{i}(0^{-1}g(201)0),& & \hskip-6pt U_{6i+5}=g^{i}(30),
\\& & & & & &
&U_{6i+6}=g^{i}(2g(03)),& &
\hskip-6pt U_{6i+7}=g^{i+1}(20), \\
& V_0=g(0)03,& &\hskip-9pt V_1=2g(2)0,& &\hskip-10pt \dots,
\hskip-8pt & &V_{6i+2}=g^{i}(0^{-1}g(1)0),& & \hskip-6pt
V_{6i+3}=g^{i}(0^{-1}g(03)0),\\& & & & & & &V_{6i+4}=g^{i}(1),& &
\hskip-6pt V_{6i+5}=g^{i}(3),\\& & & & & & &V_{6i+6}=g^{i+1}(0),&
& \hskip-6pt V_{6i+7}=g^{i+1}(0^{-1}g(2)0).
\end{aligned}$$
Now we verify that $$w=\prod_{i=0}^\infty U_i V_i
=\prod_{i=0}^\infty U_i =\prod_{i=0}^\infty V_i,$$ from which it follows that $w$ is self-shuffling. It suffices to show that each of the above products is fixed by $g$. Indeed, straightforward computations show that
$$\prod_{i=0}^\infty U_i = g^2(0) g^2(121) g^3(121) \cdots \, ,$$
which is fixed by $g$:
$$\begin{aligned}
g\left(\prod_{i=0}^\infty U_i\right) &= g\left(g^2(0) g^2(121) g^3(121) \cdots \right) = g^3(0) g^3(121) g^4(121) \cdots \\
&= g^2(0121) g^3(121) g^4 (121) \cdots = g^2(0) g^2(121) g^3(121) \cdots =\prod_{i=0}^\infty U_i, \end{aligned}$$
hence $\prod_{i=0}^\infty U_i$ is fixed by $g$ and thus $w=\prod_{i=0}^\infty U_i$. In a similar way we show that $w=\prod_{i=0}^\infty V_i=\prod_{i=0}^\infty U_i V_i$.
Square-free self-shuffling words on three letters
=================================================
We remark that we can immediately produce a square-free self-shuffling word over $\Sigma_3$ from $g^\omega(0)$: Charlier et al. [@CKPZ] noticed that the property of being self-shuffling is preserved by the application of a morphism. Furthermore, Brandenburg [@Brandenburg] showed that the morphism $f : \Sigma_4^* \rightarrow \Sigma_3^*$, defined by $$\begin{aligned}
f(0) &= 010201202101210212, \\
f(1) &= 010201202102010212, \\
f(2) &= 010201202120121012, \\
f(3) &= 010201210201021012,\end{aligned}$$ is square-free. Therefore, the word $f(g^\omega(0))$ is a ternary square-free self-shuffling word, from which we can produce a multitude of others by applying square-free morphisms from $\Sigma_3^*$ to $\Sigma_3^*$.
A word with non self-shuffling shift orbit closure
==================================================
In this section we provide a positive answer to the question from [@CKPZ] whether there exists a word for which no element of its shift orbit closure is self-shuffling.
The *Hall word* $\mathcal{H}=012021012102\cdots$ is defined as the fixed point of the morphism $h(0) =012, h(1)= 02, h(2) =1$. Sometimes it is referred to as a *ternary Thue-Morse* word. It is well known that this word is square-free. We show that no word in the shift orbit closure $S_\mathcal{H}$ of the Hall word is self-shuffling. More generally, we show that if $x$ is a shuffle of $y$ and $z$ for $x,y,z\in S_\mathcal{H}$, then they are pairwise different.
\[prop\_xyy\] There are no words $x, y$ in the shift orbit closure of the Hall word such that $x \in \mathscr{S}
(y,y)$.
Suppose the converse, i.e., there exist words $x, y\in
S_\mathcal{H}$ such that
$$x = \prod_{i=0}^{\infty} U_i V_i, \qquad y= \prod_{i=0}^{\infty} U_i = \prod_{i=0}^{\infty} V_i.$$
Define the set $X$ of infinite words as follows:
$$X=\{x \in S_\mathcal{H} \, \mid \, x \in \mathscr{S} (y,y) \mbox{ for some } y\in S_\mathcal{H} \}.$$ In other words, $X$ consists of words in $S_\mathcal{H}$ which can be introduced as a shuffle of some word $y$ in $S_\mathcal{H}$ with itself. Now suppose, for the sake of contradiction, that $X$ is non empty, and consider $x\in X$ with the first block $U_0$ of the smallest possible positive length. We remark that such $x$ and corresponding $y$ are not necessarily unique.
We can suppose without loss of generality that $y$ starts with $0$ or $10$. Otherwise, we exchange $0$ and $2$, consider the morphism $0 \mapsto 1, 1 \mapsto 20, 2\mapsto 210$, and the argument is symmetric.
It is not hard to see from the properties of the morphism $h$ that removing every occurrence of $1$ from $y$ results in $(02)^\omega$. Hence the blocks in the factorizations of $y$ after removal of $1$ are of the form $(02)^i$ for some integer $i$. Thus the first letter of each block $U_i$ and $V_i$ that is different from $1$ is $0$, and the last letter different from $1$ is $2$.
Then, $U_i$ and $V_i$ are images by the morphism $h$ of factors of the fixed point of $h$. Therefore, there are words $x', y'\in
S_{\mathcal{H}}$ such that $x=h(x'), y=h(y'), U_i=h(U'_i),
V_i=h(V'_i),$ and $x'= \prod_{i=0}^\infty U'_i V'_i$, $y' = \prod_{i=0}^\infty U'_i =
\prod_{i=0}^\infty V'_i $.
Notice that the first block $U_0$ cannot be equal to $1$. Indeed, otherwise $x$ starts with $11$, which is impossible, since $11$ is not a factor of the fixed point of $h$.
Clearly, taking the preimage decreases the lengths of blocks in the factorization (except for those equal to $1$), and since $U_0\neq 1$, the length of the first block in the preimage is smaller, i.e., $|U_0'|<|U_0|$. This is a contradiction with the minimality of $|U_0|$.
There are no self-shuffling words in the shift orbit closure of $\mathcal{H}$.
With a similar argument we can prove the following:
\[prop\_xxy\] There are no words $x, y$ in the shift orbit closure of $\mathcal{H}$ such that $x \in \mathscr{S} (x,y)$.
First we introduce a notation $x\in \mathscr{S}_2
(y,z)$, meaning that there exists a shuffle starting with the word $z$ (i.e., $U_0=\varepsilon$, $V_0 \neq \varepsilon$). Next, $x \in
\mathscr{S} (x,y)$ implies that there exists $z$ in the same shift orbit closure such that $z \in \mathscr{S}_2 (z,y)$. Indeed, one can remove the prefix $U_0$ of $x$ to get $z$, i.e., $z=(U_0)^{-1}x$, and keep all the other blocks $U_i$, $V_i$ in the shuffle product.
Define the set $Z$ of infinite words as follows:
$$Z=\{z \in S_{\mathcal{H}} \, \mid \, z \in \mathscr{S}_2 (z,y) \mbox{ for some } y\in S_\mathcal{H} \}.$$ In other words, $Z$ consists of words in $S_\mathcal{H}$ which can be introduced as a shuffle of some word $y$ in $S_\mathcal{H}$ with $z$ starting with the block $V_0$. Now consider $z\in Z$ with the first block $V_0$ of the smallest possible length. We remark that such $z$ and a corresponding $y$ are not necessarily unique.
As in the proof of Proposition \[prop\_xyy\], the shuffle cannot start with a block of length $1$. Again, if we remove every occurrence of $1$ in $y$ (and in $z$), we get $(02)^\omega$ or $(20)^\omega$; moreover, since $V_0$ contains letters different from $1$, the first letter different from $1$ is the same in $y$ and $z$. So, without loss of generality we assume that both $y$ and $z$ without $1$ are $(02)^\omega$, and the blocks $U_i$ and $V_i$ without $1$ are integer powers of $02$. Then, $U_i$ and $V_i$ are images by the morphism $h$ of factors of $\mathcal{H}$. Therefore, there are words $z', y'\in
S_\mathcal{H}$ such that $z=h(z'), y=h(y'), U_i=h(U'_i),
V_i=h(V'_i),$ and $z'= \prod_{i=0}^\infty (U'_i V'_i) = \prod_{i=0}^\infty V'_i$, $y' = \prod_{i=0}^\infty
U'_i $ (i.e., $z'\in Z$).
As in the proof of Proposition \[prop\_xyy\], since $V_0\neq 1$, the length of the first block in the preimage is smaller, i.e., $|V_0'|<|V_0|$. This is again a contradiction with the minimality of $|V_0|$.
So, we proved that if there are three words $x, y, z$ in the shift orbit closure of the fixed point of $h$ such that $x \in
\mathscr{S} (y,z)$, then they should be pairwise distinct. Now we are going to prove that for any infinite word there exist three different words in its shift orbit closure such that $x \in \mathscr{S} (y,z)$.
An infinite word $x$ is called *recurrent*, if each its prefix occurs infinitely many times in it.
\[prop\_xyz\] Let $x$ be a recurrent infinite word. Then there exist two words $y, z$ in the shift orbit closure of $x$ such that $x \in \mathscr{S}
(y,z)$.
We build the shuffle inductively.
Start from any prefix $U_0$ of $x$. Since $x$ is recurrent, each of its prefixes occurs infinitely many times in it. Find another occurrence of $U_0$ in $x$ and denote its position by $i_1$. Put $V_0= x[|U_0|, i_1+|U_0|-1]$.
At step $k$, suppose that the shuffle of the prefix of $x$ is built: $$x[0, \Sigma_{l=0}^{k-1}(|U_l|+|V_l|)-1]=\prod_{l=0}^{k-1} U_l V_l, \quad
y[0, \Sigma_{l=0}^{k-1}|U_l|-1]=\prod_{l=0}^{k-1} U_l, \quad z[0,
\Sigma_{l=0}^{k-1} |V_l|-1]=\prod_{l=0}^{k-1} V_l,$$ such that $\prod_{l=0}^{k-1} U_l$ is the suffix of $x[0,
\Sigma_{l=0}^{k-1}(|U_l|+|V_l|)-1]=\prod_{i=0}^{k-1} U_l V_l$ starting at position $i_{k-1}$, and $\prod_{l=0}^{k-1} V_l$ is the suffix of $x[0,
\Sigma_{l=0}^{k-1}(|U_l|+|V_l|)-1]=\prod_{i=0}^{k-1} U_l V_l$ starting at position $j_{k-1}$.
Find another occurrence of $\prod_{l=0}^{k-1} V_l$ in $x$ at some position $j_k > j_{k-1}$. We can do it since $x$ is recurrent. Put $U_k= x[\Sigma_{l=0}^{k-1}(|U_l|+|V_l|), j_k-1+\Sigma_{l=0}^{k-1}|V_l|]$. We note that $\prod_{l=0}^{k} U_l$ is a factor of $x$ by the construction; more precisely, it occurs at position $i_{k-1}$.
Find an occurrence of $\prod_{l=0}^{k} U_l$ at some position $i_k>i_{k-1}$, put $V_k= x[\Sigma_{l=0}^{k-1}(|U_l|+|V_l|) +
|U_k|, i_k-1+\Sigma_{l=0}^{k}|U_l|]$. As above, $\prod_{l=0}^{k} V_l$ is a factor of $x$ by the construction since it occurs at position $j_{k-1}$. Moreover, both $\prod_{l=0}^{k} U_l$ and $\prod_{l=0}^{k} V_l$ are suffixes of $x[0,
\Sigma_{l=0}^{k}(|U_l|+|V_l|)-1]=\prod_{i=0}^{k} U_l V_l$.
Continuing this line of reasoning, we build the required factorization.
Since each infinite word contains a recurrent (actually, even a uniformly recurrent) word in its shift orbit closure, we obtain the following corollary:
\[col:xyz\] Each infinite word $w$ contains words $x, y, z$ in its shift orbit closure such that $x \in \mathscr{S} (y,z)$.
The following example shows that the recurrence condition in Proposition \[prop\_xyz\] cannot be omitted:
Consider the word $3\mathcal{H} = 3012021\cdots$ which is obtained from $\mathcal{H}$ by adding a letter $3$ in the beginning. Then the shift orbit closure of $3\mathcal{H}$ consists of the shift orbit closure of $\mathcal{H}$ and the word $3\mathcal{H}$ itself. Assuming $3\mathcal{H}$ is a shuffle of two words in its shift orbit closure, one of them is $3\mathcal{H}$ (there are no other $3$’s) and the other one is something in the shift orbit closure of $\mathcal{H}$, we let $y$ denote this other word. Clearly, the shuffle starts with $3\mathcal{H}$, and cutting the first letter $3$, we get $\mathcal{H}\in
\mathscr{S}(\mathcal{H},y)$, a contradiction with Proposition \[prop\_xxy\].
There also exist examples where each letter occurs infinitely many times:
The following word: $$x=012001120001112\cdots0^k1^k2\cdots$$ does not have two words $y,z$ in its shift orbit closure such that $x \in
\mathscr{S} (y,z)$. The idea of the proof is that the shift orbit closure consists of words of the following form: $1^*20^{\omega}$, $0^*1^{\omega}$, $x$ itself and all their right shifts. Shuffling any two words of those types, it is not hard to see that there exists a prefix of the shuffle which contains too many or too few occurrences of some letter compare to the prefix of $x$. We leave the details of the proof to the reader.
By Corollary \[col:xyz\], there are $x,y,z$ in the shift orbit closure of $\mathcal{H}$ such that $x\in \mathscr{S}(y,z)$. To conclude this section, we give an explicit construction of two words in the shift orbit closure of $\mathcal{H}$ which can be shuffled to give $\mathcal{H}$. We remark though that this construction gives a shuffle different from the one given by Corollary \[col:xyz\]. Let: $$h:\left\{
\begin{array}{lll}
0&\mapsto&012\\
1&\mapsto&02\\
2&\mapsto&1\\
\end{array}\right. \text{ and ~ }
h':\left\{
\begin{array}{lll}
0&\mapsto&210\\
1&\mapsto&20\\
2&\mapsto&1.\\
\end{array}\right.$$
By definition, the shift orbit closure of the Hall word is closed under $h$. Moreover this shift orbit closure is also closed under $h'$, since the factors of the Hall word are closed under the morphism $0\to 2, 1\to 1, 2\to 0$. $$h'\circ h:\left\{
\begin{array}{lll}
0&\mapsto&210201\\
1&\mapsto&2101\\
2&\mapsto&20\\
\end{array}\right.
h\circ h':\left\{
\begin{array}{lll}
0&\mapsto&102012\\
1&\mapsto&1012\\
2&\mapsto&02\\
\end{array}\right.
h^2:\left\{
\begin{array}{lll}
0&\mapsto&012021\\
1&\mapsto&0121\\
2&\mapsto&02\\
\end{array}\right.
h'^2:\left\{
\begin{array}{lll}
0&\mapsto&120210\\
1&\mapsto&1210\\
2&\mapsto&20.\\
\end{array}\right.$$ Note that if $w$ is an infinite word, then $2 (h\circ
h')(w) = (h'\circ h)(w)$ and $0h'^2(w)=h^2(w)$.
$h^\omega(0) \in \mathscr{S}
(h^2((h'^2)^\omega(1)),h'^3(h^\omega(0)))$.
Let $$U_0=01, \, U_1=h'(0), \, U_2=h'(1), \, V_0=h'(1),$$ and for every $i\ge 0$, $$U_{i+3} = h'^2(h^i(1)) \text{ and } V_{i+1}=h'^2(h^i(1)).$$ Let furthermore $$u=\prod_{i=0}^\infty U_i, \, v=\prod_{i=0}^\infty V_i, \text{ and } w=\prod_{i=0}^\infty U_i V_i \, .$$ We show that $w= h^\omega(0)$, $u=h^2((h'^2)^\omega(1))$ and $v=h'^3(h^\omega(0))$.
Note that $2h(h'(h^\omega(0)))= h'(h^\omega(0))$, thus $h'(h^\omega(0))=\prodi h^i(2)$. Then we have $$v=20
\prod_{i=0}^\infty h'^2(h^i(1)) = h'^2 \left (\prod_{i=0}^\infty
h^i(2)\right )= h'^3(h^\omega(0)).$$ Moreover, $$\begin{aligned}
u &= 01 210 20 \prod_{i=0}^\infty h'^2(h^i(1))= 01210h'^2\left (\prodi h^i(2)\right ) \\
&= 01210h'^3\left(h^\omega(0)\right)= 01h'(0h'^2(h^\omega(0)))=
01h'(h^\omega(0))=0 h'(2h^\omega(0)).\end{aligned}$$ Since $h'^2(2h^\omega(0)) = 20 h'^2(h^\omega(0))= 2 h^\omega(0)$, the word $2h^\omega(0)$ is the fixed point $(h'^2)^\omega(2)$ of $h'^2$, and then $h'(2h^\omega(0))$ is the fixed point $(h'^2)^\omega(1)$. Thus $u = 0 (h'^2)^\omega(1) =
h^2((h'^2)^\omega(1))$. Finally: $$w=0120210121020\prod_{i=0}^\infty h'^2(h^i(021))
=012 021 h(021) h^2\left (\prod_{i=0}^\infty h^i(021) \right)
= 012 \prod_{i=0}^\infty h^i(021).$$
Applying the morphism $h$ to the second expression for $w$, we get $$h(w)= 012021 h\left (\prod_{i=0}^\infty h^i(021) \right ) = 012
\prod_{i=0}^\infty h^i(021).$$
Thus $w= h^\omega(0)$ since $h$ is injective.
Conclusion and open question
============================
We showed that infinite square-free self-shuffling words exist. The natural question that arises now is whether we can find infinite self-shuffling words subject to even stronger avoidability constraints: For this we recall the notion of *repetition threshold* $RT(k)$, which is defined as the least real number such that an infinite word over $\Sigma_k$ exists, that does not contain repetitions of exponent greater than $RT(k)$. Due to the collective effort of many researchers (see [@CurrieR11; @Rao11] and references therein), the repetition threshold for all alphabet sizes is known and characterized as follows: $$RT(k) =
\begin{cases}
\frac{7}{4} & \text{if } k=3 \\
\frac{7}{5} & \text{if } k=4 \\
\frac{k}{k-1} & \text{else}.
\end{cases}$$ A word $w \in \Sigma_k^\omega$ without factors of exponent greater than $RT(k)$ is called a *Dejean word*. Charlier et al. showed that the Thue-Morse word, which is a binary Dejean word, is self-shuffling [@CKPZ].
Do there exist self-shuffling Dejean words over non-binary alphabets?
[10]{} F.-J. Brandenburg. Uniformly Growing $k$th Power-Free Homomorphisms. *Theor. Comput. Sci.* 23: 69–82, 1983.
E. Charlier, T. Kamae, S. Puzynina, L. Q. Zamboni. Infinite self-shuffling words. *J. Comb. Theory, Ser. A,* 128: 1–40, 2014.
J. Currie, N. Rampersad. A proof of Dejean’s conjecture. *Math. Comput.* 80(274): 1063–1070, 2011.
J. Currie, K. Saari. Square-free Words with Square-free Self-shuffles. *Electr. J. Comb.,* 21(1): P1.9 (2014).
T. Harju. A Note on Square-Free Shuffles of Words. *LNCS* 8079 (WORDS 2013): 154–160.
T. Harju, M. Müller. Square-Free Shuffles of Words. CoRR abs/1309.2137 (2013)
M. Lothaire. *Algebraic combinatorics on words.* Cambridge University Press, 2002.
M. Rao. Last cases of Dejean’s conjecture. *Theor. Comput. Sci.* 412(27): 3010–3018, 2011.
[^1]: Supported in part by the DFG under grant 582014.
|
---
abstract: 'It is known that quasiperiodic hypermeander of spiral waves almost certainly produces a bounded trajectory for the spiral tip. We analyse the size of this trajectory. We show that this deterministic question does not have a physically sensible deterministic answer and requires probabilistic treatment. In probabilistic terms, the size of the hypermeander trajectory proves to have an infinite expectation, despite being finite with probability one. This can be viewed as a physical manifestation of the classical “St. Petersburg paradox” from probability theory and economics.'
author:
- 'V. N. Biktashev'
- 'I. Melbourne'
date:
title: 'St. Petersburg paradox for quasiperiodically hypermeandering spiral waves'
---
Rotating spiral waves are a class of self-organized patterns observed in a large variety of spatially extended thermodynamically nonequilibrium systems with oscillatory or excitable local dynamics, of physical, chemical or biological nature [@Zhabotinsky-Zaikin-1971; @Allessie-etal-1973; @Alcantara-Monk-1974; @Carey-etal-1978; @Gorelova-Bures-1983; @Murray-etal-1986; @Schulman-Seiden-1986; @Madore-Freedman-1987; @Jakubith-etal-1990; @Lechleiter-etal-1991; @Frisch-etal-1994; @Yu-etal-1999; @Agladze-Steinbock-2000; @Kastberger-etal-2008]. Of particular practical importance are spiral waves of electrical excitation in the heart muscle, where they underlie dangerous arrhythmias [@Alonso-etal-2016-PR]. Very soon after their experimental discovery in Belousov-Zhabotinsky reaction, it was noticed that rotation of spiral waves is not necessarily steady, but their tip can describe a complicated trajectory, “meander” [@Winfree-1973]. Subsequent mathematical modelling allowed a more detailed classification of possible types of rotation of spiral waves in ideal conditions: steady rotation like a rigid body, when the tip of the spiral travels along a perfect circle; meander, when the solution is two-periodic and the tip traces a trajectory resembling a roulette (hypocycloid or epicyloid) trajectory; and more complicated patterns, dubbed “hypermeander”[@Roessler-Kahlert-1979; @Zykov-1986; @Winfree-1991]. Often different types of meander may be observed in the same model at different values of parameters [@Winfree-1991], including cardiac excitation models (see [fig. \[patterns\]]{}). The question of the spatial extent of the spiral tip path can be of practical importance. Here we discuss this question for quasiperiodic hypermeander.
![ Snapshots of anticlockwise rotating spiral waves of electrical excitation, together with traces of their tips, in a reaction-diffusion model of guinea pig ventricular tissue, (a) classical meander in a model with standard parameters [@Biktashev-Holden-1996], (b) hypermeander in the same model with parameters changed to represent Long QT syndrome [@Biktashev-Holden-1998a]. []{data-label="patterns"}](fig01.pdf)
#### The equations of motion of the meandering spiral tip
may be derived by the standard procedure of rewriting the underlying partial differential equations as a skew product [@Barkley-1994; @Fiedler-etal-1996; @Biktashev-etal-1996; @Sandstede-etal-1997; @Biktashev-Holden-1998; @Golubitsky-etal-2000; @Nicol-etal-2001; @Ashwin-etal-2001; @Roberts-etal-2002; @Beyn-Thummler-2004; @Foulkes-Biktashev-2010; @Hermann-Gottwald-2010; @Gottwald-Melbourne-2013]. Consider the $\l$-component reaction-diffusion system on the plane, $$\partial_\t \u = \D\nabla^2\u+\f(\u),
\quad \u(\r,\t)\in{\mathbb{R}}^\l,
\quad \r\in{\mathbb{R}}^2,$$ as a flow in the phase space which is an infinite-dimensional space of functions ${\mathbb{R}}^2\to{\mathbb{R}}^\l$. The symmetry group is the Euclidean group $\G$ of transformations of the plane $\g:{\mathbb{R}}^2\to{\mathbb{R}}^2$ acting on ${\mathbb{R}}^2$ by translations and rotations and thereby acting on functions $\u:{\mathbb{R}}^2\to{\mathbb{R}}^\l$ by $\u(\r)\mapsto \u(\g^{-1}\r)$.
Such systems with symmetry, or “equivariant dynamical systems” can be cast into a *skew product* form $$\dot \X = \fx(\X), \qquad \dot \g = \g\gx(\X),$$ on $\Xspace\times \G$, where the dynamics on the symmetry group $\G$ is driven by the “shape dynamics” on a cross-section $\Xspace$ transverse to the group directions. Here, $\g\gx(\X)$ denotes the action of the group element $\g\in \G$ on vectors $\gx(\X)$ lying in the Lie algebra of $\G$; $\fx$ and $\gx$ are defined by components of the vector field along $\Xspace$ and orbits of $\G$ respectively.
The shape dynamics $\dot \X=\fx(\X)$ on the cross-section $\Xspace$ is a dynamical system devoid of symmetries. Substituting the solution $\X(\t)$ for the shape dynamics into the $\dot \g$ equation yields the nonautonomous finite-dimensional equation $\dot \g=\g\gx(\X(\t))$ to be solved for the group dynamics.
For the Euclidean group $\G$ consisting of planar translations $\tipp$ and rotations $\tipangle$, the equations become $$\label{tipt}
\dot \X = \fx(\X), \qquad
\dot\tipangle = \Tiprot(\X) , \qquad
\dot\tipp = \Tipv(\X) \; e^{i\tipangle} .$$ The variables $\tipp$ and $\tipangle$ can be interpreted as position and orientation of the tip of the spiral, then $\X(\t)$ describes the evolution in the frame comoving with the tip [@Biktashev-etal-1996; @Foulkes-Biktashev-2010]. Standard low-dimensional attractors in $\Xspace$ produce the classical tip meandering patterns through the $\dot\tipp$ equation, namely an equilibrium produces stationary rotation, a limit cycle produces the two-frequency flower-pattern meander, and more complicated attractors produce “hypermeander”. Hypermeander produced by chaotic base dynamics is asymptotically a deterministic Brownian motion [@Biktashev-Holden-1998; @Nicol-etal-2001]. Quasiperiodic base dynamics produce another kind of hypermeander, with tip trajectories almost certainly bounded, but exhibiting unlimited directed motion at a dense set of parameter values [@Nicol-etal-2001]. Similar dynamics may be observed when a spiral with two-periodic meander is subject to periodic external forcing [@Mantel-Barkley-1996].
Our aim is to characterize the size of a quasiperiodic meandering trajectory when it is finite.
#### The mathematical problem.
We assume $\m$-frequency quasiperiodic dynamics in the base system, $\m\ge2$, with $\X=\phase\in{\mathbb{T}}^\m=({\mathbb{R}}/2\pi{\mathbb{Z}})^\m$ being coordinates on the invariant $\m$-torus, so that the shape dynamics $\dot\X=\fx(\X)$ becomes $$\begin{aligned}
\dot \phase = \freq , \label{qp}\end{aligned}$$ where $\freq\in{\mathbb{R}}^\m$ is a set of irrationally related frequencies [^1]. The $\dot \g$ equations become $$\dot\tipangle = \Tiprot(\phase) , \qquad
\dot\tipp = \Tipv(\phase) \; e^{i\tipangle} .
\label{extension}$$ Equations (\[qp\],\[extension\]) comprise a closed system describing the trajectory of the quasiperiodic meandering spiral tip.
#### The ${\mathbb{R}}^1$-extension of the quasiperiodic dynamics.
First we illustrate our main idea for the simpler case where the orientation angle $\tipangle$ is absent and the position $\tipp$ is one-dimensional. The shape dynamics remains as in (\[qp\]) with $\phase\in{\mathbb{T}}^\mq$, $\mq\ge2$. Then a point with coordinate $\tipp\in{\mathbb{R}}^1$ moves according to $$\dot\tipp = \veloc(\phase)
= \sum\limits_{\n\in{\mathbb{Z}}^\mq} \veloc_\n e^{i \n\cdot\phase }, \qquad
\dot\phase = \freq . \label{xeqn}$$ Termwise integration gives $$\tipp(\t) = \tipp(0)+\veloc_0 \t + {\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^\mq} \frac{-i\veloc_\n}{\n\cdot\freq} \left(e^{i\n\cdot\freq\t}-1\right) ,$$ where the prime denotes summation over $\n\ne0$. Consider the infinite sum here, defining the deviation of $\tipp$ from steady motion, $\devn_\t(\freq)=\tipp(\t)-\tipp(0)-\veloc_0 \t$. For an arbitrarily chosen $\freq$, its components are almost certainly incommensurate, and, moreover, Diophantine. So the denominators in the infinite sum are nonzero, but many of them are very small; nevertheless they decay slowly with ${\left\lVert\n\right\rVert}=\left(\n_1^2+\dots+\n_\mq^2\right)^{1/2}$. This is compensated by the fact that if the function $\veloc(\phase)$ is sufficiently smooth, its Fourier coefficients $\veloc_\n$ in the numerators quickly decay with ${\left\lVert\n\right\rVert}$. As a result, the infinite sum remains bounded for $\t\ge0$, for $\veloc(\phase)$ sufficiently smooth and almost all $\freq$ [@Nicol-etal-2001].
So if we consider the trajectories in the frame moving with the velocity $\veloc_0$, we know they are typically confined to a finite space. Now we ask how large they can be. The size of a finite piece of trajectory may be measured in various ways, say by the departure from the initial point $\devn_\t(\freq)=\tipp(\t)-\tipp(0)$, its time average, $\meansizeT{\T}(\freq)=\T^{-1}\int_0^\T\devn_\t(\freq)\,{\mathrm{d}}\t$, and the corresponding variance, $\mssizesqT{\T}(\freq)=\T^{-1}\int_0^\T{\left\lvert \devn_\t(\freq)-\meansizeT{\T}(\freq) \right\rvert}^2\,{\mathrm{d}}\t$. For instance, as $\T\to\infty$ we obtain $$\mssizesq(\freq) = {\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^\mq}
\frac{{\left\lvert \veloc_{\n} \right\rvert}^2}{(\n\cdot\freq)^2} . \label{Delta-val}$$ By the above arguments, for almost any vector $\freq$, this expression is finite. However, as typically all $\veloc_\n$ are nonzero, expression [(\[Delta-val\])]{} is infinite for all $\freq$ for which the denominator is zero, and for $\mq\ge2$, this is a dense set. That is, the function $\mssize(\freq)$ is almost everywhere defined and finite, but is everywhere discontinuous. The latter property implies that for any physical purpose, questions about the value of the function at a particular point are meaningless, as any uncertainty in the arguments, no matter how small, causes a non-small, in fact infinite, uncertainty in the value of the function.
Hence, a deterministic view on the function $\mssize(\freq)$ is inadequate, and we are forced to adopt a probabilistic view. Suppose we know $\freq$ approximately, say, its probability density is uniformly distributed in $\Ball=\Ball_{\dfreqq}(\freq_0)$, a ball of radius $\dfreqq$ centered at $\freq_0$ [^2]. The expectation of the trajectory size is then $${\mathrm{E}\left[\mssize\right]}
= \frac{1}{\mesBall} \int\limits_{\Ball} \mssize(\freq) \;{\mathrm{d}}\freq
= \frac{1}{\mesBall} \int\limits_{\Ball} \left(
{\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^\mq} \frac{{\left\lvert \veloc_{\n} \right\rvert}^2}{(\n\cdot\freq)^2}
\right)^{1/2} {\mathrm{d}}\freq , $$ where $\mesBall={\mathop{\mathrm{Vol}}_{\mq}(\Ball)}$. The set of hyperplanes $\n\cdot\freq=0$, $\n\in{\mathbb{Z}}^\mq$ is dense so there is an infinite set of $\n\in{\mathbb{Z}}^\mq$ whose hyperplanes $\n\cdot\freq=0$ cut through $\Ball$. For any such $\n$, we have $${\mathrm{E}\left[\mssize\right]}
\geq \frac{1}{\mesBall} \int\limits_{\Ball}
{\left\lvert \frac{\veloc_{\n}}{\n\cdot\freq} \right\rvert} {\mathrm{d}}\freq.$$ Then, for some $\A,\eps>0$ depending on $\n$, we have $$\int_{\Ball} \frac{{\mathrm{d}}\freq}{|\n\cdot\freq|}>\A\int_{-\eps}^{\eps}\,\frac{{\mathrm{d}}{\z}}{|\z|}=+\infty.$$ Typically, ${\left\lvert \veloc_{\n} \right\rvert}>0$ for all such $\n$, therefore we have ${\mathrm{E}\left[\mssize\right]}=+\infty$.
That is, the deviation from steady motion is almost certainly finite, but its average expected value is infinite.
#### The quasiperiodic hypermeander trajectories.
We now return to the equations (\[qp\],\[extension\]) governing quasiperiodic hypermeander. Consider first the $\dot\phase$, $\dot\tipangle$ subsystem $$\dot\phase = \freq , \qquad
\dot\tipangle = \Tiprot(\phase) . \label{rot}$$ This has the form of [(\[xeqn\])]{} with $\mq=\m$, $\tipp=\tipangle$, $\veloc=\Tiprot$. Proceeding as for ${\mathbb{R}}^1$-extensions, we obtain $\tipangle=\tipangle_0+\Tiprot_0 \t + \Tipangle(\phase)$, where $\Tipangle(\phase) = - i {\sideset{}{'}\sum}_{\n\in{\mathbb{Z}}^\m} \Tiprot_\n
\left(e^{i\n\cdot\phase}-1\right)/\n\cdot\freq$. Substituting into the $\dot\tipp$ equation, we obtain $$\dot\tipp=\Tipv(\phase)e^{i\tipangle_0+\Tipangle(\phase)}e^{i\Tiprot_0 \t}
=\Tipv(\phase)e^{i(\tipangle_0+\Tipangle(\phase)+\phase_{\m+1})},$$ where $\phase_{\m+1}\in{\mathbb{T}}^1$ satisfies the equation $\dot\phase_{\m+1}=\Tiprot_0$. Hence the evolution of $\dot\tipp$ is governed by the skew product equations $$\dot\phasex = \freqx , \qquad \dot\tipp = \vvv(\phasex) , \label{transl}$$ where $\freqx=(\freq, \Tiprot_0)\in{\mathbb{R}}^{\m+1}$, $\phasex=(\phase,\phasen)\in{\mathbb{T}}^{\m+1}$ and $$\vvv(\phasex)=\Tipv(\phase) \, e^{i(\tipangle_0+\Tipangle(\phase)+\phasen)}.
\label{translv}$$ System [(\[transl\])]{} has a similar form to [(\[xeqn\])]{} (separately for the real and imaginary parts of $\tipp$), except that now $\mq=\m+1$, $\phase\in{\mathbb{T}}^\m$ and $\phasex\in{\mathbb{T}}^{\m+1}$ Also, we notice that due to [(\[translv\])]{}, Fourier components $\vvv_\n$ are nonzero only for $\n_{\m+1}=\pm1$, which implies that $\vvv_0=0$. Physically speaking, due to the rotation of the meandering tip, its average spatial velocity is always zero. Hence, the function $\mssize(\freqx)$ in this case is just the size of the trajectory, defined as the root mean square of the distance of the tip from the centroid of the trajectory.
Based on the results of the previous paragraph, we conclude from here our main result: for hypermeandering spirals, the long-term average of the displacement of the tip from its centroid is a random quantity, which takes finite values with probability one, but has an infinite expectation. This result is proved rigorously in [@Melbourne-Biktashev-2018]. The rest of our results below are at the physical level of rigour.
#### The asymptotic distribution of the trajectory size
is fairly generic for typical systems. Consider when the trajectory size $$\begin{aligned}
&&
\mssize(\freqx)
= \left(
{\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^{\m+1}} \frac{{\left\lvert \Tipv_{\n}(\freqx) \right\rvert}^2}{(\n\cdot\freqx)^2}
\right)^{1/2}
\label{int-of-hyp-x}\end{aligned}$$ is large. This requires that at least one of the terms in the infinite sum is large. It is most likely that the largest term by far exceeds all the others. So, the tail of the distribution of $\mssize$ can be understood via the distribution of individual terms $\termS_\n(\freqx)={\left\lvert \Tipv_{\n}(\freqx) \right\rvert}^2/(\n\cdot\freqx)^2$. Clearly, ${\mathbb{P}\left[\termS_\n>\x^2\right]} \propto \x^{-1}$ as $\x\to+\infty$ as long as $\{\n\cdot\freqx=0\}\cap\Ball\ne\emptyset$, and the distribution of $\mssize$ corresponds to the distribution of the square root of the largest of such terms. Hence, for a typical continuous distribution of $\freqx$, we expect $$\label{distrib-law}
\F(\x) \equiv
{\mathbb{P}\left[\mssize>\x\right]}\propto \x^{-1}, \quad\textrm{as}\quad \x\to+\infty.$$
#### Growth rate of the trajectory size.
In practice we can observe the trajectory only for a finite, even if large, time interval $\T$. Let us see how the expectation of the trajectory size grows with $\T$. Consider, for instance, the departure from the initial point, $\devn_\T$. The exact expression for its square is $$\begin{aligned}
{\left\lvert \devn_\T(\freqx) \right\rvert}^2 &=
{\sideset{}{'}\sum}\limits_{\ni,\nii\in{\mathbb{Z}}^{\m+1}} \frac{{{\Tipv}^*}_{\ni}\Tipv_{\nii}}{(\ni\cdot\freqx) (\nii\cdot\freqx)}
\\ & \mbox{} \times
\left( e^{-i\ni\cdot\freqx\T}-1\right) \left( e^{i\nii\cdot\freqx\T}-1\right) .\end{aligned}$$ Secular growth of the expectation of this series is due to resonant terms, i.e. those with $\ni$ parallel to $\nii$. If $\vvv(\freqx)$ is smooth and $\Tipv_\n$ quickly decay, then the main contribution is by principal resonances $\ni=\nii$. This gives an approximation $${\left\lvert \devn_\T \right\rvert}^2 \approx {\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^{\m+1}}
\frac{2{\left\lvert \Tipv_{\n} \right\rvert}^2}{(\n\cdot\freqx)^2}
\left[1 - \cos\left(\n\cdot\freqx\T\right)\right] .$$ To evaluate the corresponding expectation, $${\mathrm{E}\left[{\left\lvert \devn_\T \right\rvert}^2\right]} = \frac{1}{\mesBall}
\int\limits_{\freqx\in\Ball}
{\left\lvert \devn_\T \right\rvert}^2
\,{\mathrm{d}}\freqx ,$$ where $\mesBall={\mathop{\mathrm{Vol}}_{\m+1}(\Ball)}$, we substitute $\z=\n\cdot\freqx\T$ and let $\sig_\n = {\mathop{\mathrm{Vol}}_{\m}\left( \{ \freqx \,|\, \n\cdot\freqx=0 \} \cap \Ball \right)}$. This leads to $$\label{devn-growth}
{\mathrm{E}\left[\devn_\T^2\right]} \approx \C_1\T,
\quad
\C_1 = \frac{2\pi}{\mesBall}
{\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^{\m+1}}
\frac{{\left\lvert \Tipv_{\n} \right\rvert}^2 \sig_{\n}}{{\left\lVert\n\right\rVert}}.$$ Detailed calculations are given in the Supplementary materials, where we also show that under similar assumptions, $$\label{exp-growth}
{\mathrm{E}\left[\mssizesqT\T\right]} \approx \C_2 \T,
\quad
\C_2 = \frac{\pi}{3\mesBall}
{\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^{\m+1}}
\frac{{\left\lvert \Tipv_{\n} \right\rvert}^2 \sig_{\n}}{{\left\lVert\n\right\rVert}}.$$
#### Numerical illustration.
[Fig. \[traj\]]{}(a) shows a snapshot of a spiral wave solution, together with a piece of the corresponding tip trajectory, for the FitzHugh-Nagumo model [^3], $$\begin{aligned}
& u_\t=20(u-u^3/3-v)+\nabla^2u, \nonumber\\
& v_\t=0.05(u+1.2-0.5v). \label{fhn}\end{aligned}$$ [Fig. \[traj\]]{}(b) shows longer pieces of the tip trajectory, which illustrates the key feature of hypermeander: the area occupied by the trajectory can keep growing for a very long time. We have crudely emulated these dynamics by a system (\[tipt\],\[qp\],\[extension\]) [^4] with $$\begin{aligned}
&& \m=2, \quad
\Tipv(\phase)=\left(0.6-0.2\beta-0.2\alpha\beta\right)^{-1} - 1,
\nonumber\\
&&
\Tiprot(\phase)
=\left(0.675+0.1\alpha+0.05\beta+0.5\alpha^2+0.5\alpha\beta
\right.\nonumber\\ &&\hspace{3em}\left. \mbox{}
+0.2\alpha^3+0.6\alpha^2\beta\right)^{-1}- 1,
\nonumber\\
&&\alpha=\cos(\phase_1)+0.05\tanh(30\cos(\phase_2)),
\quad
\beta=\sin(\phase_1),
\nonumber\\
&& \freq_1=0.354, \quad \freq_2\in[0.475,0.525]. \label{bm18}\end{aligned}$$ This was done in the spirit of [@Ashwin-etal-2001] with the base dynamics replaced by an explicit two-periodic flow, but with the view to (i) mimic the actual meander pattern in the PDE model, and (ii) provide sufficient nonlinearity to ensure abundance of combination harmonics in [(\[int-of-hyp-x\])]{}. [Fig. \[traj\]]{}(c) shows pieces of a trajectory of this “caricature” model. One can see the same key feature, that the apparent size of the trajectory very much depends on the interval of observation; however the details are very sensitive to the choice of parameters, including $\freq_2$.
[Fig. \[stats\]]{}(a) illustrates the approach of $\mssizeT{\T}(\freq_2)$ to an everywhere discontinuous function as $\T\to\infty$. This was obtained for $10^5$ values of $\freq_2$ randomly chosen in the shown interval. For smaller $\T$ one can see well shaped individual peaks associated with the poles of $\mssize(\freq_2)$ corresponding to the resonances with the highest $\Tipv_{\n}$; for larger $\T$, more of such peaks become pronounced, and they grow stronger. [Fig. \[stats\]]{}(b) shows the empirical distribution of the trajectory sizes for the pieces of trajectories of the same $10^5$ simulations, of different lengths. We see that for larger $\T$, the distribution approaches the theoretical prediction [(\[distrib-law\])]{}. Finally, [fig. \[stats\]]{}(c) shows the growth of two empirical estimates of trajectory size with time, in agreement with [(\[exp-growth\])]{} and [(\[devn-growth\])]{}, including the predicted approximate ratio of 6 between them.
#### In conclusion,
quasiperiodic hypermeander of spiral waves has paradoxical properties. Even though described by deterministic equations, with no chaos involved, the question of the size of the tip trajectory does not have a meaningful deterministic answer and requires probabilistic treatment. In probabilistic terms, although the tip trajectory is confined with probability one, the expectation of its size, however measured, is infinite. There it is similar to the “St. Petersburg lottery”, in which a win is almost certainly finite, but its expectation is infinite [@Bernoulli-1738; @Sommer-1954]. The realistic price for a ticket in this lottery is nevertheless finite and modest; the resolution of this pardox relevant to us is that high wins require unrealistically long games [@Feller-1968 Section X.4]. In our case, the dependence of the trajectory size, whether defined via mean square displacement ${\left\lvert \devn_\T \right\rvert}^2$ or variance $\mssizesqT\T$, on any parameter affecting the frequency ratios becomes more and more irregular as $\T\to\infty$, and the expectations ${\mathrm{E}\left[{\left\lvert \devn_\T \right\rvert}^2\right]}$ and ${\mathrm{E}\left[\mssizesqT\T\right]}$ defined as averages over parameter variations, grow linearly in $\T$ even though the individual trajectories are bounded. Note that this is different from the linear growth for the mean square displacement of chaotically hypermeandering spirals [@Biktashev-Holden-1998; @Ashwin-etal-2001] which is for averages over initial conditions.
Practical applications of the theory are most evident for re-entrant waves in cardiac tissue, underlying dangerous cardiac arrhythmias. However implications may be also expected in any physics where the theory involves differential equations with quasiperiodic coefficients. One example may be provided by evolution of tracers in quasi-periodic fluid flows [@Boatto-Pierrehumbert-1999]. On a more speculative level, extension from ODEs in time to PDEs in spatial variables may provide insights into properties of quasicrystals [@Janssen-Janner-2015] or quasiperiodic dissipative structures [@Subramanian-etal-2016]. Note that properties of quasicrystals, among other things, include superlubricity [@Koren-Duerig-2016] and superconductivity [@Kamiya-etal-2018], still awaiting full theoretical treatment.
*Acknowledgements* VNB was supported by EPSRC Grant EP/N014391/1 (UK), NSF Grant PHY-1748958, NIH Grant R25GM067110, and the Gordon and Betty Moore Foundation Grant 2919.01 (USA). IM was supported by European Advanced Grant ERC AdG 320977 (EU).
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Supplementary material for\
“title”\
by V. N. Biktashev and I. Melbourne author
===========================================
Details of the derivation of the trajectory size asymptotics
------------------------------------------------------------
The exact expression for the square of the departure from the initial point is $${\left\lvert \devn_\T(\freqx) \right\rvert}^2 = {\left\lvert \tipp(\T) - \tipp(0) \right\rvert}^2
= {\left\lvert
{\sideset{}{'}\sum}\limits_{\n\in{\mathbb{Z}}^{\m+1}} \frac{-i\Tipv_{\n}}{\n\cdot\freqx}
\left( e^{i\n\cdot\freqx\T}-1\right)
\right\rvert}^2 .$$ Then for its expectation we have $${\mathrm{E}\left[{\left\lvert \devn_\T \right\rvert}^2\right]} = \frac{1}{\mesBall} \int\limits_\Ball
{\left\lvert \devn_\T(\freqx) \right\rvert}^2
\,{\mathrm{d}}\freqx
= \frac{1}{\mesBall}
{\sideset{}{'}\sum}\limits_{\ni,\nii}
\Tipv_{\ni}{{\Tipv}^*}_{\nii}
\termD_{\ni\nii}(\T),$$ where $$\termD_{\ni\nii}(\T)
=
\int\limits_\Ball
\frac{e^{-i\ni\cdot\freqx\T}-1}{\ni\cdot\freqx}
\, \frac{e^{i\nii\cdot\freqx\T}-1}{\nii\cdot\freqx}
\,{\mathrm{d}}\freqx .$$ Let us investigate the behaviour of the coefficients $\termD_{\ni\nii}(\T)$ in the limit $\T\to\infty$. We have to consider separately the cases when the two zero-denominator hyperplanes cut or do not cut through $\Ball$. Recall that $
\sig_\n \equiv {\mathop{\mathrm{Vol}}_{\m} \{ \freqx \,|\, \freqx\in\Ball \;\&\;
\n\cdot\freqx=0 \}}
$, and $
{\left\lVert\n\right\rVert} \equiv \left( \sum\limits_{\j=1}^{\m+1} \n_\j^2 \right)^{1/2}
$. We write $\n'\parallel\n''$ when vectors $n'$ and $n''$ are parallel (linearly dependent), and $\n'\nparallel\n''$ otherwise.
- For $\sig_{\ni}=0$, $\sig_{\nii}=0$, the coefficients are bounded: $${\left\lvert \termD_{\ni\nii}(\T) \right\rvert} \le
4\mesBall
\left(\min\limits_{\freqx\in\Ball}{\left\lvert \ni\cdot\freqx \right\rvert}\right)^{-1}
\left(\min\limits_{\freqx\in\Ball}{\left\lvert \nii\cdot\freqx \right\rvert}\right)^{-1}
={O\left(1\right)} .$$
- For $\sig_{\nii}=0$, $\sig_{\ni}\ne0$, we use a change of variables in the space $\left\{\freqx\right\}={\mathbb{R}}^{\m+1}$; namely, $\z=\ni\cdot\freqx\T\in{\mathbb{R}}$, and $\ziii\in{\mathbb{R}}^{\m}$ for the unscaled coordinates in $\ni^\perp$. In coordinates $(\z,\ziii)$, the domain $\Ball$ is stretched in the $\z$ direction and, as $\T\to\infty$, tends to an infinite cylinder with the axis along the $\z$ axis and the base of measure $\sig_\ni$. This gives $${\left\lvert \termD_{\ni\nii}(\T) \right\rvert} \le
\left(\min\limits_{\freqx\in\Ball}{\left\lvert \nii\cdot\freqx \right\rvert}\right)^{-1}
\int\limits_\Ball {\left\lvert
\frac{e^{i\ni\cdot\freqx\T}-1}{\ni\cdot\freqx}
\right\rvert}\,{\mathrm{d}}\freqx
=
\left(\min\limits_{\freqx\in\Ball}{\left\lvert \nii\cdot\freqx \right\rvert}\right)^{-1}
\iint\limits_{\freqx\in\Ball} {\left\lvert
\frac{e^{i\z}-1}{\z/\T}
\right\rvert}
\frac{{\mathrm{d}}\z}{{\left\lVert\ni\right\rVert}\T}
\,{\mathrm{d}}\ziii$$$$=
\left(\min\limits_{\freqx\in\Ball}{\left\lvert \nii\cdot\freqx \right\rvert}\right)^{-1}
\frac{\sig_{\ni}}{{\left\lVert\ni\right\rVert}}
\int\limits_{-{O\left(\T\right)}}^{{O\left(\T\right)}} {\left\lvert \frac{e^{i\z}-1}{\z} \right\rvert} \,{\mathrm{d}}\z
= {O\left(\ln(\T)\right)},$$ and similarly for $\sig_{\ni}=0$, $\sig_{\nii}\ne0$.
- For $\sig_{\ni}\ne0$, $\sig_{\nii}\ne0$, and $\ni
\nparallel \nii$, we use variables $\zi=\ni\cdot\freqx\T\in{\mathbb{R}}$, $\zii=\nii\cdot\freqx\T\in{\mathbb{R}}$, and $\ziii\in{\mathbb{R}}^{\m-1}$ for the unscaled coordinates in ${\mathop{\mathrm{span}}}(\ni,\nii)^\perp$. Then $$\termD_{\ni\nii} (\T)= \iiint\limits_{\freqx\in\Ball}
\dfrac{e^{-i\zi}-1}{\zi/\T}
\,
\dfrac{e^{i\zii}-1}{\zii/\T}
\,
\dfrac{{\mathrm{d}}\zi {\mathrm{d}}\zii}{{\left\lVert\ni\right\rVert}{\left\lVert\nii\right\rVert}\sin\left(\widehat{\ni,\nii}\right) \T^2}
\,
{\mathrm{d}}\ziii
={O\left(1\right)}.$$ Here $\widehat{\ni,\nii}$ is the angle between vectors $\ni$ and $\nii$.
- For $\sig_{\ni}\ne0$, $\sig_{\nii}\ne0$, and $\ni
\parallel \nii$, we set $\ni=\ai\n$, $\nii=\aii\n$, where $\n\in{\mathbb{Z}}^{\m+1}$ is their GCD vector and $\ai,\aii\in{\mathbb{Z}}\setminus\{0\}$ (see Proposition \[GCD\] below). In this case the hyperplanes $\ni\cdot\freqx=0$, $\nii\cdot\freqx=0$ and $\n\cdot\freqx=0$ coincide, and correspondingly $\sig_\ni=\sig_\nii=\sig_\n$.
We use $\z=\n\cdot\freqx\T\in{\mathbb{R}}$, and $\ziii\in{\mathbb{R}}^{\m}$ for the unscaled coordinates in $\n^\perp$. That gives
$$\termD_{\ni\nii} (\T)
=
\iint\limits_{\freqx\in\Ball}
\frac{\left(e^{i\ai\z}-1\right) \left(e^{-i\aii\z}-1\right)}{\ai\aii (\z/\T)^2}
\,\dfrac{{\mathrm{d}}\z {\mathrm{d}}\ziii}{{\left\lVert\n\right\rVert}\T}
=
\frac{\T\sig_\n}{\ai\aii{\left\lVert\n\right\rVert}}
\int\limits_{-{O\left(\T\right)}}^{{O\left(\T\right)}}
\left[
e^{i(\ai-\aii)\z}
-e^{i\ai\z}
-e^{-i\aii\z}
+1
\right]
\frac{{\mathrm{d}}\z}{\z^2} .$$ Now, $$\label{Dint}
{\int\limits_{-\infty}^{\infty}}\left[
e^{i(\ai-\aii)\z}
-e^{i\ai\z}
-e^{-i\aii\z}
+1
\right]
\frac{{\mathrm{d}}\z}{\z^2}
=
\I(\ai) + \I(\aii) - \I(\ai-\aii) ,$$ where $$\label{Iint}
\I(\alpha)={\int\limits_{-\infty}^{\infty}}(1-\cos(\alpha\z)) \,\frac{{\mathrm{d}}{\z}}{\z^2}
= \pi {\left\lvert \alpha \right\rvert} ,$$ and therefore $$\termD_{\ni\nii} (\T)
\approx
\frac{ \pi \sig_\n \left({\left\lvert \ai \right\rvert} + {\left\lvert \aii \right\rvert} - {\left\lvert \ai-\aii \right\rvert}\right) }{{\left\lVert\n\right\rVert} \ai\aii} \T
\quad
\textrm{as}
\quad
\T\to\infty.$$ For the principal resonances $\ai=\aii=1$ we have $$\termD_{\n\n} (\T)
\approx
2\pi
\frac{\sig_\n}{{\left\lVert\n\right\rVert}} \T
\quad
\textrm{as}
\quad
\T\to\infty ,$$ giving the estimate [(\[devn-growth\])]{}.
The computations for other statistics are similar in technique, if slightly longer. The raw second moment, i.e. the expectation of the time-average of the square departure from initial point, is $${\mathrm{E}\left[\mspathsqT\T\right]} =
\frac{1}{\T\mesBall}\int\limits_{\Ball} \int\limits_{0}^{\T}
{\left\lvert \devn_\t(\freqx) \right\rvert}^2 \,{\mathrm{d}}\t\,{\mathrm{d}}\freqx
=
\frac{1}{\mesBall}
{\sideset{}{'}\sum}\limits_{\ni,\nii\in{\mathbb{Z}}^{\m+1}}
\Tipv_{\ni}{{\Tipv}^*}_{\nii}
\termP_{\ni,\nii}(\T),$$ where, for $\ni\nparallel\nii$, $$\termP_{\ni\nii}(\T)
=
\int\limits_\Ball
\frac{1}{(\ni\cdot\freqx)(\nii\cdot\freqx)}
\left[
\frac{e^{i(\ni-\nii)\cdot\freqx \T}-1}{i(\ni-\nii)\cdot\freqx \T}
-\frac{e^{i\ni\cdot\freqx\T}-1}{i\ni\cdot\freqx \T}
-\frac{e^{-i\nii\cdot\freqx\T}-1}{-i\nii\cdot\freqx \T}
+1
\right]
\,
\,{\mathrm{d}}\freqx ,$$
and for $\ni\parallel\nii$, $\ni/\ai=\nii/\aii=\n$,
$$\termP_{\ni\nii}(\T)
= \frac{1}{\ai\aii\T}
\left[ \I(\ai)+\I(-\aii)-\I(\ai-\aii) \right]$$ where $$\I(\al) = \int\limits_{\Ball}
\frac{
1 + i\al\n\cdot\freqx\T - \left( \al\n\cdot\freqx\T \right)^2/2
- e^{i\al\n\cdot\freqx\T}
}{
i \al(\n\cdot\freqx)^3
}
\,{\mathrm{d}}\freqx .$$
Reasoning as in the previous case, we conclude that all the terms are ${O\left(\ln(\T)\right)}$ as $\T\to\infty$, except for those with $\ni\parallel\nii$, $\sig_\n\ne0$, which grow as ${O\left(\T\right)}$. Using, as before, the variables $\z=\n\cdot\freqx\T\in{\mathbb{R}}$ and $\ziii\in{\mathbb{R}}^{\m}\sim\n^\perp$, we get
$$\I(\al)
=
\frac{\sig_\n\T^2{\left\lvert \al \right\rvert}}{{\left\lVert\n\right\rVert}} \int\limits_{-{O\left(\T\right)}}^{{O\left(\T\right)}}
\frac{
1 + i\al\z - \left( \al\z \right)^2/2 - e^{i\al\z}
}{
i\z^3
}
\, {\mathrm{d}}\z
\approx
\frac{\sig_\n\T^2{\left\lvert \al \right\rvert}}{{\left\lVert\n\right\rVert}} {\int\limits_{-\infty}^{\infty}}\frac{\z - \sin(\z)}{\z^3}
\, {\mathrm{d}}\z
= \frac{\sig_\n\T^2{\left\lvert \al \right\rvert}}{{\left\lVert\n\right\rVert}} \frac{\pi}{2},$$ so $$\termP_{\ni\nii} \approx
\frac{\pi\sig_\n( {\left\lvert \ai \right\rvert}+{\left\lvert \aii \right\rvert}-{\left\lvert \ai-\aii \right\rvert})}{2{\left\lVert\n\right\rVert}\ai\aii}
\T
\quad
\textrm{as}
\quad
\T\to\infty.$$ For the principal resonances, $\ai=\aii=1$, this simplifies to $$\termP_{\n\n} \approx
\pi \frac{\sig_\n}{{\left\lVert\n\right\rVert}}
\T
\quad
\textrm{as}
\quad
\T\to\infty.$$
The expectation of the square of the time-averaged departure from initial point, i.e. of the length of the position vector of the apparent centroid in time $\T$, is $${\mathrm{E}\left[ {\left\lvert \meansizeT{\T} \right\rvert}^2 \right]}
=
\frac{1}{\mesBall} \int\limits_\Ball {\left\lvert
\frac1\T \int\limits_0^\T \devn_\T(\freqx) \,{\mathrm{d}}\T
\right\rvert}^2 \,{\mathrm{d}}\freqx
=
\frac{1}{\mesBall}
{\sideset{}{'}\sum}\limits_{\ni,\nii\in{\mathbb{Z}}^{\m+1}}
\Tipv_{\ni}{{\Tipv}^*}_{\nii}
\termM_{\ni,\nii}(\T),$$ where $$\termM_{\ni\nii}(\T)=
\int\limits_{\Ball}
\frac{
\left(e^{i\ni\cdot\freqx\T}-1-i \ni\cdot\freqx\T\right)
\left(e^{-i\nii\cdot\freqx\T}-1+i \nii\cdot\freqx\T\right)
}{
(\ni\cdot\freqx)^2(\nii\cdot\freqx)^2\T^2
}
\,{\mathrm{d}}\freqx .$$ As before, important terms are those with $\ni/\ai=\nii/\aii=\n$, $\sig_\n\ne0$, for which we use $\z=\n\cdot\freqx\T$, $\ziii\in{\mathbb{R}}^{\m}\sim\n^\perp$, and get
$$\termM_{\ni\nii}(\T) \approx
\frac{\sig_\n\T}{{\left\lVert\n\right\rVert}\ai^2\aii^2}
\termM(\ai,\aii)
\quad
\textrm{as}
\quad
\T\to\infty,$$ where the integral $$\termM(\ai,\aii)
=
{\int\limits_{-\infty}^{\infty}}\left(e^{i\ai\z}-1-i \ai\z\right)
\left(e^{-i\aii\z}-1+i \aii\z\right)
\,\frac{{\mathrm{d}}\z}{\z^4}$$ can be calculated using differentiation by parameters. We have $${\frac{\partial^2 \termM}{\partial \ai\partial \aii}}
=
{\int\limits_{-\infty}^{\infty}}\left(e^{i\ai\z}-1\right)
\left(e^{-i\aii\z}-1\right)
\,\frac{{\mathrm{d}}\z}{\z^2}
=
\pi\left( {\left\lvert \ai \right\rvert}+{\left\lvert \aii \right\rvert}-{\left\lvert \ai-\aii \right\rvert} \right),$$ using the result (Dint,Iint) obtained above. Hence, $$\termM(\ai,\aii)
=
\pi \iint \left( {\left\lvert \ai \right\rvert}+{\left\lvert \aii \right\rvert}-{\left\lvert \ai-\aii \right\rvert} \right) \,{\mathrm{d}}\ai{\mathrm{d}}\aii$$$$= \pi\left(
\frac12\aii\ai{\left\lvert \ai \right\rvert}
+
\frac12\ai\aii{\left\lvert \aii \right\rvert}
+
\frac16(\ai-\aii)^2{\left\lvert \ai-\aii \right\rvert}
\right)
+ \arbF(\ai) + \arbG(\aii),$$ where functions $\arbF$ and $\arbG$ can be determined from boundary conditions. Consider $$\termM(\ai,0)=0
=
\frac{\pi}{6}\ai^2{\left\lvert \ai \right\rvert}
+ \arbF(\ai) + \arbG(0),$$$$\termM(0,\aii)=0
=
\dfrac{\pi}{6}\aii^2{\left\lvert \aii \right\rvert}
+ \arbF(0) + \arbG(\aii) .$$ We observe that $\arbF(0)=\arbG(0)=0$ is an admissible choice, which leads to $$\termM(\ai,\aii)
= \frac{\pi}{6}\left(
(3\aii-\ai)\ai{\left\lvert \ai \right\rvert}
+
(3\ai-\aii)\aii{\left\lvert \aii \right\rvert}
+
(\ai-\aii)^2{\left\lvert \ai-\aii \right\rvert}
\right)$$ and consequently $$\termM_{\ni\nii}(\T)\approx
\frac16 \pi\sig_\n
\frac{(3\aii-\ai)\ai{\left\lvert \ai \right\rvert}
+
(3\ai-\aii)\aii{\left\lvert \aii \right\rvert}
+
(\ai-\aii)^2{\left\lvert \ai-\aii \right\rvert}
}{{\left\lVert\n\right\rVert}\ai^2\aii^2}
\T
\quad\textrm{as}\quad\T\to\infty.$$ For the principal resonances, $\ai=\aii=1$, this gives $$\termM_{\ni\nii}(\T) \approx
\frac{2\pi}{3}
\,
\frac{\sig_\n}{{\left\lVert\n\right\rVert}}
\T
\quad\textrm{as}\quad\T\to\infty.$$ Hence the central second moment, i.e. the expectation of the time-average of the square departure from the apparent centroid is $${\mathrm{E}\left[\mssizesqT\T\right]}
=
\frac{1}{\mesBall} \int\limits_\Ball {\left\lvert
\frac1\T \int\limits_0^\T \left(
\devn_\T(\freqx)
-
\meansizeT{\T}(\freqx)
\right)\,{\mathrm{d}}\T
\right\rvert}^2 \,{\mathrm{d}}\freqx
=
{\mathrm{E}\left[\mspathsqT\T - {\left\lvert \meansizeT{\T} \right\rvert}^2\right]}
=
\frac{1}{\mesBall}
{\sideset{}{'}\sum}\limits_{\ni,\nii\in{\mathbb{Z}}^{\m+1}}
\Tipv_{\ni}{{\Tipv}^*}_{\nii}
\termS_{\ni,\nii}(\T),$$ where for the principal resonances we have $$\termS_{\n\n}(\T)=
\termP_{\n\n}(\T)-
\termM_{\n\n}(\T)
\approx
\frac{\pi}{3}
\,
\frac{\sig_\n}{{\left\lVert\n\right\rVert}}
\T ,$$ which gives the estimate [(\[exp-growth\])]{}.
\[GCD\] Let $\ni,\nii\in{\mathbb{Z}}^\m\setminus\{0\}$ be linearly dependent. Then there exist $\ai,\aii\in{\mathbb{Z}}\setminus\{0\}$ and $\n\in{\mathbb{Z}}^\m\setminus\{0\}$ such that $\ai$ and $\aii$ are coprime and $\ni=\ai\n$, $\nii=\aii\n$.
Since both vectors are nonzero, we have $\nii=\al\ni$ for a nonzero scalar $\al$. We must have $\al\in{\mathbb{Q}}\setminus\{0\}$ since it is a ratio of the corresponding components of $\ni$ and $\nii$. Let $\al=\aii/\ai$ with $\ai,\aii\in{\mathbb{Z}}\setminus\{0\}$ coprime. By writing $\ai\nii=\aii\ni$ we observe that all components of $\nii$ are divisible by $\aii$ and all components of $\ni$ are divisible by $\ai$. Hence $\nii/\aii=\ni/\ai=\n\in{\mathbb{Z}}^\m\setminus\{0\}$, as required.
[^1]: As discussed earlier, various types of spiral behaviour are associated with various types of dynamics (steady-state, periodic, quasiperiodic, chaotic) in the base equation $\dot\X=\fx(\X)$. All these types of dynamics are known to occur with positive probability. In particular, KAM theory predicts the existence of quasiperiodic dynamics. In this paper, we take the point of view that the base dynamics is known to be quasiperiodic (in accordance with the observations in [@Roessler-Kahlert-1979; @Zykov-1986; @Winfree-1991] and analyse the consequent behaviour in the full system of equations.
[^2]: When quasiperiodicity arises via KAM theory, near onset, phaselocking leads to a complicated structure for the positive measure set $\Freqset$ of frequencies $\freq$ corresponding to quasiperiodic dynamics. The integration should then be over $\Freqset\cap\Ball$ rather than the whole ball $\Ball$. However, writing $\Freqset_\param$ to indicate dependence on a parameter $\param\to0$, we have in this situation that ${\mathop{\mathrm{Vol}}_{\mq}(\Freqset_\param\cap\Ball)}\to {\mathop{\mathrm{Vol}}_{\mq}\Ball}$ and hence $\lim_{\param\to0}\int_{\Freqset_\param\cap\Ball}
{\left\lvert \n\cdot\freq \right\rvert}^{-1}\,{\mathrm{d}}\freq=+\infty$. So we obtain the same conclusion in the limit as $\param\to0$ as before.
[^3]: This was simulated by second-order center space ($h_x=2/9$), forward Euler in time ($h_\t=1/125$) differencing in a box $60\times60$ with Neumann boundaries. The tip of the spiral was defined by $u=v=0$.
[^4]: This was simulated with forward Euler, with $h_\t=0.3$. Crudeness of the method was required to make massive simulations; this does not affect the conclusions since this is a caricature model anyway.
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abstract: |
Influence maximization is among the most fundamental algorithmic problems in social influence analysis. Over the last decade, a great effort has been devoted to developing efficient algorithms for influence maximization, so that identifying the “best” algorithm has become a demanding task. In SIGMOD’17, Arora, Galhotra, and Ranu reported benchmark results on eleven existing algorithms and demonstrated that there is no single state-of-the-art offering the best trade-off between computational efficiency and solution quality.
In this paper, we report a high-level experimental study on three well-established algorithmic approaches for influence maximization, referred to as Oneshot, Snapshot, and Reverse Influence Sampling (RIS). Different from Arora et al., our experimental methodology is so designed that we examine the *distribution* of random solutions, characterize the relation between the *sample number* and the actual solution quality, and avoid *implementation dependencies*. Our main findings are as follows: **1.** For a sufficiently large sample number, we obtain a unique solution regardless of algorithms. **2.** The average solution quality of Oneshot, Snapshot, and RIS improves at the same rate up to scaling of sample number. **3.** Oneshot requires more samples than Snapshot, and Snapshot requires fewer but larger samples than RIS. We discuss the time efficiency when *conditioning* Oneshot, Snapshot, and RIS to be of identical accuracy. Our conclusion is that Oneshot is suitable only if the size of available memory is limited, and RIS is more efficient than Snapshot for large networks; Snapshot is preferable for small, low-probability networks.
author:
- Naoto Ohsaka
bibliography:
- 'main.bib'
subtitle: 'A High-level Experimental Study on Three Algorithmic Approaches'
title: The Solution Distribution of Influence Maximization
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abstract: 'The tunneling Hamiltonian describes a particle transfer from one region to the other. While there is no particle storage in the tunneling region itself, it has associated certain amount of energy. We name the corresponding flux [*energy reactance*]{} since, like an electrical reactance, it manifests itself in time-dependent transport only. Noticeably, this quantity is crucial to reproduce the universal charge relaxation resistance for a single-channel quantum capacitor at low temperatures. We show that a conceptually simple experiment is capable of demonstrating the existence of the energy reactance.'
author:
- María Florencia Ludovico
- Liliana Arrachea
- Michael Moskalets
- David Sánchez
title: Probing the energy reactance with adiabatically driven quantum dots
---
[*Motivation*]{}. A very exciting experimental activity is lately taking place in search of controlling on-demand quantum coherent charge transport in the time domain. The recent burst of activity started with the experimental realization of quantum capacitors in quantum dots under ac driving [@Gabelli:2006eg], single particle emitters [@Feve:2007jx], and was followed by the generation of quantum charged solitons over the Fermi sea (levitons) [@Dubois:2013ul]. A controlled manipulation of flying single electrons [@Hermelin:2011du; @McNeil:2011ex; @Bertrand:2016ik] and their time-resolved detection [@Fletcher:2012te] have already been reported [@Splettstoesser:2017jd]. These marvelous developments, along with the identically impressive progress in the field of fast thermometry [@zgi17; @pek13; @gas15], are opening an avenue towards the study and control of the concomitant time-dependent energy flow in the quantum realm.
The relevant systems are characterized by small (nanoscale) components confining a small number of particles in contact to macroscopic reservoirs. This puts the description of the energy transport and heat generation beyond the scope of usual thermodynamical approaches, motivating a number of formal theoretical developments in statistical mechanics [@Kosloff:2013cr] and condensed matter physics [@Ludovico:2016hh]. At the heart of this problem, there is the proper definition of the quantum heat current in the time domain. The concept of heat looks very intuitive and anyone can provide a definition for it. Formally, it is a clear and well established concept in macroscopic systems close to equilibrium. However, its accurate definition at the nanoscale and in situations away from equilibrium is a deep and subtle issue, in particular due to the coupling between a nanosystem and macroscopic reservoirs; see, e.g., Refs. . In fact, while charge and energy are concepts obeying strict fundamental conservation laws, the definition of heat implies the proper identification of a portion of the total energy.
![Schematic of our proposal. The quantum $RC$ circuit consists of a quantum dot (blue disk) coupled to a fermionic reservoir (light blue region) with well defined temperature $T$ and chemical potential $\mu$. Electrons can be transferred between the dot and the reservoir (black curve). The dot is capacitively coupled to a gate terminal where an ac potential of amplitude $V$ and frequency $\Omega$ is applied. A floating contact is also attached to the dot. The temperature $T_c$ and chemical potential $\mu_c$ of the floating contact adjust themselves to cancel both the charge and heat current flowing through it, thus allowing for an experimental test of the energy reactance, namely, the variation of the stored energy at the tunneling region between the floating contact and the dot (gray line).[]{data-label="fig1"}](fig1.pdf){width="40.00000%"}
An appealing scenario to address this problem from the theoretical point of view is a periodically driven single level in contact to an electron reservoir. This is the most basic and meaningful setup to analyze the interplay of charge and energy dynamics. At the same time, this is the simplest model for a quantum $RC$ circuit [@Buttiker:1993wh], which has been experimentally realized [@Gabelli:2006eg]. A sketch is presented in Fig. \[fig1\], where we stress that the driven level represents a quantum dot. The nonequilibrium ingredient is provided by the time-dependent gate voltage $V(t)= V \cos\left( \Omega t \right)$ locally applied to the single level. The reservoir is an electron gas with temperature $T$ and chemical potential $ \mu$, and the strength of the coupling between the two subsystems is arbitrary. The setup also includes a “floating contact”, which we will discuss in detail later on.
The effect of the periodic driving is twofold. On one hand, it induces a charge current that flows between the dot and the reservoir as a function of time. On the other hand, it performs a work on the system, thus injecting energy that is ultimately dissipated as heat deep inside the reservoir. Importantly, due to charge conservation, the electronic current is defined as the change in time of the electron number either at the reservoir or on the dot. No contribution of the tunneling region on the charge current exists. In contrast, the energy delivered by the external ac source is temporarily stored in three different parts of the setup: the dot, the reservoir and also in the dot-reservoir tunneling region. The role of the tunneling region is typically disregarded in classical thermodynamics because it is a surface term that is negligible when both the system and the reservoir are macroscopic [@bellac]. Yet, in the nanoscale setup studied here the amount of energy stored in the dot is comparable to that of the tunneling region and the latter can no longer be neglected.
In a recent work [@us1] we have coined the name of [*energy reactance*]{} to characterize the energy temporarily stored at the tunneling region. This is a thermal analogue of an electrical reactance (due to electrical capacitance and inductance), which manifests itself in a time-dependent setup only. We have argued that it is physically meaningful to take the energy reactance into account as a contribution to the time-dependent heat current flowing into the reservoir. We have shown that this is in full agreement with the laws of thermodynamics [@us2; @Ludovico:2016hh]. While some recent works raised some concerns [@Esposito:2014wu; @Ochoa:2016ee], other works arrived at conclusions similar to our analysis [@Wu:2008gk; @rossello; @Bruch:2015ux]. The aim of the present work is to make one step further by proposing a measurement scheme that is able to test the effect of the energy reactance onto a time-dependent heat flux.
*Proposed experiment and predictions.* The setup is sketched in Fig. \[fig1\], where we introduce a floating contact attached (e.g., via tunneling) to the quantum dot. When a periodic gate voltage $V(t)$ is applied, charge and heat currents enter not only the reservoir but also the floating contact. The latter can adjust its chemical potential $\mu_{c}$ and temperature $T_{c}$ to maintain zero charge and heat currents flowing into it. We will focus on slow “adiabatic” driving, which corresponds to a driving period much larger than any characteristic time scale for the system. Assuming that the charge and energy relaxation rate of the floating contact is much faster than any other characteristic time, $\mu_{c}$ and $T_{c}$ will change instantaneously to prevent charge and heat accumulation on the floating contact. In contrast, the reservoir is a massive electrode that keeps its temperature and chemical potential constant independently of the ac potential. In practice, this can be achieved grounding the reservoir as indicated in Fig. \[fig1\]. Its temperature variations would be suppressed if the reservoir has in addition a large heat capacity.
The evolution of the chemical potential and temperature of the floating contact as the dot is aidabatically driven can be sensed by means of a voltage probe and a thermometer [@Engquist:1981wz; @dubi; @benenti; @sanchez; @bergfield; @Battista:2013ew; @thermo; @probe; @ye], as indicated in the figure. We predict different behaviors for $\mu_{c}$ and $T_{c}$ depending on whether the energy reactance is considered or not in the heat flux into the floating contact. In this way, the proposed experiment would help to discern on the proper definition of the heat current and test the existence of the energy reactance.
The results are the following: (i) By defining the heat flux into the floating contact, taking into account the energy reactance as in Ref. , the temperature of the floating contact does not vary with time. The outcome is $$\begin{aligned}
T_{c} = T \,,
\label{main}\end{aligned}$$ where $T$ is the background temperature. The chemical potential of the contact $ \mu_{c}(t)$ does vary with time in a periodic fashion with a period dictated by the electrical current flowing through the dot. (ii) We demonstrate that any other definition of the heat current, that does not properly account for the energy reactance, necessarily leads to a change in both quantities, $T_{c}(t)$ and $\mu_{c}(t)$ as functions of time.
[*Heat current into the floating contact and quantum energy reactance.*]{} Let the rates of change for the charge and the internal energy stored in the floating contact due to exchanges with the rest of the device be, respectively, $\dot{N}_c(t) $ and $\dot{U}_c(t)$. Similarly, the rate of change for the energy stored at the tunneling region between the dot and the floating contact is denoted by $\dot{U}_{T_c}(t)$. The meaningful definition for the instantaneous heat current entering the floating contact is [@us1] \[heat\] \_c(t)=\_c(t)+-\_c(t)\_c(t). The energy reactance, $\dot{U}_{T_c}(t)/2$, contributes to the heat flux only instantaneously and as such vanishes when averaged over one driving period.
From the theoretical point of view, the energy reactance is necessary to derive an instantaneous Joule law for the heat current into a (single-channel) floating contact at low temperatures, $\dot{Q}_c(t)= R_q [\dot{N}_c(t)]^2$, with the universal charge relaxation resistance, the Büttiker resistance, $R_{q}= h/2e^2$ [@Gabelli:2006eg; @Buttiker:1993wh]. The energy reactance is also necessary to both reconcile the relation between the Green function and the scattering matrix formalisms [@Arrachea-floquet] for the instantaneous heat current [@us1] and also to obtain correct frequency parity properties of the response functions [@rossello].
[*Temperature and chemical potential of the floating contact.*]{} Our goal is to explicitly show that the definition of Eq. (\[heat\]) can be verified by measuring the temperature and chemical potential of the floating contact. The latter is a small conductor that we assume to be in thermal equilibrium at every instant of time such that both temperature and chemical potential adjust themselves to satisfy the condition of vanishing charge and vanishing heat current, i.e., $\mu_c(t)$ and $T_c(t)$ simultaneously fulfill $\dot{Q}_c(t)=\dot{N}_c(t)=0$. The local equilibrium condition is justified in the adiabatic regime (very low driving frequency $\Omega$), mostly accessible in experiments [@Gabelli:2006eg]. Deviation of the floating contact temperature and chemical potential from their stationary values are denoted by $\delta T_c(t)=T_c(t)-T$ and $\delta \mu_c(t)= \mu_c(t)-\mu$, respectively. In the adiabatic regime, these quantities are small, $\delta T_c(t) ,\delta \mu_c(t) \propto \hbar\Omega$. As a consequence, we can evaluate both charge and heat fluxes in linear response in these quantities (while the amplitude of the ac driving potential is arbitrary).
Following Refs. we expand the fluxes ${\bf J} (t)\equiv \left(\dot{N}_c(t),\;\dot{Q}_c(t)\right)$ in the affinities ${\bf X} =\left( \delta \mu_c(t), \; \delta T_c(t), \;\hbar \Omega \right)$, with coefficients $\Lambda_{ij}(t)$ as \[linearresp\] J\_i(t)=\_[j=1]{}\^[3]{}\_[ij]{}(t) X\_j (t), where $i=1,2$ ($j=1,2,3$) label the different components of the vectors ${\bf J}$ and ${\bf X}$, respectively. The coefficients of the above expansion are response functions evaluated with the frozen Hamiltonian at time $t$ and have the following physical interpretation: $\Lambda_{11}$ and $\Lambda_{22}$ are the usual electric and thermal conductances. On the other hand, $\Lambda_{12}$ (related to the Seebeck effect) and $\Lambda_{21}$ (related to the Peltier effect) capture the thermoelectric transport, and they satisfy the reciprocity relation $\Lambda_{21}=T\Lambda_{12}$ [@onsager; @buttiker; @matthews; @revcasati]. Finally, ${\Lambda}_{13}$ and ${\Lambda}_{23}$ describe, respectively, the generation of charge and heat currents by the ac driving. They also obey Onsager relations with the coefficients entering the work flux (not considered here) [@Ludovico:2015uw]. Explicit expressions of these coefficients will be supplied below for the specific model.
Here, we notice that the conditions of vanishing fluxes to the floating contact amounts to finding the solution of the $2 \times 2$ linear set of equations $\sum_{j=1}^2 \Lambda_{ij}X_j =- \Lambda_{i3} \; \hbar \Omega,\;i=1,2$. The solutions are $$\begin{aligned}
\delta \mu_c(t) & = & \frac{\Lambda_{12} \Lambda_{23}- \Lambda_{13} \Lambda_{22}}{\mbox{det}\,{\bf \Lambda}^{\prime}}\; \hbar \Omega ,\nonumber\\
\label{solset} \\
\delta T_c(t) & = & \frac{\Lambda_{13} \Lambda_{21}- \Lambda_{11} \Lambda_{23}}{\mbox{det}\,{\bf \Lambda}^{\prime}} \;\hbar \Omega,
\nonumber\end{aligned}$$ where ${\mbox{det}\,{\bf \Lambda}^{\prime}}$ corresponds to the determinant of the $2\times 2$ matrix determined by the condition $j\neq 3$.
The coefficients $\Lambda$ can be calculated for the system considered in Fig. \[fig1\] following Refs. (see details in Ref. ) \[coeff1\] \_[ij]{}(t) =
(t,) \_f d& j3\
\
-(-)\^[i-1]{} \_f(t,) \_f d& j=3,
The distinction between $j\neq 3$ and $j=3$ is important. In the former case, the response depends on the instantaneous transmission probability $\mathcal{T}(t,\veps)$ for electrons traversing the quantum dot between the reservoir and the floating contact. Physically, this corresponds to dc transport. In the latter case, the response is a function of the time derivative of the potential applied to the gate, $\dot V = - \Omega V \sin\left( \Omega t \right)$, and the instantaneous local density of states of the dot, $\rho_f(t,\veps)$. Physically, this is pumping and, as such, of ac nature. Both coefficients are time dependent because the system adiabatically reacts to the instantaneous ac driving potential [@mos12]. Finally, in Eq. $f$ is the Fermi-Dirac distribution of the reservoir, while $\Gamma_c=|w_c|^2 \rho_c$ and $\Gamma=|w|^2 \rho$ are the hybridization functions with $w_c$ the dot-floating contact couplings and $w$ the dot-reservoir couplings. The density of states of the floating contact is $\rho_c$ and that of the reservoir is $\rho$.
Interestingly, we readily find that the coefficients of Eq. (\[coeff1\]) satisfy [*i*]{}) $\Lambda_{13} \Lambda_{21}- \Lambda_{11} \Lambda_{23}=0$ and [*ii*]{}) $\Lambda_{j3}=-\Lambda_{j1}\frac{\dot{V}}{\Gamma\Omega}$ with $j=1,2$, leading to the solution $$\label{sol}
\delta T_c(t)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\delta\mu_c(t) = \frac{\hbar}{\Gamma}e \dot{V}\,.$$ From Eq. (\[solset\]) and the relation [*ii*]{}), we see that the above results do not actually depend on coupling to the floating contact. We have checked that Eq. is valid for any temperature $T$, provided that the adiabaticity condition $\Gamma, \Gamma_c \gg \hbar\Omega$ is satisfied [@brouwer; @spletti08]. This is true even for temperatures close to zero, in which case the second order contributions in the affinities should be added to Eq. (\[linearresp\]) [@sm].
In summary, the floating contact fulfills the conditions of vanishing heat and charge fluxes by changing $\delta \mu_c(t)$ in time according to Eq. (\[sol\]) while keeping its temperature constant and equal to the background temperature, as indicated in Eq. (\[main\]).
![Deviation of the temperature of the floating contact, $\delta\tilde{T}_c$, as a function of time for different background temperatures $T$. The ac potential is $20\,\Gamma\cos(\Omega t)$ with $\hbar\Omega=0.07\Gamma$. The hybridization between the floating contact and the quantum level is $\Gamma_c=0.6\Gamma$. All energies are expressed in units of the hybridization $\Gamma$ with the reservoir. The temperature of the floating contact displays oscillations that depend on the background temperature. As $T$ increases, the oscillations become less pronounced and the maxima positions deviate from the moment when the level is aligned with the chemical potential of the reservoir, which in this case corresponds to $t\Omega/2\pi=0.25$ and $t\Omega/2\pi=0.75$.[]{data-label="fig2"}](fig2.pdf){width="52.00000%"}
[*Examine the energy reactance*]{}. We would like to stress now that Eq. (\[sol\]), in particular, the prediction of a constant temperature of the floating contact expressed in Eq. (\[main\]), constitutes a proof for the existence of the energy reactance $\dot{U}_{T_c}(t)/2$ and the definition of the heat current as in Eq. (\[heat\]). This can be easily understood by noticing that we would arrive at completely different conclusions on the behavior of the temperature of the floating contact if we consider a definition of the heat flux that does not take into account the energy reactance.
As a proof, let us analyze the consequence of adopting a commonly used definition, that does not take into account the energy reactance. This corresponds to the following expression for the heat flux into the floating contact, \[heatilde\] \_c(t)=\_c(t)-\_c(t)\_c(t). We need to recalculate the coefficients $\Lambda_{2j}(t)$ by using the above equation. We denote the so defined coefficients by $\tilde{\Lambda}_{2,j}(t)$. From Eq. (\[solset\]), where we replace $\Lambda_{2,j}(t) \rightarrow \tilde{\Lambda}_{2,j}(t),\; j=1,2,3$, we find the floating contact temperature $\delta\tilde{T}_c(t)$ and chemical potential $\delta \tilde{\mu}_c(t)$. In contrast to Eq. (\[sol\]), now we find that both the temperature $\delta\tilde{T}_c(t)$ and the chemical potential $\delta\tilde{\mu}_c(t)$ of the floating contact change in time. In the case of the chemical potential, $\delta\tilde{\mu}_c$ evolves in time in a different pattern from that described by Eq. (\[sol\]). We turn to focus on the behavior of the temperature $\delta\tilde{T}_c$, which is shown in Fig. \[fig2\]. It is worth noting that the amplitude of the $\delta\tilde{T}_c$ oscillations decreases as $T$ increases, which shows that the two definitions of the heat current agree in the high temperature limit. These results show that the role of the energy reactance is particularly relevant in the quantum regime. In the classical high-temperature limit the temperature of the floating contact is independent of time, either with the heat current defined as in Eq, (\[heat\]) or with the definition of Eq. (\[heatilde\]).
[*Conclusion.*]{} We have shown that the behavior of the time-resolved chemical potential and temperature of a floating contact coupled to an adiabatically driven quantum dot is strongly sensitive on the definition of the instantaneous heat flux. For this reason, sensing these quantities would provide an experimental test for the relevance of the energy reactance introduced in Ref. [@us1] as a component of the time-dependent heat flux.
Specifically, for an adiabatically driven quantum dot with a single active level coupled to a single reservoir, we have shown that: (i) If the energy reactance is taken into account, then the temperature of the floating contact is constant and equal to that of the reservoir, while its chemical potential follows the time derivative of the driving potential, $\dot{V}$, as expressed in Eq. (\[sol\]). Instead, (ii) if the energy reactance is not taken into account, these two quantities follow a nonuniversal and rather cumbersome time-dependent pattern.
The experiment we propose is close to the scope of present-day experimental techniques. In fact, typical level spacing for quantum dots is around $100 $ $\mu$eV [@kouw]. Thus, by keeping driving amplitudes below this energy, we would basically have a single active level. On the other hand, typical parameters for single particle emitters have $\Gamma \simeq 1 \mu$eV ($\simeq 1$ GHz) and are operated at frequencies $\Omega \simeq 0.1$ GHz [@Gabelli:2006eg], which satisfy the adiabatic condition $\hbar \Omega < \Gamma $. As a consequence, a fast thermometer [@zgi17] is able to follow temperature changes of the floating contact on the nanosecond scale. Experiments are typically performed at temperatures close to $T \sim 100$ mK. For this temperature, the oscillations in the temperature shown in Fig. \[fig2\] have an amplitude of $\delta \tilde{T}_c \simeq 10$ mK.
We emphasize that the question about the role of the energy reactance in the definition of a time-dependent heat flux is a fundamental one. It is not restricted to slowly driven systems of noninteracting electrons but is also relevant for interacting models, for fast drivings, and for weakly and strongly coupled systems. So far this question has been addressed only theoretically. The present proposal shows that a thermometer probe response will experimentally demonstrate the existence of the energy reactance.
[*Acknowlegments*]{}. This work was supported by MINECO under Grant No. FIS2014-52564, UBACyT, CONICET and MINCyT, Argentina. LA thanks the support of the Alexander von Humboldt Foundation. MM thanks the support and the hospitality of the Aalto University, Finland.
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See Supplementary Material for details of the parameters entering the model, the calculation of the charge and heat currents, and explicit expressions of the coefficients in the linear and second order expanssions.
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Probing the energy reactance with adiabatically driven quantum dots: Supplementary Information {#probing-the-energy-reactance-with-adiabatically-driven-quantum-dots-supplementary-information .unnumbered}
==============================================================================================
This supporting document describes in detail the derivation of the time-dependent linear response coefficients for particle and heat currents flowing into the floating contact.
Theoretical model
-----------------
We consider a simple setup, with all the necessary ingredients to analyze the dynamical energy transfer and to probe the energy reactance. It is the most basic model for a quantum capacitor, which consists in a periodically driven single level (quantum dot) coupled to an electron bath, the reservoir, with temperature $T_r=T$ and chemical potential $\mu_r=\mu$. The time-dependent driving is provided by the application of an oscillatory gate voltage of the form $V(t)=V{\cos}(\Omega t)$.
To test the effect of the energy reactance on the time-dependent heat flux, we introduce a floating contact, which is coupled to the driven level. In order to be electrically and thermally isolated from the environment at every time, the floating contact instantaneously adjusts its chemical potential $\mu_c$ and temperature $T_c$.
The Hamiltonian of the full system, the quantum capacitor together with the floating contact, can be separated into three contributions, H(t)=H\_[QC]{}(t)+H\_c+H\_[T\_c]{}. The Hamiltonian $H_{QC}$ represents the quantum capacitor, which contains three elements: the single level, an electron bath (the reservoir, denoted by the letter $r$), and the coupling between the two. Then, H\_[QC]{}(t)=\_[k]{}+(\_0+V(t))d\^d, where $\veps_{k}^r$ is the energy band of the reservoir and $w$ is the coupling amplitude to the driven level. The energy $\veps_0$ corresponds to the bare level, which for simplicity will be considered aligned with the chemical potential of the reservoir, i.e. $\veps_0=\mu$ . The operator $c^\dagger_{k} (c_{k})$ create (destroy) an electron with a wavevector ${k}$ in the reservoir, while $d^\dagger$ and $d$ are associated to the degrees of freedom of the single level.
Similarly, the floating contact is represented by the Hamiltonian, H\_c=\_[q]{}\^c\_[q]{} a\^\_[q]{} a\_[q]{}, while in this case, the operators $a_q$ and $a_q^\dagger$ are responsible, respectively, for the creation and destruction of an electron in the floating contact with an energy $\veps^c_{q}$. The coupling between the level and the floating contact can be written as H\_[T\_c]{}=\_[q]{} w\^c (d\^a\_q+ a\_q\^d).
### Particle and heat currents
Now, the aim is to compute the particle and heat fluxes entering the floating contact. The time variation of the particles present in the floating contact is given by the exact expression, $\dot{N}_c=\frac{i}{\hbar}\langle [H, N_c]\rangle$. To compute the time-dependent heat current as in Eq. (\[heat\]), or by adopting the definition (\[heatilde\]), we define the internal energy stored in the floating contact and in the tunneling region as $\dot{U}_\beta=\frac{i}{\hbar}\langle [H, H_\beta]\rangle$, with $\beta=c, T_c$.
In Refs. [@us:prb] and [@Ludovico:entropy], we presented that the evolution of the expectation value of an observable (e.g the number of particles or the energies) can be obtained by recourse to Keldysh non-equilibrium Green’s functions. In this way, the different currents can be computed in terms of the retarded Green function $G^R(t,t')=-i\theta (t-t')\langle \{ d(t), d^\dagger (t')\}\rangle$ of the single level. For a time-periodic driving it is convenient to use the Floquet-Fourier representation [@Arrachea-floquet-1], G\^R(t,t’)=\_[n=-]{}\^\^\_[-]{} e\^[-i(t-t’)]{}e\^[-int]{}[G]{}(n,).
As we showed in detail in Ref. [@us:prb] for a single driven level connected to many reservoirs, the particle current entering one of them, as for example the floating contact, is \[chargesm\] \_c(t)=\_l e\^[-ilt]{}\_c {i[G]{}\^\*(-l,)-\_n\_[=r,c]{}\_(l+n,\_n)[G]{}\^\*(n,\_n)}, where $\alpha=r$ corresponds to the reservoir of the capacitor, and $\alpha=c$ is the floating contact. Some of the energies are shifted by a multiple of the energy quantum $\hbar \Omega$ as $\veps_n=\veps-n\hbar\Omega$. We have introduced the Fermi-Dirac distribution of the reservoir labeled by $\alpha$, $f_\alpha(\veps)=[e^{(\veps-\mu_\alpha)/{k_B T_\alpha}}+1]^{-1}$, with $k_B$ being the Boltzmann constant. The hybridizations with the reservoirs are $\Gamma_c=\vert w^c\vert^2 \rho_c$ for the floating contact and $\Gamma=\vert w\vert^2 \rho$ for the reservoir, with $\rho_c=\sum_{k\in c}2\pi \delta(\veps-\veps_k^c)$ and $\rho=\sum_{k\in m}2\pi \delta(\veps-\veps_k^m)$ being the density of states of the floating contact and the reservoir, respectively. We are considering the wide band limit, in which $\Gamma_c$ and $\Gamma$ are constant functions.
In the same work [@us:prb], and also in Ref. [@Ludovico:entropy] for a more general setup, we showed that according to the definition in Eq. (\[heat\]) the heat flux is $$\begin{aligned}
\label{heat1sm}
\dot{Q}_c(t) &=&\sum_l \int \frac{d\veps}{h} e^{-il\Omega t}\Gamma_c \Bigg\{i{\cal G}^*(-l,\veps)(\veps_{\frac{l}{2}}-\mu_c)\left[ f_c(\veps)-f_c(\veps_l)\right]\nonumber\\
&& \left.-\sum_n\sum_{\alpha=r,c}(\veps_{-\frac{l}{2}}-\mu_c)\left[f_c(\veps)-f_\alpha(\veps_n)\right]\Gamma_\alpha {\cal G}(l+n,\veps_n){\cal G}^*(n,\veps_n)\right\}.\end{aligned}$$
If we adopt the definition in Eq. (\[heatilde\]), which does not take into account the energy reactance $\dot{U}_{T_c}(t)/2$, then $$\begin{aligned}
\label{heat2sm}
\dot{\tilde{Q}}_c(t) &=& \sum_l \int \frac{d\veps}{h} e^{-il\Omega t}\Gamma_c \Bigg\{i{\cal G}^*(-l,\veps)\bigg[ (\veps-\mu_c)( f_c(\veps)- f_c(\veps_l))+l\hbar\Omega f_c(\veps_l) \bigg]\nonumber\\
& &-\sum_n\sum_{\alpha=r,c}\bigg[(\veps-\mu_c) (f_c(\veps)-f_\alpha(\veps_n))-\frac{l}{2}\hbar \Omega f_\alpha(\veps_n)\bigg]\Gamma_\alpha {\cal G}(l+n,\veps_n){\cal G}^*(n,\veps_n)\Bigg\}.\end{aligned}$$
Here, we stress that all the above expressions for both particle and heat fluxes are exact, in the sense that they are valid for arbitrary values of the driving frequency, amplitude and temperature.
Linear response coefficients ${\bf\Lambda}$ and ${\bf{\tilde\Lambda}}$
----------------------------------------------------------------------
In what follows, we focus on the adiabatic regime, in which the driving frequency $\Omega$ is very low. As we presented in our previous works [@us:prb; @Ludovico:entropy], to which we refer the reader for further details, we can expand the Floquet components ${\cal G}(n,\veps)$ up to first order in $\Omega$ as
\[lowfreq\] [G]{}(n,)\~\^\_0e\^[int]{}.
Here $\tau=2\pi/\Omega$ is the driving period, and \[frozeng\] G\^f(t,)=(-\_0-V(t)-i)\^[-1]{} is the frozen Green’s function, which corresponds to the equilibrium solution of the Dyson equation [@dyson-1] at a given [*frozen*]{} time $t$.
Within the adiabatic regime, the departures of the temperature and chemical potential of the floating contact from those of the reservoir, $\delta T_c$ and $\delta\mu_c$, are proportional to $\hbar\Omega$. Hence, we can also evaluate Eqs. (\[chargesm\]), (\[heat1sm\]) and (\[heat2sm\]) within linear response in these quantities by expanding \[fermi1\] f\_c(\_n)\~f()-\_f n-\_f\_c-\_fT\_c where $f_r(\veps)=f(\veps)$ is the Fermi distribution of the reservoir.
Then, by using the expansions for slow driving (\[lowfreq\]) and (\[fermi1\]) in the expressions of the charge and heat fluxes, we can compute the linear response coefficients ${\bf \Lambda}$ as defined in Eq. (\[linearresp\]) of the main text
\_[ij]{}(t) =
(t,) \_f d& j3\
\
-(-)\^[i-1]{} \_f(t,) \_f d& j=3,
where $\rho_f(t,\veps)=\vert G^f(t,\veps)\vert^2(\Gamma+\Gamma_c)$ is the total [frozen]{} density of states of the quantum dot, and $\mathcal{T}(t,\veps)=\vert G^f(t,\veps)\vert^2\Gamma_c\Gamma$ is the transmission probability.
The same procedure can be applied to the heat flux in Eq. (\[heat2sm\]), whose definition does not take into account the energy stored in the tunneling region. In this case, we replace $\Lambda^{2j}\rightarrow \tilde{\Lambda}^{2j}$, so that \_c(t)=\_[j=1]{}\^3\_[2j]{}(t)X\_j(t). For this different definition of the heat, we find that $\tilde{\Lambda}_{21}(t)=\Lambda_{21}(t)$ and $\tilde{\Lambda}_{22}(t)=\Lambda_{22}(t)$, while the coefficient describing the pumping of heat changes as \[crl\] \_[23]{}(t)= -\_f(t,).
Chemical potential and temperature of the floating contact in the zero temperature limit
----------------------------------------------------------------------------------------
In the very low temperature limit of the reservoir, when $T\rightarrow 0$, an analysis of the fluxes within linear response turns out not to be appropriate anymore since second order contributions could become dominant. In this case, instead of Eq. (\[linearresp\]), the fluxes should be expanded as J\_i=\_[j=1]{}\^[3]{}(\_[ij]{}X\_j+\_[mj]{}L\^i\_[mj]{}X\_mX\_j), where $\overrightarrow{{\bf L}}$ is a vector composed by matrices which capture the second order terms. The extreme situation occurs at $T=0$, in which absolutely all the linear response coefficients of the heat $\dot{Q}_c$ vanish ($\Lambda_{2j}=0$). In what follows, we focus on that case and compute both heat and particle fluxes entering the floating contact, in order to study if Eq. (\[sol\]) remain valid when the temperature of the reservoir is close to zero. For the heat flux we find \_c\^[T=0]{}(t)=L\^2\_[22]{}(t)T\_c(t)\^2+L\_[13]{}\^2(t)\_c(t)+L\_[33]{}\^2(t)()\^2, where $$\begin{aligned}
\label{cheatto}
L^2_{22}(t)& = & -\frac{\pi^2}{3h}\mathcal{T}(t,\mu)\nonumber\\
L^2_{13}(t)& = & -\frac{\Gamma_c\Gamma\;\dot{V}}{(\Gamma+\Gamma_c) h\Omega}\frac{\rho_f(t,\mu)^2}{2},\\
L^2_{33}(t)& = & -\frac{\dot{V}}{\Gamma \Omega}L^2_{13}(t)\nonumber\end{aligned}$$ and all other coefficients are zero, $L_{11}^2=L_{12}^2=L_{23}^2=0$. However for the particle current, unlike the heat flux, only the coefficient $\Lambda_{12}=0$ at zero temperature while first order contributions in $\delta \mu_c$ and $\hbar\Omega$ remain. Thus, we can express the particle current at lowest order in the affinities as \_c\^[T=0]{}(t)=\_[11]{}(t)\_c(t)+\_[13]{}(t)+L\_[22]{}\^1(t)T\_c(t)\^2, with $$\begin{aligned}
\label{cchargeto}
L^1_{22}(t)& = & -\frac{\pi^2}{3h}\partial_\veps\mathcal{T}(t,\mu).\end{aligned}$$ Here, it is worth to mention that $L_{12}^1=L_{23}^1=0$, so that $L_{22}^1\delta T_c^2$ is the only lowest order contribution in $\delta T_c$.
Similarly, the chemical potential and the temperature of the floating contact can also be found by the condition $\dot{N}_c^{T=0}=\dot{Q}_c^{T=0}=0$. In this occasion, the vanishing fluxes condition leads to a set of equation which is linear en $\delta T_c^2$ and $\delta \mu_c$, with solution $$\begin{aligned}
&&\delta T_c(t)^2 = \frac{ L^2_{13}\Lambda_{13}- \Lambda_{11} L^2_{33}}{(\Lambda_{11}L_{22}^2-L^1_{22}L^2_{13}\hbar\Omega)} \;(\hbar\Omega)^2,\;\nonumber \\\nonumber\\\nonumber\\
&&\delta \mu_c(t) = \frac{L^1_{22} L^2_{33}(\hbar\Omega)^2- L^2_{22}\Lambda_{13} \hbar \Omega}{(\Lambda_{11}L_{22}^2-L^1_{22}L^2_{13}\hbar\Omega)}.\end{aligned}$$
From Eqs. (\[cheatto\]) and (\[coeff1\]), it is easy to notice that $L^2_{13}\Lambda_{13}- \Lambda_{11} L^2_{33}=0$, and then T\_c(t) = 0. Moreover, the relation between $L_{13}^2$ and $L_{33}^2$ in Eq. (\[cheatto\]) and $\Lambda_{13}=-\frac{\dot{V}}{\Gamma \Omega}\Lambda_{11}$ in (\[coeff1\]), lead to the solution \_c(t) = +(\^2).
Therefore, we find that Eq. (\[sol\]) obtained for $k_BT\gg\hbar\Omega$ within linear response regime, remain valid at zero temperature for which higher order contributions take place. This is a strong result, that shows the universality of the behavior of $\delta T
_c$ and $\delta \mu_c$ for any temperature of the reservoir.
[99]{} M. F. Ludovico, M. Moskalets, D. Sánchez, and L. Arrachea, Dynamics of energy transport and entropy production in ac-driven quantum electron systems, Phys. Rev. B [**94**]{}, 035436 (2016). M. Ludovico, L. Arrachea, M. Moskalets, and D. Sánchez, Periodic Energy Transport and Entropy Production in Quantum Electronics, Entropy [**18**]{}, 419 (2016). L. Arrachea, Exact Green’s function renormalization approach to spectral properties of open quantum systems driven by harmonically time-dependent fields, Phys. Rev. B [**75**]{}, 035319 (2007). L. Arrachea and M. Moskalets, Relation between scattering matrix and Keldysh formalisms for quantum transport driven by time-periodic fields, Phys. Rev. B [**74**]{}, 245322 (2006). M. F. Ludovico, J.S. Lim, M. Moskalets, L. Arrachea, and D. Sánchez, Dynamical energy transfer in ac-driven quantum systems, Phys. Rev. B [**89**]{}, 161306 (2014).
|
---
abstract: |
We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology [@Giles2015Acta] to calculate expectations with respect to the invariant measures of ergodic SDEs. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that, when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform in time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, one can approximate expectations with respect to the invariant measure in an unbiased way without the need of a Metropolis-Hastings step. In addition, a root mean square error of $\mathcal{O}(\varepsilon)$ is achieved with $\mathcal{O}(\varepsilon^{-2})$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which however can be computationally intensive when applied to large data sets. Finally, we present a multi-level version of the recently introduced Stochastic Gradient Langevin (SGLD) method [@welling2011bayesian] built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $\mathcal{O}(\varepsilon^{-2}|\log
{\varepsilon}|^{3})$, in contrast to the complexity $\mathcal{O}(\varepsilon^{-3})$ of currently available methods. Numerical experiments confirm our theoretical findings.
author:
- 'Lukasz Szpruch[^1]'
- 'Sebastian J. Vollmer[^2]'
- 'Konstantinos C. Zygalakis[^3]'
- 'Michael B. Giles[^4]'
bibliography:
- 'biblio.bib'
- 'abd\_biblio.bib'
- 'complete.bib'
- 'complete1.bib'
- 'referencesB.bib'
- 'particles.bib'
- 'v\_stab.bib'
- 'refs.bib'
title: Multilevel Monte Carlo methods for the approximation of invariant distribution of Stochastic Differential Equations
---
Introduction
============
We consider a probability density $\pi(x)\propto e^{U(x)}$ on ${\mathbb{R}}^d$ with the unknown normalising constant. A typical task is the computation of the following quantity $$\label{eq:expi}
\pi(g):={\mathbb{E}}_\pi g = \int_{{\mathbb{R}}^d} g(x) \pi(dx), \quad g\in L^1(\pi).$$ Even if $\pi(dx)$ is given in an explicit form, quadrature methods, in general, are inefficient in high dimensions. On the other hand probabilistic methods scale very well with the dimension and are often the method of choice. With this in mind, we explore the connection between dynamics of stochastic differential equations (SDEs) $$\label{eq:langevin}
dX_{t}= \nabla U(X_{t})dt+\sqrt{2}dW_{t}, \quad X_{0} \in \IR^{d},$$ and the target probability measure $\pi(dx)$. The key idea is that under appropriate assumptions on $U(x)$ one can show that the solution to is ergodic and has $\pi(dx)$ as its unique invariant measure [@Has80]. However, there exist a limited number of cases where analytical solutions to are available and therefore numerical approximations are used [@KlP92].
The numerical analysis approach is to discretize . Extra care is required when $U$ is not globally Lipschitz [@MSH02; @DT02; @RT96; @TSS00; @HJP11]. One drawback of he numerical analysis approach is that it might be the case that even though is geometrically ergodic, the corresponding numerical discretization might not be [@RT96]. The numerical analysis approach also introduces bias because the numerical invariant measure does not coincide with the exact one in general [@TaT90; @AVZ14], resulting hence in a biased estimation of $\pi(g)$ in . Furthermore, if one uses the Euler-Maruyama method to discretize , then computational complexity[^5] of $\mathcal{O}(\varepsilon^{-3})$ is required for achieving a root mean square error of order $\mathcal{O}(\varepsilon)$ in the approximation of $\eqref{eq:expi}$. Furthermore, even if one mitigates the bias due to numerical discretization by a series of decreasing time steps in combination with an appropriate weighted time average of the quantity of interest [@DP02], the computational complexity still remains $\mathcal{O}(\varepsilon^{-3})$ [@Teh2016sgld].
An alternative way of sampling from $\pi$ exactly, so that it does not face the bias issue introduced by pure discretization of , is by using the Metropolis-Hastings algorithm [@Ha70]. We will refer to this as the computational statistics approach. The fact that the Metropolis Hastings algorithm leads to asymptotically unbiased samples of the probability measure is one of the reasons why it has been the method of choice in computational statistics. Moreover, unlike the numerical analysis approach, computational complexity of $\mathcal{O}(\varepsilon^{-2})$ is now required for achieving root mean square error of order $
\mathcal{O}(\varepsilon)$ in the (asymptotically unbiased) approximation of $\eqref{eq:expi}$ (see Table \[tab:sum\]). I like the table but would delete comments, also we need to be careful with unbiasing.
We notice that MLMC [@Giles2015Acta] and the unbiasing scheme [@Rhee2012; @Rhee2015] are able to achieve the $\mathcal{O}(\varepsilon^{-2})$ complexity for computing expectations of SDEs on a fixed time interval $[0,T]$, despite using biased numerical discretistations. We are interested in extending this approach to the case of ergodic SDEs on the time interval $[0, \infty)$, see also discussion in [@Giles2015Acta].
[1]{}[@|@|>p[0.1]{}|>p[0.15]{}|>p[0.15]{}|>p[0.14]{}|>p[0.14]{}|]{} [\
]{} Name & Standard MCMC & Time average of $m$-order methods & MLMC for SGLD & MLMC framework for SDE[\
]{} Comment & e.g. RWM, HMC, MALA if geometrically ergodic & if discretised SDE is geometrically ergodic & [\
]{}
Cost & $\varepsilon^{-2}$ & $\varepsilon^{-\frac{2(2m+1)}{2m}}$ & $\varepsilon^{-2}\left|\log\varepsilon\right|^{3}$ & $\varepsilon^{-2}$[\
]{} Bias & by choice of burn-in is asymptotically neglegible & vanishing for appropriately decreasing stepsize otherwise & cannot be debiased & can be debiased by randomisation[\
]{} Reference for Cost & [@explicitbdd; @latuszynski2012nonasymptotic] & [@MST10][@teh2014sgldB] & [\
]{} Parallel computation& & [\
]{} [\
]{} Per step cost & $N$ & & [\
]{}
A particular application of is when one is interested in approximating the posterior expectations for a Bayesian inference problem. More precisely, if the data $y_{i}$ are i.i.d $U(x)$ becomes $$\nabla U(x)=\nabla\log{\pi_{0}(x)}+\sum_{i=1}^{N}\nabla\log{\pi(y_{i}|x)},\label{eq:bay_iid}$$ with $\pi_{0}(x)$ being the prior distribution of $x$. When dealing with problems where the number of data items $N \gg 1$ is large, both the standard numerical analysis and the MCMC approaches suffer due to the high computational cost associated with calculating the likelihood terms $\nabla \log{\pi(y_{i}|x)}$ over each data item $y_{i}$. One way to circumvent this problem is the stochastic gradient Langevin algorithm (SGLD) introduced in [@welling2011bayesian], which replaces the sum of the $N$ likelihood terms by an appropriately reweighted sum of $s \ll N$ terms. This leads to the following recursion formula $$\label{eq:SGLD}
x_{k+1}=x_{k}+h\left(\nabla\log{\pi_{0}(x_{k})}+\frac{N}{s}\sum_{i=1}^{s}\nabla\log{\pi(y_{\tau_{i}^{k}}\vert x_{k})}\right)+\sqrt{2h}\xi_{k}$$ where $\xi_{k}$ is a standard Gaussian random variable on $\mathbb{R}^{d}$ and $\tau^{k}=(\tau_{1}^{k},\cdots,\tau_{s}^{k})$ is a random subset of $[N]=\{1,\cdots,N\}$, generated for example by sampling with or without replacement from $[N]$. Notice, that this corresponds to a noisy Euler discretisation, which for fixed $N,s$ still has computational complexity $\mathcal{O}(\varepsilon^{-3})$ as discussed in [@Teh2016sgld; @teh2014sgldB]. In this article, we are able to show that careful coupling between fine and coarse paths allows the application of the MLMC framework and hence reduction of the computational complexity of the algorithm to $\mathcal{O}(\varepsilon^{-2}(\log{\varepsilon})^{3})$.
In summary the main contributions of this paper are:
1. Extension of the MLMC framework to the time interval $[0,\infty)$ for when $U$ is strongly concave.
2. A convergence theorem that allows the estimation of the MLMC variance using uniform in time estimates in the Wasserstein metric for a variety of different numerical methods.
3. A new method for (unbiasedly) estimating expectations with respect to the invariant measures without the need of accept/reject steps (as in MCMC). The methods we propose can be better parallelised than MCMC.
4. The application of this scheme to stochastic gradient Langevin dynamics (SGLD) which reduces the complexity of $\varepsilon^{-3}$ to $\varepsilon^{-2}\left|\log\varepsilon\right|^{3}$ much closer to the standard $\varepsilon^{-2}$ complexity of MCMC.
The rest of the paper is organised as follows. In Section \[sec:prel\] we describe the standard MLMC framework, discuss the contracting properties of the true trajectories of and describe an algorithm for applying MLMC with respect to time $T$ for the true solution of . In Section \[sec:mlmc\] we present the new algorithm, as well as a framework that allows proving its convergence properties for a numerical method of choice. In Section \[sec:ex\] we present two examples of suitable numerical methods, while in Section \[sec:sgld\] we describe a new version of SGLD with complexity $\varepsilon^{-2}\left|\log\varepsilon\right|^{3}$. We conclude in Section \[sec:num\] where a number of relevant numerical experiments are described.
Preliminaries {#sec:prel}
=============
In Section \[subsec:mlmc\] we revise the classic, finite time, MLMC framework, while in Section \[subsec:contract\] we state key the asymptotic properties of solutions of when $U$ is strongly concave.
MLMC with fixed terminal time T. {#subsec:mlmc}
--------------------------------
Fix $T>0$ and consider the problem of approximating ${\mathbb{E}}[g(X_T)]$ where $X_T$ is a solution of the SDE and $g:{\mathbb{R}}^d\rightarrow {\mathbb{R}}$. A classical approach to this problem consists of constructing a biased (bias arise due to time-discretization) estimator of the form $$\label{eq:MC}
\frac{1}{N}\sum_{i=1}^N g((x_T^{M})^{(i)})$$ where $\{x_T^{M}\}$, is a discrete time approximation of over $[0,T]$ with $M$ number of time steps. A central limit theorem for the estimator has been derived in [@MR1384358], and it was shown its computational complexity is $\mathcal{O}(\varepsilon^{-3})$, for the root-mean-square error $\mathcal{O}(\varepsilon)$ (as opposed to $\mathcal{O}(\varepsilon^{-2})$ that can be obtained if we could sample $X_{T}$ without the bias). The recently developed Multilevel Monte Carlo approach allows recovering optimal complexity $\mathcal{O}(\varepsilon^{-2})$, despite the fact that the estimator used builds on biased samples. This is achieved by exploiting the following identity [@Giles2015Acta; @MR2187308] $$\label{eq:telescoping}
{\mathbb{E}}[g_L] = {\mathbb{E}}[g_0] + \sum_{\ell=1}^{L} {\mathbb{E}}[ g_{\ell} - g_{\ell-1} ],$$ where $g_\ell:=g(x_T^{M_\ell})$ with $\{x_T^{M_\ell}\}$, $\ell=0\ldots L$, is the discrete time approximation of over $[0,T]$ with $M_l$ number of time steps, $M_\ell h_\ell= T$. This identity leads to the following unbiased estimator of ${\mathbb{E}}[g(x_T^{M_L})]$, $$\begin{aligned}
\frac{1}{N_0} \sum_{i=1}^{N_0} g_0^{(i,0)} + \sum_{\ell=1}^{L}\left\{ \frac{1}{N_\ell} \sum_{i=1}^{N_\ell}
( g_{\ell}^{(i,\ell)} - g_{\ell-1}^{(i,\ell)} ) \right\}, \end{aligned}$$ where $g_{\ell}^{(i,\ell)}= g((x_T^{M_\ell})^{(i)})$ are independent samples at level $\ell$. The inclusion of the level $l$ in the superscript $(i,\ell)$ indicates that independent samples are used at each level $l$. The efficiency of MLMC lies in the coupling of $g_{\ell}^{(i,\ell)}$ and $g_{\ell-1}^{(i,\ell)}$ that results in small $\mathrm{Var}[g_{\ell} - g_{\ell-1} ]$. In particular, for SDE one can use the same Brownian path to simulate $g_{\ell}$ and $g_{\ell-1}$ which, through the strong convergence property of the scheme, yields an estimate for $\mathrm{Var}[g_{\ell} - g_{\ell-1} ]$.
By solving a constrained optimization problem (cost&accuracy) one can see that reduced computational cost (variance) arises since the MLMC method allows one to efficiently combine many simulations on low accuracy grids (at a corresponding low cost), with relatively few simulations computed with high accuracy and high cost on very fine grids. It is shown in Giles [@Giles2015Acta] that under the assumptions[^6] $$\begin{aligned}
\label{ml_ass}
\bigl|{\mathbb{E}}[g- g_{\ell} ]|\leq c_1h_\ell^{\alpha},\quad \mathrm{Var}[g_{\ell} - g_{\ell-1} ]\leq c_2 h_\ell^{\beta}, \end{aligned}$$ for some $\alpha\geq 1/2,$ $\beta>0,$ $c_1>0$ and $c_2>0,$ the computational complexity of the resulting multi-level estimator with the accuracy $\varepsilon$ is proportional to $$\begin{aligned}
\mathcal{C}=
\begin{cases}
\varepsilon^{-2}, & \beta>\gamma, \\
\varepsilon^{-2}\log^2(\varepsilon), & \beta=\gamma, \\
\varepsilon^{-2-(1-\beta)/\alpha}, & 0<\beta <\gamma
\end{cases}\end{aligned}$$ where the cost of the algorithm is of order $h^{-\gamma}$. Typically, constants $c_{1},c_{2}$ grow exponentially as they follow from classical finite time weak and strong convergence analysis of the numerical schemes. The aim of this paper is to establish bounds uniformly in time, i.e. $$\begin{aligned}
\label{uml_ass}
\sup_{T>0}\bigl|{\mathbb{E}}[g - g_{\ell} ]|\leq c_1 h_\ell^{\alpha},\quad
\sup_{T>0}\mathrm{Var}[g_{\ell} - g_{\ell-1} ]\leq c_2 h_\ell^{\beta}. \end{aligned}$$
The reader may notice that in the regime when $\beta>\gamma$, the computationally complexity of $\varepsilon^{-2}$ coincides with that of an unbiased estimator. However, the MLMC estimator as defined here is still biased, with the bias being controlled by the choice of level parameter $L$. In Section \[subsec:unbias\] we will demonstrate that in fact it is possible eliminate the bias by a clever randomisation trick.
Properties of ergodic SDEs with strongly concave drifts {#subsec:contract}
-------------------------------------------------------
Consider the SDE and let $U$ satisfy the following condition
- For any $x,y \in {\mathbb{R}}^d$ there exists constant $m$ s.t $$\begin{aligned}
\label{eq:ing}
\left\langle \nabla U(y) - \nabla U(x),y-x\right\rangle & \leq - m | x-y| ^{2},\end{aligned}$$
which is also known as a one-side Lipschitz condition. Condition **HU0** is satisfied for strongly concave potential, i.e when for any $x,y \in {\mathbb{R}}^d$ there exists constant $m$ s.t $$U(y) \leq U(x)+\left\langle \nabla U(x),y-x\right\rangle - \frac{m}{2}| x-y| ^{2}.$$ Condition **HU0** ensures the contraction needed to establish uniform in time estimates. For the transparency of the exposition we introduce the following flow notation for the solution to , starting at $X_{0}=x$ $$\label{eq:flow}
\psi_{s,t,W}(x) := x + \int_s^t\nabla U(X_{r})dr + \int_s^t \sqrt{2}dW_{r},\quad x\in\IR^{d}.$$ The theorem below demonstrates that solutions to driven with the same Brownian motion, but with different initial conditions enjoy a exponential contraction property.
\[thm:contract\] Let $(W(t))_{t\geq 0})$ be a standard ${\mathbb{R}}^d$ Brownian Motion. Take random variable $Y_0,X_0\in \IR^{d}$ and define $X_T=\psi_{0,T,W}(X_0)$ and $Y_T=\psi_{0,T,W}(Y_0)$. Then $$\label{eq:contract_true}
\IE |X_{T}-Y_{T}|^{2} \leq \IE |X_{0}-Y_{0}|^{2} e^{-2mT}$$
The result follows from Itô’s formula. Indeed we have $$\frac{1}{2}e^{2mt} |X_{t}-Y_{t}|^{2}=\frac{1}{2}|X_{0}-Y_{0}|^{2}
+\int_{0}^{t}\left( m e^{2ms } |X_{s}-Y_{s}|^{2} + e^{2ms}\left\langle \nabla U(X_{s}) - \nabla U(Y_{s}),X_{s}-Y_{s}\right\rangle \right) ds.$$ Assumption **HU0** yields $${\mathbb{E}}|X_{t}-Y_{t}|^{2} \leq e^{-2mT}\IE |X_{0}-Y_{0}|^{2},$$ as required.
#### Digression on $s$-Wasserstein norm.
The $s$-Wasserstein distance between measures $\nu_{1}$ and $\nu_{2}$ defined on a Polish space $E$, is given by $$\begin{aligned}
\mathcal{W}_{s}(\nu_{1},\nu_{2}) & = & \left(\inf_{\pi\in\Gamma(\nu_{1},\nu_{2})}\int_{E\times E}d^{s}(x,y)\pi(dx,dy)\right)^{\frac{1}{s}},\end{aligned}$$ with $\Gamma(\nu_{1},\nu_{2})$ being the set of couplings of $\nu_{1}$ and $\nu_{2}$ (all measures on $E\times E$ with marginals $\nu_{1}$ and $\nu_{2}$) and $d(x,y)=| x-y| $.
We denote $\mathcal{L}(\psi_{0,t,W}(x))= P_{t}(x,\cdot)$. That is $P_{t}$ is the transition kernel of the SDE . Since the choice of the same driving BM in is an example of coupling implies $$\mathcal{W}_{2}\left(P_{t}(x,\cdot),P_{t}(y,\cdot)\right)\leq| x-y| \exp\left(-mt\right)\label{eq:wassersteincontracting}$$ Consequently $P_{t}$ only has one unique invariant distribution and thus the process is ergodic [@weakHarris]. Although optimality of the coupling in the definition of Wasserstein norm is an appealing property, for the practical considerations one should only consider coupling that are feasible to implement. That is the philosophy we follow in this paper. For a more general discussion of implementable couplings we refer the reader to [@Agapiou2014unbiasing; @giles2014antithetic].
For the MLMC in $T$, coupling with the same Brownian motion is not enough, as in general solutions to SDEs are $1/2$-Hölder continuous, [@krylov2008controlled], i.e for any $t>s>0$ there exists a constant $C>0$ such that $$\label{eq:holder}
{\mathbb{E}}|X_t - X_s |^2 \le C |t-s|$$ and it is well known that this bound is sharp. As we shall see later this bound will not lead to an efficient MLMC implementation. However, by suitable coupling of the SDE solutions on time intervals of length $T$ and $S$, $T > S$, respectively, we will be able to take advantage of the exponential contraction property obtained in Theorem \[thm:contract\]. This idea is clearly demonstrated in Figure \[fig:couplings\] and described for the context of simulating SDEs below.
[0.43]{} ![Shifted couplings[]{data-label="fig:couplings"}](correctcouple.png "fig:"){width="\textwidth"}
[0.43]{} ![Shifted couplings[]{data-label="fig:couplings"}](wrongcouple.png "fig:"){width="\textwidth"}
To couple processes with different terminal times $T_i$ and $T_j$, $i\neq j$, we exploit the time homogeneous Markov property of the flow . More precisely, we construct a pair of solutions to $(X^{(f,\ell)},X^{(c,\ell)})$, $i\geq0 $ which we refer to as fine and coarse paths, such that $$\label{eq:law}
\mathcal{L}(X^{(f,\ell)})=\mathcal{L}(X_{T_{\ell}}), \quad \mathcal{L}(X^{(c,\ell)})=\mathcal{L}(X_{T_{\ell-1}}), \quad \forall i\geq 0,$$ and $${\mathbb{E}}|X^{(f,\ell)} - X^{(c,\ell)} |^2 \leq {\mathbb{E}}|X_{T_\ell} - X_{T_{\ell-1}} |^2.$$ Following [@Rhee2012; @Rhee2015; @Agapiou2014unbiasing; @Giles2015Acta] we propose to
- First obtain solution to over $[0,T_\ell - T_{\ell-1}]$. We take $X^{(f,\ell)}(0) = \psi_{0,(T_\ell - T_{\ell-1}),\tilde{W}}(X(0)) $
- Next couple fine and coarse paths on the remaining time interval $[0, T_{\ell-1}]$ using the same Brownian motion (W) i.e $$X^{(f,\ell)}(T_{\ell-1}) = \psi_{0,T_{\ell-1},W}(X^{(f,\ell)}(0)),
\quad
X^{(c,\ell)}(T_{\ell-1}) = \psi_{0,T_{\ell-1},W}(X(0)).$$ Since $\nabla U(\cdot)$ in is time homogenous and consequently so is corresponding transition kernel $\mathcal{L}(\psi_{0,t,W}(x))= P_{t}(x,\cdot)$ , condition holds.
Theorem \[thm:contract\] yields $$\label{eq:sdetime}
\IE|X^{(f,\ell)}(T_{\ell-1}) - X^{(c,\ell)}(T_{\ell-1})|^{2} \leq \IE|X^{(f,\ell)}(0)-X(0)|^{2} e^{-2mT_{\ell-1}}.$$ Take $\rho>1$ and define $$\label{eq:T_choice_true}
T_{\ell} := \frac{\log{2}}{2m} \rho (\ell+1) \quad \forall \ell\geq 0.$$ In our case $g_{\ell}^{(i,\ell)}=g((X^{(f,\ell)}(T_{\ell-1}))^{(i)})$ and $g_{\ell-1}^{(i,\ell)} =g((X^{(c,\ell)}(T_{\ell-1}))^{(i)})$ and we assume that $g$ is globally Lipschitz. Hence $$\begin{aligned}
\IE|g(X^{(f,\ell)}(T_{\ell-1})) - g(X^{(c,\ell)}(T_{\ell-1}))|^{2} &\leq Lip(g)^{2}\IE|X^{(f,\ell)}(T_{\ell-1}) - X^{(c,\ell)}(T_{\ell-1})|^{2} \\
&\leq Lip(g)^{2}\IE|X^{(f,\ell)}(0)-X(0)|^{2} e^{-2mT_{\ell-1}}\\
& \leq Lip(g)^{2}\IE|X^{(f,\ell)}(0)-X_{0}|^{2} 2^{-\rho \ell} \\
& \leq Lip(g)^{2}C | T_{\ell}-T_{\ell-1} |2^{-\rho \ell}.
\quad \forall i\geq 0, \label{eq:variance_true_decay}\end{aligned}$$ where the last inequality follows from .
MLMC in $T$ for SDEs with biased samples {#sec:mlmc}
========================================
Having described a coupling algorithm with good contraction properties, we now present the main algorithm in Section \[subsec:alg\_descr\]. We then in Section \[subsec:unbias\] discuss an approach that allows to obtain unbiased estimators for the quantities of interest. Finally, in Section \[subsec:conv\_anal\] we present a general numerical analysis framework for proving the convergence of our algorithm.
Description of the algorithm {#subsec:alg_descr}
----------------------------
We now focus on the numerical discretization of the Langevin equation . In particular, we are interested in coupling the dicretizations of based on step size $h_{\ell}$ and $h_{\ell-1}$ with $h_{\ell}=h_{0}2^{-\ell}$. Furthermore, as we are interested in computing expectations with respect to the invariant measure $\pi(dx)$ we also increase the time endpoint $T_{\ell}\uparrow\infty$ which is chosen such that $\frac{T_{\ell}}{h_{0}}, \frac{T_{\ell}}{h_{\ell}}\in\mathbb{N}$. We illustrate the main idea using two generic discrete time stochastic processes $(x^h_k)_{k\in \mathbb{N}},(y^h_k)_{k\in \mathbb{N}}$ which can be defined as $$\label{eq:genpro}
x^h_{k+1}= S^f_{h,\xi_k}(x_k^h), \quad y^h_{k+1}=S^c_{h,\tilde{\xi}_k}(y_k^h)$$ where the operators $S^{f},S^{c}: {\mathbb{R}}^d \times {\mathbb{R}}_+ \times {\mathbb{R}}^{d\times m} \rightarrow {\mathbb{R}}^d $ are Borel measurable, and $\xi,\tilde{\xi}$ are random input to the algorithms (in the simplest case Gaussian random variables). For example for the Euler discretisation we have $$S^{Euler}_{h,\xi}(x)=x-h\nabla U(x)+\sqrt{2h} \xi.$$ We also use the notation $P_{h}(x,\cdot)=\mathcal{L}\left(S_{h,\xi}(x)\right)$ with $\xi\sim\mathcal{N}(0,I)$ for the corresponding Markov kernel.
The coupling arises by evolving both fine and course paths jointly, over a time interval of length $T_{\ell-1}$, by doing two steps for the finer level (with the time step $h_\ell$) and one on the coarser level (with the time step $h_{i-1}$) using the discretization of the same Brownian path. In particular, let $$K_{h_{\ell}}((x,y),\cdot)=\mathcal{L}\left(S^{f}_{h_{\ell},\xi_{2}}\circ S^{f}_{h_{\ell},\xi_{1}}(x),S^{c}_{h_{\ell-1},\frac{1}{\sqrt{2}}(\xi_{1}+\xi_{2})}(y)\right),$$ Notice that as in Section \[subsec:mlmc\] the coupling arises due to the same Gaussian random variables being used on the fine and coarse level. From now on when the processes $(x^h_{\cdot}),(y^h_{\cdot})$ are coupled we will use notation $(x^f_{\cdot}),(x^c_{\cdot})$ for $$\begin{aligned}
x_{k+\frac{1}{2}}^{f} & =S_{\frac{h}{2},\xi_{k+\frac{1}{2}}}^{f}\left(x_{k+\frac{1}{2}}^{f}\right),\label{eq:numOperator}\\
x_{k+1}^{c} & =S_{h,\tilde{\xi}_{k+1}}^{c}(x_{k}^{c}) \label{eq:numOperator2}.\end{aligned}$$ The algorithm generating $({x^f_k })_{2k\in \mathbb{N}}$ and $({x^c_k })_{k\in \mathbb{N}}$ is presented in Algorithm \[alg:CouplingLangevinDiscretisation\].
1. Set $x_{0}^{(f,\ell)}=x_{0}$, then simulate according to $P_{h_{\ell}}$ up to $x_{\frac{T_{\ell}-T_{\ell-1}}{h_{\ell}}}^{(f,\ell)}$;
2. Set $x_{0}^{(c,\ell)}=x_{0}$ and $x_{0}^{(f,\ell)}=x_{\frac{T_{\ell}-T_{\ell-1}}{h_{\ell}}}^{(f,\ell)}$, then simulate $(x_{\cdot}^{(f,\ell)},x_{\cdot}^{(c,\ell)})$ jointly as $$\label{eq:langevincoupling}
\left(x_{k+1}^{(f,\ell)},x_{k+1}^{(c,\ell)}\right)=\left(S^{f}_{h_{\ell},\xi_{k,2}}\circ S^{f}_{h_{\ell},\xi_{k,1}}(x_{k}^{(f,\ell)}),S^{c}_{h_{\ell-1},\frac{1}{\sqrt{2}}\left(\xi_{k,1}+\xi_{k,2}\right)}(x_{k}^{(c,\ell)})\right).$$
3. Set $$\Delta^{(i)}_{\ell}:=g\left( \left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(f,\ell)} \right)^{(i)}\right)-g\left(\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(c,\ell)}\right)^{(i)}\right)$$
Unbiasing Scheme {#subsec:unbias}
----------------
We are interested in calculating $$\lim_{\ell\rightarrow\infty}\IE(g_\ell)=\pi(g),$$ where $g^{(i,\ell)}_\ell:=g\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(i)}\right)$. As in the MLMC construction we take $$\mathbb{E}(\Delta_{\ell})=\begin{cases}
\mathbb{E}(g_\ell)-\mathbb{E}(g_{\ell-1}) & \ell\ge1\\
\mathbb{E}(g_\ell) & \ell=0
\end{cases}\label{eq:telescopicMarkov}$$ It is clear that $$\pi(g)=\sum_{\ell=1}^{\infty}\IE(\Delta_{\ell}).$$ MLMC methods would result in truncating this sum at some level $L$, where the choice of deterministic $L$ corresponds to time $T_{L}$ and is dictated by the error/bias tolerance. It turns out that this truncation can be randomised. This idea goes back to von Neumann and has been systematically analysed in [@Rhee2012; @Rhee2015; @ViholaUnbiased2015] and further extended in[@Agapiou2014unbiasing]. The construction goes as follows, consider $$Z=\sum_{j=0}^{N}\frac{\Delta_{j}}{\mathbb{P}(N\ge j)}$$ with $N$ an independent integer valued random variable. Provided that we can apply Fubini’s theorem, $Z$ is unbiased which can be seen in the following calculation $${\mathbb{E}}\left[\sum_{j=0}^{N}\frac{\Delta_{j}}{P(N\geq j)}\right]={\mathbb{E}}\left[\sum_{j=0}^{\infty}\frac{1_{\{N\geq j\}}\Delta_{i}}{P(N\geq j)}\right]=\sum_{j=0}^{\infty}{\mathbb{E}}(\Delta_{j})= \pi(g).$$ The aim is to choose the distribution of $N$ such that $Z$ has finite second moment (Fubini’s theorem applies and the estimator has finite variance) and finite expected computing time (it is viable in the computational setting). Let $c_{l}$ be the expected computing time of $\Delta_{l}$, then the expected computing time $c$ of $Z$ satisfies $$\label{eq:u1}
\mathbb{E}(c)=\mathbb{E}\left(\sum_{j=0}^{N}c_{j}\right)=\sum_{j=0}^{\infty}c_{j}P(N\geq j).$$ The second moment of $Z$ is finite as long as $$\sum_{i\leq l}\frac{| \Delta_{i}| _{2}| \Delta_{l}| _{2}}{\mathbb{P}\left(N\ge i\right)}<\infty.\label{eq:unbiasingmomentcondition}$$ see Proposition 6, [@Rhee2012; @Rhee2015]. This method can also be applied in the present setting. Taking $h_\ell=2^{-\ell}$ in one can see that for to be finite and to hold one needs $\beta>\gamma$ (MLMC variance vs cost ).
Convergence analysis {#subsec:conv_anal}
--------------------
We will now present a general theorem for estimating the bias and the variance between levels given by in the MLMC set up. We refrain from prescribing the exact dynamics of $(x^h_k)_{k\geq 0}, (y^h_k)_{k\geq 0}$ in , as we seek general conditions allowing the construction of uniform in time approximations of in the Wasserstein norm. The advantage of working in this general setting is that if we wish to work with more advanced numerical scheme than the Euler-Maruyama method (implicit, projected, adapted, randomise etc) it will be enough to verify simple conditions to see how the performance of the complete algorithm will be affected without the need of reproving all estimates again. We discuss specific methods that satisfy our assumptions in Section \[sec:ex\].
### Uniform estimates in time
\[def:bias\] Let $g\in C^r_K(\mathbb{R})$ (r-times differentiable function with compact support). We say that process $(x^h_k)_{k\in \mathbb{N}}$ converges weakly uniformly in time with order $\alpha>0$ to the solution of SDEs , if there exists a constant $c>0$ such that $$\sup_{t\geq 0}|\mathbb{E}[g(X_t)] - {\mathbb{E}}[g(x^h_{\lfloor t/h \rfloor})]|\leq c h^{\alpha}.$$
The difference in Definition \[def:bias\] from the normal weak convergence definition is that here the weak error estimate needs to hold uniformly in time. The assumption of smoothness of $g$ could be dropped by using Malliavin calculus techniques.
We are ready to extend the definition for the MLMC variance. However, as operators $S^f$ and $S^c$ in - need not be the same, extra care is required. This extra flexibility allows analysing various coupling ideas. Furthermore, for notational simplicity we have dropped the dependence of the numerical scheme on $h$ as it should be clear that for a fixed $h$, ${x^f_k }$ runs with time-step $h/2$ and ${x^c_k }$ runs with time-step $h$. We now have the following definition for the MLMC variance.
Let the operators in Equation satisfy that for all $x$ $$\begin{aligned}
& \mathcal{L}\left(S^{f}_{h,\xi}(x)\right)=\mathcal{L}\left(S_{h,\tilde{\xi}}^{c}(x)\right).\label{eq:W1}\end{aligned}$$ We say that the MLMC variance is of order $\beta>0$ if there exists a constant $c_V>0$ s.t. $$\label{eq:W2}
\sup_{t\geq 0}\mathbb{E}|g(x^{c}_{\lfloor t/h\rfloor}) - g(x^{f}_{\lfloor t/h\rfloor})|^2 \leq c_V h^{\beta}.$$
Notice that the condition trivially holds if $S^{f}=S^{c}$ i.e. the case where the same numerical method is applied both in the fine and the coarse level.
### Statement of sufficient conditions
We now discuss the necessary conditions imposed on a numerical method applied to that ensure that the estimates hold uniformly in time as described above. In doing so, it would be instructive to decompose the global error analysis to the one step error and the regularity of the scheme. In order to do so, we introduce the notation $x^h_{k,x_s} $ for the process at time $k$ with initial condition $x_s$ at time $s<k$. If it is clear from the context which initial condition is used we just write $x^h_{k}$. We also define the conditional expectation operator as ${\mathbb{E}}_n[\cdot]:={\mathbb{E}}[\cdot | \mathcal{F}_{n}]$.
We now have the following definition
\[def:reg2\] We will say that one step operator $S : {\mathbb{R}}^d \times {\mathbb{R}}_+ \times {\mathbb{R}}^{d\times m} \rightarrow {\mathbb{R}}^d $ is $L^2$-regular **uniformly in time** if for any $x_n,y_n\in {\mathbb{R}}^d$, $\cF_n$-measurable random variables there exists constants $K,C>0$ and random variables $Z_{n+1}$, $\mathcal{R}_{n+1} \in \cF_{n+1}$ and $\mathcal{H}_{n} \in \cF_{n+1}$, such that $$\begin{aligned}
S_{h,\xi_{n+1}}(x_n) - S_{h,\xi_{n+1}}(y_n) = x_n - y_n + Z_{n+1} \end{aligned}$$ and $$\begin{aligned}
{\mathbb{E}}_n[ |S_{h,\xi_{n+1}}(x_n) - S_{h,\xi_{n+1}}(y_n) |^2] \le &(1 - Kh)|x_n - y_n |^2 + \mathcal{R}_{n+1} \\
{\mathbb{E}}_n[|Z_{n+1}|^2] \le & \mathcal{H}_{n} |x_n - y_n |^2 h, \end{aligned}$$ where $$\begin{aligned}
\label{eq:cond_sum}
| \mathcal{R}_{n} | &\le C(1 + |x_n|^q + |y_n|^q) h^{2\beta+1}, \quad q>0 \\
| \mathcal{H}_{n} | &\le C(1 + |x_n|^q + |y_n|^q)^{1/2}, \quad q>0 \nonumber\end{aligned}$$ and $C>0$.
We now introduce the set of the assumptions needed for the proof of the main convergence theorem
Consider two process $({x^f_k })_{2k\in \mathbb{N}}$ and $({x^c_k })_{k\in \mathbb{N}}$ obtained from the recursive application of the the operators $S^f_{h,\xi}(\cdot)$ and $S^c_{h,\xi}(\cdot)$ as defined in . We assume that
- For $q>1$ there exist functions $H_k^f:=H(k,f,q)$ and $H_k^c:=H(k,c,q)$ $$\begin{aligned}
{\mathbb{E}}|{x^f_k }|^q \leq H_k^f \quad \text{and} \quad {\mathbb{E}}|{x^c_k }|^q \leq H_k^f, \end{aligned}$$ with $ \sup_{k\geq 0}H_k^f<\infty$ and $ \sup_{k\geq 0}H_k^c<\infty$
- For any $x\in{\mathbb{R}}^d$ $$\mathcal{L}\left(S^{f}_{h,\xi}(x)\right)=\mathcal{L}\left(S_{h,\tilde{\xi}}^{c}(x)\right).$$
- The operator $S^f_{h,\xi}(\cdot)$ is $L^2$ regular uniformly in time.
\[rem:H2\] In the proof of the theorem condition $\textbf{H2}$ is applied with $x_n=x_n^f$ and $y_n=x_n^c$ (see definition ). In this context, random variables $Z_{n+1}(x_n^c, x_n^f)$, $\mathcal{R}_{n+1}((x_n^c, x_n^f))$ and $\mathcal{H}_{n+1}((x_n^c, x_n^f))$ are (measurable) functions of $(x_n^c, x_n^f)$. Consequently, under the condition **H0** we have $$\begin{aligned}
\label{eq:cond_sum2}
\sup_{n\geq 1}{\mathbb{E}}\left[ \sum_{i=1}^{n} \mathcal{R}_i \right] &\le C_{\mathcal{R}} h^{2\beta} \\
\sup_{n\geq 1}{\mathbb{E}}\left[ \mathcal{H}^2_n\right] &\le C_{\mathcal{H}}. \nonumber\end{aligned}$$ Furthermore the roles of $S^f_{h,\xi}(\cdot)$ and $S^c_{h,\xi}(\cdot)$ in **H2** can be swapped.
Below we present the main convergence result of this section. Using derived here estimates we can immediately estimate the rate of decay of MLMC variance.
\[th:convergence\] Take $({x^f_n })_{2n\in \mathbb{N}}$ and $({x^c_n })_{n\in \mathbb{N}}$ with $h\in(0,1]$ and assume that **H0**-**H2** hold. Moreover, assume that there exists constants $c_{s}>0,c_{w}>0$ and $\alpha \geq \frac{1}{2}$, $\beta \geq 0, p\geq 1$ with $\alpha \geq \frac{\beta}{2}$ such that for all $n\geq 1$ $$\label{eq:weak}
|{\mathbb{E}}_{n-1}({x^c_{n,{x^c_{n-1} }} } - {x^f_{n,{x^c_{n-1} }} })| \leq c_{w}(1+ |{x^c_{n-1} }|^p)h^{\alpha+1},$$ and $$\label{eq:strong}
{\mathbb{E}}_{n-1}[|{x^c_{n,{x^c_{n-1} }} } - {x^f_{n,{x^c_{n-1} }} } |^2] \leq c_{s}(1+|{x^c_{n-1} }|^{2p})h^{\beta+1}.$$ Fix $\zeta \in (0,1)$. Then the global error is bounded by $$\begin{aligned}
{\mathbb{E}}[({x^c_{T/h,x_{0}} } - {x^f_{T/h,y_0} } )^2] \leq & |x_0-y_0 |^2 e^{- K\zeta T} + \sum_{j=1}^{n}e^{ (j-(n-1))K\zeta h}{\mathbb{E}}(\mathcal{R}_{j-1}) \\
& + h^{\beta+1}\sum_{j=1}^{n} C_j(\zeta) e^{(j-(n-1))K\zeta h},
\end{aligned}$$ where $C_n(\zeta):= \left(c_{s}H_{1,n}+
\frac{(4c_{w}+2)[4c_{w}H_{1,n}+c_{s}(H_{3,n}+2 H_{2,n})]}{4(1-\zeta) K}\right) $ with $H_{1,n}:=(1+{\mathbb{E}}[|{x^c_{n-1} }|^{2p}]), H_{2,n}:=(1+{\mathbb{E}}[|{x^c_{n-1} }|^{4p}])$, $H_{3,n}:= {\mathbb{E}}[\mathcal{H}_{n-1}] $.
We begin using the following identity $$\begin{aligned}
{x^c_{n,x_{0}} } - {x^f_{n,y_{0}} } =& {x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^f_{n-1} }} } \\
=& ({x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } ) + ( {x^f_{n, {x^c_{n-1} } } } - {x^f_{n, {x^f_{n-1} } } }).\end{aligned}$$ We will be able to deal with the first term in the sum by using equations and , while the second term will be controlled because of the $L^{2}$ regularity of the numerical scheme. In particular by squaring both sides in the equality above we have $$\begin{aligned}
|{x^c_{n,y_{0}} } - {x^f_{n,x_{0}} }|^2 =
|{x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } |^2 + |{x^f_{n, {x^c_{n-1} } } } - {x^f_{n, {x^f_{n-1} } } }|^2 +2 \langle {x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } }
, {x^c_{n-1} } - {x^f_{n-1} }+ Z_{n}\rangle.\end{aligned}$$ where in the last line we have used Assumption **H2**. Applying conditional expectation operator to both sides of the above equality $$\begin{aligned}
{\mathbb{E}}_{n-1}[|{x^c_{n,y_{0}} } - {x^f_{n,x_{0}} }|^2] = &
{\mathbb{E}}_{n-1}[|{x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } |^2]
+ {\mathbb{E}}_{n-1}[| {x^f_{n, {x^c_{n-1} } } } - {x^f_{n, {x^f_{n-1} } } }|^2] \\
& + 2 \langle {x^c_{n-1} } -{x^f_{n-1} }, {\mathbb{E}}_{n-1}[{x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } ] \rangle
+2 {\mathbb{E}}_{n-1} \langle Z_{n}, {x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } \rangle \\
\end{aligned}$$ Applying Cauchy-Schwarz inequality, and using the weak error estimate leads to $$\begin{aligned}
{\mathbb{E}}_{n-1}[|{x^c_{n,y_{0}} } - {x^f_{n,x_{0}} }|^2] \leq &
{\mathbb{E}}_{n-1}[|{x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } |^2]
+ {\mathbb{E}}_{n-1}[| {x^f_{n, {x^c_{n-1} } } } - {x^f_{n, {x^f_{n-1} } } }|^2] \\
& + 2 c_{w} h^{\alpha+1} | {x^c_{n-1} } -{x^f_{n-1} }| (1+ |{x^c_{n-1} }|^p) \\
& + 2 {\mathbb{E}}_{n-1}[ |Z_{n}|^2]^{1/2} {\mathbb{E}}_{n-1}[| {x^c_{n, {x^c_{n-1} } } } - {x^f_{n, {x^c_{n-1} } } } |^2]^{1/2}.
\end{aligned}$$ By assumptions **H0**-**H2**, and the strong error estimate we have $$\begin{aligned}
{\mathbb{E}}_{n-1}[|{x^c_{n,y_{0}} } - {x^f_{n,x_{0}} }|^2] \leq &
c_{s}(1+|{x^c_{n-1} }|^{2p})h^{\beta +1}
+ |{x^c_{n-1} }-{x^f_{n-1} }|^2 (1 - K h ) + \mathcal{R}_{n-1}\\
& + 2 c_{w} h^{\alpha+1} | {x^c_{n-1} } -{x^f_{n-1} }| (1+ |{x^c_{n-1} }|^p) \\
& + 2 \Big( {\mathbb{E}}_{n-1}[\mathcal{H}_{n}] |{x^c_{n-1} }-{x^f_{n-1} }|^2 h \Big)^{1/2}
\Big( c_{s}(1+|{x^c_{n-1} }|^{2p})h^{\beta +1}\Big)^{1/2} \\
\leq &
c_{s}(1+|{x^c_{n-1} }|^{2p})h^{\beta +1}
+ |{x^c_{n-1} }-{x^f_{n-1} }|^2 (1 - K h ) + \mathcal{R}_{n-1}\\
& + 2 c_{w} h^{\alpha+1} | {x^c_{n-1} } -{x^f_{n-1} }| (1+ |{x^c_{n-1} }|^p) \\
& + 2 \Big( |{x^c_{n-1} }-{x^f_{n-1} }|^2 h \Big)^{1/2}
\Big( c_{s} {\mathbb{E}}_{n-1}[\mathcal{H}_{n} (1+|{x^c_{n-1} }|^{2p}) ] h^{\beta +1}\Big)^{1/2}.
\end{aligned}$$ while taking expected values and applying Cauchy-Schwarz inequality and the fact that $\alpha \geq\frac{\beta}{2} $ and $h<1$ (and hence $h^{\alpha+1}\leq h^{\frac{\beta}{2} +1} $) gives $$\begin{aligned}
{\mathbb{E}}[|{x^c_{n,y_{0}} } - {x^f_{n,x_{0}} }|^2] \leq &
c_{s}(1+{\mathbb{E}}[|{x^c_{n-1} }|^{2p}])h^{\beta +1}
+ {\mathbb{E}}[|{x^c_{n-1} }-{x^f_{n-1} }|^2] (1 - K h ) + {\mathbb{E}}[\mathcal{R}_{n-1}]\\
& + 4 c_{w} {\mathbb{E}}[| {x^c_{n-1} } -{x^f_{n-1} }|^2 h ]^{1/2} {\mathbb{E}}[(1+ |{x^c_{n-1} }|^{2p}) h^{\beta+1}]^{1/2} \\
& + 2 {\mathbb{E}}\Big[ |{x^c_{n-1} }-{x^f_{n-1} }|^2 h \Big]^{1/2}
{\mathbb{E}}\Big[ c_{s}\mathcal{H}_{n-1}(1+|{x^c_{n-1} }|^{2p}) h^{\beta+1} \Big]^{1/2}.\end{aligned}$$ Now Young’s inequality gives that for any $\varepsilon>0$ $$\begin{aligned}
{\mathbb{E}}[|{x^c_{n-1} } -{x^f_{n-1} }|^2 h ]^{1/2} {\mathbb{E}}[(1+|{x^c_{n-1} }|^{2p})h^{\beta +1}]^{1/2}
& \leq \varepsilon {\mathbb{E}}[({x^c_{n-1} } -{x^f_{n-1} })^2]h + \frac{1}{4\varepsilon}{\mathbb{E}}[(1+|{x^c_{n-1} }|^{2p})]h^{\beta +1} \\
{\mathbb{E}}\Big[ |{x^c_{n-1} }-{x^f_{n-1} }|^2 h \Big]^{1/2}
{\mathbb{E}}\Big[ c_{s}\mathcal{H}_{n-1}(1+|{x^c_{n-1} }|^{2p}) h^{\beta +1} \Big]^{1/2}
& \leq \varepsilon {\mathbb{E}}\Big[ |{x^c_{n-1} }-{x^f_{n-1} }|^2 \Big] h +
\frac{1}{4\varepsilon} {\mathbb{E}}\Big[ c_{s}\mathcal{H}_{n-1}(1+|{x^c_{n-1} }|^{2p})\Big]h^{\beta +1}.\end{aligned}$$ while $${\mathbb{E}}\Big[ \mathcal{H}_{n-1}(1+|{x^c_{n-1} }|^{2p})\Big]
\leq \frac{1}{2} {\mathbb{E}}\Big[ |\mathcal{H}_{n-1}|^2\Big] + {\mathbb{E}}\Big[ (1+|{x^c_{n-1} }|^{4p})\Big].$$ We now denote $\gamma_n:={\mathbb{E}}[|{x^c_{n,y_{0}} } - {x^f_{n,x_{0}} }|^2]$ and $H_{1,n}:=(1+{\mathbb{E}}[|{x^c_{n-1} }|^{2p}]), H_{2,n}:=(1+{\mathbb{E}}[|{x^c_{n-1} }|^{4p}])$, $H_{3,n}:= {\mathbb{E}}[\mathcal{H}_{n-1}] $. Whence $$\gamma_n \leq
\left(c_{s}H_{1,n}+\frac{4c_{w}H_{1,n}+c_{s}(H_{3,n}+ 2 H_{2,n})}{4\varepsilon} \right) h^{\beta +1}+{\mathbb{E}}[\mathcal{R}_{n-1}]
+ \gamma_{n-1} (1 -[K-(4c_{w}+2) \varepsilon] h )\\$$ We now choose $\varepsilon=\frac{(1-\zeta) K}{(4c_{w}+2)}$, $\zeta \in (0,1)$ and define $C_n(\zeta):= \left(c_{s}H_{1,n}+
\frac{(4c_{w}+2)[4c_{w}H_{1,n}+c_{s}(H_{3,n}+2 H_{2,n})]}{4(1-\zeta) K}\right) $ which gives $$\gamma_n \leq \left(1-K\zeta h \right)\gamma_{n-1} + C_n(\zeta) h^{\beta+1}+{\mathbb{E}}(\mathcal{R}_{n-1}).$$ We complete the proof by Lemma below.
\[lem:gronwall\] Let $a_n, g_n, c \geq 0$, $n \in \mathbb{N}$ be given. Moreover, assume that $1+\lambda>0$. Then, if $a_n \in \mathbb{R}$, $n \in \mathbb{N}$, satisfies $$a_{n+1} \leq a_n(1+ \lambda) + g_{n+1} + c, \quad n=0,1, \ldots,$$ then $$a_n \leq a_0 e^{n\lambda} + c \frac{e^{n\lambda}-1}{\lambda} +
\sum_{j=0}^{n-1} g_{j+1} e^{((n-1) - j)\lambda}, \qquad n=1, \ldots.$$
### Optimal choice of parameters
Theorem \[th:convergence\] is fundamental in terms of applying the MLMC as it guarantees that the estimate for the variance in holds. In particular, we have the following Lemma.
Assume all the assumption from \[th:convergence\] hold. \[lem:optimal\] Let $g(\cdot)$ be a Lipschitz function. Define $$h_\ell=2^{-\ell}, \quad T_{\ell} \sim -\frac{\beta}{K\zeta} \left(\log{h_{0}}+\ell\log{2} \right), \quad \forall \ell\geq 0.$$ Then resulting MLMC variance is given by. $$\text{Var}[\Delta_{\ell}] \leq 2^{-\beta \ell}, \quad \Delta_{\ell}=g\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(f,\ell)}\right)-g\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(c,\ell)}\right)$$
Since $g$ is a Lipschitz function it is enough to prove the bound for the mean square difference between the coarse and the fine path. In particular Theorem \[th:convergence\] implies $$\begin{aligned}
{\mathbb{E}}\left[\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(f,\ell)}-x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(c,\ell)}\right)^{2}\right] & \leq & {\mathbb{E}}\left|x_{\frac{T_{\ell}-T_{\ell-1}}{h_{\ell}}}
^{h_{\ell}}-x_{0}\right|^{2}\exp\left\{ -K\zeta T_{\ell-1}\right\} +C_{\mathcal{R}}h^{\beta}\\
& & +h^{\beta}{\mathbb{E}}\left(c_{s}H_{1}+\frac{(4c_{w}+2)[4c_{w}H_{1}+c_{s}(H_{3}+2H_{2})]}{4(1-\zeta)K}\right)\frac{2h}{K}\left(1-\exp\left\{ -K\zeta
T_{\ell-1}\right\} \right)\end{aligned}$$ where we have used $H_{1}:=\sup_{n\geq0}(H_{1,n}) ,H_{2}:=\sup_{n\geq0}(H_{2,n}),H_{3}:=\sup_{n\geq0}{\mathbb{E}}[\mathcal{H}_{n-1}]$. Assumption **H0** implies that ${\mathbb{E}}|x_{\frac{T_{\ell}-T_{\ell-1}}{h_{\ell}}}^{h_{\ell}}-x_{0} |^{2}\leq C$, for all $\ell$, which together with the fact that the numerical moments are bounded implies that there exist constants such that $${\mathbb{E}}\left[\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(f,\ell)}-x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(c,\ell)}\right)^{2}\right]\leq C_{1}\exp\left\{ -K\zeta T_{\ell-1}\right\} +C_{2}h_{\ell}^{\beta}.$$ In order to balance terms we choose $T_{\ell-1}=-\frac{1}{K\zeta}\log h_{\ell}^{\beta}=-\frac{\beta}{K\zeta}(\log_{h_{0}}+\ell\log2)$ and by ignoring the dependence on the constants $C_{1},C_{2}$ we obtain $${\mathbb{E}}\left[\left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(f,\ell)}-x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(c,\ell)}\right)^{2}\right]\leq C2^{-\ell\beta},$$ which completes the proof.
\[rem:complexity\] Unlike in the standard MLMC complexity theorem [@Giles2015Acta] where the cost of simulating single path is of order $\mathcal{O}(h_\ell^{-1})$, here we have $\mathcal{O}(h_\ell^{-1}|\log{h}|)$. This is due to the fact that terminal times are increasing with levels. For the case $h_\ell=2^{-\ell}$ this results in cost per path $\mathcal{O}(2^{-\ell}l)$ and does not exactly fit the complexity theorem in [@Giles2015Acta]. Clearly in the case when MLMC variance decays with $\beta >1$ we still recover the optimal complexity of order $\mathcal{O}(\varepsilon^{-2})$. However, in the case $\beta =1$ following the proof by Giles one can see that the complexity becomes $\mathcal{O}(\varepsilon^{-2}|\log{\varepsilon}|^{3})$.
In the proof above we have assumed that $K$ is independent of $h$, while we have also used crude bounds in order not to deal directly with all the individual constants, since these would be dependent on the numerical schemes used. In the example that follows we discuss an explicit calculation in the case of the Euler method.
In the case of the Euler-Maryama method as we see from the analysis[^7] in Section \[subsec:Euler\] $K=2m'-L^{2}h_{\ell}$, while $\mathcal{R}_{n}=0,\mathcal{H}_{n}=L$. Here $L$ is the Lipschitz constant of the drift $\nabla U(x)$. This implies that $$\begin{aligned}
{\mathbb{E}}[({x^c_{T/h,y_{0}} } - {x^f_{T/h,x_0} } )^2] \leq & |x_0-y_0 |^2 e^{- K\zeta T}
+ h^{\beta+1}\sum_{j=1}^{n} C_j(\zeta) e^{(j-(n-1))K\zeta h},
\end{aligned}$$ where $C_n(\zeta):= \left(c_{s}+
\frac{(4c_{w}+2)(4c_{w}+2c_{s}L)}{4(1-\zeta) K}\right)H_{1,n} $ with $H_{1,n}:=(1+{\mathbb{E}}[|{x^c_{n-1} }|^{2}])$. Furthermore, as we can see in Lemma \[lem:int\] we have that $${\mathbb{E}}[|x_{n}|^2] \leq {\mathbb{E}}|x_{0}|^2 \exp\{-(2 m' - L^2h) nh \} + 2 (b | \nabla U(0) |^2+h) \frac{1-\exp\{-(2 m' - L^2h) nh \}}{(2 m' - L^2h) h }$$ which gives $$\begin{aligned}
{\mathbb{E}}[({x^c_{T/h,y_{0}} } - {x^f_{T/h,x_0} } )^2] \leq C e^{- K\zeta T} + 2Ch^{\beta}(b | \nabla U(0) |^2+h) \sum_{j=1}^{n} \frac{1-e^{-K jh}}{K }e^{(j-(n-1))K\zeta h}
\end{aligned}$$ We now have that $$\sum_{j=1}^{n} \frac{1-e^{-K jh}}{K }e^{(j-(n-1))K\zeta h}=\frac{1}{K}\left(\frac{e^{2hK\zeta}(1-e^{K\zeta T})}{e^{hK\zeta}-1}-\frac{e^{hK(2\zeta-1)}e^{-K\zeta T} (-1+e^{K(\zeta-1)T})}{e^{hK(\zeta-1)}-1}\right)$$ The equation above is illustrative of the fact that as $T \rightarrow \infty$ the term in the sum converges to an explicit constant. Furthermore, it is trivial to see that with the same choice of $T$ as in the Lemma \[lem:optimal\] one obtains the desirable variance decay.
Examples of suitable methods {#sec:ex}
============================
In this section we present two (out of many) numerical schemes that fulfil the conditions of Theorem \[th:convergence\]. In particular, we need to verify that our scheme is $L^2$ regular in time, it has bounded numerical moments as in $\textbf{H0}$ and finally that it satisfies the one-step error estimates -.
Euler-Maryama method {#subsec:Euler}
--------------------
We start by considering the explicit Euler scheme $$\label{eq:explicit_Euler}
S_{h,\xi}^{f}(x)=x+h\nabla U(x) +\sqrt{2h}\xi,$$ while $S^{f}=S^{c}$, we are using the same numerical method for the fine and coarse paths. In order to be able to recover the integrability and regularity conditions we will need to impose further assumptions on the potential[^8] $U$. In particular, additionally to assumption **HU0** we assume, that
- There exists constant $L$ such that for any $x,y \in {\mathbb{R}}^d$ $$| \nabla U(x)-\nabla U(y)| \leq L| x-y|$$
As a consequence of this assumption we have $$\label{eq:HU1_con}
| \nabla U(x)| \leq L| x| + | \nabla U(0) |$$ We can now prove the $L^{2}$-regularity in time of the scheme.
#### $L^2$- regularity {#l2--regularity .unnumbered}
Since regularity is a property of the numerical scheme itself and it doesn’t relate with the coupling between fine and coarse levels, for simplicity of notation we prove things directly for $$x_{{n+1,x_{n}}}= S^{f}_{h,\xi_{k}}(x_{n}) .$$ In particular, the following Lemma holds.
Let **HU0** and **HU1** hold. Then the explicit Euler scheme is $L^2$ regular, i.e. and there exists constants $K\in {\mathbb{R}}$ and $L>0$ s.t $$\begin{aligned}
{\mathbb{E}}_{n-1}[|x_{n,x_{n-1}} - x_{n,y_{n-1}}|^2]\leq & (1 - (2m-L^2h)h ) |x_{n-1}-y_{n-1} |^2 \\
{\mathbb{E}}_k[|Z_n|^2] \leq & h^2 L^2 |x_{n-1} - y_{n-1} |^2 \end{aligned}$$
The difference between the Euler scheme taking values $x_{n-1}$ and $y_{n-1}$ at time $n-1$ is given by $$x_{n,x_{n-1}} - x_{n,y_{n-1}} = x_{n-1} - y_{n-1} + h( \nabla U(x_{n-1}) - \nabla U(y_{n-1})).$$ This, along with **HU0** and **HU1** leads to $$\begin{aligned}
{\mathbb{E}}_{n-1}[(x_{n,x_{n-1}} - x_{n,y_{n-1}})^2]= &
|x_{n-1} - y_{n-1}|^2
+ 2h \left\langle \nabla U(x_{n-1}) - \nabla U(y_{n-1}),x_{n-1}-y_{n-1}\right\rangle \\
& + | \nabla U(x_{n-1}) - \nabla U(y_{n-1}) |^{2} h^2 \\
\leq & |y_{n-1}-x_{n-1}|^2 (1 - 2mh + L^2 h^2) \\
= & |y_{n-1}-x_{n-1}|^2 (1 - (2m-L^2h)h ). \end{aligned}$$ This proofs the first part of the lemma. Next, due to **HU1** $$\begin{aligned}
{\mathbb{E}}_{n-1}[|Z_n|^2] = & h^2 {\mathbb{E}}_{n-1}[| \nabla U(x_{n-1}) - \nabla U(y_{n-1}) |^2] \\
\leq & h^2 L^2 |x_{n-1} - y_{n-1} |^2. \end{aligned}$$
#### Integrability {#integrability .unnumbered}
In the Lipschitz case we only require mean-square integrability This will become apparent when we analyse one-step error and and will hold with $p=1$
\[lem:int\] Let **HU0** and **HU1** hold. Then, $$\begin{aligned}
{\mathbb{E}}[|x_{n}|^2] \leq {\mathbb{E}}|x_{0}|^2 \exp\{-(2 m' - L^2h) nh \} + 2 (b | \nabla U(0) |^2+h) \frac{1-\exp\{-(2 m' - L^2h) nh \}}{(2 m' - L^2h) h }\end{aligned}$$
We have $$|x_{n}|^2 = |x_{n-1}|^2 + |\nabla U(x_{n-1})|^2 h^2 +2h \xi^{T}\xi + 2 h x^T_{n-1} \nabla U(x_{n-1}) +\sqrt{2h}x^T_{n-1} \xi+2 h\sqrt{2h} \xi^T
\nabla U(x_{n-1})$$ and hence $$\IE |x_{n}|^2 \leq \IE |x_{n-1}|^2 (1 - 2 m' h + L^2 h^2) + 2b | \nabla U(0) |^2+2dh.$$ where we have used the fact that implies $$\begin{aligned}
\left\langle x, \nabla U(x) \right\rangle & \leq - m' | x| ^{2} + 2b | \nabla U(0) |^2. \end{aligned}$$ for some[^9] $m'>0$ and $b>0$. We can now use Lemma \[lem:gronwall\] $$\begin{aligned}
{\mathbb{E}}|x_{n}|^2 \le & {\mathbb{E}}|x_{0}|^2 \exp\{-(2 m' - L^2h) nh \} + 2 (b | \nabla U(0) |^2+dh) \frac{1-\exp\{-(2 m' - L^2h) nh \}}{(2 m' - L^2h) h } \\\end{aligned}$$ The proof for $q>2$ can be done in similar way by using the binomial theorem.
#### One-step errors estimates {#one-step-errors-estimates .unnumbered}
Having proved $L^{2}$-regularity and integrability for the Euler scheme, we are now left with the task of proving equation and for Euler schemes coupled as in Algorithm \[alg:CouplingLangevinDiscretisation\] . It is enough to prove the results for $n=1$. We note that both ${x^f_0 }={x^c_0 }=x$ and we have the following Lemma.
\[lem:onestep\] Let **HU0** and **HU1** hold. Then the weak one-step distance between Euler schemes with time steps $\frac{h}{2}$ and $h$, respectively, is given $$\label{eq:weak_euler}
| {\mathbb{E}}[ {x^f_{1,x} }- {x^c_{1,x} } ] | \leq \frac{h^{3/2}}{2} L \left( {\mathbb{E}}\left[ \frac{\sqrt{h}}{2} \left(L|x| + | \nabla U(0) |\right)\right] + \sqrt{\frac{2d}{\pi}} \right).$$ The one-step $L^2$ distance can be estimated as $$\label{eq:strong_euler}
\mathbb{E}| {x^f_{1,x} }- {x^c_{1,x} } | ^{2}
\leq h^3 \frac{L^2}{4}\left(\frac{h}{2} ( |x|^{2} + | \nabla U(0) |^2 ) + d \right)$$ If in addition to **HU0** and **HU1**, $U\in C^3$ and[^10] $$| \partial^{2}U(x) | + | \partial^{3} U(x) | \leq C, \quad \forall x\in {\mathbb{R}}^d,$$ then the order of weak error bound can be improved, i.e $$\label{eq:weak_euler1}
| {\mathbb{E}}[ {x^f_{1,x} }- {x^c_{1,x} } ] | \leq C h^2 {\mathbb{E}}\left[
(|x | + h|x |^2 + |\nabla U(0)| + h |\nabla U(0)|^2 + d |) \right].$$
We calculate $$\begin{aligned}
\label{eq:euler_basis}
& & {x^f_{1,x} }- {x^c_{1,x} } \nonumber \\
& = & x+\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}+\frac{h}{2}\nabla U\left(x+\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}\right) +\sqrt{h}\xi_{2}
-\left(x+h\nabla U(x)+\sqrt{h}\left(\xi_{1}+\xi_{2}\right)\right) \nonumber\\
& = & \frac{h}{2}\nabla U \left(x+\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}\right)-\frac{h}{2}\nabla U(x)\end{aligned}$$ It then follows from **HU1** that $$| {\mathbb{E}}[ {x^f_{1,x} }- {x^c_{1,x} } ] | \leq \frac{h^{3/2}}{2} L {\mathbb{E}}| \frac{\sqrt{h}}{2}\nabla U(x)+\xi_{1}|.$$ Furthermore, if we use , a triangle equality and the fact that $\IE | \xi_{1} |=\sqrt{\frac{2d}{\pi}}$, we obtain . If we now assume that $U\in C^3$, then for $\delta=\alpha x + (1-\alpha)(x+\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1})$, $\alpha \in [0,1]$, we write $$\begin{aligned}
\nabla U \left(x+\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}\right) &= \nabla U(x)
+ \sum_{|\alpha|=2}\partial^{\alpha} U(x) \left(\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}\right)^{\alpha} \\
&+ \frac{1}{2}\sum_{|\alpha|=3}\partial^{\alpha} U(\delta) \left(\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}\right)^{\alpha},\end{aligned}$$ where we used multi-index notation. Consequently $$\begin{aligned}
& {\mathbb{E}}\left[ \nabla U \left(x+\frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1} \right)-\nabla U(x) \right ] \\
& \leq
C h^2 {\mathbb{E}}\left[
(|x | + h|x |^2 + |\nabla U(0)| + h |\nabla U(0)|^2 + |\xi_{1}|^{2} |) \right].\end{aligned}$$ which together with $\IE |\xi^{2}_{1} |=d$ gives . Equation trivially follows from by observing that $$\begin{aligned}
& & \mathbb{E}| {x^f_{1,x} }- {x^c_{1,x} } | ^{2}\\
& \leq & L^{2}\frac{h^{2}}{4}\mathbb{E}| \frac{h}{2}\nabla U(x)+\sqrt{h}\xi_{1}| ^{2}\\
& \leq & h^3 \frac{L^2}{4}\left(\frac{h}{2} ( |x|^{2} + | \nabla U(0) |^2 ) + d \right)\end{aligned}$$
In the case of log-concave target the bias of MLMC using the Euler method can be explicitly quantified using the results from [@durmus2016ula]. z
Non-Lipschitz setting {#subsec:Euler_implicit}
---------------------
In the previous subsection we found out that in order to analyse the regularity and the one-step error of the explicit Euler approximation, we had to impose an additional assumption about $\nabla U(x)$ being globally Lipschitz. This is necessary since in the absence of this condition Euler method is shown to be transient or even divergent [@RT96; @Hutzenthaler2014]. However, in many applications of interest this is a rather restricting condition. An example of this, is the potential [^11] $$U(x) = -\frac{x^4}{4} -\frac{x^2}{2}.$$ A standard way to deal with this is to use either an implicit scheme or specially designed explicit schemes [@hutzenthaler2012numerical; @szpruch2013v].
### Implicit Euler method
Here we will focus on the implicit Euler scheme $$x_{n}=x_{n-1}+h\nabla U(x_{n})+\sqrt{2h}\xi_{n}$$ We will assume that Assumption **HU0** holds and moreover replace **HU1** with
- Let $k\geq 1$. For any $x,y \in {\mathbb{R}}^d$ there exists constant $L$ s.t $$| \nabla U(x)-\nabla U(y)| \leq L(1+ |x|^{k-1} + |y|^{k-1} )| x-y|$$
As a consequence of this assumption we have $$\label{eq:HU1_cona}
| \nabla U(x)| \leq L| x|^k + | \nabla U(0) |$$
Observe that **HU0** implies that $$\begin{aligned}
\label{eq:ing}
\left\langle \nabla U(x),x\right\rangle & \leq - \frac{m}{2} | x| ^{2} + \frac{1}{2} | \nabla U(0)|^2 \quad \forall x\in {\mathbb{R}}^d\end{aligned}$$
#### Integrability {#integrability-1 .unnumbered}
The integrability uniformly in time can be easily deduced from the results in [@MR3011916; @MR2972586]. Nevertheless, for the convenience of the reader we will present the analysis of the regularity of the scheme, where the effect of the implicitness of the scheme on the regularity should become quickly apparent.
#### $L^2$- regularity {#l2--regularity-1 .unnumbered}
Let **HU0** and **HU1’** hold. Then an implicit Euler scheme is $L^2$ regular, i.e. and there exists constants $K\in {\mathbb{R}}$ and $L_Z>0$ s.t $$\begin{aligned}
{\mathbb{E}}_{n-1}[(x_{n,x_{n-1}} - x_{n,y_{n-1}})^2]\leq & (1 - 2mh ) (y_{n-1} - x_{n-1} )^2 + \mathcal{R}_{n-1},\end{aligned}$$ and $$\sum_{k=0}^{\infty}\mathcal{R}_k \leq 0.$$ Moreover $$\begin{aligned}
{\mathbb{E}}_{n-1} [ |Z_n|^2 ] \leq & h^2(1 - 2mh ) L(1+ {\mathbb{E}}_{n-1}[|x_{n}|^{k-1}] + {\mathbb{E}}_{n-1}[|y_n|^{k-1}] )^2| x_{n-1}-y_{n-1} |^2.\end{aligned}$$
The difference between the implicit Euler scheme taking values $x_{n-1}$ and $y_{n-1}$ time $n-1$ is given by $$x_{n,x_{n-1}} - x_{n,y_{n-1}} = x_{n-1} - y_{n-1} + h( \nabla U(x_{n}) - \nabla U(y_{n})).$$ This, along with **HU0** and **HU1** leads to $$\begin{aligned}
| x_{n,x_{n-1}} - x_{n,y_{n-1}} |^2= &
| x_{n-1} - y_{n-1} |^2
+ 2h \left\langle \nabla U(x_{n}) - \nabla U(y_{n}),x_{n}-y_{n}\right\rangle \\
& - | \nabla U(x_{n}) - \nabla U(y_{n}) |^{2} h^2 \\
\leq & | x_{n-1} - y_{n-1} |^2
- 2mh | x_{n,x_{n-1}}-x_{n,y_{n-1}} |^2 \\\end{aligned}$$ This implies $$\begin{aligned}
| x_{n,x_{n-1}} - x_{n,y_{n-1}} |^2
\leq | x_{n-1}-y_{n-1}|^2 \frac{1}{ 1+ 2mh}
\leq | x_{n-1}-y_{n-1}|^2 \left(1 - \frac{2mh}{1+2mh}\right).\end{aligned}$$ Next we take $$\begin{aligned}
| x_{n,x_{n-1}} - y_{n,y_{n-1}} |^2
\leq & | x_{n-1} - y_{n-1} |^2
- 2mh | x_{n}-y_{n} |^2 \\
= & ( 1 - 2mh )| x_{n-1} - y_{n-1} |^2 - 2mh( | x_{n}-y_{n} |^2 - | x_{n-1}-y_{n-1} |^2 ).\end{aligned}$$ In view of we define $$\mathcal{R}_k := - 2mh( | x_{k}-y_{k} |^2 - | x_{k-1}-y_{k-1} |^2 ),$$ and noticed that $$\sum_{k=1}^{n}\mathcal{R}_k = -2mh | x_{n}-y_{n} |^2 \leq 0.$$ Hence by remark the proof of the first statement in the lemma is complete. Now, due to **HU1’** $$\begin{aligned}
|Z_n|^2 = & h^2 | \nabla U(x_{n}) - \nabla U(y_{n}) |
\leq h^2 L(1+ |x_{n}|^{k-1} + |y_n|^{k-1} )^2| x_n-y_n |^2 \\
\leq & h^2\left(1 - \frac{2mh}{1+2mh} \right) L(1+ |x_{n}|^{k-1} + |y_n|^{k-1} )^2| x_{n-1}-y_{n-1} |^2.\end{aligned}$$ Observe that $$\begin{aligned}
{\mathbb{E}}_{n-1}[| x_{n} |^2]= &
{\mathbb{E}}_{n-1}[ | x_{n-1} |^2
+ 2h \left\langle \nabla U(x_{n}) ,x_{n} \right\rangle
- | \nabla U(x_{n}) |^{2} h^2 ]
\leq | x_{n-1} |^2
- m h | x_{n}|^2 + h| \nabla U (0)|^2. \end{aligned}$$ Consequently $$\begin{aligned}
{\mathbb{E}}_{n-1}[| x_{n} |^2] = &\frac{1}{1+mh}\left( |x_{n-1} |^2 + h| \nabla U (0)|^2 \right).\end{aligned}$$ This in turn implies that $$\mathcal{H}_{n-1}={\mathbb{E}}_{n-1} \left[ \left(1 - \frac{2mh}{1+2mh} \right)(1+ |x_{n}|^{k-1} + |y_n|^{k-1}) \right]
\leq C_{\mathcal{H}} (1 + |x_{n-1}|^{k-1} + |y_{n-1}|^{k-1})).$$ Due to uniform integrability of the implicit Euler scheme, holds.
#### One-step errors estimates {#one-step-errors-estimates-1 .unnumbered}
Having established integrability, estimating the one-step error follows exactly the same line of the argument as in Lemma \[lem:onestep\] and therefore we skip it.
Bayesian inference using MLMC SGLD
==================================
\[sec:sgld\]
The main computational tasks in Bayesian statistics is the approximation of expectations with respect to the posterior. The a priori uncertainty in a parameter $x$ is modelled using a probability density $\pi_{0}(x)$ called the prior. Here, we consider the where for a fixed parameter $x$ the data $\left\{ \data_{i}\right\} _{i=1,\dots,N}$ is supposed to be i.i.d. with density ${\pi(\data|x)}$. By Bayes’ rule the posterior $$\pi(x)\propto\pi_{0}(x)\prod_{i=1}^{N}{\pi(\data_{i}|x)}$$ this distribution is invariant for the Langevin equation with the choice $${\nabla U(x)=\nabla\log{\pi_{0}}(x)+\sum_{i=1}^{N}\nabla\log{\pi(\data_{i}|x)}}.\label{eq:SGLDU}$$ Provided that the appropriate assumptions are satisfied for $U$ we can thus use Algorithm \[alg:CouplingLangevinDiscretisation\] with Euler method or implicit Euler to approximate expectations with respect to $\pi$. For large $N$ the sum in Equation (\[eq:SGLDU\]) becomes the bottleneck. One way to deal with this is to replace the gradient by a a lower cost stochastic approximation. In the following, we extend the MLMC approach to the SGLD recursion in Equation (\[eq:SGLD\]) $$x_{k+1}=x_{k}+h\left(\nabla\log{\pi_{0}(x_{k})}+\frac{\nobs}{\sobs}\sum_{i=1}^{\sobs}\nabla\log{\pi(\data_{\tau_{i}^{k}}\vert x_{k})}\right)+\sqrt{2h}\xi_{k}$$ where we take $\tau_{j}^{k}\iid\mathcal{U}\left(\{1,\dots,N\}\right)\text{ for }j=1,\dots,s$ the uniform distribution on $1,\dots,N$ which corresponds to sampling $s$ items with replacement from $1,\dots,N$. Notice that each step only costs $s$ instead of $N$. We introduce a notation for the SGLD step as $$x_{k+1}=S_{h,\xi_{k},\tau_{k}}(x_{k}).$$ Since the SGLD is an explicit scheme, we need to make the assumptions:
\[assu:SGLDlipschitz\]Lipschitz conditions for prior and likelihood. $$\begin{aligned}
| \nabla\log\pi\left(\data_{i}\mid x\right)-\nabla\log\pi\left(\data_{i}\mid y\right)| & \leq & L| x-y| \text{ for all }i,x,y\\
| \nabla\log\pi_{0}\left(x\right)-\nabla\log\pi_{0}\left(y\right)| & \leq & L_{0}| x-y| \text{ for all }x,y.\end{aligned}$$ Convexity conditions for prior and likelihood: There are $m_{0}\ge$ and $m(d)\ge0$ [$$\begin{aligned}
\log\pi_{0}(y) & \le & \log\pi_{0}(x)+\left\langle \nabla\log\pi_{0}\left(x\right),y-x\right\rangle -\frac{m_{0}}{2}| x-y| ^{2}\\
\log\pi\left(\data_{i}\mid y\right) & \le & \log\pi\left(\data_{i}\mid x\right)+\left\langle \nabla\log\pi\left(\data_{i}\mid x\right),y-x\right\rangle -\frac{m_{\data_{i}}}{2}| x-y| ^{2}\end{aligned}$$]{} with $\inf_{i}m_{0}+m_{\data_{i}}>0.$ Let $m=m_{0}+N\inf_{i}m_{\data_{i}}$.
[We note that these conditions imply that $U$ in Equation satisfies the assumption we posed on the Euler method.]{}
\[lem:sgldRegular\]The recursion of the SGLD is uniformly $L^{2}$-regular $$\begin{aligned}
{\mathbb{E}}[|x_{n,x_{n-1}-x_{n,y_{n-1}}}|^{2}] & \leq & (1-(2m-\left(L_{0}+NL\right)^{2}h)h)|x_{n-1}-y_{n-1} |^{2}\end{aligned}$$ and has uniformly bounded moments $$\sup_{n\ge0}{\mathbb{E}}[|x_{n}|^{q}]<\infty.$$
This follows straight from Lemmas \[lem:int\] and \[lem:onestep\].
The only ingredient that changes is the one step difference estimate. We consider
$$\begin{aligned}
{x^f_{1,x} } & =S_{\frac{h}{2},\xi_{,2},\ssetfb}\circ S_{\frac{h}{2},\xi_{1},\ssetfa}(x)\nonumber \\
{x^c_{1,x} } & =S_{h,\frac{1}{\sqrt{2}}\left(\xi_{1}+\xi_{2}\right),\ssetc}(x).\label{eq:couplingSGLD}\end{aligned}$$
We note that we have already coupled the noise between the fine and coarse path in the traditional MLMC way. One question that naturally occurs now is if and how should one choose to couple between the subsampling of data. In particular, in order for the telescopic sum to be respected one needs to have that $$\mathcal{L}\left(\ssetfa\right)=\mathcal{L}\left(\ssetfb\right)=\mathcal{L}\left(\ssetc\right).\label{eq:SGLDTelescopic}$$ [We first take $s$ independent samples $\ssetfa$ on the first fine-step and another $s$ independent s-samples $\ssetfb$ on the second fine-step. The following three choices of $\ssetc$ ensure that equation (\[eq:SGLDTelescopic\]) holds.]{}
- an independent sample of $\left\{ 1,\dots,N\right\} $ without replacement denoted as $\ssetc_{\text{ind}}$ called independent coupling;
- a draw of $s$ samples without replacement from $\left(\ssetfa,\ssetfb\right)$ denoted as $\ssetc_{\text{union}}$ called union coupling;
- the concatination of a draw of $\frac{s}{2}$ samples without replacement from $\ssetfa$ and a draw of $\frac{s}{2}$ samples without replacement from $\ssetfb$ (provided that $s$ is even) $\ssetc_{\text{strat}}$ called [stratified coupling]{}.
[We stress that any of these couplings can be used in Algorithm \[alg:CouplingSGLDs\] and hence we have use the generic symbol $\ssetc$ to denote this.]{}
\[h\]
1. Set $x_{0}^{(f,\ell)}=x_{0}$, then simulate according to $S_{h_{\ell},\xi,\tau}(x)$ for $\frac{T_{\ell}-T_{\ell-1}}{h_{\ell}}$ steps with independent random input;
2. set $x_{0}^{(c,\ell)}=x_{0}$ and $x_{0}^{(f,\ell)}=x_{\frac{T_{\ell}-T_{\ell-1}}{h_{\ell}}}^{h_{\ell}}$, then simulate $(x_{\cdot}^{(f,\ell)},x_{\cdot}^{(c,\ell)})$ jointly according to $$\left(x_{k+1}^{(f,\ell)},x_{k+1}^{(c,\ell)}\right)=\left(S_{h_{\ell},\xi_{k,2},\ssetfb_{k}}\circ S_{h_{\ell},\xi_{k,1},\ssetfa_{k}}(x_{k}^{(f,\ell)}),S_{h_{\ell-1},\frac{1}{\sqrt{2}}\left(\xi_{k,1}+\xi_{k,2}\right),\ssetc_{k}}(x_{k}^{(c,\ell)})\right).$$
3. set $${
\Delta^{(i)}_{\ell}:=g\left( \left(x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(f,\ell)} \right)^{(i)}\right)-g\left( \left( x_{\frac{T_{\ell-1}}{h_{\ell-1}}}^{(c,\ell)}\right)^{(i)} \right)}$$
[In order to simplify the one step proof we need to introduce the following notation. In particular, we set $\left(\ssetfa,\ssetfb\right)=\left(\ssetf_{1:\sobs},\ssetf_{(s+1):2\sobs}\right)$, while ]{}
$$\begin{aligned}
\gli(x,i) & = & \begin{cases}
\nabla\log\pi_{0}(x) & i=0\\
\nabla\log\pi\left(\data_{i}\mid x\right) & \text{otherwise}
\end{cases}\\
\gle(x,\tau): & = & c(x,0)+\frac{N}{\left|\tau\right|}\sum_{i=1}^{\left|\tau\right|}c(x,\tau_{i}).\end{aligned}$$
where $\left|\tau\right|$ is cardinality of vector $\tau$. [We will now prove one step estimates for the SGLD. We stress here that in terms of the estimate nothing changes and one obtains $\alpha=1$. However, the subsampling has an important effect on the value of the parameter $\beta$ and hence this is the one we focus here. Furthermore, we prove our estimate in the case of the union and independent coupling but the same result would hold in terms of $\beta$ for the stratified coupling, with the only difference being the constant multiplying the $h^{2}$ term.]{}
\[lem:sgldONEstep\](One step error bound) Suppose Assumption \[assu:SGLDlipschitz\] is satisfied, then $$\begin{aligned}
& & {\mathbb{E}}| S_{\frac{h}{2},\xi_{,2},\ssetfb}\circ S_{\frac{h}{2},\xi_{1},\ssetfa}(x)-S_{h,\frac{1}{\sqrt{2}}\left(\xi_{1}+\xi_{2}\right),\ssetc_{\text{ind}}}(x)| ^{2}\\
& \leq & M+4\left(\frac{h}{2}\right)^{2}\left(\frac{1}{s}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle +6\frac{N}{s}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}\right)\\
& & {\mathbb{E}}| S_{\frac{h}{2},\xi_{,2},\ssetfb}\circ S_{\frac{h}{2},\xi_{1},\ssetfa}(x)-S_{h,\frac{1}{\sqrt{2}}\left(\xi_{1}+\xi_{2}\right),\ssetc_{\text{union}}}(x)| ^{2}\\
& \leq & M+4\left(\frac{h}{2}\right)^{2}\frac{1}{s}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \end{aligned}$$ with $M=2\left(\frac{h}{2}\right)^{2}(NL+L_{0})^{2}\left(h+\frac{h^{2}}{4}\frac{\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}+\frac{\sobs(\sobs-1)}{\nobs^{2}}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \right)$
We will first perform the calculations that both approaches above have in common. First we introduce an intermediate term at $x_{\frac{1}{2}}=x+\frac{h}{2}\gle\left(x,\ssetf_{1:\sobs}\right)+\sqrt{h}\xi$
$$\begin{aligned}
& & {\mathbb{E}}| S_{\frac{h}{2},\xi_{,2},\ssetfb}\circ S_{\frac{h}{2},\xi_{1},\ssetfa}(x)-S_{h,\frac{1}{\sqrt{2}}\left(\xi_{1}+\xi_{2}\right),\ssetc}(x)| \nonumber \\
& = & \left(\frac{h}{2}\right)^{2}{\mathbb{E}}| \gle(x_{\frac{1}{2}},\ssetf_{(s+1):2\sobs})+\gle\left(x,\ssetf_{1:\sobs}\right)-2\gle\left(x,\ssetc\right)| ^{2}\nonumber \\
& \le & 2\left(\frac{h}{2}\right)^{2}{\mathbb{E}}| \gle(x,\ssetf_{(s+1):2\sobs})+\gle\left(x,\ssetf_{1:\sobs}\right)-2\gle\left(x,\ssetc\right)| ^{2}\label{eq:sgldonestep1}\\
& & +2\left(\frac{h}{2}\right)^{2}{\mathbb{E}}| \gle(x_{\frac{1}{2}},\ssetf_{(s+1):2\sobs})-\gle(x,\ssetf_{(s+1):2\sobs})| ^{2}.\label{eq:onestepsgld2}\end{aligned}$$
We focus first on (\[eq:onestepsgld2\]) using Assumption \[assu:SGLDlipschitz\] $$\begin{aligned}
& & {\mathbb{E}}| \gle(x_{\frac{1}{2}},\ssetf_{(s+1):2\sobs})-\gle(x,\ssetf_{(s+1):2\sobs})| ^{2}\\
& \leq & (NL+L_{0})^{2}{\mathbb{E}}| x-x_{\frac{1}{2}}| ^{2}\\
& \leq & (NL+L_{0})^{2}\left(h+\frac{h^{2}}{4}{\mathbb{E}}_{\tau}| \gle\left(x,\ssetf_{(s+1):2\sobs}\right)| ^{2}\right)\end{aligned}$$ where we introduced operator ${\mathbb{E}}_{\tau}[\cdot]$ to highlight the fact that expectation is taken with respect to the distribution of the vector $\tau$. $$\begin{aligned}
& & {\mathbb{E}}_{\tau}| \gle\left(x,\ssetf_{(s+1):2\sobs}\right)| ^{2}\\
& = & | \gli(x,0)| ^{2}+\sum_{i=1}^{N}2\left\langle \gli(x,0),\gli(x,i)\right\rangle +\left(\frac{N}{s}\right)^{2}\underbrace{{\mathbb{E}}_{\tau}| \sum_{i=1}^{s}\gli(x,\ssetf_{i})| ^{2}}_{\frac{\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}+\frac{\sobs(\sobs-1)}{\nobs^{2}}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle }.\end{aligned}$$
Now we focus on (\[eq:sgldonestep1\])
$$\begin{aligned}
& & {\mathbb{E}}| \gle(x,\ssetf_{(s+1):2\sobs})+\gle\left(x,\ssetf_{1:\sobs}\right)-2\gle\left(x,\ssetc\right)| ^{2}\\
& = & \left(\frac{\nobs}{\sobs}\right)^{2}| \underbrace{\left(\sum_{j=1}^{\sobs}\gli(x,\ssetf_{j})\right)}_{T_{1}}+\underbrace{\left(\sum_{j=\sobs+1}^{2\sobs}\gli(x,\ssetf_{j})\right)}_{T_{2}}-\underbrace{2\sum_{i=1}\gli(x,\ssetc_{i})}_{T_{3}}| \\
& = & \left(\frac{\nobs}{\sobs}\right)^{2}\mathbb{E}\left(| T_{1}| ^{2}+| T_{2}| ^{2}+| T_{3}| ^{2}+2\left\langle T_{1},T_{2}\right\rangle -2\left\langle T_{1},T_{3}\right\rangle -2\left\langle T_{2},T_{3}\right\rangle \right)\end{aligned}$$
We find $$\begin{aligned}
{\mathbb{E}}| T_{1}| ^{2}+| T_{2}| ^{2} & = & 2 \left(\frac{\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}+\frac{\sobs(\sobs-1)}{\nobs^{2}}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \right)\\
{\mathbb{E}}| T_{3}| ^{2} & = & 4\left(\frac{\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}+\frac{\sobs(\sobs-1)}{\nobs^{2}}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \right)\\
{\mathbb{E}}\left\langle T_{1},T_{3}\right\rangle +{\mathbb{E}}\left\langle T_{2},T_{3}\right\rangle & = & 2\left(\frac{\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}+\frac{\sobs(\sobs-1)+\sobs^{2}}{\nobs^{2}}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \right)\end{aligned}$$ Putting everything together yields $$\begin{aligned}
& & \mathbb{E}\left(| T_{1}| ^{2}+| T_{2}| ^{2}+| T_{3}| ^{2}+2\left\langle T_{1},T_{2}\right\rangle -2\left\langle T_{1},T_{3}\right\rangle -2\left\langle T_{2},T_{3}\right\rangle \right)\label{eq:tterms}\\
& = & \sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \underbrace{\left(6\frac{\sobs(\sobs-1)}{\nobs^{2}}-4\frac{\sobs(\sobs-1)+\sobs^{2}}{\nobs^{2}}+2\frac{\sobs^{2}}{\nobs^{2}}\right)}_{-\frac{2\sobs}{\nobs^{2}}}\nonumber \\
& & +\left(\underbrace{(1+1+4+2-4)}_{2}\frac{\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}\right).\nonumber \end{aligned}$$
For the **independent coupling**, the only terms that change are ${\mathbb{E}}\left\langle T_{1},T_{3}\right\rangle +{\mathbb{E}}\left\langle T_{2},T_{3}\right\rangle $ $$\begin{aligned}
{\mathbb{E}}\left\langle T_{1},T_{3}\right\rangle +{\mathbb{E}}\left\langle T_{2},T_{3}\right\rangle & = & 4\left(\frac{\sobs^{2}}{\nobs^{2}}\sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \right)\end{aligned}$$ So overall we get for Equation [(\[eq:tterms\]) ]{} $$\begin{aligned}
& & \mathbb{E}\left(| T_{1}| ^{2}+| T_{2}| ^{2}+| T_{3}| ^{2}+2\left\langle T_{1},T_{2}\right\rangle -2\left\langle T_{1},T_{3}\right\rangle -2\left\langle T_{2},T_{3}\right\rangle \right)\\
& = & \sum_{i,j=1}^{\nobs}\left\langle \gli(x,i),\gli(x,j)\right\rangle \underbrace{\left(6\frac{\sobs(\sobs-1)}{\nobs^{2}}-8\frac{\sobs^{2}}{\nobs^{2}}+2\frac{\sobs^{2}}{\nobs^{2}}\right)}_{\text{-}\frac{6\sobs}{\nobs^{2}}}\\
& & +\left(\frac{6\sobs}{\nobs}\sum_{i=1}^{\nobs}| \gli(x,i)| ^{2}\right)\end{aligned}$$
Notice that the term $M$ is quite similar to the corresponding term for the Euler Method. However, in contrast to the Euler method the additional term is of order $h^{2}$. This seems sharp because of line irrespective of the choice coupling no additional $h$ will appear. This is also verified by simulation. [However, as already discuss in terms of the estimate nothing changes in terms of $\alpha$, since if we were to take the expectation inside in we would obtain 0.]{}
The Algorithm \[alg:CouplingSGLDs\] based on the coupling given in Equation and subsampling coupling $\tau_{\text{union}}$ or $\tau_{\text{strat}}$ with appropriately chosen $t_{i}$ has complexity ${\varepsilon^{-2}|\log(\varepsilon)|^3}.$
Because of Lemma \[lem:sgldONEstep\] and Lemma \[lem:sgldRegular\] [we can apply the results of Section \[subsec:conv\_anal\]. In particular, if we choose $T_{\ell}$ according to Lemma \[lem:optimal\] we thus have $\beta=1$ in Theorem \[th:convergence\] and then the complexity follows from Remark \[rem:complexity\]]{} .
Numerical Investigations {#sec:num}
========================
[In this Section we perform numerical simulations that illustrate our theoretical findings. We start by studying an Ornstein-Uhlenbeck process in Section \[subsec:OU\] using the explicit Euler method, while in Section \[subsec:log\] we study a Bayesian logistic regression model using the SGLD.]{}
Ornstein Uhlenbeck process {#subsec:OU}
--------------------------
We consider the SDE $$\label{eq:OU}
dX_{t}=-\kappa X_{t}dt+\sqrt{2}dW_{t},$$ and its dicretization using the Euler method $$\label{eq:OU_num}
x_{n+1}=S_{h,\xi}(x_{n}), \quad S_{h,\xi}(x)=x-h\kappa x+\sqrt{2h}\xi.$$ Equation is ergodic with its invariant measure being $N(0,\kappa^{-1})$. Furthermore, it is possible to show that the Euler method is similarly ergodic with its invariant measure [@Zyg11] being $N\left(0,\frac{2}{2\kappa-\kappa^{2}h}\right)$. In Figure \[fig:OU1\], we plot the outputs of our numerical simulations using Algorithm \[alg:CouplingLangevinDiscretisation\]. The parameter of interest here is the variance of the invariant measure $\kappa^{-1}$ which we try to approximate for different mean square error tolerances $\varepsilon$. More precisely, in Figure \[fig:OU1\] we see allocation of samples for various levels with respect to $\varepsilon$, while in Figure \[fig:OU1\]b we compare the computational cost of the algorithm as a function of the parameter $\varepsilon$. As we can see the computational complexity grows as $\mathcal{O}(\varepsilon^{-2})$ as predicted by our theory (Here $\alpha=\beta=2$ in and ).
[0.32]{} ![MLMC results for for $g(x)=x^{2}$ and $\kappa=0.4$.[]{data-label="fig:OU1"}](levels_vs_eps.pdf "fig:"){width="\textwidth"}
[0.32]{} ![MLMC results for for $g(x)=x^{2}$ and $\kappa=0.4$.[]{data-label="fig:OU1"}](cost_vs_eps.pdf "fig:"){width="\textwidth"}
[0.32]{} ![MLMC results for for $g(x)=x^{2}$ and $\kappa=0.4$.[]{data-label="fig:OU1"}](invariant_variance.pdf "fig:"){width="\textwidth"}
Finally, in Figure \[fig:fig3\] we plot the approximation of the variance $\kappa^{-1}$ from our algorithm. Note that this coincides with the choice $g(x)=x^{2}$ since the mean of the invariant measure is 0. As we can see as $\varepsilon$ becomes smaller we get perfect agreement with the true value of the variance as expected.
Bayesian logistic regression {#subsec:log}
----------------------------
In the following we present numerical simulations for a binary Bayesian logistic regression model. In this case the data $\data_{i}\in\{-1,1\}$ is modelled by $$p(\data_{i}\vert\iota_{i},x)=\logit(y_{i}x^{t}\cova_{i})\label{eq.logistic}$$ where $\logit(z)=\frac{1}{1+\exp(-z)}\in[0,1]$ and $\cova_{i}\in\mathbb{R}^{d}$ are fixed covariates. We put a Gaussian prior $\mathcal{N}(0,C_{0})$ on $x$, for simplicity we use $C_{0}=I$ subsequently. By Bayes’ rule the posterior $\pi$ satisfies $$\pi(x)\propto\exp\left(-\frac{1}{2}| x|_{C_{0}}^{2}\right)\prod_{i=1}^{N}\logit(y_{i}x^{T}\cova_{i}).$$ We consider $d=3$ and $N=100$ data points and choose the covariate to be $$\cova=\left(\begin{array}{ccc}
\cova_{1,1} & \cova_{1,2} & 1\\
\cova_{2,1} & \cova_{2,2} & 1\\
\vdots & \vdots & \vdots\\
\cova_{100,1} & \cova_{100,2} & 1
\end{array}\right)$$ for a fixed sample of $\cova_{i,j}\overset{\text{i.i.d.}}{\sim}\mathcal{N}\left(0,1\right)$ for $i=1,\dots100$.
[[In Algorithm \[alg:CouplingSGLDs\] we can choose the starting position $x_0$. It is reasonable to start the path of the individual SGLD trajectories at the mode of the target distribution (heuristically this makes the distance ${\mathbb{E}}{| x_{0}^{(c,\ell)}-x_{0}^{(f,\ell)} |}$ in step 2 in Algorithm \[alg:CouplingSGLDs\] small). That is we set the $x_0$ to be the maximum a posteriori estimator (MAP)]{}]{} $$x_{0}=\text{argmax}\:\exp\left(-\frac{1}{2}| x |_{C_{0}} ^{2}\right)\prod_{i=1}^{N}\logit(y_{i}x^{T}\cova_{i})$$ which is approximated using the Newton-Raphson method. [Our numerical results are described in Figure \[fig:Logistic\]]{}. [In particular,]{} [[in Figure \[fig:fig4\] we illustrate the behaviour of the coupling by plotting an estimate of the average distance during the joint evolution in step 2 of Algorithm \[alg:CouplingSGLDs\]. The behaviour in this figure agrees qualitatively with the statement of Theorem \[th:convergence\], as $T$ grows there is an initial exponential decay up to an additive constant. For the simulation we used $h_0=0.02$, $T_\ell=3(\ell+1)$ and $s=20$.]{}]{} Furthermore, in Figure \[fig:fig5\] we plot $\text{CPU-time}\times \varepsilon^2 $ against $\varepsilon$ for the estimation of the mean. The objective here is to estimate the mean square distance from the MAP estimator $x_0$ and the posterior that is $\int{| x-x_0 |}^2 \pi(x)dx$. Again, after some initial transient where $\text{CPU-time}\times \varepsilon^2$ decreases, we see that we get a quantitive agreement with our theory since the $\text{CPU-time}\times \varepsilon^2$ increases in a logarithmic way in the limit of $\varepsilon$ going to zero.
[0.49]{} ![(a) Illustration of the joint evolution in step 2 of Algorithm \[alg:CouplingSGLDs\] for the union coupling, (b) Cost of MLMC (sequential CPU time) SGLD for Bayesian Logistic Regression for decreasing accuracy parameter $\varepsilon$ and different couplings[]{data-label="fig:Logistic"}](coupledlevels.png "fig:"){width="\textwidth"}
[0.49]{} ![(a) Illustration of the joint evolution in step 2 of Algorithm \[alg:CouplingSGLDs\] for the union coupling, (b) Cost of MLMC (sequential CPU time) SGLD for Bayesian Logistic Regression for decreasing accuracy parameter $\varepsilon$ and different couplings[]{data-label="fig:Logistic"}](logisticregression "fig:"){width="\textwidth"}
Acknowledgement
===============
KCZ was supported by a grant from the Simons Foundation. Part of this work was done, during the author’s stay at the Newton Institute for the program Stochastic Dynamical Systems in Biology: Numerical Methods and Applications. SJV has supported by EPSRC under grant EP/N000188/1.
[^1]: L.Szpruch@ed.ac.uk
[^2]: vollmer@stats.ox.ac.uk
[^3]: K.Zygalakis@ed.ac.uk
[^4]: Mike.Giles@maths.ox.ac.uk
[^5]: In this paper the computational complexity is measured in terms of the expected number of random number generations and arithmetic operations.
[^6]: Recall $h_{\ell}$ is the time step used in the discretization of the level $l$.
[^7]: As we will see there $m' \leq m$ depending on the size of $\nabla U(0)$
[^8]: this restriction will be alleviated in Section \[subsec:Euler\_implicit\] by means of more advanced integrators
[^9]: If $\nabla U(0)=0$ then $m'=m$. Otherwise $m'<m$ (implication of Young’s inequality).
[^10]: Thanks to integrability conditions we could easily extend the analysis to the case where the derivatives are bounded by a polynomial of x.
[^11]: One also may consider the case of products of distribution functions, where after taking the $\log$ one ends up with a polynomial in the different variables.
|
---
abstract: 'Sobolev loss is used when training a network to approximate the values and derivatives of a target function at a prescribed set of input points. Recent works have demonstrated its successful applications in various tasks such as distillation or synthetic gradient prediction. In this work we prove that an overparametrized two-layer relu neural network trained on the Sobolev loss with gradient flow from random initialization can fit any given function values and any given directional derivatives, under a separation condition on the input data.'
author:
- Jorio Cocola
- Paul Hand
bibliography:
- 'references.bib'
title: |
Global Convergence of Sobolev Training\
for Overparametrized Neural Networks
---
**Acknowledgements.** PH is supported in part by NSF CAREER Grant DMS-1848087.
|
---
abstract: 'Computer simulation provides an automatic and safe way for training robotic control policies to achieve complex tasks such as locomotion. However, a policy trained in simulation usually does not transfer directly to the real hardware due to the differences between the two environments. Transfer learning using domain randomization is a promising approach, but it usually assumes that the target environment is close to the distribution of the training environments, thus relying heavily on accurate system identification. In this paper, we present a different approach that leverages domain randomization for transferring control policies to unknown environments. The key idea that, instead of learning a single policy in the simulation, we simultaneously learn a family of policies that exhibit different behaviors. When tested in the target environment, we directly search for the best policy in the family based on the task performance, without the need to identify the dynamic parameters. We evaluate our method on five simulated robotic control problems with different discrepancies in the training and testing environment and demonstrate that our method can overcome larger modeling errors compared to training a robust policy or an adaptive policy.'
author:
- |
Wenhao Yu & C. Karen Liu & Greg Turk\
School of Interactive Computing\
Georgia Institute of Technology, GA\
`wyu68@gatech.edu, {karenliu,turk}@cc.gatech.edu`
bibliography:
- 'iclr2019\_conference.bib'
title: Policy Transfer with Strategy Optimization
---
Introduction
============
Recent developments in Deep Reinforcement Learning (DRL) have shown the potential to learn complex robotic controllers in an automatic way with minimal human intervention. However, due to the high sample complexity of DRL algorithms, directly training control policies on the hardware still remains largely impractical for agile tasks such as locomotion.
A promising direction to address this issue is to use the idea of transfer learning which learns a model in a source environment and transfers it to a target environment of interest. In the context of learning robotic control policies, we can consider the real world the target environment and the computer simulation the source environment. Learning in simulated environment provides a safe and efficient way to explore large variety of different situations that a real robot might encounter. However, due to the model discrepancy between physics simulation and the real-world environment, also known as the Reality Gap [@boeing2012leveraging; @koos2010crossing], the trained policy usually fails in the target environment. Efforts have been made to analyze the cause of the Reality Gap [@neunert2017off] and to develop more accurate computer simulation [@TanRSS18] to improve the ability of a policy when transferred it to real hardware. Orthogonal to improving the fidelity of the physics simulation, researchers have also attempted to cross the reality gap by training more capable policies that succeed in a large variety of simulated environments. Our method falls into the second category.
To develop a policy capable of performing in various environments with different governing dynamics, one can consider to train a *robust policy* or to train an *adaptive policy*. In both cases, the policy is trained in environments with randomized dynamics. A robust policy is trained under a range of dynamics without identifying the specific dynamic parameters. Such a policy can only perform well if the simulation is a good approximation of the real world dynamics. In addition, for more agile motor skills, robust policies may appear over-conservative due to the uncertainty in the training environments. On the other hand, when an adaptive policy is used, it learns to first identify, implicitly or explicitly, the dynamics of its environment, and then selects the best action according to the identified dynamics. Being able to act differently according to the dynamics allows the adaptive policy to achieve higher performance on a larger range of dynamic systems. However, when the target dynamics is notably different from the training dynamics, it may still produce sub-optimal results for two reasons. First, when a sequence of novel observations is presented, the learned identification model in an adaptive policy may produce inaccurate estimations. Second, even when the identification model is perfect, the corresponding action may not be optimal for the new situation.
In this work, we introduce a new method that enjoys the versatility of an adaptive policy, while avoiding the challenges of system identification. Instead of relating the observations in the target environment to the similar experiences in the training environment, our method searches for the best policy directly based on the task performance in the target environment.
Our algorithm can be divided to two stages. The first stage trains a family of policies, each optimized for a particular vector of dynamic parameters. The family of policies can be parameterized by the dynamic parameters in a continuous representation. Each member of the family, referred to as a *strategy*, is a policy associated with particular dynamic parameters. Using a locomotion controller as an example, a strategy associated with low friction coefficient may exhibit cautious walking motion, while a strategy associated with high friction coefficient may result in more aggressive running motion. In the second stage we perform a search over the strategies in the target environment to find the one that achieves the highest task performance.
We evaluate our method on three examples that demonstrate transfer of a policy learned in one simulator DART, to another simulator MuJoCo. Due to the differences in the constraint solvers, these simulators can produce notably different simulation results. A more detailed description of the differences between DART and MuJoCo is provided in Appendix \[app:DART\_mj\_diff\]. We also add latency to the MuJoCo environment to mimic a real world scenario, which further increases the difficulty of the transfer. In addition, we use a quadruped robot simulated in Bullet to demonstrate that our method can overcome actuator modeling errors. Latency and actuator modeling have been found to be important for Sim-to-Real transfer of locomotion policies [@TanRSS18; @neunert2017off]. Finally, we transfer a policy learned for a robot composed of rigid bodies to a robot whose end-effector is deformable, demonstrating the possiblity of using our method to transfer to problems that are challenging to model faithfully.
Related Work
============
While DRL has demonstrated its ability to learn control policies for complex and dynamic motor skills in simulation [@schulman2015trust; @schulman2017proximal; @peng2018deepmimic; @peng2017deeploco; @YuSIGGRAPH2018; @heess2017emergence], very few learning algorithms have successfully transferred these policies to the real world. Researchers have proposed to address this issue by optimizing or learning a simulation model using data from the real-world [@TanRSS18; @tan2016simulation; @deisenroth2011pilco; @HA2015; @Abbeel2005]. The main drawback for these methods is that for highly agile and high dimensional control problems, fitting an accurate dynamic model can be challenging and data inefficient.
Complementary to learning an accurate simulation model, a different line of research in sim-to-real transfer is to learn policies that can work under a large variety of simulated environments. One common approach is *domain randomization*. Training a robust policy with domain randomization has been shown to improve the ability to transfer a policy [@TanRSS18; @tobin2017domain; @rajeswaran2016epopt; @pinto2017robust]. @tobin2017domain trained an object detector with randomized appearance and applied it in a real-world gripping task. @TanRSS18 showed that training a robust policy with randomized dynamic parameters is crucial for transferring quadruped locomotion to the real world. Designing the parameters and range of the domain to be randomized requires specific knowledge for different tasks. If the range is set too high, the policy may learn a conservative strategy or fail to learn the task, while a small range may not provide enough variation for the policy to transfer to real-world.
A similar idea is to train an adaptive policy with the current and the past observations as input. Such an adaptive policy is able to identify the dynamic parameters online either implicitly [@learndexmanipulation18; @peng2018sim] or explicitly [@YuRSS17] and apply actions appropriate for different system dynamics. Recently, adaptive policies have been used for sim-to-real transfer, such as in-hand manipulation tasks [@learndexmanipulation18] or non-prehensile manipulation tasks [@peng2018sim]. Instead of training one robust or adaptive policy, @zhang2018learning trained multiple policies for a set of randomized environments and learned to combine them linearly in a separate set of environments. The main advantage of these methods is that they can be trained entirely in simulation and deployed in real-world without further fine-tuning. However, policies trained in simulation may not generalize well when the discrepancy between the target environment and the simulation is too large. Our method also uses dynamic randomization to train policies that exhibit different strategies for different dynamics, however, instead of relying on the simulation to learn an identification model for selecting the strategy, we propose to directly optimize the strategy in the target environment.
A few recent works have also proposed the idea of training policies in a source environment and fine-tune it in the target environment. For example, @cully2015robots proposed MAP-Elite to learn a large set of controllers and applied Bayesian optimization for fast adaptation to hardware damages. Their approach searches for individual controllers for discrete points in a behavior space, instead of a parameterized family of policies as in our case, making it potentially challenging to be applied to higher dimensional behavior spaces. @rusu2016sim used progressive network to adapt the policy to new environments by designing a policy architecture that can effectively utilize previously learned representations. @chen2018hardware learned an implicit representation of the environment variations by optimizing a latent policy input for each discrete instance of the environment. They showed that fine-tuning on this learned policy achieved improved learning efficiency. In contrast to prior work in which the fine-tuning phase adjusts the neural network weights in the target environment, we optimize only the dynamics parameters input to the policy. This allows our policies to adapt to the target environments with less data and to use sparse reward signal.
Background
==========
We formulate the motor skill learning problem as a Markov Decision Process (MDP), $\mathcal{M}=(\mathcal{S}, \mathcal{A}, r, \mathcal{P}, p_0, \gamma)$, where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space, $r: \mathcal{S} \times \mathcal{A} \mapsto \mathbb{R}$ is the reward function, $\mathcal{P}: \mathcal{S} \times \mathcal{A} \mapsto \mathcal{S}$ is the transition function, $p_0$ is the initial state distribution and $\gamma$ is the discount factor. The goal of reinforcement learning is to find a control policy $\pi: \mathcal{S} \mapsto \mathcal{A}$ that maximizes the expected accumulated reward: $J_\mathcal{M}(\pi) = \mathbb{E}_{\tau=(\vs_0, \va_0, \dots, \vs_T)} \sum_{t=0}^{T} \gamma^t r(\vs_t, \va_t)$, where $\vs_0 \sim p_0$, $\va_t \sim \pi(\vs_t)$ and $\vs_{t+1}=\mathcal{P}(\vs_t, \va_t)$. In practice, we usually only have access to an observation of the robot that contains a partial information of the robot’s state. In this case, we will have a Partially-Observable Markov Decision Process (POMDP) and the policy would become $\pi: \mathcal{O} \mapsto \mathcal{A}$, where $\mathcal{O}$ is the observation space.
In the context of transfer learning, we can define a source MDP $\mathcal{M}^s$ and a target MDP $\mathcal{M}^t$ and the goal would be to learn a policy $\pi^s$ for $\mathcal{M}^s$ such that it also works well on $\mathcal{M}^t$. In this work, $\mathcal{P}$ is regarded as a parameterized space of transition functions, $\vs_{t+1}=\mathcal{P}_{\mu}(\vs_t, \va_t)$, where $\mu$ is a vector of physical parameters defining the dynamic model (friction coefficient). The transfer learning in this context learns a policy under $\mathcal{P}^s$ and transfers to $\mathcal{P}^t$, where $\mathcal{P}^s \neq \mathcal{P}^t$.
Methods
=======
We propose a new method for transferring a policy learned in simulated environment to a target environment with unknown dynamics. Our algorithm consists of two stages: learning a family of policy and optimizing strategy.
Learning a Family of Policies
-----------------------------
The first stage of our method is to learn a family of policies, each for a particular dynamics $\mathcal{P}^s_{\mu}(\cdot)$. One can potentially train each policy individually and interpolate them to cover the space of $\mu$ [@stulp2013learning; @da2012learning]. However, as the dimension of $\mu$ increases, the number of policies required for interpolation grows exponentially.
Since many of these policies are trained under similar dynamics, our method merges them into one neural network and trains the entire family of policies simultaneously. We follow the work by @YuRSS17, which trains a policy $\pi: (\vo, \mathbf{\mu}) \mapsto \va$ that takes as input not only the observation of the robot $\vo$, but also the physical parameters $\mathbf{\mu}$. At the beginning of each rollout during the training, we randomly pick a new set of physical parameters for the simulation and fix it throughout the rollout. After training the policy this way, we obtain a family of policies that is parameterized by the dynamics parameters $\mu$. Given a particular $\mu$, we define the corresponding policy as $\pi_\mu: \vo \mapsto \va$. We will call such an instantiated policy a *strategy*.
Optimizating Strategy
---------------------
The second stage of our method is to search for the optimal strategy in the space of $\mathbf{\mu}$ for the target environment. Previous work learns a mapping between the experiences under source dynamics $\mathcal{P}_\mu^s$ and the corresponding $\mu$. When new experiences are generated in the target environment, this mapping will identify a $\mathbf{\mu}$ based on similar experiences previously generated in the source environment. While using experience similarity as a metric to identify $\mu$ transfers well to a target environment that has the same dynamic parameter space [@YuRSS17], it does not generalize well when the dynamic parameter space is different.
Since our goal is to find a strategy that works well in the target environment, a more direct approach is to use the performance of the task, the accumulated reward, in the target environment as the metric to search for the strategy:
$$\mu^*=\argmax_\mu J_{\mathcal{M}^t}(\pi_{\mu}).
\label{so_eq}$$
Solving Equation \[so\_eq\] can be done efficiently because the search space in Equation \[so\_eq\] is the space of dynamic parameters $\mu$, rather than the space of policies, which are represented as neural networks in our implementation. To further reduce the number of samples from the target environment needed for solving Equation \[so\_eq\], we investigated a number of algorithms, including Bayesian optimization, model-based methods and an evolutionary algorithm (CMA). A detailed description and comparison of these methods are provided in Appendix \[app:strag\_opt\].
We chose Covariance Matrix Adaptation (CMA) [@hansen1995adaptation], because it reliably outperforms other methods in terms of sample-efficiency. At each iteration of CMA, a set of samples are drawn from a Gaussian distribution over the space of $\mu$. For each sample, we instantiate a strategy $\pi_\mu$ and use it to generate rollouts in the target environment. The fitness of the sample is determined by evaluating the rollouts using $J_{\mathcal{M}^t}$. Based on the fitness values of the samples in the current iteration, the mean and the covariance matrix of the Gaussian distribution are updated for the next iteration.
Experiments
===========
To evaluate the ability of our method to overcome the reality gap, we train policies for four locomotion control tasks (hopper, walker2d, half cheetah, quadruped robot) and transfer each policy to environments with different dynamics. To mimic the reality gap seen in the real-world, we use target environments that are different from the source environments in their contact modeling, latency or actuator modeling. In addition, we also test the ability of our method to generalize to discrepancies in body mass, terrain slope and end-effector materials. Figure \[fig:tasks\] shows the source and target environments for all the tasks and summarizes the modeled reality gap in each task. During training, we choose different combinations of dynamic parameters to randomize and make sure they do not overlap with the variations in the testing environments. For clarity of exposition, we denote the dimension of the dynamic parameters that are randomized during training as $dim(\mathbf{\mu})$. For all examples, we use the Proximal Policy Optimization (PPO) [@schulman2017proximal] to optimize the control policy. A more detailed description of the experiment setup as well as the simulated reality gaps are provided in Appendix \[app:exp\_detail\]. For each example presented, we run three trials with different random seeds and report the mean and one standard deviation for the total reward.
![The environments used in our experiments. Environments in the top row are source environments and environments in the bottom row are the target environments we want to transfer the policy to. (a) Hopper from DART to MuJoCo. (b) Walker2d from DART to MuJoCo with latency. (c) HalfCheetah from DART to MuJoCo with latency. (d) Minitaur robot from inaccurate motor modeling to accurate motor modeling. (e) Hopper from rigid to soft foot.[]{data-label="fig:tasks"}](images/tasks){width="0.9\linewidth"}
Baseline Methods {#ssec:baseline}
----------------
We compare our method, Strategy Optimization with CMA-ES (SO-CMA) to three baseline methods: training a robust policy (Robust), training an adaptive policy (Hist) and training a Universal Policy with Online System Identification (UPOSI) [@YuRSS17]. The robust policy is represented as a feed forward neural network, which takes as input the most recent observation from the robot, i.e. $\pi_{robust}: \vo \mapsto \va$. The policy needs to learn actions that work for all the training environments, but the dynamic parameters cannot be identified from its input. In contrast, an adaptive policy is given a history of observations as input, i.e. $\pi_{adapt}: (\vo_{t-h}, \dots, \vo_t) \mapsto \va_t$. This allows the policy to potentially identify the environment being tested and adaptively choose the actions based on the identified environment. There are many possible ways to train an adaptive policy, for example, one can use an LSTM network to represent the policy or use a history of observations as input to a feed-forward network. We find that for the tasks we demonstrate, directly training an LSTM policy using PPO is much less efficient and reaches lower end performance than training a feed-forward network with history input. Therefore, in our experiments we use a feed-forward network with a history of $10$ observations to represent the adaptive policy $\pi_{adapt}$. We also compare our method to UPOSI, which decouples the learning of an adaptive policy into training a universal policy via reinforcement learning and a system identification model via supervised learning. In theory UPOSI and Hist should achieve similar performance, while in practice we expect UPOSI to learn more efficiently due to the decoupling. We adopt the same training procedure as done by @YuRSS17, and use a history of $10$ observations as input to the online system identification model. For fair comparison, we continue to train the baseline methods after transferring to the target environment, using the same amount of samples SO-CMA consumes in the target environment. We refer this additional training step as ‘fine-tuning’. In addition to the baseline methods, we also compare our method to the performance of policies trained directly in the target environments, which serves as an ‘Oracle’ benchmark. The Oracle policies for Hopper, Walke2d, HalfCheetah and Hopper Soft was trained for $1,000,000$ samples in the target environment as in @schulman2017proximal. For the quadruped example, we run PPO for $5,000,000$ samples, similar to @TanRSS18. We detail the process of ‘fine-tuning’ in Appendix \[app:training\]
Hopper DART to MuJoCo
---------------------
In the first example, we build a single-legged robot in DART similar to the Hopper environment simulated by MuJoCo in OpenAI Gym [@brockman2016openai]. We investigate two questions in this example: 1) does SO-CMA work better than alternative methods in transferring to unknown environments? and 2) how does the choice of $dim(\mathbf{\mu})$ affect the performance of policy transfer? To this end, we perform experiments with $dim(\mathbf{\mu})=2$, $5$ and $10$. For the experiment with $dim(\mathbf{\mu})=2$, we randomize the mass of the robot’s foot and the restitution coefficient between the foot and the ground. For $dim(\mathbf{\mu})=5$, we in addition randomize the friction coefficient, the mass of the robot’s torso and the joint strength of the robot. We further include the mass of the rest two body parts and the joint damping to construct the randomized dynamic parameters for $dim(\mathbf{\mu})=10$. The specific ranges of randomization are described in Appendix \[app:training\].
We first evaluate how the performance of different methods varies with the number of samples in the target environment. As shown in Figure \[fig:hopper\_single\], when $dim(\mathbf{\mu})$ is low, none of the four methods were able to transfer to the MuJoCo Hopper successfully. This is possibly due to there not being enough variation in the dynamics to learn diverse strategies. When $dim(\mathbf{\mu})=5$, SO-CMA can successfully transfer the policy to MuJoCo Hopper with good performance, while the baseline methods were not able to adapt to the new environment using the same sample budget. We further increase $dim(\mathbf{\mu})$ to $10$ as shown in Figure \[fig:hopper\_single\] (c) and find that SO-CMA achieved similar end performance to $dim(\mathbf{\mu})=5$, while the baselines do not transfer well to the target environment.
We further investigate whether SO-CMA can generalize to differences in joint limits in addition to the discrepancies between DART and MuJoCo. Specifically, we vary the magnitude of the ankle joint limit in $[0.5, 1.0]$ radians (default is $0.785$) for the MuJoCo Hopper, and run all the methods with $30,000$ samples. The result can be found in Figure \[fig:hopper\_range\]. We can see a similar trend that with low $dim(\mathbf{\mu})$ the transfer is challenging, and with higher value of $dim(\mathbf{\mu})$ SO-CMA is able to achieve notably better transfer performance than the baseline methods.
![Transfer performance vs Sample number in target environment for the Hopper example. Policies are trained to transfer from DART to MuJoCo.[]{data-label="fig:hopper_single"}](images/hopper_single_new){width="\linewidth"}
![Transfer performance for the Hopper example. Policies are traiend to transfer from DART to MuJoCo with different ankle joint limits (horizontal axis). All trials run with total sample number of $30,000$ in the target environment.[]{data-label="fig:hopper_range"}](images/hopper_range_new){width="\linewidth"}
Walker2d DART to MuJoCo with latency
------------------------------------
In this example, we use the lower body of a biped robot constrained to a $2D$ plane, according to the Walker2d environment in OpenAI Gym. We find that with different initializations of the policy network, training could lead to drastically different gaits, e.g. hopping with both legs, running with one legs dragging the other, normal running, etc. Some of these gaits are more robust to environment changes than others, which makes analyzing the performance of transfer learning algorithms challenging. To make sure the policies are more comparable, we use the symmetry loss from @YuSIGGRAPH2018, which leads to all policies learning a symmetric running gait. To mimic modeling error seen on real robots, we add a latency of $8$ms to the MuJoCo simulator. We train policies with $dim(\mu)=8$, for which we randomize the friction coefficient, restitution coefficient and the joint damping of the six joints during training. Figure \[fig:walker\_range\] (a) shows the transfer performance of different method with respect to the sample numbers in the target environment.
We further vary the mass of the robot’s right foot in $[2, 9]$kg in the MuJoCo Walker2d environment and compare the transfer performance of SO-CMA to the baselines. The default foot mass is $2.9$ kg. We use in total $30,000$ samples in the target environment for all methods being compared and the results can be found in Figure \[fig:walker\_range\] (b). In both cases, our method achieves notably better performance than Hist and UPOSI, while being comparable to Robust.
![Transfer performance for the Walker2d example. (a) Transfer performance vs sample number in target environment on flat surface. (b) Transfer performance vs foot mass, trained with $30,000$ samples in the target environment.[]{data-label="fig:walker_range"}](images/walker_range_new){width="0.8\linewidth"}
HalfCheetah DART to MuJoCo with delay
-------------------------------------
In the third example, we train policies for the HalfCheetah environment from OpenAI Gym. We again test the performance of transfer from DART to MuJoCo for this example. In addition, we add a latency of $50$ms to the target environment. We randomize $11$ dynamic parameters in the source environment consisting of the mass of all body parts, the friction coefficient and the restitution coefficient during training, i.e. $dim(\mu)=11$. The results of the performance with respect to sample numbers in target environment can be found in Figure \[fig:halfcheetah\] (a). We in addition evaluate transfer to environments where the slope of the ground varies, as shown in Figure \[fig:halfcheetah\] (b). We can see that SO-CMA outperforms Robust and Hist, while achieves similar performance as UPOSI.
![Transfer performance for the HalfCheetah example. (a) Transfer performance vs sample number in target environment on flat surface. (b) Transfer performance vs surface slope, trained with $30,000$ samples in the target environment.[]{data-label="fig:halfcheetah"}](images/halfcheetah_new){width="0.8\linewidth"}
Quadruped robot with actuator modeling error
--------------------------------------------
As demonstrated by @TanRSS18, when a robust policy is used, having an accurate actuator model is important to the successful transfer of policy from simulation to real-world for a quadruped robot, Minitaur (Figure \[fig:tasks\] (d)). Specifically, they found that when a linear torque-current relation is assumed in the actuator dynamics in the simulation, the policy learned in simulation transfers poorly to the real hardware. When the actuator dynamics is modeled more accurately, in their case using a non-linear torque-current relation, the transfer performance were notably improved.
In our experiment, we investigate whether SO-CMA is able to overcome the error in actuator models. We use the same simulation environment from @TanRSS18, which is simulated in Bullet [@pybullet]. During the training of the policy, we use a linear torque-current relation for the actuator model, and we transfer the learned policy to an environment with the more accurate non-linear torque-current relation. We use the same $25$ dynamic parameters and corresponding ranges used by @TanRSS18 for dynamics randomization during training. When applying the robust policy to the accurate actuator model, we observe that the quadruped tends to sink to the ground, similar to what was observed by @TanRSS18. SO-CMA, on the other hand, can successfully transfer a policy trained with a crude actuator model to an environment with more realistic actuators(Figure \[fig:minitaur\_hoppersoft\] (a)).
![Transfer performance for the Quadruped example (a) and the Soft-foot Hopper example (b). []{data-label="fig:minitaur_hoppersoft"}](images/minitaur_hoppersoft_new){width="0.8\linewidth"}
Hopper rigid to deformable foot
-------------------------------
Applying deep reinforcement learning to environments with deformable objects can be computationally inefficient [@clegg2018learning]. Being able to transfer a policy trained in a purely rigid-body environment to an environment containing deformable objects can greatly improve the efficiency of learning. In our last example, we transfer a policy trained for the Hopper example with rigid objects only to a Hopper model with a deformable foot (Figre \[fig:tasks\] (e)). The soft foot is modeled using the soft shape in DART, which uses an approximate but relatively efficient way of modeling deformable objects [@jain2011controlling]. We train policies in the rigid Hopper environment and randomize the same set of dynamic parameters as in the in the DART-to-MuJoCo transfer example with $\dim(\mu)=5$. We then transfer the learned policy to the soft Hopper environment where the Hopper’s foot is deformable. The results can be found in Figure \[fig:minitaur\_hoppersoft\] (b). SO-CMA is able to successfully control the robot to move forward without falling, while the baseline methods fail to do so.
Discussions
===========
We have demonstrated that our method, SO-CMA, can successfully transfer policies trained in one environment to a notably different one with a relatively low amount of samples. One advantage of SO-CMA, compared to the baselines, is that it works consistently well across different examples, while none of the baseline methods achieve successful transfer for all the examples.
We hypothesize that the large variance in the performance of the baseline methods is due to their sensitivity to the type of task being tested. For example, if there exists a robust controller that works for a large range of different dynamic parameters $\mu$ in the task, such as a bipedal running motion in the Walker2d example, training a Robust policy may achieve good performance in transfer. However, when the optimal controller is more sensitive to $\mu$, Robust policies may learn to use overly-conservative strategies, leading to sub-optimal performance (e.g. in HalfCheetah) or fail to perform the task (e.g. in Hopper). On the other hand, if the target environment is not significantly different from the training environments, UPOSI may achieve good performance, as in HalfCheetah. However, as the reality gap becomes larger, the system identification model in UPOSI may fail to produce good estimates and result in non-optimal actions. Furthermore, Hist did not achieve successful transfer in any of the examples, possibly due to two reasons: 1) it shares similar limitation to UPOSI when the reality gap is large and 2) it is in general more difficult to train Hist due to the larger input space, so that with a limited sample budget it is challenging to fine-tune Hist effectively.
We also note that although in some examples certain baseline method may achieve successful transfer, the fine-tuning process of these methods relies on having a dense reward signal. In practice, one may only have access to a sparse reward signal in the target environment, e.g. distance traveled before falling to the ground. Our method, using an evolutionary algorithm (CMA), naturally handles sparse rewards and thus the performance gap between our method (SO-CMA) and the baseline methods will likely be large if a sparse reward is used.
Conclusion
==========
We have proposed a policy transfer algorithm where we first learn a family of policies simultaneously in a source environment that exhibits different behaviors and then search directly for a policy in the family that performs the best in the target environment. We show that our proposed method can overcome large modeling errors, including those commonly seen on real robotic platforms with relatively low amount of samples in the target environment. These results suggest that our method has the potential to transfer policies trained in simulation to real hardware.
There are a few interesting directions that merit further investigations. First, it would be interesting to explore other approaches for learning a family of policies that exhibit different behaviors. One such example is the method proposed by @eysenbach2018diversity, where an agent learns diverse skills without a reward function in an unsupervised manner. Another example is the HCP-I policy proposed by @chen2018hardware, which learns a latent representation of the environment variations implicitly. Equipping our policy with memories is another interesting direction to investigate. The addition of memory will extend our method to target environments that vary over time. We have investigated in a few options for strategy optimization and found that CMA-ES works well for our examples. However, it would be desired if we can find a way to further reduce the sample required in the target environment. One possible direction is to warm-start the optimization using models learned in simulation, such as the calibration model in @zhang2018learning or the online system identification model in @YuRSS17.
Differences between DART and MuJoCo {#app:DART_mj_diff}
===================================
DART [@lee2018DART] and MuJoCo [@todorov2012mujoco] are both physically-based simulators that computes how the state of virtual character or robot evolves over time and interacts with other objects in a physical way. Both of them have been demonstrated for transferring controllers learned for a simulated robot to a real hardware [@TanRSS18; @tan2016simulation], and there has been work trying to transfer policies between DART and MuJoCo [@wulfmeier2017mutual]. The two simulators are similar in many aspects, for example both of them uses generalized coordinates for representing the state of a robot. Despite the many similarities between DART and MuJoCo, there are a few important differences between them that makes transferring a policy trained in one simulator to the other challenging. For the examples of DART-to-MuJoCo transfer presented in this paper, there are three major differences as described below:
1. Contact Handling
Contact modeling is important for robotic control applications, especially for locomotion tasks, where robots heavily rely on manipulating contacts between end-effector and the ground to move forward. In DART, contacts are handled by solving a linear complementarity problem (LCP) [@tancontact], which ensures that in the next timestep, the objects will not penetrate with each other, while satisfying the laws of physics. In MuJoCo, the contact dynamics is modeled using a complementarity-free formulation, which means the objects might penetrate with each other. The resulting impulse will increase with the penetration depth and separate the penetrating objects eventually.
2. Joint Limits
Similar to the contact solver, DART tries to solve the joint limit constraints exactly so that the joint limit is not violated in the next timestep, while MuJoCo uses a soft constraint formulation, which means the character may violate the joint limit constraint.
3. Armature
In MuJoCo, a diagonal matrix $\sigma \mathbb{I}_n$ is added to the joint space inertia matrix that can help stabilize the simulation, where $\sigma \in \mathbb{R}$ is a scalar named Armature in MuJoCo and $\mathbb{I}_n$ is the $n \times n$ identity matrix. This is not modeled in DART.
To illustrate how much difference these simulator characteristics can lead to, we compare the Hopper example in DART and MuJoCo by simulating both using the same sequence of randomly generated actions from an identical state. We plot the linear position and velocity of the torso and foot of the robot, which is shown in Figure \[fig:mj\_dart\_comp\]. We can see that due to the differences in the dynamics, the two simulators would control the robot to reach notably different states even though the initial state and control signals are identical.
![Comparison of DART and MuJoCo environments under the same control signals. The red curves represent position or velocity in the forward direction and the green curves represent position or velocity in the upward direction.[]{data-label="fig:mj_dart_comp"}](images/mj_dart_comp){width="1.0\linewidth"}
[llll]{} & & &\
\
Hopper &$11$ & $3$ & $s_t^{vel}-0.001||a_t||_2^2+1$\
Walker2d &$17$ &$6$ & $s_t^{vel}-0.001||a_t||_2^2+1$\
HalfCheetah &$17$ &$6$ & $s_t^{vel}-0.1||a_t||_2^2+1$\
Quadruped &$12$ &$8$ & $s_t^{vel} \Delta t-0.008 \Delta t |a_t \cdot \dot{q}_t|$\
Experiment Details {#app:exp_detail}
==================
Experiment Settings
-------------------
We use Proximal Policy Optimization (PPO) implemented in OpenAI Baselines [@baselines] for training all the policies in our experiments. For simulation in DART, we use DartEnv [@dartenv], which implements the continuous control benchmarks in OpenAI Gym using PyDart [@pydart]. For all of our examples, we represent the policy as a feed-forward neural network with three hidden layers, each consists of $64$ hidden nodes.
Environment Details
-------------------
The observation space, action space and the reward function used in all of our examples can be found in Table \[env\_spec\]. For the Walker2d environment, we found that with the original environment settings in OpenAI Gym, the robot sometimes learn to hop forward, possibly due to the ankle being too strong. Therefore, we reduce the torque limit of the ankle joint in both DART and MuJoCo environment for the Walker2d problem from $[-100,100]$ to $[-20, 20]$. We found that with this modification, we can reliably learn locomotion gaits that are closer to a human running gait.
Below we list the dynamic randomization settings used in our experiments. Table \[hopper\_dr\_spec\], Table \[walker\_dr\_spec\] and Table \[cheetah\_dr\_spec\] shows the range of the randomization for different dynamic parameters in different environments. For the quadruped example, we used the same settings as in @TanRSS18.
[ll]{} &\
\
Friction Coefficient &$[0.2, 1.0]$\
Restitution Coefficient &$[0.0, 0.3]$\
Mass & $[2.0,15.0]$kg\
Joint Damping & $[0.5, 3]$\
Joint Torque Scale & $[50\%, 150\%]$\
[ll]{} &\
\
Friction Coefficient &$[0.2, 1.0]$\
Restitution Coefficient &$[0.0, 0.8]$\
Joint Damping &$[0.1, 3.0]$\
[ll]{} &\
\
Friction Coefficient &$[0.2, 1.0]$\
Restitution Coefficient &$[0.0, 0.5]$\
Mass & $[1.0,15.0]$kg\
Joint Torque Scale & $[30\%, 150\%]$\
Simulated Reality Gaps
----------------------
To evaluate the ability of our method to overcome the modeling error, we designed six types of modeling errors. Each example shown in our experiments contains one or more modeling errors listed below.
1. [DART to MuJoCo]{}
For the Hopper, Walker2d and HalfCheetah example, we trained policies that transfers from DART environment to MuJoCo environment. As discussed in Appendix \[app:DART\_mj\_diff\], the major differences between DART and MuJoCo are contacts, joint limits and armature.
2. [Latency]{}
The second type of modeling error we tested is latency in the signals. Specifically, we model the latency between when an observation $o$ is sent out from the robot, and when the action corresponding to this observation $a=\pi(o)$ is executed on the robot. When a policy is trained without any delay, it is usually very challenging to transfer it to problems with delay added. The value of delay is usually below $50$ms and we use $8$ms and $50$ms in our examples.
3. [Actuator Modeling Error]{}
As noted by @TanRSS18, error in actuator modeling is an important factor that contributes to the reality gap. They solved it by identifying a more accurate actuator model by fitting a piece-wise linear function for the torque-current relation. We use their identified actuator model as the ground-truth target environment in our experiments and used the ideal linear torque-current relation in the source environments.
4. [Foot Mass]{}
In the example of Walker2d, we vary the mass of the right foot on the robot to create a family of target environments for testing. The range of the torso mass varies in $[2, 9]$kg.
5. [Terrain Slope]{}
In the example of HalfCheetah, we vary the slope of the ground to create a family of target environments for testing. This is implemented as rotating the gravity direction by the same angle. The angle varies in the range $[-0.18, 0.0]$ radians.
6. [Rigid to Deformable]{}
The last type of modeling error we test is that a deformable object in the target environment is modeled as a rigid object in the source environment. The deformable object is modeled using the soft shape object in DART. In our example, we created a deformable box of size $0.5m \times 0.19m \times 0.13m$ around the foot of the Hopper. We set the stiffness of the deformable object to be $10,000$ and the damping to be $1.0$. We refer readers to @jain2011controlling for more details of the softbody simulation.
Policy Training {#app:training}
---------------
For training policies in the source environment, we run PPO for $500$ iterations. In each iteration, we sample $40,000$ steps from the source environment to update the policy. For the rest of the hyper-parameters, we use the default value from OpenAI Baselines [@baselines]. We use a large batch size in our experiments as the policy needs to be trained to work on different dynamic parameters $\mu$.
For fine-tuning of the Robust and Adaptive policy in the target environment, we sample $2,000$ steps from the target environment at each iteration of PPO, which is the default value used in OpenAI Baselines. Here we use a smaller batch size for two reasons: 1) since the policy is trained to work on only one dynamics, we do not need as many samples to optimize the policy in general and 2) the fine-tuning process has a limited sample budget and thus we want to use a smaller batch size so that the policy can be improved more. In the case where we use a maximum of $50,000$ samples for fine-tuning, this amounts to $50$ iterations of PPO updates. Furthermore, we use a maximum rollout length of $1,000$, while the actual length of the rollout collected during training is general shorter due to the early termination, e.g. when the robot falls to the ground. Therefore, with $50,000$ samples in total, the fine-tuning process usually consists of $100\sim300$ rollouts, depending on the task.
Strategy Optimization with CMA-ES
---------------------------------
We use the CMA-ES implementation in python by [@PyCMA]. At each iteration of CMA-ES, we generate $4+\floor{3*log(N)}$ samples from the latest Gaussian distribution, where $N$ is the dimension of the dynamic parameters. During evaluation of each sample $\mu_i$, we run the policy $\pi_{\mu_i}$ in the target environment for three trials and average the returns to obtain the fitness of this sample.
Alternative Methods for Strategy Optimization {#app:strag_opt}
=============================================
In addition to CMA-ES, we have also experimented with a few other options for finding the best $\mu$ such that $\pi_\mu$ works well in the target environment. Here we show some experiment results for Strategy Optimization with Bayesian Optimization (SO-BO) and Model-based Optimization (SO-MB).
Bayesian Optimization
---------------------
Bayesian Optimization is a gradient-free optimization method that is known to work well for low dimensional continuous problems where evaluating the quality of each sample can be expensive. The main idea in Bayesian optimization is to incrementally build a Gaussian process (GP) model that estimates the loss of a given search parameter. At each iteration, a new sample is drawn by optimizing an acquisition function on the GP model. The acquisition function takes into account the exploration (search where the GP has low uncertainty) and exploitation (search where the GP predicts low loss). The new sample is then evaluated and added to the training dataset for GP.
We test Bayesian Optimization on the Hopper and Quadruped example, as shown in Figure \[fig:sobo\_compare\]. We can see that Bayesian Optimization can achieve comparable performance as CMA-ES and thus is a viable choice to our problem. However, SO-BA appears in general noisier than CMA-ES and is in general less computationally efficient due to the re-fitting of GP models.
![Comparison of SO-CMA and SO-BA for Hopper and Quadruped examples.[]{data-label="fig:sobo_compare"}](images/sobo_compare){width="1.0\linewidth"}
Model-based Optimization
------------------------
Another possible way to perform strategy optimization is to use a model-based method. In a model-based method, we learn the dynamics of the target environment using generic models such as neural networks, Gaussian process, linear functions, etc. After we have learned a dynamics model, we can use it as an approximation of the target environment to optimize $\mu$.
We first tried using feed-forward neural networks to learn the dynamics and optimize $\mu$. However, this method was not able to reliably find $\mu$ that lead to good performance. This is possibly due to that any error in the prediction of the states would quickly accumulate over time and lead to inaccurate predictions. In addition, this method would not be able to handle problems where latency is involved.
In the experiments presented here, we learn the dynamics of the target environment with a Long Short Term Memory (LSTM) network [@hochreiter1997long]. Given a target environment, we first sample $\mu$ uniformly and collect experience using $\pi_\mu$ until we have $5,000$ samples. We use these samples to fit an initial LSTM dynamic model. We then alternate between finding the best dynamic parameters $\hat\mu$ such that $\pi_{\hat\mu}$ achieves the best performance under the latest LSTM dynamic model and update the LSTM dynamic model using data generated from $\pi_{\hat\mu}$. This is repeated until we have reached the sample budget.
![Comparison of SO-CMA and SO-MB for Hopper DART-to-MuJoCo transfer.[]{data-label="fig:socma_mb"}](images/hopper_5d_SOMB){width="0.4\linewidth"}
We found that LSTM notably outperformed feed-forward networks when applied to strategy optimization. One result for Hopper DART-to-MuJoCo can be found in Figure \[fig:socma\_mb\]. It can be seen that Model-based method with LSTM is able to achieve similar performance as CMA-ES.
Model-based method provides more flexibility over CMA-ES and Bayesian optimization. For example, if the target environment changes over time, it may be desired to have $\mu$ also be time-varying. However, this would lead to a high dimensional search space, which might require significantly more samples for CMA-ES or Bayesian Optimization to solve the problem. If we can learn an accurate enough model from the data, we can use it to generate synthetic data for solving the problem.
However, there are two major drawbacks for Model-based method. The first is that to learn the dynamics model, we need to have access to the full state of the robot, which can be challenging or troublesome in the real-world. In contrast, CMA-ES and Bayesian optimization only require the final return of a rollout. Second, the Model-based method is significantly slower to run than the other methods due to the frequent training of the LSTM network.
|
---
abstract: 'This paper addresses the problem of existence of generalized Landsberg structures on surfaces using the Cartan–Kähler Theorem and a Path Geometry approach.'
author:
- 'S. V. Sabau, K. Shibuya and H. Shimada'
title: 'On the existence of Generalized Unicorns on Surfaces [^1]'
---
Introduction
============
A Finsler norm, or metric, on a real smooth, $n$-dimensional manifold $M$ is a function $F:TM\to \left[0,\infty \right)$ that is positive and smooth on $\widetilde{TM}=TM\backslash\{0\}$, has the [*homogeneity property*]{} $F(x,\lambda v)=\lambda F(x,v)$, for all $\lambda > 0$ and all $v\in T_xM$, having also the [*strong convexity*]{} property that the Hessian matrix $$g_{ij}=\frac{1}{2}\frac{\partial^2 F^2}{\partial y^i\partial y^j}$$ is positive definite at any point $u=(x^i,y^i)\in \widetilde{TM}$.\
The fundamental function $F$ of a Finsler structure $(M,F)$ determines and it is determined by the (tangent) [*indicatrix*]{}, or the total space of the unit tangent bundle of $F$ $$\Sigma_F:=\{u\in TM:F(u)=1\}$$ which is a smooth hypersurface of $TM$.\
At each $x\in M$ we also have the [*indicatrix at x*]{} $$\Sigma_x:=\{v\in T_xM \ |\ F(x,v)=1\}=\Sigma_F\cap T_xM$$ which is a smooth, closed, strictly convex hypersurface in $T_xM$.\
A Finsler structure $(M,F)$ can be therefore regarded as smooth hypersurface $\Sigma\subset TM$ for which the canonical projection $\pi:\Sigma\to M$ is a surjective submersion and having the property that for each $x\in M$, the $\pi$-fiber $\Sigma_x=\pi^{-1}(x)$ is strictly convex including the origin $O_x\in T_xM$. We point out that the strong convexity condition of $F$ implies that the fiber $\Sigma_x$ is strictly convex, but the converse is not true (see [@BCS2000] for details on this point and a counterexample).\
A generalization of this notion is the [*generalized Finsler structure*]{} introduced by R. Bryant. In the two dimensional case a generalized Finsler structure is a coframing $\omega=(\omega^1,\omega^2,\omega^3)$ on a three dimensional manifold $\Sigma$ that satisfies some given structure equations (see [@Br1995]). By extension, one can study the generalized Finsler structure $(\Sigma,\omega)$ defined in this way ignoring even the existence of the underlying surface $M$. It was pointed out by C. Robles that in the case $n>2$, there will be no such globally defined coframing on the $2n-1$-dimensional manifold $\Sigma$. The reason is that even though the orthonormal frame bundle $\mathcal{F}$ over $M$ does admit a global coframing, it is a peculiarity of the $n=2$ dimensional case that $\mathcal{F}$ can be identified with $\Sigma$ (see also [@BCS2000], p. 92-93 for concrete computations).\
A generalized Finsler structure is [*amenable*]{} if the space of leaves $M$ of the foliation $\{\omega^1=0,\omega^2=0\}$ is differentiable manifold such that the canonical projection $\pi:\Sigma\to M$ is a smooth submersion.\
In order to study the differential geometry of the Finsler structure $(M,F)$, one needs to construct the pull-back bundle $(\pi^*TM,\pi,\Sigma)$ with the $\pi$-fibers $\pi^{-1}(u)$ diffeomorphic to $T_xM$, where $u=(x,v)\in \Sigma$ (see [@BCS2000]). In general this is not a principal bundle.\
By defining an orthonormal moving coframing on $\pi^*TM$ with respect to the Riemannian metric on $\Sigma$ induced by the Finslerian metric $F$, the moving equations on this frame lead to the so-called Chern connection. This is an almost metric compatible, torsion free connection of the vector bundle $(\pi^*TM,\pi,\Sigma)$.\
The canonical parallel transport $\Phi_t:T_xM\setminus 0 \to
T_{\sigma(t)}M\setminus 0 $, defined by the Chern connection along a curve $\sigma$ on $M$, is a diffeomorphism that preserves the Finslerian length of vectors. Unlike the parallel transport on a Riemannian manifold, $\Phi_t$ is not a linear isometry in general.\
This unexpected fact leads to some classes of special Finsler metrics. A Finsler metric whose parallel transport is a linear isometry is called a [**Berwald metric**]{}, and one whose parallel transport is only a Riemannian isometry is called a [**Landsberg metric**]{} (see [@B2007] for a very good exposition).\
Equivalently, a Berwald metric is a Finsler metric whose Chern connection coincides with the Levi Civita connection of a certain Riemannian metric on $M$, in other words it is“Riemannian-metrizable". These are the closest Finslerian metric to the Riemannian ones. The connection is Riemannian, while the metric is not. However, in the two dimensional case, any Berwald structure is Riemannian or flat locally Minkowski, i.e. there are no geometrically interesting Berwald surfaces.\
Landsberg structures have the property that the Riemannian volume of the Finslerian unit ball is a constant. This remarkable property leads to a proof of Gauss-Bonnet theorem on surfaces [@BCS2000] and other interesting results.Obviously, any Berwald structure is a Landsberg one. However, there are no examples of global Landsberg structures that are not Berwald. This is one of the main open problems in modern Finsler geometry.
[**Problem.**]{} [*Do there exist Landsberg structures that are not Berwald?*]{}
The long time search for this kind of metric structures with beautiful properties, which everybody wanted to see but no one could actually get, makes D. Bao to call these metrics “unicorns".\
On the other hand, on several occasions since 2002, R. Bryant claimed that there is plenty of [*generalized*]{} Landsberg structures on manifolds that are not Berwald. Moreover, he said that there are a lot of such generalized metrics depending on two families of two variables (see [@B2007], p. 46–47).\
Even though from the first prophecy on the existence of generalized unicorns several years already passed, as far as we know, there is no proof or paper to confirm and develop further Bryant’s affirmations.\
The purpose of this paper is two folded. First, we give a proof of the existence of generalized Landsberg structures on surfaces, which are not generalized Berwald structures and discuss their local amenability.\
Namely, we prove the following
\
Secondly, using a path geometry approach we construct locally a generalized Landsberg structure by means of a Riemannian metric $g$ on the manifold of $N$-parallels $\Lambda$ (see [@Br1995] for a similar study of existence of generalized Finsler structures with $K=1$). In the case when such Riemannian metric has its Levi-Civita connection $\nabla^g$ in a Zoll projective class $[\nabla]$ on $S^2$ it follows this generalized unicorn is in fact a classical one. We conjecture that this is always possible.\
In this way, even though we haven’t explicitly computed yet the fundamental function $F:TM\to [0,\infty)$ of this Landsberg metric, our study gives an affirmative answer to the Problem posed above in the two dimensional case (see also [@Sz2008a], [@M2008], [@Sz2008b] for discussions on the existence of smooth unicorns in arbitrary dimension). Of course a proof for our conjecture in Section 9 remains to be given.
Our method is based on an [*upstairs*]{} - [*downstairs*]{} gymnastics by moving between the base manifold and the total space of a fiber bundle.\
We give here the outline of our method in order to help the reader finding his way through the paper.\
We start by assuming the existence of a generalized Landsberg structure $\{\omega^1,\omega^2,\omega^3\}$ on a 3-manifold $\Sigma$ and we perform first a coframe change (\[coframe\_change\]) by means of a function $m$ on $
\Sigma$ such that the new coframe $\{\theta^1,\theta^2,\theta^3\}$ has the properties:
1. it satisfies the structure equations (\[k\_struct\_eq\]);
2. its “geodesic foliation” $\{\theta^1=0,\theta^3=0\}$ coincides with the “indicatrix foliation” $\{\omega^1=0,\omega^2=0\}$ of the generalized Landsberg structure $(\Sigma,\omega)$.
Assuming these two conditions for $(\Sigma,\theta)$ we obtain a set of differential conditions for the function $m$ in terms of its directional derivatives with respect to the coframe $\omega$ given in Proposition 6.1, or, equivalently, in Proposition 6.2 if we start conversely.\
Based on these, one can remark the following:
1. the function $m$ is invariant along the leaves of the foliation $\{\omega^2=0,\omega^3=0\}$, therefore, if we assume that $(\Sigma,\omega)$ is normal amenable, i.e. the leaf space of $\{\omega^2=0,\omega^3=0\}$ is a 2-dimensional differentiable manifold $\Lambda$, and the quotient projection $\nu:\Sigma\to \Lambda$ is a smooth submersion, then $m$ actually lives “downstairs” on this manifold rather than “upstairs” on $\Sigma$ as initially expected;
2. If we realize $\{\theta^1,\theta^2,\theta^3\}$ as the canonical coframe of a Riemannian metric $g$ on $\Lambda$, then the function $k$ in (\[k\_struct\_eq\]) is the lift of the Gauss curvature of the Riemannian metric $g$, hence the name “curvature condition” for (\[condC\_up\]) is motivated;
3. since we have constructed from the beginning the coframe $\theta$ on $\Sigma$ such that its geodesic foliation will generate the indicatrix leafs on $\Sigma$, if we could choose a Riemannian metric $g$ “downstairs” on $\Lambda$ all of whose geodesics are embedded circles, then the amenability of $(\Sigma,\omega)$ would be guaranteed. It is known that this kind of Riemannian metric exists and they are usually called Zoll metrics (see [@B1978], [@G1976]). A more general concept is the Zoll projective structure $[\nabla]$ on $\Lambda$ (see §3.2 as well as [@LM2002]). These are projective equivalence classes of torsion free affine connections on $\Lambda$ whose geodesics are embedded circles in $\Lambda$. Moreover, under some very reasonable conditions they are metrizable by Riemannian metrics whose Levi-Civita connections $\nabla^g$ belong to the initial Zoll projective structure $[\nabla]$.
All these imply that if we start “downstairs” with a Riemannian metric $g=u^2[(dz^1)^2+(dz^2)^2]$ on $\Lambda$, for some isothermal coordinates $(z^1,z^2)\in \Lambda$, where $u$ is a smooth function on $\Lambda$, then we can construct the $g$-orthonormal oriented frame bundle $\mathcal F(\Lambda)$ with its canonical coframe, say $\{\alpha^1,\alpha^2,\alpha^3\}$.\
On the other hand, we set up a second order PDE system on $\Lambda$ for the functions $u, \bar m$ such that the lift $\widetilde m=\nu^*(\bar m)$ satisfies the conditions of Proposition 6.2. The Cartan-Kähler theorem tells us that such pairs of functions $(u,\bar{m})$ exist and they depend on 4 functions of 1 variable (Proposition 8.1). Then, by the coframe changing (\[inverse\_coframe\]) we obtain a new coframe $\widetilde \omega$ on the 3-manifold $\Sigma:=\mathcal F(\Lambda)$ which will satisfy the structure equations (\[Lands\_struct\_eq\]) of a generalized Landsberg structure. The isothermal coordinates $(z^1,z^2)$ on $\Lambda$ and a homogeneous coordinate in the fiber of $\nu:\mathcal F(\Lambda) \to \Lambda$ over a point $z\in \Lambda$ will give a local form (\[normal\_form\]).\
The following diagram shows our [*upstairs*]{}-[*downstairs*]{} gymnastics. $$\begin{matrix}
Upstairs \quad & (\Sigma,\omega)& \xrightarrow{m}& \quad (\Sigma,\theta)&\quad \equiv \quad & \quad\quad (\mathcal
F(\Lambda),\alpha)& \xrightarrow{\widetilde m}& \Sigma:=(\mathcal F (\Lambda),\widetilde \omega)\\
& & & s^*\downarrow & & \nu^*\uparrow & & \\
Downstairs & & &\quad (\Lambda,g) &\quad \equiv \quad &(\Lambda,\widetilde g) & &
\end{matrix}$$ We use extensively the Cartan-Kähler theory in this paper in order to study the existence of integral manifolds of linear Pfaffian systems associated to PDE’s upstairs as well as downstairs. The nontriviality of our generalized Landsberg structures can be achieved by choosing appropriate initial conditions for the integral submanifolds.\
The theory of exterior differential systems is one of the strongest tools to study geometric structures. E. Cartan and other mathematicians reformulated various type of geometric structures by the exterior differential systems’ terminology. However, very few essentially new results were obtained except for the work of R. Bryant, and few others (see [@Br; @et; @al; @1991], [@IL2003] and the references in these two fundamental books).\
In the present paper, the Cartan-Kähler theorem is essentially used to find the new geometric structures, namely generalized Landsberg structures. This shows the usefulness and applicability of the theory of exterior differential systems.\
For the concrete computations regarding Cartan-Kähler Theorem we have used the MAPLE package Cartan found in the Jeanne Clelland’s home page (http://math.colorado.edu/ $\tilde{ }$ jnc/). We have found it extremely useful for checking this kind of computations.
\*
Here is the structure of our paper. After a short survey of some basic notions of Finsler surfaces and generalized Finsler structures on surfaces in Section 2, we construct the linear Pfaffian exterior differential system in Section 3 whose integral manifolds are the sought structures.\
Using it we prove a local existence theorem for generalized Landsberg structures on surfaces that are not of Berwald type using the Cartan-Kähler theorem for linear Pfaffian systems in Section 4. Firstly, we assume the existence of generalized Landsberg structures on surfaces and build a linear Pfaffian system whose integral manifolds consist of the scalar invariants $I$ and $K$ of the generalized Landsberg structure considered. Then Cartan-Kähler theorem tells us also that this kind of generalized structures depend on two arbitrary functions of two variables on $\Sigma$ (§4.1, §4.2). This proves Bryant’s affirmations.\
However, this discussion holds good under the assumption that generalized Landsberg structures exist. We will show here more, namely, we will study the involutivity of a Pfaffian system on $\Sigma$ whose integral manifolds consist of the coframe $\omega$ satisfying the structure equations together with the scalar invariants $I$, $K$ satisfying the Bianchi identities . This Pfaffian system is not a linear one, so we needed to prolong, but finally, Cartan-Kähler theorem tells us that these structures depend on 3 functions of 3 variables (§4.3). The degree of freedom is in this case higher than before, including the findings in §4.1, §4.2 as partial results.\
We discuss the local amenability of these structures in Section 5.\
Since the Cartan-Kähler theory is not very popular amongst the Finsler geometers, we introduce the basic notions and results in an Appendix. For the same reason, at the first use of the Cartan-Kähler theorem for linear Pfaffian systems, we present the computations in detail. Later uses of the theorem in §4.3 and §8.2 show only the main formulas leaving the heavy computations to be verified by the reader.\
In order to obtain an amenable Landsberg structure on a 3-manifold $\Sigma$ we have considered in Section 6 a special coframe changing on $\Sigma$ constructed such that the indicatrix foliation of the initial Landsberg structure to coincide to the geodesic foliation of the new constructed structure. Moreover, this new coframe is realizable as the canonical coframe on the orthogonal frame bundle of a Riemannian surface (Section 7).\
Keeping all these in mind, by inverting the procedure in Section 7 we have constructed in Section 8 a generalized Landsberg structure, on the total space $\mathcal F(\Lambda)$ of the orthonormal frame bundle of a Riemannian surface $(\Lambda,g)$, in terms of a smooth function $\bar m$ on $
\Lambda$. The Landsberg structure is not a Berwald one if $\bar m$ is not constant.\
Finally, in Section 9, we discuss a possible way to show the existence of classical two dimensional unicorns. This problem is equivalent to the problem of finding a Riemannian metric $g$ that metrizes a Zoll projective class on $S^2$ and satisfies in the same time the PDE system (\[condL\_2\]), (\[condC\_2\]). We conjecture that this is always possible.\
Then, by construction the geodesic foliation $\{\alpha^1=0,\alpha^3=0\}$ of $g$ will foliate the 3-manifold $\mathcal F(\Lambda)$ by circles and the geodesic leaf space, say $M$, of the geodesic foliation naturally becomes a differentiable manifold and the leaf quotient mapping $\pi:
\mathcal F(\Lambda)\to M$ becomes a smooth submersion. In other words, we obtain a double fibration (see §3.1 and §3.2).\
Therefore, by our procedure it follows that this generalized Landsberg structure is amenable and its fibers $\pi^{-1}
(x)$ are compact, where $\pi:\mathcal F(\Lambda) \to M$, $x\in M$.\
A simple argument will show that this generalized Landsberg structure is actually a classical Landsberg structure on the 2-manifold $M$, provided our conjecture is true.
The authors would like to express their gratitude to Vladimir Matveev who pointed out an error in a previous version of the paper. We also thank to David Bao, Gheorghe Pitis and Colleen Robles for many useful discussions. We are also indebted to Keizo Yamaguchi for his valuable advises. Finally, we are grateful to the referee who pointed out the importance of the amenability of the generalized Landsberg structure and for many other helpful suggestions.
Riemann–Finsler surfaces
========================
We are going to restrict ourselves for the rest of the paper to the two dimensional case. To be more precise, our manifold $\Sigma$ will be always 3-dimensional, and the manifold $M$ will be 2-dimensional, in the case it exists.\
A 3-dimensional manifold $\Sigma$ endowed with a coframing $
\omega=(\omega^1,\omega^2,\omega^3)$ which satisfies the structure equations $$\label{finsler_struct_eq}\begin{split}
d\omega^1&=-I\omega^1\wedge\omega^3+\omega^2\wedge\omega^3\\
d\omega^2&=-\omega^1\wedge\omega^3\\
d\omega^3&=K\omega^1\wedge\omega^2-J\omega^1\wedge \omega^3
\end{split}$$ will be therefore called a [*generalized Finsler surface*]{}, where $I$, $J$, $K$ are smooth functions on $\Sigma$, called the invariants of the generalized Finsler structure $
(\Sigma,\omega)$ (see [@Br1995] for details).\
As long as we work only with generalized Finsler surfaces, it might be possible that this generalized structure is not realizable as a classical Finslerian structure on a surface $M$. This imposes the following definition [@Br1995].\
A generalized Finsler surface $(M,\omega)$ is said to be [*amenable*]{} if the leaf space $
\mathcal{M}$ of the codimension 2 foliation defined by the equations $\omega^1=0$, $\omega^2=0$ is a smooth surface such that the natural projection $\pi:\Sigma\to \mathcal{M}$ is a smooth submersion.\
As R. Bryant emphasizes in [@Br1995] the difference between a classical Finsler structure and a generalized one is global in nature, in the sense that [*every generalized Finsler surface structure is locally diffeomorphic to a classical Finsler surface structure.* ]{}\
The following fundamental result can be also found in [@Br1995]\
\
In the same source it is pointed out that if for example the $\{\omega^1=0,\ \omega^2=0\}$ leaves are not compact, or even in the case they are, if they are ramified, or if the curves $\Sigma_x$ winds around origin in $T_xM$, in any of these cases, the generalized Finsler surface structure is not realizable as a classical Finsler surface.\
An illustrative example found in [@Br1995] is the case of an amenable generalized Finsler surface such that the invariant $I$ is constant, however $I$ is not zero. This kind of generalized structure is not realizable as a Finsler surface because $I\neq 0$ means that the leaves of the foliation $\{\omega^1=0, \ \omega^2=0\}$ are not compact. Indeed, in the case $I^2<4$, the $\pi$-fibers $\Sigma_x$ are logarithmic spirals in $T_xM$.\
Let us return to the general theory of generalized Finsler structures on surfaces. By taking the exterior derivative of the structure equations (\[finsler\_struct\_eq\]) one obtains the [*Bianchi equations of the Finsler structure*]{}: $$ \begin{split}
& J=I_2\\
& K_3+KI+J_2=0,
\end{split}$$ where we denote by $I_i$ the directional derivatives with respect to the coframing $\omega$, i.e. $$df=f_1\omega^1+f_2\omega^2+f_3\omega^3,$$ for any smooth function $f$ on $\Sigma$.\
Taking now one more exterior derivative of the last formula written above, one obtains the Ricci identities with respect to the generalized Finsler structure $$\begin{split}
& f_{21}-f_{12}=-Kf_3\\
& f_{32}-f_{23}=-f_1\\
& f_{31}-f_{13}=If_1+f_2+Jf_3.
\end{split}$$
1. Remark first that the structure equations of a Riemannian surface are obtained from (\[finsler\_struct\_eq\]) by putting $I=J=0$.
2. Since $J=I_2$, one can easily see that the necessary and sufficient condition for a generalized Finsler structure to be non-Riemannian is $I\neq 0$.
on $\Sigma$ is a generalized Finsler structure $(M,\omega)$ such that $J=0$, or equivalently, $I_2=0$.\
Remark that such a generalized structure is characterized by the structure equations $$\label{Lands_struct_eq}\begin{split}
d\omega^1&=-I\omega^1\wedge\omega^3+\omega^2\wedge\omega^3\\
d\omega^2&=-\omega^1\wedge\omega^3\\
d\omega^3&=K\omega^1\wedge\omega^2,
\end{split}$$ and Bianchi identities $$\label{Lands_Bianchi}\begin{split}
dI & =I_1\omega^1 \qquad \qquad +I_3\omega^3 \\
dK & =K_1\omega^1+K_2\omega^2-KI\omega^3,
\end{split}$$ where $\omega=(\omega^1$, $\omega^2$, $\omega^3)$ is a coframing on a certain 3-dimensional manifold $\Sigma$, and $I$ and $K$ are smooth functions defined on $\Sigma$. We will see that we actually need more, so we assume that the functions $I$ and $K$ are analytic on $\Sigma$.\
It is also useful to have the Ricci identities [@BCS2000] for the invariants $I$ and $K$. Indeed, taking first into account that $$K_{31}=-I_1K-IK_1,\quad K_{32}=-IK_2,\quad K_{33}=K(I^2-I_3),$$ we obtain $$\begin{aligned}
& I_{12}=KI_3, & K_{21}&-K_{12}=IK^2 \\
& I_{32}=-I_1, & K_{23}&=K_1-IK_2 \\
& I_{31}-I_{13}=II_1, & K_{13}&=-(2K_1I+KI_1+K_2).\end{aligned}$$ We are interested in studying the existence of non-trivial generalized Landsberg structures on $\Sigma$, i.e. generalized Landsberg structures that are not of Berwald type.\
Recall the following definition.
A [*generalized Berwald structure*]{} is a generalized Finsler structure characterized by the structure equations (\[Lands\_struct\_eq\]), and $$dI \equiv 0 \quad \mod\quad \omega^3,$$ or, equivalently, $$I_1=I_2=0,\qquad I_3\neq 0.$$ The reason we called Berwald structures (generalized or not) on surfaces [*trivial*]{} is given in the following rigidity theorem.\
[@Sz1981]\
In other words, the only possible Berwald structures are either the flat locally Minkowski ones, or the Riemannian ones. Therefore the term [*non-trivial*]{} in the present paper addresses a Landsberg structure that is not locally Minkowski, nor Riemannian. Both of these are well studied trivial examples of Landsberg surfaces.\
\
It is interesting to remark that $I_1=0$ is not the only condition that makes a Lansdsberg to become a Berwald one.
Indeed, using the structure and the Ricci equations one can easily see that if for a Landsberg structure on a surface at least one of the following relation is satisfied $$ I_3=0, \quad K_2=0,$$ then that structure must be a Berwald one.\
Remark also that the condition $$K_1=0$$ does not necessarily imply triviality. In fact, all the generalized Landsberg structures in this paper satisfy this condition.
Path Geometries
===============
Path geometries of a generalized Landsberg structure
----------------------------------------------------
Recall from [@Br1997] that a [*(classical) path geometry*]{} on a surface $M$ is a foliation $\mathcal P$ of the projective tangent bundle $\mathbb P (TM)$ by contact curves, each of which is transverse to the fibers of the canonical projection $\pi:\mathbb P (TM)\to M$.\
Namely, let $\gamma:(a,b)\to M$ be a smooth, immersed curve, and let us denote by $\hat{\gamma}:(a,b)\to \mathbb{P}(TM)$ its canonical lift to the projective tangent bundle $\pi:\mathbb{P}(TM)\to M$. Then, the fact that the canonical projection $\pi$ is a submersion implies that, for each line $L\in \mathbb{P}(TM)$, the linear map $$\pi_{*,L}:T_L \mathbb{P}(TM)\to T_xM,$$ is surjective, where $\pi(L)=x\in M$. Therefore $$E_L:=\bigl(\pi_{*,L}\bigr)^{-1}(L)\subset T_L \mathbb{P}(TM)$$ is a 2-plane in $T_L \mathbb{P}(TM)$ that defines a contact distribution and therefore a contact structure on $\mathbb{P}(TM)$.\
A curve on $\mathbb{P}(TM)$ is called [*contact curve*]{} if it is tangent to the contact distribution $E$. Nevertheless, the canonical lift $\hat{\gamma}$ to $\mathbb{P}(TM)$ of a curve $\gamma$ on $M$ is a contact curve.\
A [*local path geometry*]{} on $M$ is a foliation $\mathcal P$ of an open subset $U\subset \mathbb P (TM)$ by contact curves, each of which is transverse to the fibers of $\pi:\mathbb P (TM)\to M$.\
In the case there is a surface $\Lambda$ and a submersion $l:\mathbb P (TM) \to \Lambda$ whose fibers are the leaves of $\mathcal P$, then the path geometry will be called [*amenable*]{}.\
More generally, a [*generalized path geometry*]{} on a 3-manifold $\Sigma$ is a pair of transverse codimension 2 foliations $(\mathcal P, \mathcal Q)$ with the property that the (unique) 2-plane field $D$ that is tangent to both foliations defines a contact structure on $\Sigma$.\
In the case when there is a surface $\Lambda_{\mathcal P}$ and a submersion $l_{\mathcal P}:\mathbb P (TM) \to \Lambda_{\mathcal P}$ whose fibers are the leaves of the foliation $\mathcal P$, then the generalized path geometry $(\mathcal P, \mathcal Q)$ will be called $\mathcal P$-[*amenable*]{}. A $\mathcal Q$-[*amenable*]{} generalized path geometry $(\mathcal P, \mathcal Q)$ is defined in a similar way.\
One can easily see that a classical path geometry on $\Sigma=\mathbb P (TM)$ is a special case of generalized path geometry where the second foliation $ \mathcal Q$ is taken to be the fibers of the canonical projection $\pi:\mathbb
P (TM)\to M$.\
In the case of a Landsberg structure on a 3-manifold $\Sigma$, we can define two kinds of generalized path geometries.\
We can consider
1. $ \mathcal P := \{\omega^1=0,\omega^3=0\}$ the [*“geodesic” foliation*]{} of $\Sigma$, i.e. the leaves are curves on $\Sigma$ tangent to $\hat e_2$;
2. $ \mathcal Q := \{\omega^1=0,\omega^2=0\}$ the [*“indicatrix” foliation*]{} of $\Sigma$, i.e. the leaves are curves on $\Sigma$ tangent to $\hat e_3$;
3. $ \mathcal R := \{\omega^2=0,\omega^3=0\}$ the [*“normal” foliation*]{} of $\Sigma$, i.e. the leaves are curves on $
\Sigma$ tangent to $\hat e_1$.
We can consider now the generalized path geometries $$ \mathcal G_1=(\mathcal P, \mathcal Q),\qquad \mathcal G_2=(\mathcal R, \mathcal Q).$$ Remark that on the case of $\mathcal G_1$, the 2-plane field $D_1=<\hat e_2,\hat e_3>$ defines indeed a contact structure on $\Sigma$. To verify this we need a contact 1-form $\eta$ on $\Sigma$ such that $D_1=\ker \eta$. By definition it follows that $\eta$ has to be $$ \eta=A\omega^1$$ and we have $$ \eta\wedge d\eta=A^2\omega^1\wedge\omega^2\wedge\omega^3.$$ Therefore $\eta$ is a contact form on $\Sigma$ if and only if $A\neq 0$, so $\mathcal G_1$ is a well defined path geometry on $\Sigma$.\
We can do the same discussion for $\mathcal G_2$, where the 2-plane field is $D_2=<\hat e_1,\hat e_3>$. As above, we look for $\eta$ such that $D_2=\ker \eta$, so we must have $$ \eta=A\omega^2,$$ and a simple computation shows that again $$ \eta\wedge d\eta=A^2\omega^1\wedge\omega^2\wedge\omega^3.$$ Therefore, again $\eta$ is a contact form on $\Sigma$ if and only if $A\neq 0$, and again $\mathcal G_2$ is a well defined path geometry on $\Sigma$.\
Recall also from the same [@Br1997] that [*every generalized path geometry is always identifiable with a local path geometry on a surface*]{}. Indeed, for a $u\in \Sigma$, let $U\subset \Sigma$ be an open neighborhood of $u$ on which the foliation $\mathcal Q$ is locally amenable, i.e. there exist a smooth surface $M$ and a smooth surjective submersion $\pi:U\to M$ such that the fibers of $\pi$ are the leaves of $Q$ restricted to $U$. Remark that this is always possible (for example due to Frobenius theorem) and that $M$ and $\pi$ are uniquely determined by $U$ up to a diffeomorphism.\
A natural smooth map $\nu:U\subset \Sigma\to \mathbb P (TM)$, which makes the following diagram commutative, $$\begin{split}
& \qquad \ \ \quad \nu \\
& U\subset \Sigma \longrightarrow \mathbb P (TM) \\
\pi & \downarrow \qquad \swarrow \\
& M
\end{split}$$ can be defined as follows $$\nu(u)=\pi_{*,u}(T_u\mathcal P),$$ for any $u\in U$. This application is well defined because $\pi_{*,u}(T_u\mathcal P)$ is a 1-dimensional subspace of $T_{\pi(u)}M$, and therefore an element of $\mathbb P (T_{\pi(u)}M)$.\
For the generalized path geometry $\mathcal G_1=(\mathcal P, \mathcal Q)$ we put $$ \nu_1:U\subset \Sigma\to \mathbb P (TM),\qquad \nu_1(u)=\pi_{*,u}(\hat e_2),$$ and for the generalized path geometry $\mathcal G_2=(\mathcal R, \mathcal Q)$ we put $$ \nu_2:U\subset \Sigma\to \mathbb P (TM),\qquad \nu_2(u)=\pi_{*,u}(\hat e_1).$$ Remark that because the foliations $\mathcal P$, $\mathcal Q$ and $\mathcal R$ are all transverse to each other, it follows again that $\pi_{*,u}(T_u \mathcal P)$ and $\pi_{*,u}(T_u \mathcal R)$ are 1-dimensional subspaces in $T_{\pi(u)}M$, i.e. $\nu_1$, $\nu_2$ are immersions and therefore local diffeomorphisms.
Zoll projective structures
--------------------------
A classical example of a path geometry on a 3-manifold is the path geometry of a Riemannian metrizable Zoll projective structure. This is not only an example of path geometry, but it will be very useful in the construction of a non-trivial Landsberg structure.\
Recall that a Riemannian metric $g$ on a smooth manifold $\Lambda$ is called a [*Zoll metric*]{} if all its geodesics are simple closed curves of equal length. See [@B1978] for basics of Zoll metrics as well as [@G1976] for the abundance of Zoll metrics on $S^2$.\
We will use in the present paper a more general notion, namely Zoll projective structures. Our exposition follows closely [@LM2002].\
If $\nabla$ is a torsion free affine connection on a smooth manifold $\Lambda$, then the projective class $[\nabla]$ of $\nabla$ is called a [*Zoll projective structure*]{} if the image of any maximal geodesic of $
\nabla$ is an embedded circle $S^1\subset \Lambda$.\
Given a Zoll projective structure $[\nabla]$ on $\Lambda$, the canonical lift of its geodesics will provide the geodesic foliation $\mathcal P$ on the projectivized tangent bundle $\mathbb P(T\Lambda)$ which foliates $\mathbb P(T\Lambda)$ by circles. Let $M$ be the leaf space of the geodesic foliation $\mathcal P$ of a Zoll projective structure.\
It can be shown that any Zoll projective structure $[\nabla]$ on a compact orientable surface $\Lambda$ is [*tame*]{}, namely each leaf of its geodesic foliation on $\mathbb P(T\Lambda)$ has a neighborhood which is diffeomorphic to $\mathbb R^2\times S^1$, such that each leaf corresponds to a circle of the form $\{u\}\times S^1$, for any $u\in \mathbb P(T\Lambda)$.\
This implies further that the leaf space $M$ of a Zoll projective structure $[\nabla]$ on a compact orientable surface $\Lambda$ has a canonical structure of differentiable manifold such that the quotient map $\pi:\mathbb P(T
\Lambda) \to M$ becomes a submersion. We obtain therefore the following [*double fibration*]{} of a Zoll projective structure. $$\begin{split}
& \qquad \mathbb P (T\Lambda) \\
& \nu \swarrow \qquad \searrow \pi\\
& \Lambda\qquad\qquad \quad M
\end{split}$$ Let us assume from now $\Lambda=S^2$. It is natural to ask when a given Zoll projective structure $[\nabla]$ on $S^2$ can be represented by the Levi-Civita connection of a Riemannian metric $g$ on $\Lambda=S^2$.\
The answer is given in Theorem 4.8. of [@LM2002], p. 512. We are not going to state or to prove this theorem here because it will take too much space to define all the notions that are involved. Instead, we are going to describe the construction of the Riemannian metric $g$ that represents a Zoll projective structure, in the case such a metric exists. It is clear from [@LM2002] that the set of Riemannian metrizable Zoll projective structures is not empty, so we can assume the existence of Riemannian metrizable Zoll projective structures $[\nabla]$ on $S^2$.\
Let us consider the isothermal local coordinates $(z^1,z^2)$ on $S^2$ induced from the Zoll projective structure (the concrete construction can be found in [@LM2002], p. 513), and let $$g=u^2\Bigl[ (dz^1)^2+(dz^2)^2 \Bigr],$$ be the metric on $S^2$ in these coordinates, where $u$ is a smooth function. If one puts $$\gamma=d\log u,$$ then the Levi-Civita connection $\nabla^g$ of the Riemannian metric $g$ belongs to the Zoll projective structure $
[\nabla]$ if $$\label{Gamma}\Gamma_{kl}^j=\gamma_k\delta_l^j+\gamma_l\delta_k^j-\gamma^j\delta_{kl},$$ where $\gamma=\gamma_1dz^1+\gamma_2dz^2$, and $\Gamma_{kl}^i$ are the Christoffel symbols of the Zoll projective structures $[\nabla]$, i.e. $$\Gamma_{jk}^i=\Bigl< dz^i,\nabla_{\frac{\partial}{\partial z^j}}\frac{\partial}{\partial z^k}\Bigr>$$ for a connection $\nabla$ in the Zoll projective structure $[\nabla]$, and $\gamma^j=g^{ji}\gamma_i$.\
It follows that for a given Zoll projective structure $[\nabla]$ we obtain $$\label{small_gamma}\gamma_i=\frac{1}{2}\Bigl(\Gamma_{i1}^1+\Gamma_{i2}^2 \Bigr),\qquad i=1,2.$$ If we denote by $R$ the Gauss curvature of $g$, then taking into account that $\gamma_i=\frac{1}{u}\frac{\partial u}{\partial z^i}$, it follows $$R=-\frac{1}{u^2}\textrm{div} \gamma,$$ where we put $\textrm{div} \gamma=\frac{\partial\gamma_1}{\partial z^1}+\frac{\partial\gamma_2}{\partial z^2}$.\
If we denote by $\{\alpha^1,\alpha^2,\alpha^3\}$ the canonical coframe on the bundle of $g$-orthonormal frames on $\Lambda$ then $\mathcal G=(\mathcal P, \mathcal Q)$ is a path geometry on $\mathbb P(T\Lambda)$, where $\mathcal P:=\{\alpha^1=0,\alpha^2=0\}$ is the geodesic foliation of $g$ and $\mathcal Q:=\{\alpha^1=0,\alpha^3=0\}$.
The Cartan–Kähler theory
========================
A linear Pfaffian system on generalized Landsberg surfaces
----------------------------------------------------------
This section and the following one are motivated by Bryant’s prophecy on the existence of generalized unicorns that we mentioned already in Introduction. Since the Finsler geometry community is familiarized with his statements, we will give here our interpretation of it. We point out however, that the discussion following hereafter does not imply the existence of non-trivial generalized unicorns. This will be shown only in section 4.3 in a different setting.\
In order to make use of the Cartan-Kähler theory, we are going to construct an exterior differential system associated to the coframe $(\omega^1,\omega^2,\omega^3)$ that satisfies (\[Lands\_struct\_eq\]), (\[Lands\_Bianchi\]).\
In this section we [*assume*]{} the existence of three linear independent one forms $(\omega^1,\omega^2,\omega^3)$ on the 9-dimensional manifold $\widetilde{\Sigma}=\Sigma\times \mathbb R^2\times \mathbb R^4$ that satisfy the structure equations (\[Lands\_struct\_eq\]), where we consider the free coordinates $(I,K)\in \mathbb R^2$, and $(I_1,I_3,K_1,K_2)\in \mathbb R^4$, and study the degree of freedom of the scalar functions $I$ and $K$.\
First, we consider the following 1-forms $$\label{Lands_Pfaffian}\begin{split}
\theta^1: & = dI-I_1\omega^1-I_3\omega^3 \\
\theta^2: & = dK -K_1\omega^1-K_2\omega^2+KI\omega^3,
\end{split}$$ and let us denote by $\mathcal{I}$ the differential ideal generated by $\{\theta^1,\theta^2\}$. We also denote $$\begin{split}
\Omega & :=\omega^1\wedge\omega^2\wedge\omega^3, \\
J & :=\{\theta^a,\omega^i\}, \\
I& :=\{\theta^a\},
\end{split}$$ where [*a*]{}=1,2, [*i*]{}=1,2,3.\
We will use the same letter $I$ for the invariant of a (generalized) Finsler structure as well as for the set of 1-forms $\theta^1,\theta^2$. We hope that this will not lead to any confusion.\
In order to use the Cartan–Kähler theory we are going to consider the pair $(I,J)$ as an EDS with independence condition on a certain manifold $\widetilde{\Sigma}$ to be determined later. We consider $dI$ and $dK$ as linearly independent 1-forms on the manifold $\widetilde{\Sigma}$.\
By exterior differentiation of $\{\theta^1,\theta^2\}$ we obtain $$\begin{split}
d\theta^1 & =-dI_1\wedge \omega^1-dI_3\wedge \omega^3-I_3K\omega^1\wedge \omega^2-I_1\omega^2\wedge
\omega^3+II_1\omega^1\wedge\omega^3\\
d\theta^2 & =-dK_1\wedge\omega^1-dK_2\wedge\omega^2+IK^2\omega^1\wedge\omega^2+(IK_2-
K_1)\omega^2\wedge\omega^3\\
& +(2IK_1+I_1K+K_2)\omega^1\wedge\omega^3+K\theta^1\wedge\omega^3+I\theta^2\wedge\omega^3.
\end{split}$$ Let us remark that the above formulas can be rewritten as $$\begin{split}
d\theta^1 & \equiv (-dI_1+I_3K\omega^2-II_1\omega^3)\wedge\omega^1+
(-dI_3-I_1\omega^2)\wedge\omega^3 \quad \mod\ \{I\} \\
d\theta^2 & \equiv \Bigl[-dK_1-IK^2\omega^2-(2IK_1+I_1K+K_2)\omega^3\Bigr]
\wedge\omega^1\\
& +\Bigl[-dK_2-(IK_2-K_1)\omega^3\Bigr]\wedge\omega^2 \quad \mod\ \{I\}.
\end{split}$$ It follows that we can write $$d\theta^a \equiv \pi^a_i\wedge \omega^i\quad \mod\ \{I\},$$ where [*a*]{}=1,2, [*i*]{}=1,2,3.\
The 1-forms matrix $(\pi^a_i)$ has the following non-vanishing entries: $$\label{pi_matrix}
\begin{split}
\pi_1^1 & = -dI_1+I_3K\omega^2-II_1\omega^3,\\
\pi_3^1 & = -dI_3-I_1\omega^2,\\
\pi_1^2 & = -dK_1-IK^2\omega^2-(2IK_1+I_1K+K_2)\omega^3\\
\pi_2^2 & = -dK_2-(IK_2-K_1)\omega^3.
\end{split}$$ By putting now $$\label{pi_vector}
\begin{split}
& \pi^1:=\pi_1^1,\qquad \pi^2:=\pi_3^1,\\
& \pi^3:=\pi_1^2,\qquad \pi^4:=\pi_2^2,
\end{split}$$ we obtain that $(I,J)$ is a linear Pfaffian system that lives on the 9 dimensional manifold $\widetilde{\Sigma}$ which has the coframing $$\{\theta^1,\theta^2,\omega^1,\omega^2,\omega^3,\pi^1,\pi^2,\pi^3,\pi^4
\}$$ that is adapted to the filtration $$I\subset J\subset T^*\widetilde{\Sigma}.$$ Since the apparent torsion was absorbed, we can write $$d\theta^a \equiv A_{\epsilon i}^a\pi^\epsilon \wedge \omega^i \qquad
\mod\ \{I\},$$ where the non-vanishing entries of $ A^a_{i\epsilon}$ are $$\label{Lands_tableau}A_{11}^1=A^1_{23}=A^2_{31}=A^2_{42}=1.$$ The 1-forms $\pi_a^i$ are sections of $T^*\widetilde{\Sigma} / J$, or, equivalently, they are components of a section of $I^*\otimes J/I$.\
From now on, by abuse of notation we will write the structure equations of the EDS as $$\begin{split}
& \theta^a=0\\
& d\theta^a \equiv A_{\epsilon i}^a\pi^\epsilon \wedge \omega^i \qquad \mod\ \{I\}\\
& \Omega=\omega^1\wedge \omega^2\wedge \omega^3\neq 0.
\end{split}$$ From (\[Lands\_tableau\]) it follows that the tableau $A$ of the linear Pfaffian system $(I,J)$ is given by $$A=
\begin{pmatrix}
a & 0 & d \\
b & c & 0
\end{pmatrix},$$ where $a,b,c,d$ are nonzero constants. Therefore, the reduced characters of the tableau $A$ are $s_1=2$, $s_2=2$, $s_3=0$, and $s_0=$rank$\ I=2$.\
The symbol $B$ of the linear Pfaffian system $(I,J)$ is then $$B=
\begin{pmatrix}
0 & e & 0 \\
0 & 0 & f
\end{pmatrix},$$ where $e,f$ are nonzero constants.
The integrability conditions
----------------------------
Let us denote by $(G_3(T\widetilde{\Sigma}),\pi,\widetilde{\Sigma})$ the Grassmannian of three planes through the origin of $T\widetilde{\Sigma}$. Then the dimension of the base manifold and the fiber over a point $p\in
\widetilde{\Sigma}$ are given by $$\dim G_3(T\widetilde{\Sigma}) =27,\qquad \dim G_3(T_p\widetilde{\Sigma}) =18,$$ respectively.\
If we denote by $p_i^a$, ($a=1,...,6$, $i=1,2,3$) the local coordinates of the fiber $G_3(T_p
\widetilde{\Sigma})$, then for a $3$-plane $E\in G_3(T_p\widetilde{\Sigma})$, that satisfies the independence condition $\omega^1\wedge\omega^2\wedge\omega^3_{\ |E}\neq 0$, by an eventual relabeling of the coordinates, equations of integral elements of $(I,J)$ are $$\label{integ_elem}\begin{split}
& \theta^b=0,\ (b=1,2)\\
& \pi^\epsilon-p^{\epsilon}_i\omega^i=0,
\end{split}
$$ where $p_i^\epsilon$, ($\epsilon=1,...,4$, $i=1,2,3$) are functions on $G_3(T_p\widetilde{\Sigma})$.\
The relations (\[integ\_elem\]) regarded as system of linear equations in $p_i^\epsilon$ are the [*first order integrability conditions*]{} of the linear Pfaffian system $(I,J)$. One can remark that in the most general case, these equations are over-determined, in the sense that there are more equations than unknowns. Therefore, in general there is likely for such linear systems to be incompatible.\
In our case, using the fact that integral elements of $\theta^a=0$ must satisfy $d\theta^a=0$ also, then using (\[Lands\_tableau\]) we obtain the solutions of (\[integ\_elem\]) as follows: $$\label{integ_manif}\begin{split}
& p_2^1=0,\qquad \qquad \qquad p_3^1=p_1^2,\\
& p_2^2=0,\\
& p_3^3=0,\qquad \qquad \qquad p_2^3=p_1^4,\\
& p_3^4=0,
\end{split}
$$ the rest of the functions, namely $p^1_1,p^2_1,p^2_3,p^3_1,p^4_1,p^4_2$, being arbitrary.\
One can see that the maximum rank of this system of functions is $d=6$, and that it is of local rank constant. In other words, $\mathcal V_3(\mathcal{I},\Omega)$ is a smooth codimension 6 submanifold of $G_3(T\widetilde{\Sigma})$, where we denoted by $\mathcal V_3(\mathcal{I},\Omega)\subset G_3(T
\widetilde{\Sigma})$ the subbundle of 3-dimensional integral elements of $\mathcal{I}$.\
Remark that $(\mathcal
V_3(\mathcal{I},\Omega),\widetilde{I})$ is the prolongation of $(\widetilde{\Sigma},\mathcal{I})$, where $\mathcal{I}$ is the exterior differential system generated by the Pfaffian system $I$. Here, $\widetilde{I}$ is the exterior differential system on $\mathcal
V_3(\mathcal{I},\Omega)$ generated by the Pfaffian system $$\widetilde{I}=\{\theta^1,\theta^2,\pi^1-p_1^1\omega^1-p_1^2\omega^3,
\pi^2-p_1^2\omega^1-p_3^2\omega^3,\pi^3-p_1^3\omega^1-p_1^4\omega^2,
\pi^4-p_1^4\omega^1-p_2^4\omega^2\},$$ i.e. $\widetilde{I}$ is the pullback to $\mathcal
V_3(\mathcal{I},\Omega)$, by the inclusion $\iota:\mathcal
V_3(\mathcal{I},\Omega)\to G_3(T_p\widetilde{\Sigma})$, of the canonical system on $G_3(T_p\widetilde{\Sigma})$.\
Moreover, since the dimension of the solution space of equations (\[integ\_manif\]) is 6, the Cartan involutivity test is satisfied: $$s_1+2s_2+3s_3=2+2\cdot2+0=6=d.$$ Using the Cartan-Kähler theorem for linear Pfaffian systems (see [@IL2003], p. 176, [@Br; @et; @al; @1991] for a more general exposition, and the Appendix), we can summarize the findings in this section in the following theorem.\
\
We emphasize the fact that the existence of analytical integral manifolds of $(I,J)$ is guaranteed only in a neighborhood $U\subset \widetilde{\Sigma}$ of $\tilde{u}$.\
Therefore, for any point $\tilde{u}\in\widetilde{\Sigma}$ chosen such that $I_1\neq 0$, the existence of an integral submanifold of $(I,J)$ passing through this point is guaranteed by Theorem 4.1. This is a non-trivial generalized Landsberg surface structure on which the independence condition $\omega^1\wedge\omega^2\wedge\omega^3\neq 0$ is satisfied. In other words, this integral submanifold can be realized as the graph of the analytical mapping $$\begin{split}
&\Sigma\to\widetilde{\Sigma},\\
&u \mapsto (u,I(u),K(u),I_1(u),I_3(u),K_1(u),K_2(u))\in \widetilde{\Sigma}.
\end{split}$$ This proves R. Bryant’s prophecy. Unfortunately, these generalized structures are not always amenable, in other words, they are not always realizable as Finsler structures on surfaces as will be seen.\
\
If we write the structure equations as $$\begin{pmatrix}
d\theta^1\\
d\theta^2
\end{pmatrix}
=
\begin{pmatrix}
\pi^1 & 0 & \pi^2 \\
\pi^3 & \pi^4 & 0
\end{pmatrix}
\wedge
\begin{pmatrix}
\omega^1\\
\omega^2\\
\omega^3
\end{pmatrix}
,$$ then we can put them in a normal form which reflects the Cartan test for involutivity.\
Indeed, if one changes the basis $\{\omega^1,\omega^2,\omega^3\}$ to $\{\widetilde{\omega}^1:=\omega^1,\widetilde{\omega}^2:=\omega^2,\widetilde{\omega}^3:=\omega^3-\omega^2\}$, then it follows $$\begin{split}
& d\theta^1\equiv {\pi}^1\wedge\widetilde{\omega}^1+{\pi}^2\wedge\widetilde{\omega}^2+{\pi}^2\wedge
\widetilde{\omega}^3\\
& d\theta^2\equiv
{\pi}^3\wedge\widetilde{\omega}^1+{\pi}^4\wedge\widetilde{\omega}^2,\qquad\qquad\qquad \mod {I}.
\end{split}$$ Therefore, in this frame, the tableau $A$ of $(I,J)$ is now given by $$A=
\begin{pmatrix}
a & d & d \\
b & c & 0
\end{pmatrix}.$$ One can now directly verify by visual inspection that, indeed, there are $s_1=2$ independent 1-forms in the first column of the tableau matrix of $(I,J)$, $s_1+s_2=4$ independent 1-forms in the first two columns, and $s_1+s_2+s_3=4$ independent 1-forms in the first three columns, i.e. in the entire matrix. This agrees with Cartan’s test for involutivity.
The existence of generalized Landsberg structures on surfaces
-------------------------------------------------------------
In the present section we are going to generalize our setting and show the existence of the coframes $\omega$ satisfying (\[Lands\_struct\_eq\]) together with the scalar functions $I$ and $K$ satisfying (\[Lands\_Bianchi\]), without using any of the assumptions in §4.1, §4.2.\
Let $\Sigma$ be again a 3-manifold, and let $\pi:\mathcal{F}(\Sigma)\to\Sigma$ be its frame bundle, namely $$\mathcal{F}(\Sigma)=\{(u,f_u) | f_u:T_u\Sigma\to V \ \textrm{linear isomorphism}\},$$ where $V$ is a 3-dimensional real vector space.\
Let $\eta$ be the tautological $V$-valued 1-form on $\mathcal{F}(\Sigma)$, defined as usual by $$\label{taut_def}
\eta_f(w)=f_u(\pi_*w),$$ where $f=(u,f_u)\in \mathcal{F}(\Sigma)$, and $w\in T_f\mathcal{F}(\Sigma)$.\
It is known that a coframe on the manifold $\mathcal{F}(\Sigma)$ is given by $(\eta^i,\alpha^i_j)$, $i,j=1,2,3$, where $\eta^i$ are the components of the $V$-valued tautological form $\eta$, and $\alpha^i_j$ are the 1-forms on $\mathcal{F}(\Sigma)$ that satisfy the structure equations $$\label{str.eq_1}
d\eta^i=-\alpha^i_j\wedge\eta^j,\qquad i,j=1,2,3.$$ Such 1-forms always exist, but without supplementary conditions, they are not unique. These forms are the connection forms of the frame bundle.\
Here, we choose a “flat type connection form”, i.e. 1-forms $\alpha^i_j$ satisfying $$\label{str.eq_2}
d\alpha^i_j=\alpha^i_k\wedge\alpha^k_j,\qquad i,j,k=1,2,3.$$ We define next the following (local) trivialization of the frame bundle $$\label{id}
\begin{split}
t:& \ \mathcal{F}(\Sigma)\to \Sigma\times GL(3,\mathbb{R})\\
&f=(u,f_u)\mapsto (u,(f_j^i)),
\end{split}$$ where for a coordinate system $(x^1,x^2,x^3)$ on $\Sigma$, and a basis $\{e_1,e_2,e_3\}$ of $V$, $(f_j^i)$ is the representation matrix of the mapping $f_u:T_u\Sigma\to\Sigma$ with respect to the bases $\{\frac{\partial}{\partial x^i}\}$ and $\{e_i\}$.\
A system of coordinates on $\Sigma\times GL(3,\mathbb{R})$ is given by $(x^i,f^i_j)$, $i,j=1,2,3$, and a coframe on the manifold $\Sigma\times GL(3,\mathbb{R})$ will be $(\omega^i,df^i_j)$, where we put $$\label{omegas}
\omega^i=f_j^idx^j.$$ We remark that the tautological 1-forms $\eta=(\eta^i)$, $i=1,2,3$, on $\mathcal{F}(\Sigma)$ correspond to the 1-forms $(\omega^i)$ under the identification (\[id\]). This can be verified by direct computation checking that the 1-forms $\omega^i$ in (\[omegas\]) satisfy (\[taut\_def\]).\
Moreover, if we put $$\beta_j^i=d(f_k^i)(f^{-1})_j^k,\qquad i,j,k=1,2,3,$$ then the 1-forms $(\beta_j^i)$ on $\mathcal{F}(\Sigma)$ correspond to the “connection forms” $(\alpha_j^i)$. Indeed, a straightforward computation shows that the $\beta_j^i$’s defined above verify the structure equations (\[str.eq\_1\]), (\[str.eq\_2\]).\
With these preparations in hand, we move on to the study of the existence of a coframe $\omega$ and the scalars $I$, $K$ on the 3-manifold $\Sigma$ that satisfy (\[Lands\_struct\_eq\]) and (\[Lands\_Bianchi\]), respectively.\
In order to do this, we consider the 18-dimensional manifold $$\widetilde{\Sigma}=\mathcal{F}(\Sigma)\times \mathbb{R}^6$$ with the coframe $$\{\eta^1,\eta^2,\eta^3,(\alpha_j^i)_{i,j=1,2,3},\theta^1,\theta^2,\pi^1,\pi^2,\pi^3,\pi^4\},$$ where $\pi^1,\pi^2,\pi^3,\pi^4$ are the 1-forms in , .\
We consider the 1-forms $$\label{forms1_eta}
\begin{split}
\Theta^1 & =d\eta^1+I\eta^1\wedge\eta^3-\eta^2\wedge\eta^3\\
\Theta^2 & =d\eta^2+\eta^1\wedge\eta^3\\
\Theta^3 & =d\eta^3-K\eta^1\wedge\eta^2
\end{split}$$ and $$\label{forms2_eta}
\begin{split}
\theta^1 & =dI-I_1\eta^1-I_3\eta^3\\
\theta^2 & =dK-K_1\eta^1-K_2\eta^2+KI\eta^3,
\end{split}$$ obtaining in this way the exterior differential system $$\widetilde{\mathcal I}=\{\Theta^1,\Theta^2,\Theta^3,\theta^1,\theta^2\}$$ with independence condition $$\Omega=\eta^1\wedge\eta^2\wedge\eta^3\neq 0.$$
Let us remark that any element $E\in G_3(T\widetilde{\Sigma})$ such that $\Omega |_{E}\neq 0$ is defined by $$\begin{split}
\alpha^i_{j\ |E} & =A^i_{jk}(E)\eta^k_{\ |E}\\
\theta^i_{\ |E} & =B_k^i(E)\eta^k_{\ |E}\\
\pi^i_{\ |E} & =C_k^i(E)\eta^k_{\ |E},
\end{split}$$ where $(A^i_{jk})_{i,j,k=1,2,3}$, $(B_k^i)_{i=1,2;k=1,2,3}$, $(C_k^i)_{i=1,2,3,4;k=1,2,3}$ are smooth functions on $G_3(T\widetilde{\Sigma},\Omega)$. In other words, $(A^i_{jk},B_k^i,C_k^i)$ are the fiber coordinates of the fibration $G_3(T\widetilde{\Sigma},\Omega)\to \widetilde{\Sigma}$. This fiber is 45-dimensional.\
However, due to the identification (\[id\]) and the discussion above, we can consider the local coordinates $$(x^1,x^2,x^3,(f_j^i)_{i,j=1,2,3},I,K,I_1,I_3,K_1,K_2)\in\widetilde{\Sigma}$$ on the 18-dimensional manifold $\mathcal{F}(\Sigma)\times\mathbb{R}^6$ and identify the 1-forms $\eta^i$ with $\omega^i$ given in (\[omegas\]). Since the settings are equivalent, for simplicity, we will work in these coordinates instead of the general case described at the beginning of this subsection.\
It follows that the 1-forms (\[forms1\_eta\]), (\[forms2\_eta\]) of the exterior differential system $\widetilde{\mathcal{I}}$, can be written as $$\label{forms1_omega}
\begin{split}
\Theta^1 & =d\omega^1+I\omega^1\wedge\omega^3-\omega^2\wedge\omega^3\\
\Theta^2 & =d\omega^2+\omega^1\wedge\omega^3\\
\Theta^3 & =d\omega^3-K\omega^1\wedge\omega^2
\end{split}$$ and $$\label{forms2_omega}
\begin{split}
\theta^1 & =dI-I_1\omega^1-I_3\omega^3\\
\theta^2 & =dK-K_1\omega^2-K_2\omega^2+KI\omega^3,
\end{split}$$ with independence condition $$\Omega=\omega^1\wedge\omega^2\wedge\omega^3\neq 0,$$ where $\omega$’s are given by (\[omegas\]).\
The integral manifolds of $(\widetilde{\mathcal{I}},\Omega)$ will consist of the coframe $\{\omega^1,\omega^2,\omega^3\}$, and the functions $(I,K,I_1,I_3,K_1,K_2)$ on $\Sigma$. The projection of such integral manifold to $\Sigma$ gives a generalized Landsberg structure $(\Sigma,\omega)$.\
Let us remark that the situation is now quite different from the one in §4.1. The $\Theta$’s are 2-forms, while $\theta$’s are 1-forms, so the exterior differential system $(\widetilde{\mathcal{I}},\Omega)$ is not a linear Pfaffian system, and therefore we cannot apply the Cartan-Kähler theorem for linear Pfaffian systems as we did previously. Even there are more general versions of the Cartan-Kähler theorem, the strategy we adopt here is to prolong $\widetilde{\mathcal{I}}$ in order to obtain a linear Pfaffian system (for details see [@IL2003], p. 177).\
Let us consider the prolongation $\mathcal{V}(\widetilde{\mathcal{I}},\Omega)\subset G_3(T\widetilde{\Sigma})$ over $\widetilde{\Sigma}$, with the fiber inhomogeneous Grassmannian coordinates $\Bigl((p^i_j)_{i=1,2;j=1,2,3}, (p_{jk}^i)_{i,j,k=1,2,3}$,\
$(q_k^i)_{i=1,2,3,4;k=1,2,3}\Bigr)$, such that $$\begin{split}
\theta^i_{|_E} & =p_k^i(E)dx^k_{|_E}\\
{df_j^i}_{|_E} & =p_{jk}^i(E)dx^k_{|_E}\\
\pi^i_{|_E} & =q_k^i(E)dx^k_{|_E},
\end{split}$$ for any integral element $E$.\
Then, the equations $$\begin{split}
& \theta^i=d\theta^i=0,\qquad i=1,2,\\
& \Theta^j=d\Theta^j=0,\qquad j=1,2,3,
\end{split}$$ will give the defining equations of the prolongation $\mathcal{V}(\widetilde{\mathcal{I}},\Omega)$.\
As concrete computation, we remark first that $\theta^i=0$ will imply $p_j^i=0$, so these functions will not appear in out analysis. A similar computation as in §4.1 shows that the structure equations for $\theta$’s are $$\begin{split}
& d\theta^1\equiv \pi^1\wedge \omega^1+\pi^2\wedge \omega^2\quad \mod \{\theta,\Theta\} \\
& d\theta^2\equiv \pi^3\wedge \omega^1+\pi^4\wedge \omega^3.
\end{split}$$ These equations will give some of the $q_j^i$’s.\
The equations $\Theta^i\equiv 0$ $\mod \{\theta,\Theta\}$ will give some of the $p_{jk}^i$. The rest of the equations $d\Theta^i\equiv 0$ $\mod \{\theta,\Theta\}$ will be satisfied due to some Bianchi identities, so they will give no further conditions.\
In this way, we obtain the linear Pfaffian $\widetilde{\widetilde{\mathcal{I}}}$ on $\mathcal{V}(\widetilde{I},\Omega)$ generated by the 1-forms $$\label{large_lin_pfaff}
\{\theta^1,\theta^2,({\Theta^i_j})_{i,j=1,2,3},\Pi^1,\Pi^2,\Pi^3,\Pi^4\},$$ where $$\begin{split}
& \Theta^i_j=df^i_j-p_{jk}^idx^k,\quad i,j,k=1,2,3,\\
& \Pi^i=\pi^i-q^i_kdx^k,\quad i=1,2,3,4, \quad k=1,2,3
\end{split}$$ and we will study its involutivity by means of Cartan-Kähler theory as we did in §4.1, §4.2.\
It is easy to see that putting the conditions $d\theta^i=0$, $i=1,2$ it results 6 relations with 12 unknown functions $(q_j^i)_{i=1,2,3,4;\ j=1,2,3}$. We solve $q_3^1$, $q_2^2$, $q_3^2$ in terms of $q_1^1$, $q_2^1$, $q_1^2$, and $q_3^3$, $q_2^4$, $q_3^4$ in terms of $q_1^3$, $q_2^3$, $q_1^4$. It follows $$\begin{split}
& \quad q_2^2=\frac{1}{f_1^3}\Bigl(q_1^1f_2^1-q_2^1f_1^1+q^2_1f_1^3\Bigr),\\
& \begin{pmatrix}
q_3^1\\
q_3^2
\end{pmatrix}=
\begin{pmatrix}
-f_2^1 & -f_2^3 \\
f_1^1 & f_1^3
\end{pmatrix}^{-1}
\begin{pmatrix}
-q_2^1f_3^1 & -q_2^2f_3^2 \\
q_1^1f_3^1 & q_1^2f_3^3
\end{pmatrix},
\end{split}$$ and $$\begin{split}
& \quad q_2^4=\frac{1}{f_1^2}\Bigl(q_3^1f_2^1-q_2^3f_1^1+q^4_1f_1^2\Bigr),\\
& \begin{pmatrix}
q_3^3\\
q_3^4
\end{pmatrix}=
\begin{pmatrix}
-f_2^1 & -f_2^2 \\
f_1^1 & f_1^2
\end{pmatrix}^{-1}
\begin{pmatrix}
-q_2^1f_3^1 & -q_2^4f_3^2 \\
q_1^3f_3^1 & q_1^4f_3^2
\end{pmatrix}.
\end{split}$$ In the same way, from $\Theta^i_j=0$, $i,j=1,2,3$, we obtain 9 relations with 27 unknown functions $(p_{jk}^i)$, $i,j,k=1,2,3$. Solving 9 of them, we obtain $$\begin{split}
& p_{31}^1=p_{13}^1-I(f_3^1f_1^3-f_1^1f_3^3)+(f_3^2f_1^3-f_1^2f_3^3)\\
& p_{12}^1=p_{21}^1-I(f_2^1f_1^3-f_1^1f_2^3)+(f_2^2f_1^3-f_1^2f_2^3)\\
& p_{23}^1=p_{32}^1-I(f_3^1f_2^3-f_2^1f_3^3)+(f_3^2f_2^3-f_2^2f_3^3),
\end{split}$$ $$\begin{split}
& p_{12}^2=p_{21}^2-(f_1^1f_2^3-f_2^1f_1^3)\\
& p_{23}^2=p_{32}^2-(f_2^1f_3^3-f_3^1f_2^3)\\
& p_{31}^3=p_{13}^3-(f_3^1f_1^3-f_1^1f_3^3),
\end{split}$$ $$\begin{split}
& p_{12}^3=p_{21}^3+K(f_1^1f_2^2-f_2^1f_1^2)\\
& p_{23}^3=p_{32}^3+K(f_2^1f_3^2-f_3^1f_2^2)\\
& p_{31}^3=p_{13}^3+K(f_3^1f_1^2-f_1^1f_3^2).
\end{split}$$ Using these relations we study the involutivity of the linear Pfaffian (\[large\_lin\_pfaff\]). By similar computations as in §4.1, §4.2 we obtain that the structure equations of (\[large\_lin\_pfaff\]) are given by $$\label{large_str.eq}
d\begin{pmatrix} \theta^1 \\ \theta^2 \\ \Theta^1_1 \\ \Theta^1_2 \\ \Theta^1_3 \\ \Theta^2_1 \\ \Theta^2_2 \\ \Theta^2_3 \\ \Theta^3_1 \\ \Theta^3_2 \\ \Theta^3_3 \\ \Pi^1 \\ \Pi^2 \\ \Pi^3 \\ \Pi^4
\end{pmatrix}\equiv
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
\rho^1 & \rho^2 & \rho^3\\
\rho^2 & \rho^4 & \rho^5\\
\rho^3 & \rho^5 & \rho^6\\
\rho^7 & \rho^8 & \rho^9\\
\rho^8 & \rho^{10} & \rho^{11}\\
\rho^9 & \rho^{11} & \rho^{12}\\
\rho^{13} & \rho^{14} & \rho^{15}\\
\rho^{14} & \rho^{16} & \rho^{17}\\
\rho^{15} & \rho^{17} & \rho^{18}\\
\rho^{19} & \rho^{20} & \Phi^1\\
\rho^{21} & \Phi^2 & \Phi^3\\
\rho^{22} & \rho^{23} & \Phi^4\\
\rho^{24} & \Phi^5 & \Phi^6
\end{pmatrix}
\begin{pmatrix}
dx^1 \\ dx^2 \\ dx^3
\end{pmatrix}\qquad \mod\{\widetilde{\widetilde{\mathcal{I}}}\},$$ where $\rho^i$, $i=1,\dots,24$ are 1-forms on $\mathcal{V}(\widetilde{\mathcal{I}},\Omega)$, linearly independent from the 1-forms in (\[large\_lin\_pfaff\]), and $\Phi^j$, $j=1,\dots,6$ are linear combinations of the $\rho$’s.\
It means that the apparent torsion can be absorbed. It also can be checked that the space of integral elements at each point has dimension 38.\
On the other hand, the reduced characters of the tableau corresponding to (\[large\_str.eq\]) are $$s_1=13,\quad s_2=8,\quad s_3=3,$$ and Cartan test’s for involutivity reads $$s_1+2s_2+3s_3=38.$$ Therefore the Pfaffian system (\[large\_lin\_pfaff\]) is involutive.\
Putting all these together, and assuming that $\Sigma$ and $\alpha,\eta$ are analytic, from Cartan-Kähler theory we obtain
\
\
Since the projection of an integral manifold of the prolongation $\widetilde{\widetilde{\mathcal{I}}}$ to $\widetilde{\Sigma}$ is also an integral manifold of $\widetilde{\mathcal{I}}$, it follows
\
The non-triviality of the integral manifolds can be obtained by choosing an appropriate initial value. See the discussion at the end of §4.2.\
\
We point out that the degree of freedom of the integral manifolds of $\widetilde{\widetilde{\mathcal{I}}}$ does not equal the degree of freedom of the scalar functions $I$ and $K$. The reason is that the 3 functions of 3 variables obtained in Theorem 4.2 include the degree of freedom of the coframe $(\omega^1,\omega^2,\omega^3)$ as well.
The local amenability of generalized Landsberg structures on surfaces
=====================================================================
The notion of amenability given in Definition 2.2 has the following local version
The generalized Finsler structure $(\Sigma,\omega)$ is called [*locally*]{} [*amenable*]{} if for any point $u\in \Sigma$, there exists an open neighborhood $U\subset \Sigma$ of $u$ to which $(\Sigma,\omega)$ restricts to be amenable, i.e. $(U,\omega_{|U})$ is amenable in the sense of Definition 2.2.
We can now formulate a local version of the Theorem 2.1.
[**Theorem 5.1.**]{}
[*Proof.*]{} The proof is quite straightforward. Remark first that the differential system $\{\omega^1=0,\omega^2=0\}$ is completely integrable. Indeed, the structure equations (\[finsler\_struct\_eq\]) of a generalized Finsler structure show that $$\begin{split}
& d\omega^1 \equiv 0\qquad \qquad \mod\{\omega^1,\omega^2\}.\\
& d\omega^2 \equiv 0
\end{split}$$ It follows from Frobenius theorem that for any point $u\in \Sigma$, there exists an open neighborhood $U\subset\Sigma$ of $u$ such that the leaf space of the foliation $\{\omega^1_{|U}=0,\omega^2_{|U}=0\}$ is a differentiable manifold, say $\widetilde{U}$, such that the canonical projection $\pi:U\to \widetilde{U}$ is a smooth submersion.
From here we see immediately that $\nu:U\to T(\widetilde{U})$ is a smooth embedding.
Q. E. D.
We point out that the condition (1) in Theorem 2.1 is not necessarily true for this $U$.
Indeed, imagine for a moment the case when the generalized Finsler structure $(\Sigma,\omega)$ satisfies all the conditions in Theorem 2.1, i.e. it is a classical Finsler structure on a differentiable surface $M$ such that $\pi:\Sigma\to M$ is a smooth submersion. In this case, even though if we restrict ourselves to a small neighborhood $\widetilde{U}\subset M$, the fibers $\Sigma_x$ over $x\in \widetilde{U}$ are not changed in any way, they remain diffeomorphic to $S^1$ when we shrink the base manifold $M$.
This situation changes dramatically when we are working with a local generalized structure on $\Sigma$. Considering the neighborhood $U \subset \Sigma$ as given by the Frobenius theorem, the fibers are also [*cut off*]{}. The situation is similar with taking a neighborhood of a point on the surface of the sphere $S^2$, for example. In general, the great circles will have only some open arcs contained in this neighborhood, and there is no reason for these arcs to be compact.
Hence, the local conditions in Theorem 5.1 are not enough for $(\Sigma, \omega)$ to be classical Finsler structure on $\widetilde{U}$.
Therefore, we have
[**Corollary 5.2.**]{}
In conclusion, recall that we have proved the existence of non-trivial generalized Landsberg surfaces in Theorem 4.1. In other words, the Cartan–Kähler theorem assures us that there exists a neighborhood $U\subset \Sigma$ such that $(U,\omega_{|U})$ is a non-trivial generalized Landsberg surface.
On the other hand, since the differential system $(U,\omega^1,\omega^2)$ is completely integrable, from the discussion above it follows that, on a possible smaller open set on $\Sigma$, there exists the local coordinate system $u=(x,y,p)$ such that the leaf space of the foliation $\{\omega^1=0,\omega^2=0\}$ is a differentiable manifold.
We can therefore conclude that for a small enough $\varepsilon>0$, there exist amenable non-trivial generalized Landsberg structures $(U,\omega)$, depending on two functions of two variables, over an open disk $D=\{(x,y):x^2+y^2<\varepsilon\}\subset \widetilde{U}$ in the plane.
Finally, we emphasize that these non-trivial Landsberg generalized structures do not necessarily satisfy the condition (1) in Theorem 2.1, so they are not necessarily classical Finsler structures.
A special coframing
===================
For a nowhere vanishing smooth function $m$ on $\Sigma$, we define the 1-forms $$\label{coframe_change} \begin{split}
\theta^1 & =m\omega^2\\
\theta^2 & =\omega^3\\
\theta^3 & =m\omega^1+m_3\omega^2,
\end{split}$$ where the subscripts represent the directional derivatives with respect to the generalized Landsberg coframe $(\omega^1,\omega^2,\omega^3)$.\
Remark that $$\theta^1\wedge\theta^2\wedge\theta^3=m^2\omega^1\wedge\omega^2\wedge\omega^3,$$ therefore $\{\theta^1,\theta^2,\theta^3\}$ is a coframe on $\Sigma$ provided $m$ is nowhere vanishing smooth function on $
\Sigma$.\
An easy linear algebra exercise will show that we have $$\label{frame_change} \begin{split}
f_1 & =-\frac{m_3}{m}\hat{e}_1+\frac{1}{m}\hat{e}_2\\
f_2 & =\hat{e}_3\\
f_3 & =\frac{1}{m}\hat{e}_1,
\end{split}$$ where we have denoted by $\{f_1,f_2,f_3\}$ and $\{\hat{e}_1,\hat{e}_2,\hat{e}_3\}$ the dual frames of $\{\theta^1,\theta^2,\theta^3\}$ and $\{\omega^1,\omega^2,\omega^3\}$, respectively.\
We would like to impose conditions on the function $m$ and the invariants $I$, $K$ such that the new coframe $\theta=\{\theta^1,\theta^2,\theta^3\}$ satisfies the structure equations $$\label{k_struct_eq} \begin{split}
d\theta^1 & =\theta^2\wedge\theta^3\\
d\theta^2 & =\theta^3\wedge\theta^1\\
d\theta^3 & = k\theta^1\wedge\theta^2,
\end{split}$$ where $k$ is a smooth function on $\Sigma$ to be determined (one can see that from the third structure equation of the coframe $\theta$ that $dk\wedge \theta^1\wedge \theta^2=0$, therefore the directional derivative of $k$ with respect to $
\theta^3$ must vanish). This is a so-called [*K-Cartan structure*]{} (see [@GG2002]).\
Straightforward computations show that $$d\theta^1 =\theta^2\wedge\theta^3$$ holds if and only if $$m_1=0.$$ This is our first condition on $m$.\
It also follows that $$ I=-2\ \frac{m_3}{m},\quad K=m^2.$$\
In this case, we obtain $$k=1-\frac{m_{33}}{m}.$$ Remark that the [*Landsberg condition*]{} reads $$ I_2=0 \Longleftrightarrow m_{32}=\frac{m_2m_3}{m},$$ and the [non-triviality conditions]{} $$\begin{split}
I_1 & \neq 0 \Longleftrightarrow m_2\neq 0 \\I_3 & \neq 0 \Longleftrightarrow mm_{33}-(m_3)^2\neq 0\\K_2 & \neq 0 \Longleftrightarrow I_1 \neq 0\\
K_3 & \neq 0 \Longleftrightarrow m_3 \neq 0. \end{split}$$ We obtain therefore the following
[**Proposition 6.1.**]{}\
*Let $(\Sigma, \omega)$ be a generalized Landsberg structure on the 3-manifold $\Sigma$ and let $m:\Sigma \to \mathbb{R}$ be a smooth nowhere vanishing function satisfying the conditions*
1. the direction invariance condition $$m_1=0$$
2. the Landsberg condition $$m_{23}=\frac{m_2 m_3}{m}.$$
Then $\theta=\{\theta^1, \theta^2, \theta^3\}$, with the $\theta^i$’s given in (\[coframe\_change\]), is a coframe on the 3-manifold $\Sigma$ that satisfies the structure equations (\[k\_struct\_eq\]) with
\(3) the curvature condition $$\label{curv_cond}
k=1-\frac{m_{33}}{m}.$$
Remark that in this case, besides the conditions in the proposition above, the function $m$ will satisfy the Ricci type identities $$\begin{split}
& m_{21}=-m^2m_3\\
& m_{23}-m_{32}=0\\
& m_{31}=m_2.
\end{split}$$ Conversely, we can start with a coframe $\theta=\{\theta^1, \theta^2, \theta^3\}$ on the 3-manifold $\Sigma$ that satisfies the structure equations (\[k\_struct\_eq\]) for a function $k:\Sigma\to \mathbb{R}$ such that $k_{\theta 3}=0$. Here, we denote by $h_{\theta i}$ the directional derivatives of a smooth function $h$ with respect to the coframe $\theta$, i.e. $dh=h_{\theta 1}\theta^1+h_{\theta 2}\theta^2+h_{\theta 3}\theta^3$. Making use of a nowhere vanishing smooth function $m:\Sigma\to \mathbb{R}$, we can construct the 1-forms $$\label{coframe_change_omega}
\begin{split}
&\omega^1=\frac{1}{m}(\theta^3-\frac{m_{\theta 2}}{m}\theta^1)\\
&\omega^2=\frac{1}{m}\theta^1\\
&\omega^3=\theta^2.
\end{split}$$
By a simple straightforward computation we obtain
[**Proposition 6.2.**]{}
*Let $\theta=\{\theta^1, \theta^2, \theta^3\}$ be a coframe on the 3-manifold $\Sigma$ that satisfies the structure equations (\[k\_struct\_eq\]) for a smooth function $k:\Sigma \to \mathbb{R}$, and let $m:\Sigma\to \mathbb{R}$ be a nowhere vanishing smooth function that satisfies the conditions*
1. the direction invariance condition $$m_{\theta 3}=0,$$
2. the Landsberg condition $$\label{lands_cond_theta}
(L)\qquad m_{\theta 21}=0,$$
3. the curvature condition $$\label{curv_cond_theta}
(C)\qquad \frac{m_{\theta 22}}{m}=1-k.$$ Then $\omega=\{\omega^1,\omega^2,\omega^3\}$, with the $\omega^i$’s given in (\[coframe\_change\_omega\]), is a generalized Landsberg structure on the 3-manifold $\Sigma$ with the invariants $$I=-2\frac{m_{\theta 2}}{m},\quad K=m^2.$$
In this case, the Ricci type equations for $m$ in the coframe $\theta^1$, $\theta^2$, $\theta^3$ are $$\label{Ricci_cond_theta}
\begin{split}
& m_{\theta 12}=m_{\theta 21}=0\\
& m_{\theta 13}=-m_{\theta 2}\\
& m_{\theta 23}=m_{\theta 1}.
\end{split}$$
1. Let $(\Sigma,\omega)$ be a generalized Landsberg structure, and suppose that $U\subset \Sigma$ is an open set where the foliation $$\mathcal R = \{\omega^2=0,\omega^3=0\}$$ is amenable, i.e. the leaf space $\Lambda$ of integral curves of $\hat e_1$ in $U$ is a differentiable manifold, and $$l:U\to \Lambda$$ is a smooth submersion. Then $\theta^1$, $\theta^2$ can be regarded as the tautological 1-forms of the frame bundle and $\theta^3$ as the Levi-Civita connection of the Riemannian manifold $\Lambda$. The function $k$ plays the role of the Gauss curvature.
2. The indicatrix foliation $\mathcal Q:\{\omega^1=0,\omega^2=0\}$ of the generalized Landsberg structure $\{\omega^1,\omega^2,\omega^3\}$ coincides with the geodesic foliation $\mathcal P:\{\theta^1=0,\theta^3=0\}$ of the new coframe $\{\theta^1,\theta^2,\theta^3\}$ on $\Sigma$.
3. The normal foliation $\mathcal R:\{\omega^2=0,\omega^3=0\}$ of the generalized Landsberg structure $\{\omega^1,\omega^2,\omega^3\}$ coincides with the indicatrix foliation $\mathcal Q:\{\theta^1=0,\theta^2=0\}$ of the coframe $\{\theta^1,\theta^2,\theta^3\}$ on $\Sigma$.
4. In the case when the generalized Landsberg structure $\{\omega^1,\omega^2,\omega^3\}$ is realizable as a classical Finsler structure $(M,F)$ on a certain 2-dimensional differentiable manifold $M$ such that $\pi:\Sigma \to M$ is its indicatrix bundle, then the leaves of the normal foliation $\mathcal R:\{\omega^2=0,\omega^3=0\}$ are the (normal) lifts of some paths on $M$ called $N$-parallel or $N$-extremal curves. The geometric meaning of such curves $\gamma:
[a,b]\to M$ is that the normal vector field $N(t)$ along $\gamma(t)$, defined by $g_N(N,T)=0$, is parallel along $\gamma$. Here $T(t)$ is the tangent vector field to the curve $\gamma$, and $g$ is the Riemannian metric induced by the Finslerian structure in each tangent plane $T_xM$. It is also known that the $N$-parallels $\gamma$ are solutions of a second order differential equation on $M$ and the solution of this SODE is uniquely determined by some initial conditions $(x_0,Y_0)\in TM$ (see [@ISS2009] for details).
The geometry of quotient space $\Lambda$
========================================
The setting
-----------
In the light of our discussion in §6, we can conclude that if $U\subset \Sigma$ is an open set where the normal foliation $ \mathcal R = \{\omega^2=0,\omega^3=0\}$ is amenable, i.e. the leaf space $\Lambda$ of integral curves of $\hat e_1$ in $U$ is a differentiable manifold, $l:U\to \Lambda$ is a smooth submersion, and $m$ is a smooth function on $\Sigma$ that satisfies the conditions in Proposition 6.1., then there exist
1. a quadratic form $g$ on $\Lambda$ such that $l^*(g)=m^2(\omega^2)^2+(\omega^3)^2$;
2. a 2-form $dA$ on $\Lambda$ such that $l^*(dA)=m\omega^2\wedge\omega^3$;
3. a smooth function $\bar{m}$ on $\Lambda$ such that $l^*(\bar{m})=m$.
We can construct now a $g$-orthonormal coframe $\eta^1$, $\eta^2$ on $\Lambda$ (it may be only locally defined), i.e. there exist two 1-forms $
\eta^1$, $\eta^2$ on $\Lambda$, such that $$g=(\eta^1)^2+(\eta^2)^2,\qquad dA=\eta^1\wedge \eta^2>0.$$ This is equivalent with giving a smooth section $s$ of the orthonormal frame bundle $\nu:\mathcal F(\Lambda)
\longrightarrow \Lambda$, i.e. a [*first order adapted lift*]{} to the geometry of the Riemannian manifold $(\Lambda,g)$.\
If we denote by $\{e_1,e_2\}$ the dual frame of $\{\eta^1,\eta^2\}$ it follows that $\{e_{1\ |z},e_{2\ |z}\}$ is a $g$-orthonormal basis of $T_z\Lambda$, and $(z,e_{1\ |z},e_{2\ |z})\in \mathcal F(\Lambda)$ is a frame on the manifold $
\Lambda$ at each point $z\in \Lambda$.\
There exist two smooth functions, say $a$ and $b$, on $\Lambda$ such that $$\begin{split}
& d\eta^1=a\eta^1\wedge\eta^2\\
& d\eta^2=b\eta^1\wedge \eta^2.
\end{split}$$ By straightforward computation, it also follows that there exists a 1-form, say $\eta^3$, on $\Lambda$, such that $$\begin{split}
& d\eta^1=\eta^2\wedge\eta^3\\
& d\eta^2=\eta^3\wedge \eta^1,
\end{split}$$ and therefore we must have $$\eta^3=-a\eta^1-b\eta^2.$$ One can easily check that if $\{\widetilde{\eta}^1,\widetilde{\eta}^2\}$ is another $g$-orthonormal frame, then it follows $d\widetilde{\eta}^3=d\eta^3$.\
By straightforward computation we obtain further $$d\eta^3=R\eta^1\wedge\eta^2,$$ where $R=a_2-a^2-b_1-b^2$, where $a_i$, $b_i$ means directional derivatives with respect to the coframe $\{\eta^1,\eta^2\}$.\
One can easily see that for another $g$-orthonormal frame $\{\widetilde{\eta}^1,\widetilde{\eta}^2\}$, the function $R$ remains unchanged, and therefore it depends only on $g$.\
Let us denote $$\begin{split}
s(z)=(z,f_z)
\end{split}$$ a local section of $\nu:\mathcal F(\Lambda)\to \Lambda$.\
It is then known that on $\mathcal F(\Lambda)$ there are tautological 1-forms $$\alpha^i_f\in T_f^* \mathcal F(\Lambda),\quad \alpha^i_f:=\eta^i(\nu_*w),$$ where $w\in T_f \mathcal F(\Lambda)$, and $i\in\{1,2\}$, such that $$(\nu^*_f(\eta^1),\nu^*_f(\eta^2))=(\alpha^1_f,\alpha^2_f)$$ gives a basis of semibasic forms on $\mathcal F(\Lambda)$.\
Consider now the $g$-orthonormal frame bundle $\nu:\mathcal F_{\textrm{on}}(\Lambda)\to \Lambda$ with its tautological 1-forms $\{\alpha^1,\alpha^2\}$.\
If $s:\Lambda\to \mathcal F_{\textrm{on}}(\Lambda)$ is a smooth (local) section, then $$\begin{split}
& \eta^1=s^*(\alpha^1)\\
& \eta^2=s^*(\alpha^2)
\end{split}$$ is a local coframe on $\Lambda$ such that $$g=(\eta^1)^2+(\eta^2)^2.$$ Recall that the [*“downstairs”*]{} Fundamental Lemma of Riemannian geometry tells us that there exists a unique 1-form $\eta^3$ on $\Lambda$ such that $$\begin{split}
& s^*(d\alpha^1)=s^*(\alpha^2)\wedge\eta^3\\
& s^*(d\alpha^2)=\eta^3\wedge s^*(\alpha^1)\\
& d\eta^3=Rs^*(\alpha^1)\wedge s^*(\alpha^2),
\end{split}$$ where $R:\Lambda\to \mathbb R$ is the Gauss curvature of the Riemannian surface $(\Lambda,g)$. These are the so- called [*“downstairs”*]{} structure equations of the Riemannian metric $g$ on $\Lambda$.\
We also recall the [*“upstairs”*]{} Fundamental Lemma of Riemannian geometry that states that it must exist a unique 1-form $\alpha^3$ on $\mathcal F(\Lambda)$ such that $$\begin{split}
& d\alpha^1=\alpha^2\wedge \alpha^3\\
& d\alpha^2=\alpha^3\wedge\alpha^1\\
& d\alpha^3=k\alpha^1\wedge\alpha^2,
\end{split}$$ where $k:\mathcal F(\Lambda)\to \mathbb R$ is the Gauss curvature [*“upstairs”*]{}. In our setting it must satisfy the curvature condition . It follows that $R=s^*k$. These are the [*“upstairs”*]{} structure equations of the Riemannian metric $g$ on $\Lambda$. One can also see that on $\Lambda
$ we have $$(\eta^1,\eta^2,\eta^3)=s^*(\alpha^1,\alpha^2,\alpha^3).$$\
Let us consider a flat Riemannian metric $\widetilde g$ on $\Lambda$, i.e. $\widetilde R=0$. It follows that there exist local coordinates $z=(z^1,z^2)$ on $\Lambda$, such that $$\widetilde \eta^1=dz^1,\qquad \widetilde \eta^2=dz^2,$$ and therefore $a=0$, $b=0$ because $d\widetilde \eta^1=0$, $d\widetilde \eta^2=0$.\
It follows $\widetilde \eta^3=0$ as well as $R=0$.\
We construct now the coframe $(z; dz^1,dz^2)$ on $\Lambda$ and its oriented orthonormal frame bundle $\nu:\widetilde{\mathcal F}_{\textrm{on}}(\Lambda)\to \Lambda$ with respect to the Riemannian metric $$\tilde{g}=(dz^1)^2+(dz^2)^2.$$\
In this case, the tautological 1-forms on $\widetilde{\mathcal F}_{\textrm{on}}(\Lambda)$ will have the normal form $$\begin{split}
& \widetilde \alpha^1=\cos(t) dz^1-\sin(t) dz^2\\
& \widetilde \alpha^2=+\sin(t) dz^1+\cos(t) dz^2\\
& \widetilde \alpha^3=dt,
\end{split}$$ where $t\in [0,2\pi]$ is the fiber coordinate over $z\in \Lambda$.
A more general example is the local form of a metric $g=u^2\widetilde g$ conformal to the flat case discussed above, where $u$ is a smooth function on $\Lambda$. In this case we have $g=(\eta^1)^2+(\eta^2)^2$, where $$\eta^1=u \textrm{d}z^1,\qquad \eta^2=u \textrm{d}z^2.$$ By exterior differentiation it follows $$\begin{split}
& a=-\frac{1}{u^2}\frac{\partial u}{\partial z^2}\\
& b=\ \frac{1}{u^2}\frac{\partial u}{\partial z^1}.
\end{split}$$ If we denote by $\nu:{\mathcal F}_{\textrm{on}}(\Lambda)\to \Lambda$ the bundle of $g$-oriented orthonormal frames on $\Lambda$, we obtain on ${\mathcal F}_{\textrm{on}}(\Lambda)$ the tautological 1-forms $$\begin{split}
& \alpha^1=u \widetilde \alpha^1\\
& \alpha^2=u \widetilde \alpha^2\\
& \alpha^3= \widetilde \alpha^3-*d(\log u),
\end{split}$$ where $*$ is the Hodge operator, $\widetilde \alpha^1, \widetilde \alpha^2$ and $\widetilde \alpha^3$ are the the tautological 1-forms and the Levi-Civita connection form of the flat metric $\widetilde g$, respectively.\
A straightforward computation shows that the Gauss curvature $R$ of $g$ is $$\label{Gauss_curv}
R=-\frac{1}{u^2}\Delta(\log u),$$ where $\Delta$ is the Laplace operator in the coordinates $(z^1,z^2)$.\
It follows that a local form for the coframe $( \alpha^1, \alpha^2, \alpha^3)$ is given by $$\begin{split}
& \alpha^1=u\Bigl(\cos(t) dz^1-\sin(t) dz^2\Bigr)\\
& \alpha^2=u\Bigl(\sin(t) dz^1+\cos(t) dz^2\Bigr)\\
& \alpha^3=dt-*d(\log u),
\end{split}$$ where $t\in [0,2\pi]$ is the fiber coordinate over $z\in \Lambda$. Here, we denote the pullback $\nu^*(u)$ of $u$ to $\mathcal{F}(\Lambda)$ by the same letter.
The frame bundle $\mathcal{F}(\Lambda)$
---------------------------------------
We return to our setting in §7.1, and start with an arbitrary Riemannian surface $(\Lambda,g)$ with the area 2-form $dA$ given such that $$g=(\eta^1)^2+(\eta^2)^2,\qquad dA=\eta^1\wedge \eta^2>0,$$ where $\{\eta^1,\eta^2\}$ is an $g$-orthonormal coframe on $\Lambda$, and $\{e_1,e_2\}$ is its dual frame.\
We construct as above the $g$-oriented frame bundle $\nu:\mathcal F(\Lambda)\to \Lambda$, where $(z,e_{1\ |
z},e_{2\ |z})$ is a $g$-oriented frame on $\Lambda$.Let us denote by $\hat{l}$ the mapping $$\hat{l}:\Sigma\to\mathcal F(\Lambda),\quad
u\mapsto \hat{l}(u)=\Bigl(l(u);l_{*,u}(f_{1\ |u}), l_{*,u}(f_{2\ |u}) \Bigr),$$ where $f_1$, $f_2$ are given in (\[frame\_change\]).
[**Proposition 7.1.**]{} [*The mapping $\hat{l}:\Sigma\to\mathcal F(\Lambda)$ defined above is a local diffeomorphism.*]{}
We will give the proof of this result below.\
We have therefore the commutative diagram. $$\begin{matrix}
\Sigma & \xrightarrow{\hat{l}} & \mathcal F(\Lambda)\\
& l \searrow & \downarrow \nu \\
& & \Lambda
\end{matrix}$$ Remark that due to Proposition 7.1 we can locally identify $\Sigma$ with $\mathcal F(\Lambda)$ as well as the coframes $\theta$ and $\alpha$. In order to avoid confusion we will still write $\hat{l}^*$, but we will consider all the formulas proved above for the coframe $\theta$ to hold good for $\alpha$ as well via $\hat{l}^*$.\
Let us consider now the tautological 1-forms $\{\alpha^1,\alpha^2\}$ on $\mathcal F(\Lambda)$, i.e. $$\nu^*(\eta^1)=\alpha^1,\qquad \nu^*(\eta^2)=\alpha^2,$$ or, equivalently, $$\label{alpha12_omega}\hat{l}^*(\alpha^1)=m\omega^2,\qquad \hat{l}^*(\alpha^2)=\omega^3.$$ A simple computation shows that we must also have $${l}^*(\eta^1)=m\omega^2,\qquad {l}^*(\eta^2)=\omega^3.$$
The structure equations
-------------------------
We are going to discuss the structure equations on $\mathcal F(\Lambda)$ and $\Lambda$, respectively.
We have mentioned already the [*“upstairs”*]{} structure equations on $\mathcal F(\Lambda)$. If we pullback the first two equations to $\Sigma$ by the means of $\hat{l}^*$, it follows $$\begin{split}
&d (\hat{l}^* \alpha^1)=\hat{l}^*(\alpha^2)\wedge\hat{l}^*(\alpha^3)\\
&d (\hat{l}^* \alpha^2)=\hat{l}^*(\alpha^3)\wedge\hat{l}^*(\alpha^1)
\end{split}$$ and from here, by using (\[alpha12\_omega\]) we obtain $$\label{alpha3_omega}\hat{l}^*(\alpha^3)=m\omega^1+m_3\omega^2$$ on $\Sigma$.\
Remark that $$\hat{l}^*(\alpha^1\wedge\alpha^2\wedge \alpha^3)=
m^2\omega^1\wedge\omega^2\wedge\omega^3\neq 0,$$ i.e. $\hat{l}$ is indeed a local diffeomorphism and this proves the Proposition 7.1 above.
The [*“downstairs”*]{} structure equations on $\Lambda$ are $$\begin{split}
& d\eta^1=\eta^2\wedge \eta^3\\
& d\eta^2=\eta^3\wedge \eta^1\\
& d\eta^3=R\ \eta^1\wedge \eta^2,
\end{split}$$ where $R$ is the [*“downstairs”*]{} Gauss curvature of $(\Lambda,g)$.\
We pullback the last equation above to $\mathcal F(\Lambda)$ by means of $\nu^*$. It follows $$d\alpha^3=\nu^*(R\ \eta^1\wedge \eta^2).$$ On the other hand, by exterior differentiation of (\[alpha3\_omega\]) we obtain $$\begin{split}
\hat{l}^*(d\alpha^3) & =d(m\omega^1)+d(m_3\omega^2)=(m-m_{33})\omega^2\wedge\omega^3\\
&=\frac{m-m_{33}}{m}\hat{l}^*(\alpha^1)\wedge\hat{l}^*(\alpha^2)=
(1-\frac{m_{33}}{m})l^*(\eta^1\wedge\eta^2).
\end{split}$$ It follows $$l^*(R\eta^1\wedge\eta^2)=(1-\frac{m_{33}}{m})l^*(\eta^1\wedge\eta^2),$$ and from here we obtain the following [*curvature condition*]{} on $\Sigma$: $$\label{condC_up}(C)\qquad \frac{m_{33}}{m}l^*(\eta^1\wedge\eta^2)=l^*\Bigl[(1-R)\eta^1\wedge\eta^2\Bigr].$$ We would like to express now the quantity $\frac{m_{33}}{m}$ living on $\Sigma$ as the image of a quantity living on $\Lambda$ through $l^*$.\
Recall from the general theory that if $\{e_1,e_2\}$ is an adapted frame to the geometry of the Riemannian surface $(\Lambda,g)$, this is equivalent with giving a section of the frame bundle $\nu:\mathcal F(\Lambda)\to \Lambda
$, i.e. $$s:\Lambda\to \mathcal F(\Lambda),\qquad \nu\circ s=id_\Lambda,$$ i.e. we have a so called [*first order adapted lift*]{}.\
Let us consider next an arbitrary smooth function $\bar{m}$ on $\Lambda$, and lift it [*“upstairs”*]{}, i.e. we obtain a function $\widetilde{m}=\bar{m}\circ \nu$ on $\mathcal F(\Lambda)$, such that $s^*(\widetilde{m})=\bar{m}$, and a function $m$ on $
\Sigma$ such that $$m=\hat{l}^*(\widetilde{m})=\hat{l}^*(\nu^*\bar{m})=(\nu\circ \hat{l})^*\bar{m}.$$ We take next the exterior derivative of the relation $m=l^*(\bar m)$. It follows $$\begin{split}
dm=l^*(d\bar m)=l^*(\bar m_1\eta^1+\bar m_2\eta^2)=l^*(\bar m_1)m\omega^2+l^*(\bar m_2)\omega^3,
\end{split}$$ i.e. $dm$ is a linear combination of the 1-forms $\omega^2$, $\omega^3$. This implies $$m_1=0.$$ It follows that this $m$ can be used to relate the coframes $\omega$ and $\alpha$ as in §6.1. Under these conditions, we take the exterior derivative of the relation $m=\hat{l}^*(\widetilde{m})$. It follows that $$\begin{split}
dm & =m_2\omega^2+m_3\omega^3=\hat{l}^*(\widetilde{m}_1\alpha^1+\widetilde{m}_2\alpha^2+
\widetilde{m}_3\alpha^3)\\
& =\hat{l}^*(\widetilde{m}_1)m\omega^2+\hat{l}^*(\widetilde{m}_2)\omega^3+\hat{l}^*(\widetilde{m}_3)
(m \omega^1+m_3\omega^2),
\end{split}$$ and from here, we obtain $$\begin{split}
& \hat{l}^*(\widetilde{m}_1)=\frac{m_2}{m}\\
& \hat{l}^*(\widetilde{m}_2)=m_3\\
& \hat{l}^*(\widetilde{m}_3)=0.
\end{split}$$ Remark that Proposition 7.1 together with the last condition above imply that $$\widetilde{m}_3=0.$$ By a straightforward computation we also obtain $$\hat{l}^*(\widetilde{m}_{22})=m_{33}.$$ Recall that $(\eta^1,\eta^2,\eta^3)=s^*(\alpha^1,\alpha^2,\alpha^3)$, and using now the relation $
\bar{m}=s^*(\widetilde{m})$ we have $$s^*(d\widetilde{m})=s^*(\widetilde{m}_1)\eta^1+s^*(\widetilde{m}_2)\eta^2,$$ where we have put $d\widetilde{m}=\widetilde{m}_1\alpha^1+\widetilde{m}_2\alpha^2$ on $\mathcal F(\Lambda)$ and $d\bar m=\bar m_1\eta^1+\bar m_2\eta^2$ on $\Lambda$.\
Then, it follows $$\begin{split}
& \bar m_1=s^*(\widetilde{m}_1)\\
& \bar m_2=s^*(\widetilde{m}_2).
\end{split}$$ A straightforward computation using (\[lands\_cond\_theta\]), (\[Ricci\_cond\_theta\]) pulled back through $\hat{l}^*$ shows that $$d\widetilde{m}_2=\widetilde{m}_{22}\alpha^2+\widetilde{m}_1\alpha^3,$$ and pulling this equation back through $s^*$ we get $$s^*(\widetilde{m}_{22})=\bar m_{22}+b\bar m_1,$$ where $b$ is the function on $\Lambda$ from $d\eta^2=b\eta^1\wedge\eta^2$.\
In the same way we obtain $$\begin{split}
& s^*(\widetilde{m}_{11})=\bar m_{11}-a\bar m_2,\\
& s^*(\widetilde{m}_{12})= s^*(\widetilde{m}_{21})=\bar m_{12}-b\bar m_2=\bar m_{21}+a\bar m_1,
\end{split}$$ where we take into account the Ricci type identity on $\Lambda$: $$\bar{m}_{21}-\bar{m}_{12}+a\bar m_1+b\bar m_2=0.$$ Hence, we obtain $$m_{33}=l^*(s^*(\widetilde{m}_{22}))=l^*(\bar m_{22}+b\bar m_1).$$ Using now this in (\[condC\_up\]) we are led to the following [*curvature relation on $\Lambda$*]{}: $$\label{condC_down}(C) \qquad \frac{\bar m_{22}+b\bar m_1}{\bar m}=1-R,$$ which, together with the [*Landsberg condition on $\Lambda$*]{} , namely $$\label{condL_down}(L) \qquad \bar m_{12}-b\bar m_2=\bar m_{21}+a\bar m_1=0,$$ are the fundamental relations to be satisfied by $\bar m$ on $\Lambda$.\
Remark that the non-triviality relations $m_2\neq 0$, $m_3\neq 0$ are equivalent to $$\widetilde{m}_1\neq 0,\qquad \widetilde{m}_2\neq 0$$ on $\mathcal F(\Lambda)$ or, equivalently, $$\label{condN_down}(N)\qquad \bar{m}_1\neq 0,\qquad \bar{m}_2\neq 0$$ on $\Lambda$.
Constructing local generalized unicorns
=======================================
Recovering the generalized Landsberg structure
----------------------------------------------
Conversely, one can locally construct a generalized Landsberg structure as follows. Let us consider
1. an oriented Riemannian surface $(\Lambda,g)$ of Gauss curvature $R$, and
2. a function $\bar m$ on $\Lambda$ that satisfies the PDE system (\[condC\_down\]), (\[condL\_down\]) with the non-triviality conditions (\[condN\_down\]).
Then, on the orthonormal frame bundle $\nu:\mathcal F(\Lambda)\to \Lambda$ there exist the tautological 1-forms $\alpha^1$, $\alpha^2$ and the Levi-Civita connection form $\alpha^3$ that satisfy the usual structure equations $$\label{riemann_struct_eq}
\begin{split}
d\alpha^1& =\alpha^2\wedge\alpha^3\\
d\alpha^2& =\alpha^3\wedge\alpha^1\\
d\alpha^3& =\nu^*(R)\ \alpha^1\wedge\alpha^2.
\end{split}$$ Let us construct the coframing $$\label{inverse_coframe}\begin{split}
& \bar{\omega}^1=\frac{1}{\widetilde{m}}(\alpha^3-\frac{\widetilde{m}_2}{\widetilde{m}}\alpha^1)\\
& \bar{\omega}^2=\frac{1}{\widetilde{m}}\alpha^1\\
& \bar{\omega}^3=\alpha^2,
\end{split}$$ where $\widetilde{m}=\nu^*(\bar m)$.\
It follows from Section 6, Section 7 that $\{\bar\omega^1,\bar\omega^2,\bar\omega^3\}$ is a non-trivial generalized Landsberg structure on the 3-manifold $\mathcal F(\Lambda)$ with the invariants $$I=-2\frac{\widetilde{m}_2}{\widetilde{m}},\qquad K=\widetilde{m}^2.$$ By similar computations as in Section 4 one can show by means of Cartan-Kähler theorem that the PDE system (\[condC\_down\]), (\[condL\_down\]) is involutive. We will not discuss here the most general situation, but a particular case will be described below. We recall also that a Riemannian structure on a surface depends on a function of two variables, say $u$ on $\Lambda$ (this is a consequence of the existence of isothermal coordinates on a Riemannian surface).\
Summarizing, it follows from the Cartan–Kähler theorem used in Section 4 that the degree of freedom of the scalar invariants $I$, $K$ of a generalized Landsberg structure locally depends on two arbitrary functions of two variables (see §4.1, §4.2). We point out that these two functions of two variables are in the Cartan-Kähler sense, i.e. they show the degree of freedom of $(I,K)$, but one should not think that they are exactly the functions $u$ and $\bar{m}$ used in the precedent section.\
More generally, a generalized Landsberg structure, i.e. the coframe $\{\omega^1,\omega^2,\omega^3\}$ together with the scalar invariants $I$, $K$, depends on 3 functions of 3 variables (see §4.3). A particular case is the generalized Landsberg structure (\[inverse\_coframe\]) constructed using a function $u$ on $\Lambda$, from the Riemannian structure $(\Lambda,g)$ downstairs, and a function $\bar m$ on $\Lambda$ satisfying (\[condC\_down\]), (\[condL\_down\]). We will show in the next section that the degree of freedom of the pair of functions $(u,\bar m)$ is actually 4 functions of 1 variable (see Proposition 8.1).\
Remark that our solution has a lower degree of freedom than the general solution predicted by our first use of Cartan–Kähler theorem in Section 4 due to our particular choice of the coframe changing (\[inverse\_coframe\]), so there is no contradiction with our results in Section 4.\
Remark also that our condition $\widetilde m_1=0$ implies that the directional derivative of the invariant $K$ with respect to $\widetilde \omega^1$ vanishes, in other words we are considering here an integral manifold of the linear Pfaffian system (\[Lands\_Pfaffian\]) passing through the initial condition $$(u_0,I(u_0),K(u_0),I_1(u_0),I_3(u_0),0,K_2(u_0)),$$ as explained in §4.2, where the invariants $I$, $K$ are given above.
A local form
------------
In order to construct a local form for the generalized Landsberg structure given by (\[inverse\_coframe\]), we are going to use Zoll projective structures.\
Let us start with a Riemannian metric $g=u^2[(dz^1)^2+(dz^2)^2]$ on the surface $\Lambda$ with the Christoffel symbols $\Gamma_{jk}^i$, and construct the 1-form $\gamma$ on $\Lambda$ as in , .\
By putting $\gamma=d(\log u)$, i.e. $$\label{u_ODE}\frac{1}{u}\frac{\partial u}{\partial z^i}=\gamma_i,\qquad i=1,2,$$ in some isothermal coordinates $(z^1,z^2)\in \Lambda$, it follows that the Gauss curvature of the Riemannian metric $g=u^2[(dz^1)^2+(dz^2)^2]$ will be given by $$R=-\frac{1}{u^2}\textrm{div}\gamma$$ as explained in §3.2. See also Example 7.2 for other formulas.\
On the other hand, in order to obtain a generalized Landsberg structure upstairs, we need a function $\bar{m}$ on $\Lambda$ that satisfies the conditions (\[condC\_down\]), (\[condL\_down\]) and the non-triviality conditions (\[condN\_down\]).\
If we denote by numerical subscripts the directional derivatives of $\bar{m}$ with respect to the $g$-orthonormal coframe $$\eta^1=udz^1,\quad \eta^2=udz^2,$$ and with letters the partial derivatives, then straightforward computations show the expression of first order directional derivatives $$\label{m_1st_deriv}\bar{m}_i=\frac{1}{u}\bar{m}_{z^i},\qquad i=1,2,$$ and second order directional derivatives $$\label{m_2st_deriv}\begin{split}
& \bar{m}_{11}=\frac{1}{u^2}(-\gamma_1\bar{m}_{z^1}+\bar{m}_{z^1z^1})\quad
\bar{m}_{12}=\frac{1}{u^2}(-\gamma_2\bar{m}_{z^1}+\bar{m}_{z^1z^2})\\
& \bar{m}_{21}=\frac{1}{u^2}(-\gamma_1\bar{m}_{z^2}+\bar{m}_{z^2z^1})\quad
\bar{m}_{22}=\frac{1}{u^2}(-\gamma_2\bar{m}_{z^2}+\bar{m}_{z^2z^2}).
\end{split}$$ It follows from (\[condC\_down\]), (\[condL\_down\]) that $\bar{m}$ must satisfy
1. [*The Landsberg condition*]{} $$\label{condL_2}(L)\qquad \bar{m}_{z^1z^2}=\gamma_1\bar{m}_{z^2}+\gamma_2\bar{m}_{z^1},$$
2. [*The curvature condition*]{} $$\label{condC_2}(C)\qquad \bar{m}_{z^2z^2}= -(\gamma_1m_{z1}- \gamma_2m_{z2})+u^2+\textrm{div}\gamma.$$
It follows that these two conditions can be regarded as a PDE system for $\bar{m}$ on $\Lambda$, where $\gamma$’s is given by (\[u\_ODE\]).\
The first question that arises is the involutivity of such a PDE system. We will discuss this using our favorite tool, the Cartan-Kähler theorem.\
Let $J^2(\mathbb{R}^2, \mathbb{R}^2)$ be a second order jet space of two functions on a plane. The second jet space $J^2 (\mathbb{R}^2, \mathbb{R}^2)$ has the canonical system $$C^2= \{\theta_{i}^{j}=0 \quad (i=0,1,2 ,j=1,2)\}$$ where $(z^1,z^2,\bar{m},u, \bar{m}_{z^1},\bar{m}_{z^2},
u_{z^1},u_{z^2},\bar{m}_{z^1 z^1},\bar{m}_{z^1 z^2},\bar{m}_{z^2 z^2},
u_{z^1 z^1},u_{z^1z^2},u_{z^2 z^2})$ are the coordinates on $J^2(\mathbb{R}^2, \mathbb{R}^2)$ and $$\begin{aligned}
\theta_{0}^{1}=d\bar{m}-\bar{m}_{z^1}dz^1 -\bar{m}_{z^2}dz^2 \qquad
&,&\quad \theta_{0}^{2}= du-u_{z^1}dz^1 -u_{z^2}dz^2\ , \\
\theta_{1}^{1}=d\bar{m}_{z^1}-\bar{m}_{z^1 z^1}dz^1 -\bar{m}_{z^1 z^2}dz^2
&,& \quad\theta_{1}^{2}= du_{z^1}-u_{z^1 z^1}dz^1 -u_{z^1 z^2}dz^2\ , \\
\theta_{2}^{1}= d\bar{m}_{z^2}-\bar{m}_{z^1 z^2}dz^1 -\bar{m}_{z^2 z^2}dz^2 &,& \quad\theta_{2}^{2}= du_{z^2}-u_{z^1
z^2}dz^1 -u_{z^2 z^2}dz^2\end{aligned}$$ are the canonical contact forms.\
We consider the system of PDE formed by the equations $(L),(C)$, namely, $$R=\{(L),(C) \} \subset J^2(\mathbb{R}^2, \mathbb{R}^2),\quad I= C^2 |_R , \quad
\Omega =dz^1 \wedge dz^2,$$ with coordinates $(z^1,z^2,\bar{m},u, \bar{m}_{z^1},\bar{m}_{z^2},
u_{z^1},u_{z^2},\bar{m}_{z^1 z^1},
u_{z^1 z^1},u_{z^1z^2},u_{z^2 z^2})$ on $R$.\
By a straightforward computation we find that the Pfaffian system $I$ has absorbable torsion. Moreover, its tableau is given by $$\begin{pmatrix}
0 & \qquad & 0 \\
a & \qquad & 0 \\
0 & \qquad & \frac{\bar m}{u}(b+d) \\
0 & \qquad & 0 \\
b & \qquad & c \\
c & \qquad & d
\end{pmatrix}$$ and the characters of the tableau are $s_1=4,\ s_2=0$. Since the dimension of the space of integral elements is $4=s_1+2s_2$, Cartan’s Test for involutivity implies that the system is involutive.\
Hence, in the analytic category, the Cartan-Kähler theorem implies that the solutions exist, and, roughly speaking, they depend on 4 functions of 1 variable.\
We are led in this way to the following result.
[**Proposition 8.1.**]{}\
[*The system of partial differential equations (L), (C) for two unknown functions u, $\bar m$ of two variables has solutions. Moreover, these solutions depend in Cartan-Kähler sense on 4 functions of 1 variable.*]{} We obtain therefore the following prescription for constructing generalized Landsberg structures:
$\bullet$ Start with a smooth surface $\Lambda$ with local coordinates $z^1$, $z^2$ and consider the functions $
\bar{m}, u:\Lambda\to \mathbb R$ which satisfy (\[condL\_2\]), (\[condC\_2\]). The existence of such an $\bar{m}$ and $u$ is guaranteed by the Cartan-Kähler theorem (Proposition 8.1).
$\bullet$ Denote by $g=u^2[(dz^1)^2+(dz^2)^2]$ the corresponding Riemannian metric on $\Lambda$ conformal equivalent to the flat metric, and by $R$ its Gauss curvature given by (\[Gauss\_curv\]);
$\bullet$ Construct the $g$-orthonormal frame bundle $\nu:\mathcal F(\Lambda)\to \Lambda$ with the tautological 1-forms $\alpha^1$, $\alpha^2$ and the Levi-Civita connection form $\alpha^3$;
$\bullet$ Lift the function $\bar{m}$ to $\Sigma:=\mathcal F(\Lambda)$ as $\widetilde m:=\nu^*(\bar{m})$;
$\bullet$ Construct the coframe $(\bar{\omega}^1,\bar{\omega}^2,\bar{\omega}^3)$ on $\Sigma=\mathcal
F(\Lambda)$ given by (\[inverse\_coframe\]).
Then, we have
Indeed, remark first that $\widetilde m:=\nu^*(\bar{m})$ implies $s^*(\widetilde m)=\bar{m}$, as well as $\widetilde m_3=0$ by taking the exterior derivative. Then, in the present setting, similar computations with those in §7.2 show that conditions (L) and (C) upstairs in Proposition 6.2 hold good. Computing now the structure equations of the coframe $\bar{\omega}$ and making use of (\[riemann\_struct\_eq\]) and properties in Proposition 6.2, one can easily verify that $\bar{\omega}$ is a generalized Landsberg structure on the 3-manifold $\Sigma=\mathcal{F}(\Lambda)$.\
Using the normal form from Example 7.2 in §7.1, we obtain the following normal form of this generalized unicorn: $$\label{normal_form}\begin{split}
& \bar{\omega}^1=\frac{1}{\widetilde{m}}\Bigl[dt-*d(\log u)-\frac{u\ \widetilde{m}_2}{\widetilde{m}}\Bigl(\cos(t)
dz^1-\sin(t) dz^2\Bigr)\Bigr]\\
& \bar{\omega}^2=\frac{u}{\widetilde{m}}\Bigl(\cos(t) dz^1-\sin(t) dz^2\Bigr)\\
& \bar{\omega}^3=u\Bigl(\sin(t)dz^1+\cos(t)dz^2\Bigr),
\end{split}$$ where $\widetilde{m}=\nu^*(\bar m)$, $\widetilde{m}_2=\nu^*(\frac{1}{u}\frac{\partial \bar{m}}{\partial z^2})$ and $t\in [0,2\pi]$ is the fiber coordinate over $z=(z^1,z^2)\in \Lambda$. Here, we denote again the prolongation $\nu^*(u)$ of $u$ to $\mathcal{F}(\Lambda)$ by the same letter.
Concluding remarks
==================
In the present note we have shown how is possible to construct a non-trivial generalized Landsberg structure $\{\omega^1,\omega^2,\omega^3\}$ on a 3-manifold $\Sigma$ using a Riemannian metric $g$ on a surface $\Lambda$ that basically depends on 2 functions of 1 variable, namely, $u$ and $\bar{m}$. Due to Cartan-Kähler Theorem in §8.2, we know that these functions are locally described by 4 functions of 1 variable, case included in the general solution predicted by Cartan-Kähler Theory in Section 4. A local form of it is given by (\[normal\_form\]). This generalized Landsberg structure is [*locally amenable*]{} in the sense of §5.2. Our generalized unicorn has the fundamental geometrical property that its indicatrix foliation $\{\omega^1=0,\omega^2=0\}$ coincides with the geodesic foliation $\{\alpha^1=0,\alpha^3=0\}$ of the Riemannian metric $g$ of $\Lambda$.\
However, our initial intention was to search for classical unicorns on surfaces, i.e. generalized Landsberg structures that satisfy the conditions of Theorem 2.1.\
Recall that a generalized Finsler structure is amenable if the indicatrix foliation $\mathcal Q=\{\omega^1=0,\omega^2=0\}$ is amenable, i.e. the leaf space is a differentiable manifold.\
Let us also recall that a Zoll metric on $S^2$ depends on one odd arbitrary function on one variable (see [@B1978] and [@LM2002] for details). We are lead in this way to the following
If we accept this conjecture as true, then we just have constructed a generalized Landsberg structure $\{\bar
\omega^1,\bar \omega^2,\bar \omega^3\}$ on the frame bundle $\Sigma:=\mathcal F(S^2)$ of a Riemannian surface $
(S^2,g)$ whose Levi-Civita connection $\nabla^g$ belongs to a Zoll projective structure on $S^2$, in other words, the geodesic foliation $\mathcal P=\{\alpha^1=0,\alpha^3=0\}$ of $g$ foliates the 3-manifold $\Sigma$ by circles. Remark in the same time that we had constructed our coframe $\bar \omega$ from $\alpha$ by (\[inverse\_coframe\]) such that its indicatrix foliation $\mathcal Q=\{\omega^1=0,\omega^2=0\}$ coincides with the geodesic foliation $\mathcal P=\{\alpha^1=0,\alpha^3=0\}$ of $g$. Then, by the properties of Zoll projective structure on $S^2$ described partially in §3.2 it follows that the space of geodesics, say $M$, of the metric $
(\Lambda=S^2,g)$ is a differentiable manifold, and hence, the generalized Landsberg structure $\{\bar \omega^1,\bar
\omega^2,\bar \omega^3\}$ is globally amenable. In other words, the map $\pi:\Sigma\to S^2$ is a smooth submersion. Obviously, the leaves of the indicatrix foliation $\{\bar \omega^1=0,\bar \omega^2=0\}$ are diffeomorphic to $S^1$, so they must be compact.\
Finally, in order to have a true classical unicorn, we have to show more, namely that the canonical immersion $
\iota:\Sigma\to TM$, given by $\iota(u)=\pi_{*,u}(\hat{e}_2)$ is injective on each $\pi$-fiber $\Sigma_x$, as stated in Theorem 2.1. This is not so difficult to prove. Let us denote by $$\gamma_u:[a,b]\to \Sigma$$ the geodesic flow of the Zoll projective structure $[\nabla]$ on $S^2$ through the point $u\in \Sigma$, and let us take another point, say $u_1$ on the same leaf, i.e. there exist some parameter values $s_0, s_1\in [a,b]$ such that $$\gamma_u(s_0)=u,\qquad \gamma_u(s_1)=u_1$$ on $\Sigma$.\
From §3.2 we know that the leaves $\gamma$ are closed, periodic, simple curves of same length on $
\Sigma$, i.e. for $$\gamma_u(s_0)=u \neq \gamma_u(s_1)=u_1 \Longrightarrow \hat{e}_{2\ |\gamma_u(s_0)}\neq \hat{e}_{2\ |
\gamma_u(s_1)},$$ where $\hat{e}_{2}\in T_{\gamma}\Sigma$ is thought as a vector field along $\gamma$. Applying to this the linear map $
\pi_{*,u}$ it follows $$\pi_{*,\gamma(s_0)}(\hat{e}_{2\ |\gamma_u(s_0)})\neq \pi_{*,\gamma(s_1}(\hat{e}_{2\ |\gamma_u(s_1)})$$ and therefore it follows that $\iota$ must be injective on each $\pi$-fiber $\Sigma_x$.\
Then, from Theorem 2.1 we can conclude
[*There are Landsberg structures on $M=S^2$ which are not Berwald type, provided the conjecture above is true.*]{}
Appendix. The Cartan–Kähler theorem for linear Pfaffian systems
===============================================================
We give a short outline of the main tool used in the present paper, the Cartan–Kähler theorem for linear Pfaffian systems. This theorem is presented in several textbooks, for [@Br; @et; @al; @1991], [@IL2003], [@O1995], etc., but our presentation here follows our favorite monograph [@IL2003].\
Let us denote by $\Omega^*(\Sigma)=\bigoplus_{k} \Omega^k(\Sigma)$ the space of smooth differential forms on the manifold $\Sigma$. It is a standard fact that $\Omega^*(\Sigma)$ is a graded algebra under the wedge product.\
A subspace $\mathcal{I}\subset \Omega^*(\Sigma)$ is called [*an exterior ideal*]{} or [*an algebraic ideal*]{} if it is a direct sum of homogeneous subspaces (namely, $\mathcal{I} = \bigoplus_{k} {\mathcal{I}}^k$, ${\mathcal{I}}^k \subset \Omega^k(\Sigma)$.) and it satisfies $$\omega\wedge \eta\in \mathcal{I},$$ for $\omega\in \mathcal{I}$ and [*any*]{} differential form $\eta\in \Omega^*(\Sigma)$.\
An exterior ideal is called [*a differential ideal*]{} if for any $\omega\in \mathcal{I}$, we have $d\omega\in
\mathcal{I}$ also.\
A differential ideal $\mathcal{I} \subset \Omega^*(\Sigma)$ is called an [*exterior differential system*]{} on a manifold $\Sigma$ (EDS for short).\
A set of differential forms of arbitrary degree $\{\omega^1,
\omega^2, \dots,\omega^k \}$ is said to [*generate the EDS*]{} $\mathcal{I}$ if any $\theta \in \mathcal{I}$ can be written as a finite “linear combination”, namely $$\mathcal{I} =\{ \sum_{i=1}^{k} \alpha^i \wedge \omega^i+
\sum_{i=1}^{k} \beta^i \wedge d\omega^i \ |\ \alpha^i,\beta^i
\in \Omega^*(\Sigma)
\}.$$ A [*Pfaffian system*]{} $\mathcal{I}$ on a manifold $\Sigma$ is an EDS finitely generated by 1-forms $\{\omega^1,\omega^2,\dots\omega^k\}$ only.\
For an EDS $\mathcal{I}$ on a manifold $\Sigma$, a decomposable differential $k$-form $\Omega$ (up to scale) is called the independence condition if $\Omega$ does not vanish modulo $\mathcal{I}$ on $\Sigma$.\
We denote by $(\mathcal{I}, \Omega)$ a pair of an EDS and an independence condition on a manifold $\Sigma$.\
A submanifold $f:M\to\Sigma$ is called [*an integral submanifold*]{} (or solution) of the EDS $(\mathcal{I}, \Omega)
$ if $$\begin{split}
& f^*(\theta^a)=0, \qquad \theta^a\in \mathcal{I},\\
& f^*(\Omega)\neq 0.
\end{split}$$ Remark also that $f^*(\theta)=0$ imply $f^*(d\theta)=0$.\
There is a notion of infinitesimal solution also. A $k$-dimensional subspace $E\subset T_x\Sigma$ is called [*an integral element*]{} of $(\mathcal{I}, \Omega)$ if $$\begin{split}
& \theta^a_{|_E}=0, \qquad \theta^a\in \mathcal{I},\\
& \Omega_{|_E}\neq 0.
\end{split}$$ Usually one regards $E$ as an element of the Grassmannian $G_k(T_x\Sigma)$ of $k$-planes through the origin of the vector space $T_x\Sigma$. The space of $k$-dimensional integral elements of $(\mathcal{I},\Omega)$ is usually denoted by $\mathcal{V}_k(\mathcal{I},\Omega)
$.\
Roughly speaking, a differential system will be called [*integrable*]{} if one can determine its integral manifolds of a prescribed dimension passing through each point. In the case of a Pfaffian system with the maximum degree independence condition, its integrability is guaranteed by Frobenius theorem. However, in the case when the independence condition is not the maximum degree, then one has to use more powerfull tools as the Cartan-Kähler Theorem.\
Let $(I,J)$ be a pair of a collection of $1$-forms $I=\{\theta^1, \theta^2,
\ldots, \theta^s \}$ and $J=\{\omega^1 ,\omega^2 ,\ldots ,\omega^k \}$ which are linearly independent modulo $I$.\
Remark that $(I,J)$ induces an EDS $(\mathcal{I},\Omega)$ by a Pfaffian system $\mathcal{I}$ generated by $I$ and the independence condition $\Omega=
\omega^1 \wedge \omega^2 \wedge \ldots \wedge \omega^k$.\
The pair $(I,J)$ is called a [*linear Pfaffian system*]{} if $$ d\theta^a\equiv 0\qquad \mod J,$$ for all $\theta^a$ in $I$.\
If $(I,J)$ is a linear Pfaffian system, let us denote by $\pi^{\epsilon}$, $\epsilon=1,2,\dots,\dim \Sigma-s-k$ such that $T^*\Sigma$ is locally spanned by $\theta^a,\omega^i,
\pi^\epsilon$. The coframing $\theta^a,\omega^i,\pi^\epsilon$ is called [*adapted*]{} to the filtration $I\subset J\subset T^*\Sigma$. It follows immediately that there must locally exist some functions $A_{\epsilon i}^a$ and $T_{ij}^a$ on $\Sigma$ such that $$\label{struct_eq_torsion} d\theta^a\equiv A_{\epsilon i}^a\pi^\epsilon\wedge\omega^i+T_{ij}^a\omega^i\wedge\omega^j\qquad \mod I.$$ The terms $T_{ij}^a\omega^i\wedge\omega^j$ in (\[struct\_eq\_torsion\]) are called [*apparent torsion*]{}. Apparent torsion must be normalized before prolonging the system. Namely, one have to choose, if possible, some new one forms $\tilde{\pi}^\epsilon$ such that $\tilde{T}_{ij}^a=0$, with respect to the new coframe $
\theta^a,\omega^i,\tilde{\pi}^\epsilon$ on $\Sigma$. In this case one says that [*the apparent torsion is absorbable*]{}.\
If this is not possible, then one says that there is [*torsion*]{} and in this case the system admits no integral elements.\
Remark that the functions $A_{\epsilon i}^a$ and $T_{ij}^a$ depend on the choices of the bases for $I$ and $J$. However, one can construct invariants from these functions. Indeed, for a fixed generic point $x\in \Sigma$, the [ *tableau of $(I,J)$ at x*]{} is defined as $\Sigma$ such that $$ A_x:=\{A_{\epsilon i}^aw_a\otimes v^i\ :\ 1\leq \epsilon\leq \dim \Sigma-\dim J_x\}\subseteq W\otimes V^*,$$ where $V^*:=(J/I)_x$, $W^*=I_x$, $w^a=\theta^a_x$, $v^j=\omega^j_x$. A standard argument of linear algebra shows that $A_x$ is independent of any choices.\
We fix a point $x\in \Sigma$ and denote the tableau $A_x$ simply with $A\in W\otimes V^*$. The tableau $A$ depends on the basis $b=(v^1,v^2,\dots,v^n)$ of $W$. One defines
----------------------- --- -------------------------------------------------------
$s_1(b)$ = no. of independent entries in the first col. of $A$
$s_1(b)+s_2(b)$ = no. of independent entries in the first 2 col. of $A$
…
$s_1(b)+\dots+s_n(b)$ = no. of independent entries in $A$.
----------------------- --- -------------------------------------------------------
Equivalently, one can see that [*the characters $s_1(b),s_2(b),\dots,s_n(b)$ of the tableau A*]{} do not depend actually on the choice of the basis $b$ of $W$, but only on the flag of subspaces $$ F:\ (0)=F_n\subset F_{n-1}\subset \dots F_1\subset F_0=V^*.$$ This allows us to rewrite $s_k(b)$ as $s_k(F)$. By defining $$A_k(F)=(W\otimes F_k)\cap A,$$ it follows that $$\dim A_k(F)=s_{k+1}(F)+\dots s_n(F).$$ One can easily see that $A_k(F)$ is the subspace of matrices in $A$ for which the first $k$ columns are zero with respect to the basis $b$ for $V$.\
One defines next the [*reduced characters of the tableau A*]{} as\
------- --- -------------------------------------------------------------------------------
$s_1$ = $\max\{s_1(F)\ :\ \text{all flags}\}$
$s_2$ = $\max\{s_1(F)\ :\ \text{flags with}\ s_1(F)=s_1\}$
…
$s_n$ = $\max\{s_n(F)\ :\ \text{flags with} \ s_1(F)=s_1,\dots,s_{n-1}(F)=s_{n-1}\}$.
------- --- -------------------------------------------------------------------------------
These scalars are invariants of the tableau $A$, i.e. they are independent of any choice of bases of $V$ or $W$.\
It can be shown that the reduced characters must satisfy the inequality: $$\label{dim_intergral_elem} \dim A^{(1)}\leq s_1+2s_2+\dots +ns_n,$$ where $A^{(1)}$ is the [*first prolongation of A*]{}, namely $$A^{(1)}:=(A\otimes V^*)\cap (W\otimes S^2 V^*),$$ and $S^2 V^*$ is the space of symmetric 2-tensors of $V^*$.\
We reach in this way to one of the most important notion in the theory of exterior differential systems. The tableau $A\in W\otimes V^*$ is called [*involutive*]{} if equality holds in (\[dim\_intergral\_elem\]), i.e. we have $$ \dim A^{(1)}= s_1+2s_2+\dots +ns_n.$$ This condition is also called [*Cartan test for involutivity*]{}.\
If $A$ is involutive such that $s_l\neq 0$ and $s_{l+1}= 0$, then $s_l$ is called the [*character*]{} of the system and the integer $l$ is called the [*Cartan integer*]{} of the system.\
We can give now the main tool used in this paper, the Cartan–Kähler Theorem for Linear Pfaffian systems. Even though the theorem can be formulated in general for arbitrary exterior differential systems (see [@Br; @et; @al; @1991], [@IL2003]), the version for Linear Pfaffian systems will suffice for our purposes in the present paper.
[**Theorem A.1. The Cartan–Kähler Theorem for Linear Pfaffian systems**]{}
\
Informally, one says that the solutions depend (in Cartan-Kähler sense) on $s_l$ functions of $l$ variables, where $s_l$ is the character of the system (see [@IL2003], p. 176 for the precise statement of the Theorem and other details). A linear Pfaffian system satisfying the conditions (1), and (2) in the Cartan–Kähler Theorem for linear Pfaffian systems is said to be [*involutive*]{}.\
Recall that if an EDS is not a linear Pfaffian system, then by prolongation one can linearize it and then study its involutivity by Cartan–Kähler Theorem for linear Pfaffian systems.
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Sorin V. SABAU\
School of Science, Department of Mathematics\
Tokai University,\
Sapporo, 005–8601 Japan
Kazuhiro SHIBUYA\
Graduate School of Science, Hiroshima University,\
Higashi Hiroshima, 739–8521, Japan
Hideo SHIMADA\
School of Science, Department of Mathematics\
Tokai University,\
Sapporo, 005–8601 Japan
[^1]: Mathematics Subject Classification (2000): 53B40, Primary 53C60; Secondary 53D35 .
|
---
abstract: 'We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and in that sense this paper is of expository nature. However, we put the above notions in the spotlight and provide a self-contained, purely measure-theoretic, detailed and thorough investigation of their properties, and in that aspect our paper enhances and complements the existing literature. Our work constitutes part of the necessary theoretical framework for categorical constructions involving measured and topological groupoids with Haar systems, a line of research we pursue in separate papers.'
address:
- 'Aviv Censor, Department of Mathematics, University of California at Riverside, Riverside, CA, 92521, U.S.A.'
- 'Daniele Grandini, Department of Mathematics, University of California at Riverside, Riverside, CA, 92521, U.S.A.'
author:
- 'A. Censor'
- 'D. Grandini'
title: Borel and Continuous Systems of Measures
---
Introduction
============
We first give an overview of the contents of this paper. This is followed by a discussion of the nature of our work and its relation to the existing literature.
Overview
--------
Our treatment of Borel systems of measures (BSMs) and continuous systems of measures (CSMs) in this paper is very general. Loosely speaking, a system of measures on a map $\pi:X \rightarrow Y$ is a family of measures $\lambda^{\bullet} = \{ \lambda^y \}_{y \in Y}$ on $X$, such that each $\lambda^y$ is concentrated on $\pi^{-1}(y)$. This can be made precise when the nature of $X$, $Y$ and $\pi$ is specified (e.g. topological spaces with a continuous map, Borel spaces with a Borel map), leading to appropriate assumptions on the measures $\{ \lambda^y \}$. We will denote a map $\pi:X \rightarrow Y$ admitting a system of measures $\lambda^{\bullet}$ by the diagram $\xymatrix{X\ar [rr]^{\pi}_{\lambda^{\bullet}}&&Y}$.
In the spotlight of our work are mapping properties of systems of measures. We establish terminology, notation and basic properties of systems of measures in Section \[sec:SOM\]. Then, in Section \[sec:composition\], we study *composition* of systems, corresponding to the following diagram $$\xymatrix{X\ar [rr]^{p}_{\alpha^{\bullet}}&&Y\ar [rr]^{q}_{\beta^{\bullet}}&&Z}$$ The composition $(\beta\circ\alpha)^{\bullet}$ is defined for any Borel set $E \subseteq X$ by $(\beta\circ\alpha)^{z}(E) =\int_Y \alpha^{y}(E) \ d\beta^z(y)$ (Definition \[def:composition BSM\]).
In Section \[sec:lifting\] we treat the notion of *lifting*, namely producing a system of measures $(q^*\alpha)^{\bullet}$ on $\pi_Y$ in the following pull-back diagram: $$\xymatrix{X * Y \ar [dd]_{\pi_X}\ar [rr]^{\pi_Y}&&Y\ar [dd]_{q}\\\\
X\ar [rr]^{p}_{\alpha^{\bullet}}&&Z}$$ The lifting is given by $(q^*\alpha)^y=\alpha^{q(y)}\times \delta_y$ (Definition \[def:lifting BSM\]).
Section \[sec:fibred products\] deals with the *fibred product*, which is a system of measures $(\gamma_X * \gamma_Y)^{\bullet}$ on the map $f*g$ in the following diagram: $$\xy 0;<.2cm,0cm>:
(20,20)*{X_2 * Y_2}="1";
(40,20)*{Y_2}="2";
(20,0)*{X_2}="3";
(40,0)*{Z}="4";
(10,10)*{X_1 * Y_1}="5";
(30,10)*{Y_1}="6";
(10,-10)*{X_1}="7";
(30,-10)*{Z}="8";
{"1"+CR+(.5,0);"2"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.5)+(0,1)*{\scriptstyle \pi_{Y_2}}};
{"5"+CR+(.5,0);"6"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.75)+(0,1)*{\scriptstyle \pi_{Y_1}}};
{"3"+CR+(.5,0);"4"+CL+(-10,0)**@{-}};
{"3"+CR+(9,0);"4"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.5)+(0,1)*{\scriptstyle p_2}};
{"7"+CR+(.5,0);"8"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.5)+(0,1)*{\scriptstyle p_1}};
{"1"+CD+(0,-.5);"3"+CU+(0,9.6)**@{-}} ;
{"1"+CD+(0,-9.8);"3"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.5)+(-1.5,0)*{\scriptstyle \pi_{X_2}}} ;
{"2"+CD+(0,-.5);"4"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.5)+(-1,0)*{\scriptstyle q_2}} ;
{"5"+CD+(0,-.5);"7"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.5)+(-1.5,0)*{\scriptstyle \pi_{X_1}}} ;
{"6"+CD+(0,-.5);"8"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.25)+(-1,0)*{\scriptstyle q_1}} ;
{"5"+C+(1.4,1.4);"1"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>*{\dir{>}}?>(.5)+(-2,0)*{\scriptstyle f*g}} ;
{"6"+C+(1.4,1.4);"2"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>(.5)+(-1,0)*{\scriptstyle g}?>(.5)+(1.5,0)*{\scriptstyle \gamma_Y^{\bullet}}} ;
{"7"+C+(1.4,1.4);"3"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>(.5)+(-1,0)*{\scriptstyle f}?>(.5)+(1.5,0)*{\scriptstyle \gamma_X^{\bullet}}} ;
{"8"+C+(1.4,1.4);"4"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>(.5)+(-1.5,0)*{\scriptstyle id}};
\endxy$$ The fibred product is defined by $\left(\gamma_X * \gamma_Y\right)^{(x_2,y_2)}=\gamma_X^{x_2} \times \gamma_Y^{y_2}$ (Definition \[def:fibred product BSM\]).
Section \[sec:disintegration\] explores the concept of *disintegration*, a most valuable tool in applications: If $(X,\mu)$ and $(Y,\nu)$ are measure spaces, and $f:X \rightarrow Y$ is a Borel map, then a system of measures $\gamma^{\bullet}$ on $f$ is a disintegration of $\mu$ with respect to $\nu$ if $ \mu (E) = \int_Y \gamma^y (E) d\nu (y)$ for every Borel set $E \subseteq X$.
We conclude, in section \[sec:groupoids\], with a brief discussion of systems of measures for groupoids, in particular Haar systems.
Broad perspective
-----------------
While our interest in systems of measures originated from our work with groupoids, in this paper we develop the theory from elementary principles and our approach is purely measure theoretic. This is in contrast to many references where the subject has been studied from very specialized perspectives. Systems of measures (also called $\pi$-systems or kernels) appear in various mathematical contexts, and have been investigated from different viewpoints in the literature. For example, a general introduction to the topic can be found in Bourbaki [@bourbaki], which takes a very functional analytic approach.
The primary goal we set for this paper was to collect and clarify the categorically-flavored constructions that we needed, details of which we managed to trace only in part in the functional analysis, probability and groupoid literature. We do not claim to present an exhaustive account of the literature on systems of measures.
The world of groupoids, which motivated our study, is a discipline in which systems of measures play a fundamental role. Most notably, a Haar system for a groupoid $G$ is essentially a left-invariant system of measures on the range map $r:G \rightarrow G^{(0)}$, which generalizes the notion of a Haar measure on a group. In particular, Haar systems are a crucial ingredient for integration on groupoids, for groupoid representations, and for constructing groupoid $C^*$-algebras. Beyond Haar systems, maps between groupoids naturally give rise to systems of measures as well.
In the groupoid literature, systems of measures have been studied extensively, for example by Connes in [@connes-noncommutative-integration] (using the term “kernel", noyau in French), by Muhly in [@muhly-book-unpublished] and by Renault and Anantharaman-Delaroche in [@renault-anantharaman-delaroche] (using the term “$\pi$-systems"). The scope of our current study of systems of measures was therefore restricted to mapping properties which were essential for specific applications that came up in our work. Some of the results presented here appear scattered across the literature, which is why we opted to give a self contained treatment, including all definitions, and full proofs whenever lacking precise references. We point out that some of the formulas and diagrams which we make explicit, can be found in [@renault-anantharaman-delaroche]. In fact, significant parts of the theory are implicit in, and can be non-trivially derived from the aforementioned groupoid references, as well as other works of Renault (e.g. [@renaultJOT]), Ramsay (e.g. [@ramsay71]) and others. We single out a couple of such sources which we refer the specialized reader to: The first is Appendix A.1 of [@renault-anantharaman-delaroche] on transverse measure theory, which builds on Connes’ work, starting with [@connes-noncommutative-integration]. The second is a fibred product construction beginning on page 265 of [@ramsay71]. A detailed discussion of how to extract some of our results from these is beyond the scope of this paper.
This paper provides tools and techniques that allow us to form certain categorical constructions with topological groupoids, which we shall present in separate papers. Primarily, we were interested in forming the so-called “weak pull-back" of a diagram of topological groupoids, each endowed with a Haar system and a quasi-invariant measure on its unit space [@WPB]. The weak pull-back is a key ingredient for degroupoidification à la Baez and Dolan [@baez-hoffnung-walker], which together with Christopher Walker we are currently generalizing from the discrete setting to the realm of topology and measure theory.
Systems of measures {#sec:SOM}
===================
***Throughout, we will assume all topological spaces to be second countable and $\mathbf{T_1}$. We require spaces to also be locally compact and Hausdorff whenever dealing with continuous systems of measures, as well as throughout Section \[sec:disintegration\].*** Measures will always be positive and Borel. Unless stated otherwise, continuous functions will be complex-valued, whereas Borel functions are allowed to take infinite values.\
We first recall the definition of the **support** of a Borel measure $\mu$ on a space $X$: $$supp(\mu)=\{x \in X: \mu(A)> 0\ \ \text{for every open neighborhood } A \text{ of } x \}.$$ We say that the measure $\mu$ is **concentrated** on a subset $S \subseteq X$ if $\mu(X \setminus S)=0$.
\[lem:support\] The support is a closed subset of $X$. Moreover, if $S$ is a closed subset of $X$, then $supp(\mu)\subseteq S$ if and only if the measure $\mu$ is concentrated on $S$.
Take $x\notin supp(\mu)$. Then $x$ has an open neighborhood $A$ such that $\mu(A)=0$. Furthermore, $A \cap supp(\mu) = \emptyset$. This shows that the complement of $supp(\mu)$ is open.
For the second part, note first that $supp(\mu)\subseteq S$ if and only if $$x\notin S \quad \Rightarrow \quad \exists A\subseteq X\text{ open }:\ x\in A,\ \mu(A)=0.$$ Assume that $\mu(X \setminus S)=0$. Since the complement $X \setminus S$ is open, $A=X \setminus S$ satisfies the above statement for any $x\notin S$ and it follows that $supp(\mu)\subseteq S$.\
Viceversa, assume $supp(\mu)\subseteq S$. Fix a countable basis ${\mathcal B}$ for the topology of $X$. Then the following statement is true: $$x\notin S\quad\Rightarrow\quad \exists A_x\in {\mathcal B}:\ x\in A_x,\ \mu(A_x)=0.$$ It follows that $X \setminus S \subseteq \bigcup_{x\notin S}A_x$. But this union consists of countably many distinct elements of the basis ${\mathcal B}$, so we can invoke countable subadditivity to obtain $\mu(X \setminus S)\leq \sum_{x\notin S}\mu(A_x)=0.$
Let $\pi:X \rightarrow Y$ be a Borel map. A **system of measures** on $\pi$ is a family of measures $\lambda^{\bullet} = \{ \lambda^y \}_{y \in Y}$ such that:
1. Each $\lambda^y$ is a Borel measure on $X$;
2. For every $y$, $\lambda^y$ is concentrated on $\pi^{-1}(y)$.
If the map $\pi:X \rightarrow Y$ is continuous (or proper if the spaces are $T_2$), then condition *(2)* is equivalent to
1. *For every* $y$, $supp(\lambda^y) \subseteq \pi^{-1}(y)$.
This follows immediately from Lemma \[lem:support\] since $\pi^{-1}(y)$ is a closed subset of $X$.
We will denote a map $\pi:X \rightarrow Y$ admitting a system of measures $\lambda^{\bullet}$ by the diagram $\xymatrix{X\ar [rr]^{\pi}_{\lambda^{\bullet}}&&Y}$.
Trivially, when $Y$ is a singleton $\{y\}$, a system of measures on the projection $\pi:X \rightarrow \{y\}$ is merely a Borel measure on $X$. This obvious observation will be of use in the sequel.
We will say that a system of measures $\lambda^{\bullet}$ is:
- **positive on open sets** if $\lambda^y(A) > 0$ for every $y \in Y$ and for every open set $A \subseteq X$ such that $A \cap \pi^{-1}(y) \neq \emptyset$.
- **locally bounded** if for any $x \in X$ there exists a neighborhood $U_x$ and a constant $C >0$ such that $\lambda^y(U_x) < C$ for any $y \in Y$.
A system of measures will be called *bounded on compact sets* if for any compact set $K \subseteq X$, $\lambda^{\bullet}(K)$ is a bounded function on $Y$. In general, it is not hard to see that being locally bounded implies being bounded on compact sets. If $X$ is assumed to be locally compact, the converse is also trivially true. Our discussion of this property will usually be restricted to the setting of locally compact spaces, where the two notions coincide.
\[lem:positive iff full support\] Assume that the map $\pi:X\rightarrow Y$ is continuous. A system of measures $\lambda^{\bullet}$ on $\pi$ is positive on open sets if and only if $supp(\lambda^y) = \pi^{-1}(y)$ for every $y \in Y$.
Suppose that $\lambda^{\bullet}$ is positive on open sets. For any $x\in \pi^{-1}(y)$ and any open neighborhood $A$ of $x$, we have that $A \cap \pi^{-1}(y) \neq \emptyset$ and thus $\lambda^{y}(A)>0$. Therefore, $x\in\ supp(\lambda^y)$. This proves that $\pi^{-1}(y)\subseteq\ supp(\lambda^y)$. Condition *(2’)* above implies that $supp(\lambda^y) = \pi^{-1}(y)$.
Conversely, assume that $supp(\lambda^y) = \pi^{-1}(y)$ and let $A\subseteq X$ be an open subset satisfying $A\cap \pi^{-1}(y)\neq\emptyset$. Pick $x\in A\cap \pi^{-1}(y)$. Since $x\in supp(\lambda^y)$ and $A$ is an open neighborhood of $x$, it follows that $\lambda^y(A)>0$. Therefore, $\lambda^{\bullet}$ is positive on open sets.
A system of measures $\lambda^{\bullet}$ on a continuous map $\pi:X \rightarrow Y$ will be called a **continuous system of measures** or **CSM** if for every non-negative continuous compactly supported function $0 \leq f\in C_c(X)$, the map $\displaystyle y \mapsto \int_X f(x)d \lambda^y (x)$ is a continuous function on $Y$.
Note that implicit in the above definition is the assumption on $\lambda^{\bullet}$ that $\int_X f(x)d \lambda^y (x)$ is finite for all $y$ and for any $0 \leq f\in C_c(X)$. This implies that $\int_X f(x)d \lambda^y (x)$ is finite for *any* complex-valued function $f\in C_c(X)$. Hence, a CSM can be defined, equivalently, by requiring the map $y \mapsto \int_X f(x)d \lambda^y (x)$ to be a continuous function on $Y$ for any complex-valued function $f\in C_c(X)$.
A system of measures $\lambda^{\bullet}$ on a Borel map $\pi:X \rightarrow Y$ is called a **Borel system of measures** or **BSM** if for every Borel subset $E\subseteq X$, the function $\lambda^{\bullet}(E):Y\rightarrow [0,\infty]$ given by $y\mapsto \lambda^{y}(E)$ is a Borel function.
In the sequel it will be implicit that whenever a map $\pi:X \rightarrow Y$ admits a BSM, it is a Borel map, and if it admits a CSM, it is a continuous map. Also, recall that in the CSM context, spaces are assumed to be locally compact and Hausdorff.
\[lem:equiv\_BSM\] A system of measures $\lambda^{\bullet}$ on $\pi:X\rightarrow Y$ is a BSM if and only if for every nonnegative Borel function $f:X \rightarrow [0,\infty]$, the map $\displaystyle y \mapsto \int f(x)d \lambda^y (x)$ is a Borel function on $Y$.
Assume that $ y \mapsto \int f(x) d \lambda^y (x)$ is Borel for any Borel function $f:X \rightarrow [0,\infty]$, and let $E\subseteq X$ be a Borel subset. Then the function $ y \mapsto \int \chi_{_E}(x) d \lambda^y (x)= \lambda^y(E)$ is Borel.
Now suppose $\lambda^{\bullet}$ is a BSM. The following argument is standard. If $ s=\sum_{i=1}^n r_i\chi_{_{E_i}}$ is a nonnegative simple function on $X$, then the map $y \mapsto \int s(x) d \lambda^y (x)=\sum_{i=1}^n r_i\lambda^y(E_i)$ is Borel, being a linear combination of the Borel functions $y \mapsto \lambda^y(E_i)$. Now let $f$ be any nonnegative Borel function. There exists an increasing sequence of nonnegative simple functions $s_n$ that converges to $f$ pointwise on $X$. From the Monotone Convergence Theorem, $\int f(x)d \lambda^y (x)=\int \lim_{n\longrightarrow \infty}s_n(x)d \lambda^y (x)=\lim_{n \longrightarrow \infty}\int s_n(x)d \lambda^y (x)$. Therefore the function $y \mapsto \int f(x)d \lambda^y (x)$ is a limit of Borel functions and thus Borel.
Let $\lambda^{\bullet}$ be a BSM. For any function $\displaystyle f\in \bigcap_{y\in Y}L^1(\lambda^y)$, the map $\displaystyle y \mapsto \int f(x)d \lambda^y (x)$ is Borel.
The proof is a routine argument stemming from Lemma \[lem:equiv\_BSM\]. We will denote $F_f(y) = \int_X f(x)d \lambda^y (x)$. Assume first that $f$ is real-valued. Write $f=f_+-f_-$, where $f_+,f_-$ are respectively the positive and negative parts of $f$. By Lemma \[lem:equiv\_BSM\], the functions $F_{f_+}(y)$ and $F_{f_-}(y)$ are both Borel and finite, which implies that the function $F_{f}(y) = F_{f_+}(y) - F_{f_-}(y)$ is Borel. For complex-valued $f$, write $f = f_1 + if_2$, and $F_{f}(y) = F_{f_1}(y) + iF_{f_2}(y)$ is Borel.
\[lem:CSM\_support\] Assume that $\lambda^{\bullet}$ is a CSM on $\pi:X \rightarrow Y$, and let $f \in C_c(X)$. Let $F:Y \rightarrow \mathbb{C}$ be the continuous function on $Y$ given by $\displaystyle F(y) = \int_X f(x)d \lambda^y (x)$. Then $supp(F) \subseteq \pi(supp(f))$.
Define $A=\{x\in X:f(x)\neq 0\}$ and $B=\{y\in Y:F(y)\neq 0\}$. By definition, $\overline{A} = supp(f)$ and $\overline{B} = supp(F)$. Recall that $\lambda^y$ is concentrated on $\pi^{-1}(y)$, from which it follows that $$y \notin \pi(A) \ \Rightarrow \ \pi^{-1}(y) \cap A = \emptyset \ \Rightarrow \ \forall x \in \pi^{-1}(y), f(x)=0 \ \Rightarrow \ \int_X f(x)d \lambda^y (x) = 0 \ \Rightarrow \ y\notin B.$$ Thus $B\subseteq \pi(A)$. Since $\pi$ is continuous, $\overline{A}$ is compact, and $Y$ is $T_2$, we obtain $supp(F) = \overline{B} \subseteq \overline{\pi(A)} = \pi(\overline{A}) = \pi(supp(f))$.
\[cor:CSM has compact support\] A CSM $\lambda^{\bullet}$ on $\pi:X \rightarrow Y$ satisfies that for every $f\in C_c(X)$, the map $\displaystyle y \mapsto \int_X f(x)d \lambda^y (x)$ is in $C_c(Y)$.
In the literature, the compact support of the map $y \mapsto \int_X f(x)d \lambda^y (x)$ is often included in the definition of continuity for a system of measures.
\[lem:CSM always locally bounded\] A CSM is always locally bounded.
Let $\lambda^{\bullet}$ be a continuous system of measures on the continuous map $\pi:X \rightarrow Y$ and let $K\subseteq X$ be compact. There exists a function $f \in C_c(X)$ such that $f:X\rightarrow [0, 1]$ and $f \equiv 1$ on $K$. Therefore, $\lambda^y(K) = \int_X \chi_{_K}(x) d\lambda^y(x) \leq \int_X f(x) d\lambda^y(x)$. By Lemma \[lem:CSM\_support\], the support of the continuous function $F(y) = \int_X f(x) d\lambda^y(x)$ is contained in $\pi(supp(f))$, which is compact. Therefore, $F$ is a bounded function on $Y$, and so is $\lambda^{\bullet}(K)$. Hence $\lambda^{\bullet}$ is bounded on compact sets and therefore locally bounded.
A system of measures $\lambda^{\bullet}$ on $\pi:X \rightarrow Y$ satisfying that $\lambda^{y}(X)<\infty$ for every $y\in Y$ will be called a **system of finite measures**. If $\lambda^{\bullet}$ is also a BSM, it will be called a **finite BSM**, and if $\lambda^{\bullet}$ is also a CSM, it will be called a **finite CSM**.
A system of measures $\lambda^{\bullet}$ on $\pi:X \rightarrow Y$ satisfying that $\lambda^{y}(X)=1$ for every $y\in Y$ will be called a **system of probability measures**. If $\lambda^{\bullet}$ is also a BSM, it will be called a **probability BSM**, and if $\lambda^{\bullet}$ is also a CSM, it will be called a **probability CSM**.
A system of measures $\lambda^{\bullet}$ on $\pi:X \rightarrow Y$ satisfying that every $x \in X$ has a neighborhood $U_x$ such that $\lambda^{y}(U_x)<\infty$ for every $y \in Y$, will be called a **locally finite system of measures**. If $\lambda^{\bullet}$ is also a BSM, it will be called a **locally finite BSM**.
A locally finite system of measures is, in particular, a system of locally finite measures. We deliberately chose the stronger notion, as it is needed for our purposes (in particular for Lemma \[lem:criterion for locally finite BSM\]).
Observe that a system of measures which is locally bounded, is of course locally finite. In light of Lemma \[lem:CSM always locally bounded\] we have the following immediate corollary.
\[cor:CSM is locally finite\] A CSM is always locally finite.
Before we proceed, we briefly recall the following well known facts from basic measure theory. A **Dynkin system** $\mathcal D$ is a non-empty collection of subsets of a space $X$ which is
(i) closed under relative complements, i.e. if $A,B \in \mathcal D$ and $A \subseteq B$ then $B \setminus A \in \mathcal D$;
(ii) closed under countable unions of increasing sequences, i.e. if $A_i \in \mathcal D$ and $A_i \subseteq A_{i+1}$ then $\bigcup_{i=1}^{\infty} A_i \in \mathcal D$;
(iii) contains $X$ itself.
An equivalent notion is that of a **$\mathbf{\lambda}$-system** $\mathcal D$, which is a non-empty collection of subsets of a space $X$ which is
(a) closed under complements, i.e. if $A \in \mathcal D$ then $A^c \in \mathcal D$;
(b) closed under disjoint countable unions, i.e. if $A_i \in \mathcal D$ and $A_i \cap A_j = \emptyset \ \forall i \neq j$ then $\bigcup_{i=1}^{\infty} A_i \in \mathcal D$;
(c) contains $X$ itself.
A **$\mathbf{\pi}$-system** $\mathcal P$ is a non-empty collection of subsets that is closed under finite intersections. **Dynkin’s $\mathbf{\pi}$-$\mathbf{\lambda}$ Theorem** says that if a $\pi$-system $P$ is contained in a Dynkin system $D$, then the entire $\sigma$-algebra generated by $\mathcal P$ is contained in $\mathcal D$.
For our purposes, the following definition will be useful.
\[def:pre-dynkin\] We will say that a collection $\mathcal{D}$ of subsets of $X$ ia a **pre-Dynkin system** if it satisfies the following two properties:
1. if $E,F\mbox{ and }E\cap F\in{\mathcal D}$, then $E\cup F$ and $E\setminus F \in {\mathcal D}$;
2. if ${\mathcal C} \subseteq {\mathcal D}$ is at most countable, and any finite intersection of elements in ${\mathcal C}$ belongs to $\mathcal D$, then the union of all elements of ${\mathcal C}$ belongs to $\mathcal D$.
\[lem:equivalent Dynkin\] Let $\mathcal{D}$ be a collection of subsets of a space $X$. $\mathcal{D}$ is a Dynkin system if and only if $\mathcal{D}$ is a pre-Dynkin system and $X$ belongs to $\mathcal{D}$.
Let $\mathcal{D}$ be a pre-Dynkin system on $X$ such that $X\in \mathcal{D}$. In order to prove that $\mathcal{D}$ is a Dynkin system, we verify properties (a), (b) and (c) above. Property (c) holds by assumption. For property (a), let $A \in \mathcal{D}$. Since $X\in \mathcal{D}$ and $X\cap A=A$, property (1) of a pre-Dynkin systems implies that $A^c=X\setminus A\in \mathcal{D}$, hence $\mathcal{D}$ is closed under complements. Finally, for property (b), let $\mathcal{C} = \{A_i\}_{i=1}^{\infty}\subseteq \mathcal{D}$ be a countable collection of pairwise disjoint subsets of $X$. For any finite intersection of distinct elements of $\mathcal{C}$ we have $$A_{i_1}\cap A_{i_2}\cap \dots\cap A_{i_k}=\left\{\begin{array}{lcl}A_{i_1}\in \mathcal{D}&\mbox{ if }&k=1,\\ \emptyset=X^c\in \mathcal{D}&\mbox{ if }&k>1.\end{array}\right.$$ Therefore, property (2) of a pre-Dynkin system guarantees that $\bigcup_{i=1}^{\infty} A_i \in \mathcal D$. We conclude that $\mathcal{D}$ is a $\lambda$-system and thus a Dynkin system.
We now turn to the converse. Let $\mathcal{D}$ be a Dynkin system. Clearly, $X\in \mathcal{D}$. For property (1) of a pre-Dynkin system, let $E,\ F$ and $E\cap F\in \mathcal{D}$. Since by property (i) $\mathcal{D}$ is closed under relative complements, we have that $E\setminus F=E\setminus (E\cap F)\in \mathcal{D}$. Likewise, $F\setminus E=F\setminus (E\cap F)\in \mathcal{D}.$ From property (b) it follows that $\mathcal{D}$ is closed under disjoint finite unions, and thus we have that $E\cup F=(E\setminus F)\cup (F\setminus E)\cup (E\cap F)\in \mathcal{D}.$ For property (2) of a pre-Dynkin system, observe first that property (1) implies that if we have a *finite* collection of sets in $\mathcal{D}$, satisfying that all their intersections are also in $\mathcal{D}$, then their union is in $\mathcal{D}$ as well. Now let $\mathcal{C}=\{C_i\}_{i=1}^{\infty}\subseteq \mathcal{D}$ be a *countable* collection such that any finite intersection of its elements is in $\mathcal{D}$. Denote $V_k=\bigcup_{i=1}^k C_i$, for any $i\geq 1$. Applying the observation we just made to the finite collections $\mathcal{C}_k:=\{C_1,C_2,\dots, C_k\}$, we deduce that $V_k\in \mathcal{D}$, for all $k$. Since by property (ii) $\mathcal{D}$ is closed under countable unions of increasing sequences, we conclude that $\bigcup_{i=1}^{\infty}C_i=\bigcup_{k=1}^{\infty}V_k\in\mathcal{D}$. This completes the proof.
\[prop:D contains Borel\] Let ${\mathcal D}$ be a pre-Dynkin system in $X$. If there is a countable basis ${\mathcal B}$ for the topology of $X$ such that $U_{1}\cap U_{2}\cap\dots\cap U_{n}\in {\mathcal D}$ for any $\{U_{1},U_{2},\dots, U_{n}\}\subset {\mathcal B}$, then ${\mathcal D}$ consists of all Borel subsets of $X$.
Let $A$ be an open subset of $X$. Since ${\mathcal B}$ is a countable basis, there is a sequence $\{U_{1},U_{2},\dots, U_{n},\dots\}\subset {\mathcal B}$ such that $A=\bigcup_{i=1}^{\infty} U_i.$ Since, by assumption, all finite intersections of elements of the sequence belong to ${\mathcal D}$, property *(2)* of Definition \[def:pre-dynkin\] implies that $A\in {\mathcal D}$. It follows that ${\mathcal D}$ contains all open subsets of $X$ and in particular $X\in{\mathcal D}$. Therefore, ${\mathcal D}$ is a Dynkin system, containing all open subsets. Since open subsets form a $\pi$-system, we can invoke Dynkin’s $\pi$-$\lambda$ Theorem to conclude that all Borel subsets of $X$ are in ${\mathcal D}$.
\[lem:collection D\] Let $\lambda^{\bullet}$ be a system of finite measures. The collection of subsets $${\mathcal D}=\{E\subseteq X\text{ Borel }:\ \lambda^{\bullet}(E)\text{ is a Borel function on } Y\}$$ is a pre-Dynkin system.
We will prove that ${\mathcal D}$ satisfies properties *(1)* and *(2)* of Definition \[def:pre-dynkin\]. For any $y\in Y$ we have: $${\lambda}^y(E\cup F)={\lambda}^y(E)+{\lambda}^y(F)-{\lambda}^y(E\cap F),\quad {\lambda}^y(E\setminus F)={\lambda}^y(E)-{\lambda}^y(E\cap F).$$ Therefore $\lambda^{\bullet}(E\cup F)$ and $\lambda^{\bullet}(E\setminus F)$ are Borel functions, and *(1)* follows.
If ${\mathcal C}$ is finite, then *(2)* is a consequence of an inclusion-exclusion formula as in *(1)*. Suppose now that ${\mathcal C}$ is infinite, write ${\mathcal C}=\{E_n\}_{n=1}^{\infty}$ and let $E=\bigcup_{n=1}^{\infty} E_n$. Consider the sets $$F_1:=E_1,\quad F_2:=E_1\cup E_2, \quad F_3:=E_1\cup E_2\cup E_3, \quad \dots$$ From the finite case we have that $F_n\in{\mathcal D}$ for all $n\geq 1$. Moreover, $E=\bigcup_{n=1}^{\infty} F_n$ and $\lambda^{y}(E)=\lim_{n\rightarrow\infty} \lambda^{y}(F_n)$, for every $y\in Y$. Thus $\lambda^{\bullet}(E)$ is a Borel function, being a limit of the sequence of Borel functions $\{\lambda^{\bullet}(F_n)\}$. Therefore $E\in{\mathcal D}$, proving the infinite case of *(2)*.
\[lem:criterion for finite BSM\] Let $\pi:X\rightarrow Y$ be a Borel map endowed with a system of finite measures $\lambda^{\bullet}$. Assume that there is a countable basis ${\mathcal B}$ for the topology of $X$ such that $\lambda^{\bullet}(U_{1}\cap U_{2}\cap\dots\cap U_{n})$ is a Borel function for any $\{U_{1},U_{2},\dots, U_{n}\}\subset {\mathcal B}$, $n\geq 1$. Then $\lambda^{\bullet}$ is a finite BSM.
Consider the collection ${\mathcal D}=\{E\subseteq X\text{ Borel }:\ \lambda^{\bullet}(E)\text{ is a Borel function on } Y\}$. By Lemma \[lem:collection D\] above, ${\mathcal D}$ is a pre-Dynkin system. With respect to ${\mathcal D}$, the basis ${\mathcal B}$ satisfies the condition of Proposition \[prop:D contains Borel\], which in turn implies that all Borel subsets of $X$ are in ${\mathcal D}$. Therefore, $\lambda^{\bullet}$ is a BSM.
\[lem:criterion for locally finite BSM\] Let $\pi:X\rightarrow Y$ be a Borel map endowed with a locally finite system of measures $\lambda^{\bullet}$. Assume that there is a countable basis ${\mathcal B}$ for the topology of $X$ such that $\lambda^{\bullet}(U_{1}\cap U_{2}\cap\dots\cap U_{n})$ is a Borel function for any $\{U_{1},U_{2},\dots, U_{n}\}\subset {\mathcal B}$, $n\geq 1$. Then $\lambda^{\bullet}$ is a locally finite BSM.
Let ${\mathcal B}=\{U_i\}_{i=1}^{\infty}$. Since $\lambda^{\bullet}$ is locally finite, it is straightforward to verify that the sub-collection $\{U \in \mathcal{B} ~|~ \lambda^{y}(U)<\infty \text{ for every } y \in Y \}$ is itself a basis for $X$. Therefore, we can assume that all $U_i \in \mathcal{B}$ satisfy $\lambda^{y}(U_i)<\infty$ for every $y \in Y$.
For any $i \geq 1$, consider the map $\pi_i:U_i \rightarrow Y$ given by composing the inclusion $U_i\hookrightarrow X$ with $\pi : X\rightarrow Y$. Let $\lambda^{\bullet}_i$ denote the system of measures on $\pi_i$ obtained by restricting $\lambda^{\bullet}$. Note that $\lambda^{\bullet}_i$ is a system of *finite* measures, since $\lambda_i^{y}(U_i) = \lambda^{y}(U_i)<\infty$ for every $y \in Y$. Now consider the collection $${\mathcal D}_i=\{E\subseteq U_i\text{ Borel }:\ \lambda_i^{\bullet}(E)\text{ is a Borel function on } Y\}.$$ We can apply Lemma \[lem:collection D\], which guarantees that ${\mathcal D}_i$ is a pre-Dynkin system in $U_i$. Also, the collection ${\mathcal B}_i=\left\{U_i\cap U_j\right\}_{j=1}^{\infty}$ is a basis for the topology of $U_i$. Moreover, due to our assumption on $\mathcal B$, we see that ${\mathcal B}_i$ satisfies the hypotheses of Proposition \[prop:D contains Borel\] with respect to the collection ${\mathcal D}_i$. Consequently, ${\mathcal D}_i$ consists of all Borel subsets of $U_i$.
Let $E\subseteq X$ be a Borel subset. We need to show that $\lambda^{\bullet}(E)$ is a Borel function. For any $i$, the function $\lambda_i^{\bullet}(E\cap U_i)$ is a Borel function on $Y$, since $E\cap U_i$ is a Borel subset of $U_i$ and therefore in ${\mathcal D}_i$. Therefore, for any $i$, $\lambda^{\bullet}(E\cap U_i)$ is a Borel function on $Y$. Likewise, for any $i_1,\dots,i_k$ the function $\lambda^{\bullet}(E\cap U_{i_1}\cap\dots\cap U_{i_k})$ is a Borel function on $Y$.
Next, we define $V_n= \bigcup_{i=1}^n U_i$. This is an increasing sequence of open sets $\{V_n\}_{n=1}^{\infty}$, and each $V_n$ satisfies $\lambda^y(V_n) \leq \sum_{i=1}^n \lambda^y(U_i)<\infty$ for every $y \in Y$. Since $E\cap V_n = E \cap \left(\bigcup_{i=1}^n U_i \right) = (E \cap U_1) \cup (E \cap U_2) \cup \dots \cup (E \cap U_n)$, a routine inclusion-exclusion type argument yields that for all $n$, $\lambda^{\bullet}(E\cap V_n)$ can be written as a linear combination of functions of the form $\lambda^{\bullet}(E\cap U_{i_1}\cap\dots\cap U_{i_k})$, and is therefore a Borel function on $Y$.
Finally, $\lambda^{\bullet}(E)$ is the limit of the increasing sequence of Borel functions $\lambda^{\bullet}(E\cap V_n)$, hence by the Monotone Convergence Theorem, $\lambda^{\bullet}(E)$ is a Borel function on $Y$, as required.
An immediate consequence of Lemma \[lem:criterion for locally finite BSM\] is the following.
\[cor:criterion for BSM\] Let $\pi:X\rightarrow Y$ be a Borel map endowed with a locally finite system of measures $\lambda^{\bullet}$. If $\lambda^{\bullet}(A)$ is a Borel function for any open set $A$, then $\lambda^{\bullet}$ is a locally finite BSM.
\[prop:CSM is BSM\] A CSM is a locally finite BSM.
Let $\lambda^{\bullet}$ be a CSM on $\pi:X \rightarrow Y$. By Corollary \[cor:CSM is locally finite\] $\lambda^{\bullet}$ is locally finite. By Corollary \[cor:criterion for BSM\] it is sufficient to show that $\lambda^{\bullet}(A)$ is a Borel function for any open subset $A$.
There exists an increasing sequence $\{A_n\}_{n=1}^{\infty}$ of open subsets of $A$ such that $\overline{A_n}$ is compact for every $n$, $\overline{A_n}\subset A_{n+1}$ and $\bigcup_{n=1}^{\infty}A_n=A$. Moreover, there exists a non-decreasing sequence of compactly supported continuous functions $\psi_n:X\rightarrow [0, 1]$ such that $\psi_n\equiv 1$ on $A_n$ and $supp(\psi_n)\subseteq \overline{A_{n+1}}$ for every $n$. Therefore, $\forall x\in X$, $\lim_{n\rightarrow \infty}\psi_n(x)=\chi_{_A}(x)$. It follows, by the Monotone Convergence Theorem, that $$\lambda^{y}(A)=\int_X\chi_{_A}(x)d\lambda^y(x)=\int_X\lim_{n\rightarrow \infty}\psi_n(x)d\lambda^y(x)=\lim_{n\rightarrow\infty}\int_X\psi_n(x)d\lambda^y(x).$$ Since $\forall n$, $\psi_n \in C_c(X)$ and $\lambda^{\bullet}$ is continuous, the map $y \mapsto \int_X\psi_n(x)d\lambda^y(x)$ is continuous $\forall n$. Therefore the map $y \mapsto \lambda^y(A)$ is a (monotone) limit of continuous (hence Borel) functions, and is thus a Borel function.
We omit the full proof of the following lemma, which is analogous to the proof of Lemma \[lem:criterion for locally finite BSM\], via a corresponding version of Lemma \[lem:collection D\] with ${\mathcal D}=\{E\subseteq X\text{ Borel }:\ \mu(E) = \nu(E) \}$.
\[lem:mu=nu for every E Borel\] Let $\mu$ and $\nu$ be two locally finite measures on a space $X$. Assume that there is a countable basis ${\mathcal B}$ for the topology of $X$ such that $\mu(U_{1}\cap U_{2}\cap\dots\cap U_{n})=\nu(U_{1}\cap U_{2}\cap\dots\cap U_{n})$ for any $\{U_{1},U_{2},\dots, U_{n}\}\subset {\mathcal B}$, $n\geq 1$. Then $\mu(E) = \nu(E)$ for any Borel subset $E \subseteq X$.
\[cor:mu=nu for every A open\] Let $\mu$ and $\nu$ be two locally finite measures on a space $X$. If $\mu(A) = \nu(A)$ for any open subset $A \subseteq X$, then $\mu(E) = \nu(E)$ for any Borel subset $E \subseteq X$.
We will make use of the above lemma and corollary in the sequel.
Composition of systems of measures {#sec:composition}
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The notion of composition of systems of measures appears in §1.3.a of [@renault-anantharaman-delaroche], and is also mentioned briefly in [@revuz-book] (see Definition 1.5). Consider the diagram $$\xymatrix{X\ar [rr]^{p}_{\alpha^{\bullet}}&&Y\ar [rr]^{q}_{\beta^{\bullet}}&&Z},$$ where $\alpha^{\bullet}$ is a BSM on $p:X \rightarrow Y$ and $\beta^{\bullet}$ is a system of measures on $q:Y \rightarrow Z$.
\[def:composition BSM\] We define the **composition** $(\beta\circ\alpha)^{\bullet}$ by $$(\beta\circ\alpha)^{z}(E)=\int_Y \alpha^{y}(E)\ d\beta^z(y) \ \ \ \ \ \forall z \in Z, \text{ and } E \subseteq X \text{ Borel. }$$
\[prop:composition BSM\] The composition $(\beta\circ\alpha)^{\bullet}$ is a system of measures on $q\circ p$. If $\alpha^{\bullet}$ and $\beta^{\bullet}$ are both BSMs, then $(\beta\circ\alpha)^{\bullet}$ is a BSM.
Note that for any $z\in Z$ and any Borel subset $E \subseteq X$, $(\beta\circ\alpha)^{z}(E)$ is well defined, since $\alpha^{\bullet}(E)$ is a Borel function on $Y$ and $\beta^z$ is a Borel measure on $Y$. To prove that $(\beta\circ\alpha)^{z}$ is a Borel measure on $X$, let $\left\{E_n\right\}_{n=1}^{\infty}$ be a countable family of disjoint Borel subsets of $X$. Using a standard Monotone Convergence Theorem argument with $\sum_{n=1}^{k}\alpha^{y}\left( E_n \right) \nearrow \sum_{n=1}^{\infty}\alpha^{y}\left( E_n\right)$, we obtain $$\begin{aligned}
(\beta\circ\alpha)^{z}(\bigcup_{n=1}^{\infty} E_n) &=& \int_Y \alpha^{y}(\bigcup_{n=1}^{\infty} E_n)\ d\beta^z(y)=\int_Y \sum_{n=1}^{\infty}\alpha^{y}\left( E_n\right)\ d\beta^z(y)= \nonumber \\ &=& \sum_{n=1}^{\infty}\int_Y \alpha^{y}\left( E_n\right)\ d\beta^z(y) = \sum_{n=1}^{\infty}(\beta\circ\alpha)^{z}(E_n).\nonumber\end{aligned}$$ To prove that $(\beta\circ\alpha)^{z}$ is concentrated on $(q\circ p)^{-1}(z)$, observe that if $y\in q^{-1}(z)$ then $p^{-1}(y) \subseteq (q\circ p)^{-1}(z)$. Taking complements in $X$ we get $\alpha^y(X\setminus (q\circ p)^{-1}(z)) \leq \alpha^y(X\setminus p^{-1}(y)) = 0$. Since $\beta^z$ is concentrated on $q^{-1}(z)$, we obtain $$(\beta\circ\alpha)^{z}\left(X\setminus (q\circ p)^{-1}(z)\right)=\int_Y\alpha^y\left(X\setminus (q\circ p)^{-1}(z)\right)d\beta^z(y)=0.$$ We have shown that $(\beta\circ\alpha)^{\bullet}$ is a system of measures on $q \circ p$. Now assume that both $\alpha^{\bullet}$ and $\beta^{\bullet}$ are BSMs. Let $E\subseteq X$ be a Borel subset. Since $\alpha^{\bullet}$ is a BSM, the function $\alpha^{\bullet}(E)$ is a nonnegative Borel function on $Y$. But $\beta^{\bullet}$ is a BSM as well, so from Lemma \[lem:equiv\_BSM\] we have that $z\mapsto \int_Y \alpha^{y}(E)\ d\beta^z(y)$, which is precisely the function $(\beta\circ\alpha)^{\bullet}(E)$, is a Borel function on $Z$. Therefore, $(\beta\circ\alpha)^{\bullet}$ is a BSM. This completes the proof.
\[prop:composition CSM\] If $\alpha^{\bullet}$ and $\beta^{\bullet}$ are both CSMs, then $(\beta\circ\alpha)^{\bullet}$ is a CSM.
Let $f\in C_c(X)$. We need to show that the map $z \mapsto \int_X f(x)d (\beta \circ \alpha)^z (x)$ is a continuous function on $Z$. Define $g(y) = \int_X f(x)d\alpha^y (x)$. Since $\alpha^{\bullet}$ is a CSM, Corollary \[cor:CSM has compact support\] implies that $g(y) \in C_c(Y)$. From the fact that $\beta^{\bullet}$ is a CSM we now get that the map $z \mapsto \int_Y g(y) d\beta^z(y) \in C_c(Z)$. This completes the proof, since $$\int_Y g(y) d\beta^z(y) = \int_Y \left(\int_X f(x)d\alpha^y (x) \right) d\beta^z(y) = \int_X f(x)d (\beta \circ \alpha)^z (x).$$
\[prop:composition pos and bdd\] Consider the setting of Definition \[def:composition BSM\].
1. Assume that $p$ is an open map. If $\alpha^{\bullet}$ and $\beta^{\bullet}$ are positive on open sets then so is $(\beta\circ\alpha)^{\bullet}$.
2. Assume that $p$ is a continuous map. If $\alpha^{\bullet}$ and $\beta^{\bullet}$ are locally bounded then so is $(\beta\circ\alpha)^{\bullet}$.
\(1) Fix $z \in Z$ and let $A\subseteq X$ be an open set satisfying $A\cap (q\circ p)^{-1}(z)\neq \emptyset$. We need to show that $(\beta\circ\alpha)^{z}(A) >0$. The set $p(A)\cap q^{-1}(z)$ is not empty since $$\emptyset \neq p\left(A\cap (q\circ p)^{-1}(z)\right) \subseteq p(A)\cap p(p^{-1}(q^{-1}(z))) \subseteq p(A)\cap q^{-1}(z).$$ Furthermore, $p$ is assumed to be an open map, so $p(A)$ is open in $Y$. This implies that $\beta^z(p(A))>0$, since $\beta^{\bullet}$ is positive on open sets. Obviously, for every $y\in p(A)$ there exists $x\in A$ such that $p(x)=y$, hence $p^{-1}(y)\cap A\neq\emptyset$. This implies that $\alpha^y(A)>0$ for every $y\in p(A)$, since $\alpha^{\bullet}$ is positive on open sets. We conclude that $$(\beta\circ\alpha)^{z}(A)=\int_Y \alpha^{y}(A)\ d\beta^z(y)\geq \int_{p(A)} \alpha^{y}(A)\ d\beta^z(y)>0.$$ (2) Take $x \in X$. Since $\beta^{\bullet}$ is locally bounded, there exists an open neighborhood $V$ of $p(x)$ and a constant $C_2$ such that $\beta^z(V) < C_2$ for every $z \in Z$. Since $p$ is continuous and $\alpha^{\bullet}$ is locally bounded, there exists an open neighborhood $U$ of $x$ and a constant $C_1$ such that $p(U) \subseteq V$ and $\alpha^y(U) < C_1$ for every $y \in Y$. Note that if $y \notin p(U)$, then $p^{-1}(y)\cap U=\emptyset$. Hence $\alpha^y(U)=0$ for all $y \notin p(U)$. We therefore have $$(\beta\circ\alpha)^{z}(U)=\int_Y \alpha^{y}(U)\ d\beta^z(y)=\int_{p(U)} \alpha^{y}(U)\ d\beta^z(y)\leq C_1\cdot\beta^z(p(U))\leq C_1\cdot C_2$$ for every $z\in Z$.
In general, the composition of locally finite systems of measures need not be locally finite. In order to assure local finiteness of the composition we need to require a stronger property of the system $\alpha^{\bullet}$. We omit the proof of the following Lemma, which is an obvious modification of the proof of the second part of Proposition \[prop:composition pos and bdd\].
\[lem:condition for composition locally finite\] Consider the setting of Definition \[def:composition BSM\], and assume that the map $p$ is continuous. If $\alpha^{\bullet}$ is locally bounded and $\beta^{\bullet}$ is locally finite then $(\beta\circ\alpha)^{\bullet}$ is locally finite.
We have seen in Lemma \[lem:CSM always locally bounded\] that any CSM is locally bounded. Taken together with Lemma \[lem:condition for composition locally finite\] and the fact that a locally bounded system is in particular locally finite, this implies that the composition is guaranteed to be locally finite in several more scenarios.
\[cor:conditions for composition locally finite\] Consider the setting of Definition \[def:composition BSM\], and assume that the map $p$ is continuous. Each of the following conditions implies that $(\beta\circ\alpha)^{\bullet}$ is locally finite.
1. $\alpha^{\bullet}$ is a CSM and $\beta^{\bullet}$ is locally finite.
2. $\alpha^{\bullet}$ and $\beta^{\bullet}$ are both locally bounded.
3. $\alpha^{\bullet}$ and $\beta^{\bullet}$ are both CSMs.
4. either $\alpha^{\bullet}$ or $\beta^{\bullet}$ is a CSM and the other is locally bounded.
As a particular case of Lemma \[lem:condition for composition locally finite\] we obtain the following useful result. The proof amounts to taking $Z = \{z\}$, viewing $\beta$ as a trivial system of measures on the projection $\pi:Y \rightarrow \{z\}$ and applying Lemma \[lem:condition for composition locally finite\].
\[cor:locally finite composition of BSM and measure\] Let $\alpha^{\bullet}$ be a locally bounded BSM on a continuous map $p:X \rightarrow Y$ and let $\beta$ be a locally finite measure on $Y$. For every Borel set $E \subseteq X$, define $$\mu(E) = \int_{Y} \alpha^y (E) d\beta (y) .$$ Then $\mu$ is a locally finite measure on $X$.
Lifting of systems of measures {#sec:lifting}
==============================
The concept of lifting, which we define below, is discussed in Appendix A.1 of [@renault-anantharaman-delaroche], in the broader context of transverse measure theory. Let $X$, $Y$ and $Z$ be topological spaces, and let $p:X \rightarrow Z$ and $q:Y \rightarrow Z$ be Borel maps. The usual **pullback** of $X$ and $Y$ over $Z$ is the space $$X *_Z Y = \{(x,y) \in X \times Y~:~p(x) = q(y) \}.$$ In order to lighten notation, we will usually write $X * Y$, keeping $Z$ implicit. The topology on $X * Y$ is inherited from the product topology on $X \times Y$. Consider the pullback diagram $$\xymatrix{X * Y \ar [dd]_{\pi_X}\ar [rr]^{\pi_Y}&&Y\ar [dd]_{q}\\\\
X\ar [rr]^{p}_{\alpha^{\bullet}}&&Z}$$ where $\pi_X$ and $\pi_Y$ are the obvious projections, and $\alpha^{\bullet}$ is a system of measures on $p:X \rightarrow Z$. Observe that the fibers of the map $\pi_Y$ are Cartesian products of the form $\pi_Y^{-1}(y)=p^{-1}(q(y))\times\{y\}$.\
We will assume throughout this section that $\alpha^{\bullet}$ is a *locally finite* system of measures.
\[def:lifting BSM\] The **lifting** of the locally finite system of measures $\alpha^{\bullet}$ to $\pi_Y$, denoted $(q^*\alpha)^{\bullet}$, is given by $$(q^*\alpha)^y=\alpha^{q(y)}\times \delta_y.$$ More precisely, $(q^*\alpha)^{y}(E)=(\alpha^{q(y)}\times \delta_y)(E\cap \pi_Y^{-1}(y))$ for every $y\in Y$ and every Borel set $E\subseteq X*Y$.
If $\beta^{\bullet}$ is a locally finite system of measures on $q:Y \rightarrow Z$, then the lifting $(p^*\beta)^{\bullet}$ to $\pi_X$ is defined similarly, by $(p^*\beta)^x=\delta_x\times \beta^{p(x)}.$ The properties of the lifting $(q^*\alpha)^{\bullet}$ which we state and prove below, hold for $(p^*\beta)^{\bullet}$ as well, with the obvious modifications.
\[rem:elementary open sets\] In the sequel, we will make frequent use of open sets $E \subseteq X*Y$ of the form $E=(A\times B)\cap (X*Y)$, where $A$ and $B$ are open sets in $X$ and $Y$ respectively. We will refer to these as *elementary open sets*. For any elementary open set we have $$\begin{aligned}
(q^*\alpha)^{y}(E)&=&(q^*\alpha)^{y}(E\cap \pi_Y^{-1}(y))\label{calc:lifting}\nonumber\\
&=&(q^*\alpha)^{y}\left((A\cap p^{-1}(q(y)))\times (B\cap\{y\})\right)\\
&=&\alpha^{q(y)}(A\cap p^{-1}(q(y)))\cdot \delta_y(B\cap\{y\})\nonumber\\
&=&\alpha^{q(y)}(A)\cdot \delta_y(B).\nonumber\end{aligned}$$ If $\{A_n\}_{n=1}^{\infty}$ and $\{B_m\}_{m=1}^{\infty}$ are countable bases for the topologies of $X$ and $Y$ respectively, we can set ${\mathcal B}=\{(A_n\times B_m)\cap X*Y\}_{n,m=1}^{\infty}$. This gives a countable basis ${\mathcal B}$ for the topology of $X*Y$ consisting of elementary open sets.
\[prop:lifting BSM\] The lifting $(q^*\alpha)^{\bullet}$ is a locally finite system of measures on $\pi_Y$. If $\alpha^{\bullet}$ is a BSM, then so is $(q^*\alpha)^{\bullet}$.
As a product of locally finite (hence $\sigma$-finite) Borel measures, $(q^*\alpha)^{y}$ is a well defined Borel measure for every $y\in Y$. By definition it is concentrated on $p^{-1}(q(y))\times\{y\} = \pi_Y^{-1}(y)$. Let $(x,y) \in X*Y$. Since $\alpha^{\bullet}$ is locally finite, there exists a neighborhood $U_x$ of $x$ such that $\alpha^z(U_x) < \infty$ for all $z \in Z$. By calculation (\[calc:lifting\]) above, the open neighborhood $(U_x \times Y) \cap (X*Y)$ of $(x,y)$ satisfies $(q^*\alpha)^y\left((U_x \times Y) \cap (X*Y)\right) = \alpha^{q(y)}(U_x)\cdot \delta_y(Y) = \alpha^{q(y)}(U_x) < \infty$ for every $y \in Y$, hence $(q^*\alpha)^{\bullet}$ is a locally finite system of measures.
Now assume that $\alpha^{\bullet}$ is a BSM. In order to prove that $(q^*\alpha)^{\bullet}$ is a BSM, we show first that $(q^*\alpha)^{\bullet}(E)$ is a Borel function for any elementary open set $E=(A\times B)\cap (X*Y)$. For such $E$ we have, by calculation (\[calc:lifting\]), that $(q^*\alpha)^{y}(E) = \alpha^{q(y)}(A)\cdot \delta_y(B).$ Therefore, if we denote the composition of the Borel functions $\alpha^{\bullet}(A)$ and $q$ by $\alpha^{q(\bullet)}(A)$, we can write $(q^*\alpha)^{\bullet}(E)=\alpha^{q(\bullet)}(A)\cdot \chi_{_B}$. Thus $(q^*\alpha)^{\bullet}(E)$ is a Borel function.
Finite intersections of elementary open sets are themselves elementary open sets, and thus the basis ${\mathcal B}$ as in Remark \[rem:elementary open sets\] satisfies the hypotheses of Lemma \[lem:criterion for locally finite BSM\]. We conclude that $(q^*\alpha)^{\bullet}$ is a BSM.
\[lem:continuity lemma\] Let $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$ be topological spaces and let $\gamma^{\bullet}$ be a CSM on $\phi:\mathcal{Y}\rightarrow \mathcal{Z}$. For every $\psi\in C_c(\mathcal{X}\times \mathcal{Y})$, the function $\displaystyle (x,z)\mapsto \int_{\mathcal{Y}} \psi(x,y)d\gamma^z(y)$ belongs to $C_c(\mathcal{X}\times \mathcal{Z})$.
We first show that $F(x,z)= \int_{\mathcal{Y}} \psi(x,y)d\gamma^z(y)$ has compact support. Let $\pi_{\mathcal{X}}: \mathcal{X}\times \mathcal{Y} \rightarrow \mathcal{X}$ and $\pi_{\mathcal{Y}}: \mathcal{X}\times \mathcal{Y} \rightarrow \mathcal{Y}$ denote the projections, and let $K \subseteq \mathcal{X}\times \mathcal{Y}$ be the support of $\psi$. Observe that if $(x,z)\notin \pi_{\mathcal{X}}(K)\times \phi(\pi_{\mathcal{Y}}(K))$, then $(x,y)$ does not belong to $K$ for any $y\in \phi^{-1}(z)$. Therefore, for such $(x,z)$ we have $F(x,z) = \int_{\mathcal{Y}} \psi(x,y)d\gamma^z(y)=\int_{\phi^{-1}(z)} \psi(x,y)d\gamma^z(y)=0$. Thus $\{(x,z) ~|~ F(x,z) \neq 0\}$ is contained in $\pi_{\mathcal{X}}(K)\times \phi(\pi_{\mathcal{Y}}(K))$ which is compact (hence closed), and it follows that $supp(F) \subseteq \pi_{\mathcal{X}}(K)\times \phi(\pi_{\mathcal{Y}}(K))$ is compact.
We turn to proving that $F$ is continuous on $\mathcal{X}\times \mathcal{Z}$. Fix $x_0\in \mathcal{X}$, $z_0\in \mathcal{Z}$ and $\epsilon>0$. We claim that there exists a neighborhood $A_{x_0}$ of $x_0$ such that $\sup_{y}\left|\psi(x,y)-\psi(x_0,y)\right|<2\epsilon$ for any $x \in A_{x_0}$.
Let $ y' \in \pi_{\mathcal{Y}}(K)$. Since $\psi$ is continuous, there exist open sets $A_{x_0,y'}\subset \mathcal{X}$ and $B_{x_0,y'}\subset\mathcal{Y}$ such that $(x_0, y')\in A_{x_0,y'}\times B_{x_0,y'}$, and $\left|\psi(x,y)-\psi(x_0,y')\right|<\epsilon$ for any $(x, y)\in A_{x_0,y'}\times B_{x_0,y'}$. In particular, $\left|\psi(x,y)-\psi(x_0,y)\right|\leq \left|\psi(x,y)-\psi(x_0,y')\right|+\left|\psi(x_0,y')-\psi(x_0,y)\right|<2 \epsilon$. Since $\{x_0\}\times \pi_{\mathcal{Y}}(K)$ is compact, it admits a finite cover $\bigcup_{i=1}^n \left(A_{x_0,y_i'}\times B_{x_0,y_i'} \right)$. Define $A_{x_0}=\bigcap_{i=1}^n A_{x_0,y_i'}$ and $B_{x_0}=\bigcup_{i=1}^n B_{x_0,y_i'}$. Now consider $(x,y)\in A_{x_0}\times \mathcal{Y}$. If $y\in B_{x_0}$, then $\left|\psi(x,y)-\psi(x_0,y)\right|<2\epsilon$. If $y\notin B_{x_0}$ then $(x,y), (x_0, y)\notin K$, hence $\left|\psi(x,y)-\psi(x_0,y)\right|=|0-0|=0$. Thus, for any $x\in A_{x_0}$, we have $\sup_{y}\left|\psi(x,y)-\psi(x_0,y)\right|<2\epsilon$, as claimed.
For every $x\in A_{x_0}$ and $z\in\mathcal{Z}$, $$\begin{aligned}
\left| F(x,z) - F(x_0,z) \right| &=& \left|\int_{\mathcal{Y}} \psi(x,y)d\gamma^z(y)-\int_{\mathcal{Y}} \psi(x_0,y)d\gamma^z(y)\right| \\ &=& \left|\int_{\mathcal{Y}} \left( \psi(x,y)-\psi(x_0,y) \right) d\gamma^z(y)\right| \\ &\leq&\int_{\mathcal{Y}} \left| \psi(x,y)-\psi(x_0,y)\right|d\gamma^z(y) \ < \ 2 \; \epsilon \; \gamma^z(\pi_{\mathcal{Y}}(K))\end{aligned}$$ Since $\gamma^{\bullet}$ is a CSM, by Lemma \[lem:CSM always locally bounded\] it is locally bounded, or equivalently - bounded on compact sets. It follows that for every $x\in A_{x_0}$ and $z\in\mathcal{Z}$, $$\left| F(x,z) - F(x_0,z) \right| \ < \ \epsilon \cdot C,$$ where $C$ is a constant depending only on $K$ and $\gamma^{\bullet}$.
On the other hand, by the definition of a CSM, there is a neighborhood $V_{z_0}$ of $z_0$ such that for any $z\in V_{z_0}$ $$\left| F(x_0,z) - F(x_0,z_0) \right| \ < \ \epsilon.$$
We conclude that for every $(x,z)\in A_{x_0}\times V_{z_0}$, $$\left| F(x,z) - F(x_0,z_0) \right| \leq \left| F(x,z) - F(x_0,z) \right| + \left| F(x_0,z) - F(x_0,z_0) \right| \ < \ \epsilon \ (C+1),$$ hence $F$ is continuous.
\[prop:lifting CSM\] If $\alpha^{\bullet}$ is a CSM, then so is the lifting $(q^*\alpha)^{\bullet}$.
Let $f \in C_c(X * Y)$. We need to show that the function $y \mapsto \int_{X * Y} f(x,\eta) d(q^*\alpha)^y(x,\eta)$ is continuous on $Y$. The space $X * Y$ is closed in $X \times Y$, as the inverse image of the diagonal $\Delta(Z)$ under the continuous map $(p,q)$. Therefore, by Tietze’s Extension Theorem, there exists a function $F \in C(X \times Y)$ such that $F|_{X * Y} = f$. Since we can multiply $F$ by a function $\varphi \in C_c(X \times Y)$ which satisfies $\varphi =1$ on $K = supp(f)$, we can assume, without loss of generality, that $F \in C_c(X \times Y)$.
We now apply (a symmetric version of) lemma \[lem:continuity lemma\] above, and obtain that the map $(y,z) \mapsto \int_{X \times Y} F(x,y)d\alpha^z(x)$ belongs to $C_c(Y \times Z)$. Composing with the continuous function $y \mapsto (y,q(y))$, we deduce that the map $y \mapsto \int_{X \times Y} F(x,y)d\alpha^{q(y)}(x)$ is continuous on $Y$.
Observe that $\alpha^{q(y)}$ is concentrated on $p^{-1}(q(y))$. Therefore, since $ p^{-1}(q(y))\times\{y\} \subset X*Y$, we have $$\begin{aligned}
\int_{X \times Y} F(x,y)d\alpha^{q(y)}(x) &=& \int_{p^{-1}(q(y))\times\{y\}} F(x,y)d\alpha^{q(y)}(x) \ = \ \int_{X*Y} f(x,y)d\alpha^{q(y)}(x) \\ &=& \int_{X*Y} f(x,\eta) d(\alpha^{q(y)} \times \delta_y)(x,\eta) \ = \ \int_{X*Y} f(x,\eta) d(q^*\alpha)^y(x,\eta).\end{aligned}$$ We conclude that the map $y \mapsto \int_{X*Y} f(\xi,\eta) d(q^*\alpha)^y(\xi,\eta)$ is continuous on $Y$, as required.
\[prop:lifting stability\] The properties of being positive on open sets and locally bounded are preserved under lifting.
Assume that $\alpha^{\bullet}$ is positive on open sets. In order to prove that $(q^*\alpha)^{\bullet}$ is positive on open sets, it suffices to consider only elementary open sets, since they generate the topology of $X*Y$. So fix $y \in Y$ and let $E=(A\times B)\cap (X*Y)$ be an elementary open set such that $E\cap\pi_Y^{-1}(y)\neq \emptyset$. This implies that $A\cap p^{-1}(q(y))\neq \emptyset$ and $y\in B$, hence $\alpha^{q(y)}(A)>0$ and $\delta_y(B)=1$. Using calculation (\[calc:lifting\]) above we obtain that $(q^*\alpha)^{y}(E)=\alpha^{q(y)}(A)\cdot \delta_y(B)>0$. This proves that $(q^*\alpha)^{\bullet}$ is positive on open sets.
Proving that the lifted system is locally bounded is similar to the proof that it is locally finite in Proposition \[prop:lifting BSM\].
Consider the pull-back diagram $$\xymatrix{X * Y \ar [dd]_{\pi_X}^{(p^*\beta)^{\bullet}}\ar [rr]^{\pi_Y}_{(q^*\alpha)^{\bullet}}&&Y\ar [dd]_{q}^{\beta^{\bullet}}\\\\ X\ar [rr]^{p}_{\alpha^{\bullet}}&&Z}$$ where $\beta^{\bullet}$ and $\alpha^{\bullet}$ are locally finite BSMs, and $(p^*\beta)^{\bullet}$ and $(q^*\alpha)^{\bullet}$ are their lifting to $\pi_X$ and $\pi_Y$, respectively. By Proposition \[prop:lifting BSM\], $(p^*\beta)^{\bullet}$ and $(q^*\alpha)^{\bullet}$ are also locally finite BSMs.
The above pull-back diagram is a commutative diagram of locally finite BSMs. In other words, $(\beta\circ q^*\alpha)^{\bullet}$ and $(\alpha\circ p^*\beta)^{\bullet}$ are locally finite and $$(\beta\circ q^*\alpha)^{\bullet}=(\alpha\circ p^*\beta)^{\bullet}.$$
Fix $z\in Z$ and denote $\mu = (\beta\circ q^*\alpha)^{z}$ and $\nu = (\alpha\circ q^*\beta)^{z}$. We claim that $\mu(E) = \nu(E)$ for any elementary open subset of $X*Y$. Indeed, let $E=(A\times B)\cap (X*Y)$. Then by calculation (\[calc:lifting\]) preceding Proposition \[prop:lifting BSM\], and recalling that $\beta^z$ is concentrated on $q^{-1}(z)$, we have: $$\begin{aligned}
\mu(E)=(\beta\circ q^*\alpha)^{z}(E)&=&\int_{Y}(q^*\alpha)^y(E)d\beta^z(y) \ = \ \int_{Y}\alpha^{q(y)}(A)\cdot\delta_y(B)d\beta^z(y) \ = \nonumber \\
&=&\int_{B\cap q^{-1}(z)}\alpha^{q(y)}(A)d\beta^z(y) \ = \ \int_{B}\alpha^{z}(A)d\beta^z(y) \ = \ \alpha^{z}(A)\beta^z(B). \nonumber\end{aligned}$$ Analogously, $$\begin{aligned}
\nu(E)=(\alpha\circ p^*\beta)^{z}(E) & = & \int_{X}(p^*\beta)^x(E)d\alpha^z(x) \
= \ \int_{X}\delta_x(A)\cdot\beta^{p(x)}(B)d\alpha^z(x) \ = \nonumber \\
&=& \int_{A \cap p^{-1}(z)}\beta^{p(x)}(B)d\alpha^z(x) \ = \ \int_{A}\beta^{z}(B)d\alpha^z(x) \ = \ \alpha^{z}(A)\beta^z(B). \nonumber\end{aligned}$$ Therefore $\mu(E) = \nu(E)$ for any elementary open set.
The systems $\alpha^{\bullet}$ and $\beta^{\bullet}$ are locally finite. Thus, the topology of $X*Y$ admits a basis $\mathcal{B}$ as in Remark \[rem:elementary open sets\], comprised of elementary open sets of the form $E=(A\times B)\cap (X*Y)$ satisfying that $\alpha^{z}(A)$ and $\beta^z(B)$ are both finite, for any $z \in Z$. It follows from the above calculations that the compositions $(\beta\circ q^*\alpha)^{\bullet}$ and $(\alpha\circ p^*\beta)^{\bullet}$ are locally finite systems, and moreover, $\mu$ and $\nu$ are locally finite measures. Since finite intersections of elementary open sets are themselves elementary open sets, we can apply Lemma \[lem:mu=nu for every E Borel\] with the basis ${\mathcal B}$, and conclude that $\mu(E) = \nu(E)$ for every Borel subset $E \subseteq X$. This completes the proof.
Fibred products of systems of measures {#sec:fibred products}
======================================
Fibred products are mentioned in §1.3.a in [@renault-anantharaman-delaroche]. Assume that we have two pullback diagrams: $$\xymatrix{X_i * Y_i \ar [dd]_{\pi_{X_i}}\ar [rr]^{\pi_{Y_i}}&&Y_i\ar [dd]_{q_i}\\\\
X_i\ar [rr]^{p_i}&&Z}$$ Also, let $\xymatrix{X_1\ar [rr]^{f}_{\gamma_X^{\bullet}}&&X_2}$ and $\xymatrix{Y_1\ar [rr]^{g}_{\gamma_Y^{\bullet}}&&Y_2}$ be connecting maps endowed with locally finite systems of measures, satisfying that $p_1=p_2\circ f$ and $q_1=q_2\circ g$. Putting these together we obtain the following diagram: $$\xy 0;<.2cm,0cm>:
(20,20)*{X_2 * Y_2}="1";
(40,20)*{Y_2}="2";
(20,0)*{X_2}="3";
(40,0)*{Z}="4";
(10,10)*{X_1 * Y_1}="5";
(30,10)*{Y_1}="6";
(10,-10)*{X_1}="7";
(30,-10)*{Z}="8";
{"1"+CR+(.5,0);"2"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.5)+(0,1)*{\scriptstyle \pi_{Y_2}}};
{"5"+CR+(.5,0);"6"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.75)+(0,1)*{\scriptstyle \pi_{Y_1}}};
{"3"+CR+(.5,0);"4"+CL+(-10,0)**@{-}};
{"3"+CR+(9,0);"4"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.5)+(0,1)*{\scriptstyle p_2}};
{"7"+CR+(.5,0);"8"+CL+(-.5,0)**@{-}?>*{\dir{>}}?>(.5)+(0,1)*{\scriptstyle p_1}};
{"1"+CD+(0,-.5);"3"+CU+(0,9.6)**@{-}} ;
{"1"+CD+(0,-9.8);"3"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.5)+(-1.5,0)*{\scriptstyle \pi_{X_2}}} ;
{"2"+CD+(0,-.5);"4"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.5)+(-1,0)*{\scriptstyle q_2}} ;
{"5"+CD+(0,-.5);"7"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.5)+(-1.5,0)*{\scriptstyle \pi_{X_1}}} ;
{"6"+CD+(0,-.5);"8"+CU+(0,.5)**@{-}?>*{\dir{>}}?>(.25)+(-1,0)*{\scriptstyle q_1}} ;
{"5"+C+(1.4,1.4);"1"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>*{\dir{>}}?>(.5)+(-2,0)*{\scriptstyle f*g}} ;
{"6"+C+(1.4,1.4);"2"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>(.5)+(-1,0)*{\scriptstyle g}?>(.5)+(1.5,0)*{\scriptstyle \gamma_Y^{\bullet}}} ;
{"7"+C+(1.4,1.4);"3"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>(.5)+(-1,0)*{\scriptstyle f}?>(.5)+(1.5,0)*{\scriptstyle \gamma_X^{\bullet}}} ;
{"8"+C+(1.4,1.4);"4"+C+(-1.4,-1.4)**@{-}?>*{\dir{>}}?>(.5)+(-1.5,0)*{\scriptstyle id}};
\endxy$$ where the map $f*g=X_1*Y_1\rightarrow X_2*Y_2$ is defined by $(f*g)(x_1,y_1)=(f(x_1),g(y_1))$. This is a Borel map, as the restriction of the Borel function $f \times g$ to the Borel subspace $X_1*Y_1 \subseteq X_1 \times Y_1$. Moreover, the above diagram is commutative. Observe that the fibers of the map $f*g$ are Cartesian products of the form $(f*g)^{-1}(x_2,y_2)=f^{-1}(x_2)\times g^{-1}(y_2)$.\
We will assume throughout this section that $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$ are *locally finite* systems of measures.
\[def:fibred product BSM\] The **fibred product** of the locally finite systems of measures $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$, denoted $(\gamma_X * \gamma_Y)^{\bullet}$, is defined by $$\left(\gamma_X * \gamma_Y\right)^{(x_2,y_2)}=\gamma_X^{x_2} \times \gamma_Y^{y_2}.$$ More precisely, $\left(\gamma_X * \gamma_Y\right)^{(x_2,y_2)}(E)=\left(\gamma_X^{x_2} \times \gamma_Y^{y_2}\right)(E\cap (f*g)^{-1}(x_2,y_2))$, for every $(x_2,y_2)\in X_2*Y_2$ and every Borel set $E\subseteq X_1*Y_1$.
\[prop:fibred product BSM\] The fibred product $(\gamma_X * \gamma_Y)^{\bullet}$ is a locally finite system of measures on $f * g$. If $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$ are both locally finite BSMs, then so is $(\gamma_X * \gamma_Y)^{\bullet}$.
The proof is very similar to the proof of Proposition \[prop:lifting BSM\]. As a product of locally finite Borel measures, $(\gamma_X * \gamma_Y)^{(x_2,y_2)}$ is a well defined Borel measure for every $(x_2,y_2)\in X_2*Y_2$. By definition it is concentrated on $f^{-1}(x_2)\times g^{-1}(y_2)=(f*g)^{-1}(x_2,y_2)$. A calculation analogous to (\[calc:lifting\]) in Remark \[rem:elementary open sets\] gives $$\begin{aligned}
(\gamma_X * \gamma_Y)^{(x,y)}(E)=\gamma_X^{x}(A)\cdot\gamma_Y^{y}(B)\label{calc:fibred product}\end{aligned}$$ for any elementary open set of the form $E=(A\times B)\cap (X_1*Y_1)$. Therefore, using the local finiteness of $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$, we can find for any $(x_1,y_1) \in X_1*Y_1$ a neighborhood $(U_{x_1} \times U_{y_1}) \cap (X_1*Y_1)$ satisfying $\left(\gamma_X * \gamma_Y\right)^{(x_2,y_2)}\left((U_{x_1} \times U_{y_1}) \cap (X_1*Y_1)\right) = \gamma_X^{x_2}(U_{x_1})\cdot\gamma_Y^{y_2}(U_{y_1}) <\infty$ for all $(x_2,y_2) \in X_2*Y_2$. Thus $(\gamma_X * \gamma_Y)^{\bullet}$ is a locally finite system of finite measures.
Now assume that $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$ are both locally finite BSMs. We need to prove that $(\gamma_X * \gamma_Y)^{\bullet}(E)$ is a Borel function for any Borel subset $E\subseteq X_1*Y_1$, but as in Proposition \[prop:lifting BSM\] it is sufficient to prove it for any elementary open subset $E=(A\times B)\cap (X_1*Y_1)$. The rest of the proof uses the same arguments as Proposition \[prop:lifting BSM\].
In order to prove that a fibred product of CSMs is a CSM, we first need a lemma. We remind that in the CSM context, spaces are assumed to be Hausdorff and locally compact.
\[lem:fibred cartesian continuity\] Let $\psi \in C_c(X_1 \times Y_1)$. The function $\displaystyle (\xi,\eta) \mapsto \int_{X_1 \times Y_1} \psi(x,y) d\gamma_X^{\xi} d\gamma_Y^{\eta}$ is in $C_c(X_2 \times Y_2)$.
Define a function $F$ on $X_2 \times Y_1$ by $(\xi,y)\mapsto \int_{X_1} \psi(x,y)d\gamma_X^{\xi}(x)$. Using (a symmetric version of) Lemma \[lem:continuity lemma\] with $\mathcal{X} = Y_1$, $\mathcal{Y} = X_1$ and $\mathcal{Z} = X_2$, we deduce that $F \in C_c(X_2 \times Y_1)$.
Now define a function $G$ on $X_2 \times Y_2$ by $(\xi,\eta) \mapsto \int_{Y_1} F(\xi,y)d\gamma_Y^{\eta}(y)$. Again by (a symmetric version of) Lemma \[lem:continuity lemma\] with $\mathcal{X} = X_2$, $\mathcal{Y} = Y_1$ and $\mathcal{Z} = Y_2$, we deduce that $G \in C_c(X_2 \times Y_2)$. This is what we had to prove.
\[prop:fibred continuity\] If $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$ are both CSMs, then so is the fibred product $(\gamma_X * \gamma_Y)^{\bullet}$.
The proof is similar to that of Proposition \[prop:lifting CSM\]. Let $\psi \in C_c(X_1 * Y_1)$. We need to show that the function $(\xi,\eta) \mapsto \int_{X_1 * Y_1} \psi(x,y) d(\gamma_X * \gamma_Y)^{(\xi,\eta)}(x,y)$ is continuous on$X_2 * Y_2$. As argued in the proof of Proposition \[prop:lifting CSM\], by Tietze’s Extension Theorem, there exists a function $F \in C_c(X_1 \times Y_1)$ such that $F|_{X_1 * Y_1} = \psi$.
By Lemma \[lem:fibred cartesian continuity\], the map $G:(\xi,\eta) \mapsto \int_{X_1 \times Y_1} F(x,y) d\gamma_X^{\xi}(x) d\gamma_Y^{\eta}(y)$ is in $C_c(X_2 \times Y_2)$. In fact, $G|_{X_2 * Y_2} \in C_c(X_2 * Y_2)$, since $X_2 * Y_2$ is closed in $X_2 \times Y_2$.
Note that the measure $\gamma_X^{\xi}$ is concentrated on $f^{-1}(\xi)$ and the measure $\gamma_Y^{\eta}$ is concentrated on $g^{-1}(\eta)$. Hence their product is concentrated on the set of $(x,y)$ satisfying $f(x) = \xi, g(y) = \eta$. For $(\xi,\eta) \in X_2 * Y_2$ we have $p_2(\xi) = q_2(\eta)$, so $p_2(f(x)) = q_2(g(y))$. Recalling that $p_1=p_2\circ f$ and $q_1=q_2\circ g$, we get $p_1(x) = p_2(f(x)) = q_2(g(y)) = q_1(y)$, i.e. $(x,y) \in X_1 * Y_1$. We conclude that the continuous map $G|_{X_2 * Y_2}$ satisfies $$\begin{aligned}
G|_{X_2 * Y_2}(\xi,\eta) &=& \int_{X_1 \times Y_1} F(x,y) d\gamma_X^{\xi} (x) d\gamma_Y^{\eta} (y) \ = \ \int_{X_1 * Y_1} F(x,y) d\gamma_X^{\xi} (x)d\gamma_Y^{\eta}(y) \\ &=& \int_{X_1 * Y_1} \psi(x,y) d\gamma_X^{\xi}(x) d\gamma_Y^{\eta}(y) \ = \ \int_{X_1 * Y_1} \psi(x,y) d(\gamma_X * \gamma_Y)^{(\xi,\eta)}(x,y).\end{aligned}$$ This completes the proof.
\[prop:fibred stability\] The properties of being positive on open sets and locally bounded are preserved under fibred products.
The proof is very similar to its counterpart for lifting in Proposition \[prop:lifting stability\]. Assume that $\gamma_X^{\bullet}$ and $\gamma_Y^{\bullet}$ are positive on open sets. To prove that $(\gamma_X*\gamma_Y)^{\bullet}$ is positive on open sets is suffices to consider elementary open sets. Fix $(x,y)\in X_2*Y_2$ and let $E=(A\times B)\cap (X_1*Y_1)$ be an elementary open set such that $E\cap(f*g)^{-1}(x,y)\neq \emptyset$. This implies that $A\cap f^{-1}(x)\neq \emptyset$ and $B\cap g^{-1}(y)\neq \emptyset$, hence $\gamma_X^x(A)>0$ and $\gamma_Y^y(B)>0$. Using calculation (\[calc:fibred product\]) from Proposition \[prop:fibred product BSM\] we obtain $(\gamma_X * \gamma_Y)^{(x,y)}(E)=\gamma_X^x(A)\cdot\gamma_Y^y(B)>0$.
Proving that the lifted system is locally bounded is similar to the proof that it is locally finite in Proposition \[prop:composition BSM\].
Assume that we now have for $i=$1,2,3 the following three pull-back diagrams $$\xy\xymatrix{X_i*Y_i\ar[r]\ar[d]&X_i\ar[d]_{p_i}\\
Y_i\ar[r]^{q_i}&Z}\endxy$$ where the maps $p_i$ and $q_i$ are all continuous. Furthermore, assume that we have continuous connecting maps $\xymatrix{X_1\ar [rr]^{f_1}_{\gamma_1^{\bullet}}&&X_2}$, $\xymatrix{Y_1\ar [rr]^{g_1}_{\xi_1^{\bullet}}&&Y_2}$, $\xymatrix{X_2\ar [rr]^{f_2}_{\gamma_2^{\bullet}}&&X_3}$ and $\xymatrix{Y_2\ar [rr]^{g_2}_{\xi_2^{\bullet}}&&Y_3}$, all endowed with locally finite systems of measures, satisfying that $p_1=p_2\circ f_1$, $q_1=q_2\circ g_1$, $p_2=p_3\circ f_2$ and $q_2=q_3\circ g_2$. Finally, assume that $\gamma_1^{\bullet}$ and $\xi_1^{\bullet}$ are locally bounded. This data allows us to implement the fibred product construction above, giving rise to the following diagram, which is commutative as a diagram of topological spaces and continuous maps: $$\xy \xymatrix@R=.2in@C=.2in{ & X_1*Y_1\ar[rr]^{f_1 *g_1}_{(\gamma_1 *\xi_1)^{\bullet}}\ar@{-}[dd]\ar[ddl]&&X_2*Y_2\ar[rr]^{f_2*g_2}_{(\gamma_2*\xi_2)^{\bullet}}\ar@{-}[dd]\ar[ddl]&&X_3*Y_3\ar[dddd]\ar[ddl]\\%
\hspace{0in}&&\hspace{0in}&&&\hspace{0in}\\%
X_1\ar[rr]^(.65){f_1}_(.65){\gamma_1^{\bullet}}\ar[dddd]^(.3){p_1} &{\ }^{\ }\ar[dd]&X_2\ar[rr]^(.65){f_2}_(.65){\gamma_2^{\bullet}}\ar[dddd]^(.3){p_2}&{\ }^{\ } \ar[dd]&X_3\ar[dddd]^(.3){p_3}&\vspace{-2in}\\%
&&&&&\\%
&Y_1\ar@{-}[r]^(.65){g_1}_(.65){\xi_1^{\bullet}} \ar[ddl]^{q_1} &\hspace{.06in}\ar[r]&Y_2\ar@{-}[r]^(.65){g_2}_(.65){\xi_2^{\bullet}}\ar[ddl]^{q_2}&\hspace{.06in}\ar[r]&Y_3\ar[ddl]^{q_3}\\%
&&&&&\\%
Z\ar[rr]^{\scriptstyle\rm id}&&Z\ar[rr]^{\scriptstyle\rm id}&&Z&} \endxy%$$ Loosely speaking, the following proposition states that fibred products and compositions of systems of measures, commute.
\[prop:fibred products commute with compositions\] In the above setting, $$[(\gamma_2 *\xi_2)\circ (\gamma_1*\xi_1)]^{\bullet}=[(\gamma_2\circ\gamma_1)*(\xi_2\circ\xi_1)]^{\bullet}.$$
Both $[(\gamma_2 *\xi_2)\circ (\gamma_1*\xi_1)]^{\bullet}$ and $[(\gamma_2\circ\gamma_1)*(\xi_2\circ\xi_1)]^{\bullet}$ are systems of measures on the map from $X_1*Y_1$ to $X_3*Y_3$, defined by $(x_1,y_1)\mapsto (f_2(f_1(x_1)), g_2(g_1(y_1)))$. By Proposition \[prop:fibred product BSM\], $(\gamma_1*\xi_1)^{\bullet}$ and $(\gamma_2*\xi_2)^{\bullet}$ are locally finite, the former being also locally bounded by Proposition \[prop:fibred stability\]. Thus, by Lemma \[lem:condition for composition locally finite\], $[(\gamma_2 *\xi_2)\circ (\gamma_1*\xi_1)]^{\bullet}$ is a locally finite system of measures. Moreover, by Lemma \[lem:condition for composition locally finite\], $(\gamma_2\circ\gamma_1)^{\bullet}$ and $(\xi_2\circ\xi_1)^{\bullet}$ are locally finite, implying in turn that $[(\gamma_2\circ\gamma_1)*(\xi_2\circ\xi_1)]^{\bullet}$ is locally finite by Proposition \[prop:fibred product BSM\].
Fix $(x_3,y_3)\in X_3*Y_3$. For any Borel set $E \subseteq X_1*Y_1$, define $$\mu (E) = [(\gamma_2 *\xi_2)\circ (\gamma_1*\xi_1)]^{(x_3,y_3)} (E) \quad \text{and} \quad \nu(E)=[(\gamma_2\circ\gamma_1)*(\xi_2\circ\xi_1)]^{(x_3,y_3)}(E).$$ Being extracted from locally finite systems of measures, $\mu$ and $\nu$ are locally finite measures on $X_1*Y_1$.
Next, let $E=(A\times B)\cap (X_1*Y_1)$ be an elementary open set. Using the definitions of fibred products and compositions, along with Fubini’s theorem, we get $$\begin{aligned}
\mu (E) &=& [(\gamma_2 *\xi_2)\circ (\gamma_1*\xi_1)]^{(x_3,y_3)}(E)\nonumber\\
&=&\int_{X_2*Y_2}(\gamma_1*\xi_1)^{(x_2,y_2)}(E)\ d(\gamma_2*\xi_2)^{(x_3,y_3)}(x_2,y_2)\nonumber\\
&=&\int_{Y_2}\int_{X_2}(\gamma_1*\xi_1)^{(x_2,y_2)}(E)\ d\gamma_2^{x_3}(x_2)d\xi_2^{y_3}(y_2)\nonumber\\
&=&\int_{Y_2}\int_{X_2} \gamma_1^{x_2}(A)\xi_1^{y_2}(B) d\gamma_2^{x_3}(x_2)d\xi_2^{y_3}(y_2)\nonumber\\
&=&\left(\int_{X_2}\gamma_1^{x_2}(A)d\gamma_2^{x_3}(x_2)\right)\cdot \left(\int_{Y_2} \xi_1^{y_2}(B) d\xi_2^{y_3}(y_2)\right)\nonumber\\
&=&(\gamma_2\circ \gamma_1)^{x_3}(A)\cdot(\xi_2\circ\xi_1)^{y_3}(B)\nonumber\\&=&[(\gamma_2\circ \gamma_1)*(\xi_2\circ\xi_1)]^{(x_3,y_3)}(E)\nonumber\\
&=& \nu(E).\nonumber\end{aligned}$$
Finally, let $\{A_n\}_{n=1}^{\infty}$ and $\{B_m\}_{m=1}^{\infty}$ be bases for the topology of $X_1$ and $Y_1$ respectively. The collection ${\mathcal B}=\{(A_n\times B_m)\cap(X_1 *Y_1)\}_{n,m}$ is a countable basis for the topology of $X_1 *Y_1$ consisting of elementary open sets. Moreover, we have seen that $\mu$ and $\nu$ agree on finite intersections of sets in ${\mathcal B}$, since these are also of the form $E=(A\times B)\cap (X_1*Y_1)$. We can now apply lemma \[lem:mu=nu for every E Borel\] to the basis $\mathcal{B}$ and the locally finite measures $\mu$ and $\nu$, and conclude that $\mu(E)=\nu(E)$ for any Borel set $E\subseteq X_1*Y_1$. Since $(x_3,y_3)$ was arbitrary, this completes the proof.
Disintegration {#sec:disintegration}
==============
Disintegration of measures (sometimes called decomposition) has received vast attention in the literature. The purpose of presenting it here is limited to providing versions and derivatives of the fundamental result (Theorem \[thm:disintegration\], Corollary \[cor:disintegration locally finite\] and Proposition \[prop:locally bounded disintegration\]) which are consistent with our approach and terminology and suitable for our needs. This is why we chose to quote Fabec [@fabec-book], rather than probably the most original source (von Neumann [@von-Neumann]) or alternatively more generalized versions. We do refer the reader interested in tracing the theorem historically to Ramsay ([@ramsay71], page 264), which in turn cites Mackey, Halmos, and ultimately von Neumann. Throughout this section we shall assume all spaces to be second countable, locally compact and Hausdorff.
\[def:measure-class-preserving\] Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces. We will say that a Borel map $f:X \rightarrow Y$ is **measure-preserving** if $f_*\mu = \nu$. We will say that $f$ is **measure-class-preserving** if $f_*\mu \sim \nu$.
In the above definition $f_*$ is the push-forward, defined for any Borel set $F \subset Y$ by $f_* \mu(F) = \mu (f^{-1}(F))$, and $\sim$ denotes equivalence of measures in the sense of being mutually absolutely continuous.
\[def:disintegration\] Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces, and let $f:X \rightarrow Y$ be a Borel map. A system of measures $\gamma^{\bullet}$ on $f$ will be called a **disintegration** of $\mu$ with respect to $\nu$ if $\displaystyle \mu (E) = \int_Y \gamma^y (E) d\nu (y)$ for every Borel set $E \subseteq X$.
\[lem:probability disintegration implies measure preserving\] If $\gamma^{\bullet}$ is a system of probability measures on $f$ which is a disintegration of $\mu$ with respect to $\nu$, then $f$ is measure preserving.
Since $\gamma^{\bullet}$ is a system of probability measures, $\gamma^y$ is concentrated on $f^{-1}(y)$ and $\gamma^y (f^{-1}(y)) = 1$ for any $y \in Y$. Therefore, $\gamma^y (f^{-1}(F)) = \chi_{_F}(y)$ for any Borel set $F \subseteq Y$. Thus, for any Borel set $F \subseteq Y$ we have $$f_* \mu(F) \ = \ \mu (f^{-1}(F)) \ = \ \int_Y \gamma^y (f^{-1}(F)) d\nu (y) \ = \ \int_Y \chi_{_F}(y) d\nu (y) \ = \ \nu (F),$$ so $f$ is measure preserving.
\[lem:disintegration implies measure class preserving\] Let $\gamma^{\bullet}$ be a system of measures on $f$ which is positive on open sets. If $\gamma^{\bullet}$ is a disintegration of $\mu$ with respect to $\nu$, then $f$ is measure-class-preserving.
Let $F \subseteq Y$ be a Borel set. For any $y\in Y$, we have $\gamma^y (f^{-1}(F))=\chi_{_F}(y) \cdot \gamma^y(f^{-1}(y)) = \chi_{_F}(y) \cdot \gamma^y(X)$. Therefore $$f_* \mu(F) \ = \ \mu (f^{-1}(F)) \ = \ \int_Y \gamma^y (f^{-1}(F)) d\nu (y) \ = \ \int_Y \chi_{_F}(y)\gamma^{y}(X) d\nu (y) \ = \ \int_F \gamma^{y}(X) d\nu (y).$$ This shows that $f_* \mu$ is absolutely continuous with respect to $\nu$. Moreover, since $\gamma^{\bullet}$ is positive on open sets, $\gamma^{\bullet}(X)$ is a positive function on $Y$, and thus $f_* \mu$ is equivalent to $\nu$. We conclude that $f$ is measure-class-preserving.
The converse to the previous lemmas is less trivial. The following theorem is a restatement of Theroem I.27 in [@fabec-book]. The original theroem requires $X$ to be a standard Borel space, which is a Polish space (i.e. a second countable topological space admitting a complete metric that generates the topology), together with its Borel $\sigma$-algebra. However, recall that our spaces are assumed to be locally compact, Hausdorff and second countable, hence they are standard Borel spaces. We refer the reader to a paper by Ramsay [@ramsay] for a discussion of these facts.
\[thm:disintegration\] Let $(X,\mu)$ and $(Y,\nu)$ be spaces equipped with $\sigma$-finite measures, and let $f:X \rightarrow Y$ be a measure-class-preserving Borel map. Then there exists a BSM $\gamma^{\bullet}$ on $f$ which is a disintegration of $\mu$ with respect to $\nu$. Moreover, if $\gamma_1^{\bullet},\gamma_2^{\bullet}$ are two disintegrations, then $\gamma_1^{y}=\gamma_2^{y}$ for $\nu$-almost every $y\in Y$.
\[cor:disintegration locally finite\] Let $(X,\mu)$ and $(Y,\nu)$ be spaces equipped with $\sigma$-finite measures, and let $f:X \rightarrow Y$ be a measure-class-preserving Borel map. If $\mu$ is a *locally finite* measure, then there exists a *locally finite* BSM $\alpha^{\bullet}$ on $f$ which is a disintegration of $\mu$ with respect to $\nu$. Moreover, if $\alpha_1^{\bullet},\alpha_2^{\bullet}$ are two disintegrations, then $\alpha_1^{y}=\alpha_2^{y}$ for $\nu$-almost every $y\in Y$.
By Theorem \[thm:disintegration\], there exists a BSM $\gamma^{\bullet}$ on $f$ which is a disintegration of $\mu$ with respect to $\nu$, and it is unique $\nu$-almost everywhere in $Y$. Let ${\mathcal B}=\{B_n\}_{n=1}^{\infty}$ be a countable basis for the topology of $X$. Since $\mu$ is locally finite, it is straightforward to verify that the sub-collection $\{B \in \mathcal{B} ~|~ \mu(B)<\infty \}$ is itself a basis for $X$. Therefore, we can assume that all $B_n \in \mathcal{B}$ satisfy $\mu(B_n)=\int_Y\gamma^y(B_n)d\nu(y)<\infty$. It follows that $\gamma^y(B_n)<\infty$ for $\nu$-almost all $y\in Y$.
Consider the Borel sets $Y_n= \{ y\in Y ~|~ \gamma^y(B_n)= \infty \}$. By our previous argument, the sets $Y_n$ all have $\nu$-measure zero, hence so does $\bigcup_{n=1}^{\infty}Y_n$. We denote $Y' = Y\setminus \bigcup_{n=1}^{\infty}Y_n$ and define a new BSM $\alpha^{\bullet}$ on $f$ by $$\alpha^y(E)= \begin{cases} \gamma^y(E)& y \in Y' \\ 0 & y \notin Y' \end{cases}$$ for any Borel set $E \subseteq X$. It is easy to verify that $\alpha^{\bullet}$ is indeed a BSM on $f$. Moreover, since $\alpha^{y}(E)=\gamma^y(E)$ for $\nu$-almost all $y\in Y$, it follows that $\alpha^{\bullet}$ is also a disintegration of $\mu$ with respect to $\nu$, and the uniqueness $\nu$-almost everywhere in $Y$ holds for $\alpha^{\bullet}$.
It remains to show that $\alpha^{\bullet}$ is locally finite. For any $x\in X$, let $B_n \in {\mathcal B}$ be a neighborhood of $x$. Since $$\alpha^y(B_n)= \begin{cases} \gamma^y(B_n)& y \in Y' \\ 0 & y \notin Y' \end{cases}$$ it follows that $\alpha^y(B_n)<\infty$ for all $y\in Y$. Thus $\alpha^{\bullet}$ is locally finite, and the proof is complete.
The next lemma, which is rather elementary, is required for the proof of Proposition \[prop:locally bounded disintegration\] below. Lacking a formal reference, we include the proof, which is adapted from lecture notes found on the homepage of Gabriel Nagy.
\[lem:easy Radon-Nikodym\] Let $\mu,\nu$ be finite measures on a measurable space $(Y, \Sigma)$. Then the Radon-Nikodym derivative $h=d\mu / d\nu$ exists and belongs to $L^{\infty}(Y,\nu)$ if and only if there is a constant $C\geq 0$ such that $\mu(E)\leq C\cdot \nu(E)$ for all $E \in \Sigma$.
Suppose that the Radon-Nikodym derivative $h\!=\!d\mu / d\nu$ exists and is in $L^{\infty}(Y,\nu)$. Then for all $E \in \Sigma$ we have $\mu(E)=\int_E h d\nu \leq \|h\|_{\infty}\cdot \nu(E).$
Conversely, assume that there is a constant $C$ such that $\mu(E)\leq C\cdot \nu(E)$ for all $E \in \Sigma$. A standard argument using simple functions and the Monotone Convergence Theorem, yields $\int_Y f d\mu \leq C\cdot \int_Y f d\nu$, for any measurable function $f:Y\rightarrow [0,\infty]$. It follows that the identity map $f \mapsto f$ is a continuous function $L^1(Y,\nu)\rightarrow L^1(Y,\mu).$ Moreover, we have a composition of continuous linear functions $L^2(Y,\nu) \hookrightarrow L^1(Y,\nu) \rightarrow L^1(Y,\mu) \rightarrow \mathbb{R}$ given by $f \mapsto f \mapsto f \mapsto \int_Y f\ d\mu$, which gives rise to a continuous linear functional on $L^2(Y,\nu)$. Since $L^2(Y,\nu)$ is a Hilbert space, there exists a function $h \in L^2(Y,\nu)$ such that $\int_Y fh d\nu = \int_Y f d\mu$ for any $f \in L^2(Y,\nu)$. Setting $f=\chi_{_E}$ we get $\mu(E) = \int_E h d\nu$ for any $E\in \Sigma$, hence $h = d\mu / d\nu$. Denote $A_n=\left\{y\in Y: h(y)\geq C+\frac{1}{n}\right\}$. Then $\mu(A_n)= \int_{A_n} h d\nu \geq \left(C+\frac{1}{n}\right)\nu(A_n) \geq \left(1+\frac{1}{nC}\right)\mu(A_n)$, from which it follows that $0 \geq \frac{1}{nC}\mu(A_n)$, so $\mu(A_n)=0$. By the above inequality, this implies that $\nu(A_n)=0$. Therefore $A=\bigcup_{n=1}^{\infty}A_n=\left\{y\in Y: h(y)> C\right\}$ also satisfies $\nu(A)=0$. Thus $\|h\|_{\infty}\leq C$ and in particular $h \in L^{\infty}(Y,\nu)$.
The following proposition provides a useful criterion for the existence of a disintegration which is locally bounded. Note that it requires the map $f$ to be continuous.
\[prop:locally bounded disintegration\] Let $(X,\mu)$ and $(Y,\nu)$ be spaces equipped with locally finite measures and let $f:X\rightarrow Y$ be a measure class preserving continuous map. The map $f$ admits a disintegration $\alpha^{\bullet}$ which is locally bounded if and only if for any compact set $K\subseteq X$ there exists a constant $C_{_K}$ such that for all Borel sets $E\subseteq Y$, $$\mu(K\cap f^{-1}(E))\leq C_{_K}\cdot \nu(E).$$
Recall that our spaces are always assumed to be locally compact, Hausdorff and second countable, and as such, every locally finite measure is $\sigma$-finite. By Corollary \[cor:disintegration locally finite\], $f$ admits a disintegration $\alpha^{\bullet}$ of $\mu$ with respect to $\nu$, which is unique $\nu$-almost everywhere in $Y$. Note that the system $\alpha^{\bullet}$ can be taken to be locally bounded, or equivalently bounded on compact sets, if and only if for any compact $K \subseteq X$, $\alpha^{\bullet}(K)$ is in $L^{\infty}(Y,\nu)$, i.e. essentially bounded.
For every compact set $K\subseteq X$, consider the measure $\mu_{_K}$ on $Y$ defined by $\mu_{_K}(E):=\mu(K\cap f^{-1}(E)),$ for all Borel sets $E\subseteq Y$. The measure $\mu_{_K}$ is finite since $\mu$ is locally finite, and moreover, since $f$ is measure class preserving, $\mu_{_K}$ is absolutely continuous with respect to $\nu$. Let $h_{_K} = d\mu_{_K}/d\nu$ denote the Radon-Nikodym derivative. Thus, for any Borel subset $E\subseteq Y$, we have $$\mu(K\cap f^{-1}(E))=\mu_{_K}(E)=\int_E h_{_K}(y)\ d\nu(y).$$ On the other hand, $$\mu(K\cap f^{-1}(E))=\int_Y \alpha^y(K\cap f^{-1}(E))\ d\nu(y)=\int_E \alpha^y(K)\ d\nu(y).$$ Therefore, $\int_E h_{_K}\ d\nu=\int_E \alpha^{\bullet}(K)\ d\nu$ for any $E$, hence $h_{_K}=\alpha^{\bullet}(K)$, $\nu$-almost everywhere in $Y$.
Let $\nu_{_K}$ be another measure on $Y$, defined by $\nu_{_K}(E)=\nu(E\cap f(K)).$ The measure $\nu_{_K}$ is finite, since $K$ is compact, $f$ is continuous and $\nu$ is locally finite. Moreover, $\mu_{_K}$ is absolutely continuous with respect to $\nu_{_K}$: $$\begin{aligned}
\nu_{_K}(E) \!=\!0 & \Rightarrow & \nu(E\cap f(K)) \!=\!0 \ \ \Rightarrow \ \ f_*\mu(E\cap f(K)) \!=\!0 \ \ \Rightarrow \ \ \mu(f^{-1}(E\cap f(K))) \!=\!0 \\
& \Rightarrow & \mu(f^{-1}(E) \cap f^{-1}(f(K))) \!=\!0 \ \ \Rightarrow \ \ \mu(f^{-1}(E) \cap K) \!=\!0 \ \ \Rightarrow \ \ \mu_{_K}(E) \!=\!0.\end{aligned}$$ The Radon-Nikodym derivative $d\mu_{_K} / d\nu_{_K}$ is equal $\nu_{_K}$-almost everywhere to $h_{_K}$, since $$\begin{aligned}
\mu_{_K}(E) &=& \mu(K\cap f^{-1}(E)) \ = \ \mu(K\cap f^{-1}(E)\cap f^{-1}(f(K))) \ = \ \mu(K\cap f^{-1}(E\cap f(K))) \\
&=& \int_{E\cap f(K)} h_{_K}(y)\ d\nu(y) \ = \ \int_E h_{_K}(y)\ d\nu_{_K}(y).\end{aligned}$$ In particular, $h_{_K} \!=\!0$ $\nu$-almost everywhere outside $f(K)$, since $\int_{E\cap f(K)} h_{_K}(y)\ d\nu(y) = \mu_{_K}(E) = \int_E h_{_K}(y)\ d\nu(y)$ for any Borel subset $E\subseteq Y$. It follows that $h_{_K} \in L^{\infty}(Y,\nu) \Leftrightarrow h_{_K} \in L^{\infty}(Y,\nu_{_K}).$
We now apply Lemma \[lem:easy Radon-Nikodym\] to the finite measures $\mu_{_K}$ and $\nu_{_K}$: The Radon-Nikodym derivative $d\mu_{_K} / d\nu_{_K} \in L^{\infty}(Y,\nu_{_K})$ if and only if there is a constant $C\!=\!C_{_K}$ such that $\mu_{_K}(E) \leq C_{_K} \cdot \nu_{_K}(E)$ for all $E\subseteq Y$ Borel. Equivalently: $h_{_K} \in L^{\infty}(Y,\nu)$ if and only if there is a constant $C_{_K}$ such that $\mu(K\cap f^{-1}(E))\leq C_K\cdot \nu(E\cap f(K))$. Observe that the condition $\mu(K\cap f^{-1}(E))\leq C_K\cdot \nu(E\cap f(K)), \forall E$ is equivalent to the condition $\mu(K\cap f^{-1}(E))\leq C_K\cdot \nu(E), \forall E$. Indeed, the latter implies the former by taking $E \cap f(K)$. Recalling that $h_{_K}=\alpha^{\bullet}(K)$ $\nu$-almost everywhere in $Y$, we conclude that $\alpha^{\bullet}(K) \in L^{\infty}(Y,\nu)$ if and only if there is a constant $C_{_K}$ such that $\mu(K\cap f^{-1}(E))\leq C_K\cdot \nu(E), \forall E$. This completes the proof.
Systems of measures for groupoids {#sec:groupoids}
=================================
Terminology in the groupoid literature is often a source for confusion. In this section we give a definition of Haar systems using the terminology we have adopted above, and show that it coincides with the standard definitions.
\[def:system of measures on G\] Let $G$ be a topological groupoid. A system of measures $\lambda^{\bullet}$ on the range map $r:G \rightarrow G^{(0)}$ is said to be a **system of measures on $G$**.
\[def:left\_invariant\] A system of measures $\lambda^{\bullet}$ on $G$ is called **left invariant** if for every $x \in G$ and for every Borel subset $E \subseteq G$, $$\lambda^{d(x)} (E) = \lambda^{r(x)}\left(x \cdot (E \cap G^{d(x)})\right).$$
\[lem:equivalent left invariance\] A system of measures $\lambda^{\bullet}$ on $G$ is left invariant if and only if for any $x \in G$ and every non-negative Borel function $f$ on $G$, $$\int f(xy)d\lambda^{d(x)}(y) = \int f(y)d\lambda^{r(x)}(y).$$
Assume $\lambda^{\bullet}$ is left invariant. Fix $x \in G$, and note that $ y \in x \cdot (E \cap G^{d(x)})
\Leftrightarrow x^{-1}y \in E \cap G^{d(x)}$. Therefore, for any Borel set $E$, $$\begin{aligned}
\int_G \chi_{_E}(y) d\lambda^{d(x)}(y) &=& \lambda^{d(x)}(E) \ = \ \lambda^{r(x)}\left(x \cdot (E \cap G^{d(x)})\right) \ = \ \int_G \chi_{_{x \cdot (E \cap G^{d(x)})}} (y) d\lambda^{r(x)}(y) \\ &=& \int_G \chi_{_{E \cap G^{d(x)}}} (x^{-1}y) d\lambda^{r(x)}(y) \ = \ \int_G \chi_{_E} (x^{-1}y) d\lambda^{r(x)}(y).\end{aligned}$$ Replacing $x$ with $x^{-1}$ we get $\int_G \chi_{_E}(y) d\lambda^{r(x)}(y) \ = \ \int_G \chi_{_E} (xy) d\lambda^{d(x)}(y)$. Passing, as usual, from characteristic functions to any non-negative Borel function, we obtain that for any $x \in G$ and for every non-negative Borel function $f$, $\int_G f(y) d\lambda^{r(x)}(y) \ = \ \int_G f (xy) d\lambda^{d(x)}(y)$ as claimed.
The converse is obtained by reversing the arguments.
\[lem:CSM is left invariant for any f in Cc\] A CSM $\lambda^{\bullet}$ on $G$ is left invariant if and only if for any $x \in G$ and every function $f\in C_c(G)$, $$\int f(xy)d\lambda^{d(x)}(y) = \int f(y)d\lambda^{r(x)}(y).$$
Assume first that $\lambda^{\bullet}$ is a left invariant CSM on $G$. By Proposition \[prop:CSM is BSM\], $\lambda^{\bullet}$ is a BSM, and by Lemma \[lem:equivalent left invariance\] we have that for any $x \in G$ and every non-negative Borel function $f$ on $G$, $\int f(xy)d\lambda^{d(x)}(y) = \int f(y)d\lambda^{r(x)}(y).$ In particular this holds for any non-negative $f\in C_c(G)$. The usual decomposition of a general complex-valued $f\in C_c(G)$ as $f=f_1+if_2$, and further as $f_k = (f_k)_+ -(f_k)_-$, yields the property for any $f\in C_c(G)$.
Conversely, if a CSM satisfies $\int f(xy)d\lambda^{d(x)}(y) = \int f(y)d\lambda^{r(x)}(y)$ for any $f\in C_c(G)$, then in particular the property holds for non-negative $f\in C_c(G)$. By approximating characteristic functions of open sets by continuous “bump" functions and using a standard Monotone Convergence Theorem argument, we obtain that $\int \chi_{_A}(xy)d\lambda^{d(x)}(y) = \int \chi_{_A}(y)d\lambda^{r(x)}(y)$ for any open subset $A \subseteq X$ and any $x \in G$. By means of a calculation similar to that of Lemma \[lem:equivalent left invariance\], we deduce that $\lambda^{d(x)} (A) = \lambda^{r(x)}\left(x \cdot (A \cap G^{d(x)})\right)$ for any open subset $A \subseteq X$ and any $x \in G$. Finally, we denote $\mu_x(A) = \lambda^{d(x)} (A)$ and $\nu_x(A) = \lambda^{r(x)}\left(x \cdot (A \cap G^{d(x)})\right)$, and apply Corollary \[cor:mu=nu for every A open\] to the locally finite measures $\mu_x$ and $\nu_x$. We conclude that $\mu_x(E) = \nu_x(E)$ for any Borel subset $E \subseteq X$. This holds for any $x \in G$, which implies that $\lambda^{\bullet}$ is left invariant.
\[def:Haar system\] A continuous left **Haar system** for $G$ is a system of measures $\lambda^{\bullet}$ on $G$ which is continuous, left invariant and positive on open sets.
We should point out that in the groupoid literature, the definition of a continuous left Haar system for $G$ appears different than ours at first glance. Modulo minor discrepancies between various sources (see for example standard references such as [@muhly-book-unpublished], [@paterson-book], [@renault-book] and [@renault-anantharaman-delaroche]), it is usually defined to be a family $\lambda= \{\lambda^u : u \in G^{(0)} \}$ of positive (Radon) measures on $G$ satisfying the following properties:
1. $supp(\lambda^u) = G^u$ for every $u \in G^{(0)}$;
2. (continuity) for any $f \in C_c(G)$, the function $u \mapsto \int f d\lambda^u$ on $G^{(0)}$ is in $C_c(G^{(0)})$;
3. (left-invariance) for any $x \in G$ and $f \in C_c(G)$, $$\int f(xy)d\lambda^{d(x)}(y) = \int f(y)d\lambda^{r(x)}(y).$$
However, by Lemma \[lem:positive iff full support\], Corollary \[cor:CSM has compact support\] and Lemma \[lem:CSM is left invariant for any f in Cc\], the above definition is equivalent to our Definition \[def:Haar system\].
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank John Baez, Jean Renault, Baruch Solel, Christopher Walker and most of all Paul Muhly, for enlightening conversations and useful suggestions.
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Arlan B. Ramsay, *Virtual Groups and Group Actions*, Advances in Mathematics 6 (1971) pp. 253–322.
Arlan B. Ramsay, *Polish groupoids*, in *Descriptive set theory and dynamical systems*, London Math. Soc. Lecture Note Series, volume 277, pp. 259–271, Cambridge University Press, Cambridge, 2000.
Jean Renault, *The ideal structure of groupoid crossed product [$C\sp{\ast} $]{}-algebras*, Journal of Operator Theory 25 (1991) pp. 3–36.
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D. Revuz, *Markov Chains*, $2^{nd}$ edition, North Holland, Amsterdam, 1984.
|
---
author:
- |
Mariana Orellana, Lucas. A. Cieza, Matthias R. Schreiber, Bruno Merín, Joanna M. Brown,\
Leonardo J. Pellizza, Gisela A. Romero
date: 'Accepted 22-11-2011'
title: 'Transition disks: 4 candidates for ongoing giant planet formation in Ophiuchus'
---
Introduction {#intro}
============
The infrared spectral energy distributions (SEDs) of circumstellar transitional disks reveal the presence of an optically thin inner region and an optically thick outer disk. Several mechanisms relevant to the overall evolution of circumstellar disks, and in particular to the short-lived phase when they dissipate, have been proposed to explain the so-called opacity holes of transition disks: giant planet formation, grain growth, photoevaporation, and tidal truncation in close binaries. See Williams & Cieza, 2011 for a recent review. The processes responsible for the inner holes of transition disks can tentatively be distinguished when disk masses, accretion rates, and multiplicity information are available (Najita et al. 2007; Cieza 2008). Following this approach, Cieza et al. (2010, hereafter Paper I) presented the initial results of an ongoing project to characterize a large set of transition disk candidates in nearby star-forming regions.\
Probing the structure of disks that are suspected to be forming planets is the most promising approach to understand the conditions in which planets are formed. The best indication for ongoing planet formation in disks is the detection of tidal gaps (e.g. Piétu et al. 2006) corresponding to a ring with significant decrease in the surface density (or the whole inner disk if it was depleted by accretion, e.g. Varneiére et al 2006). A spectacular confirmation has arrived with the recent detection of the first potential substellar object within the gap of the transitional disk T Chamaleontis (Huélamo et al. 2011). Inner holes and gaps have already been observed at (sub)millimeter wavelengths in a handful of objects bright enough for resolving disk structure (Piétu et al. 2006; Hughes et al. 2007, 2009; Brown et al. 2008, 2009; Andrews et al. 2009, 2010a, 2011, Isella et al 2010a, 2010b).\
Comprehensive studies of similar transition disks are necessary to increase the empirical constraints on their structures and investigate their diversity. This is the primary motivation of this work, where we apply a parametric description to the 4 planet-forming disks candidates in Ophiuchus presented in Paper I and derive the best fitting physical characteristics for their known SEDs.
The targets
===========
In Paper I, the observed SEDs were characterized by two parameters, as introduced by Cieza et al. (2007): the longest wavelength at which the observed flux is dominated by the stellar photosphere, $\lambda_{\rm turn-off}$, and the slope of the IR excess, $\alpha_{\rm excess}$, computed between $\lambda_{\rm turn-off}$ and 24 $\mu$m. The former parameter correlates with the size of the inner hole, and the latter with the sharpness of the edge of the hole, i.e., a large increase in the dust density over a small range in radii is indicated by a positive $\alpha_{\rm excess}$. Accreting disks with sharp inner holes that seem to lack stellar companions (e.g., from adaptive optics imaging) stand as the most promising candidates for ongoing (giant) planet formation. There were 4 cases fulfilling these criteria in Paper I from a sample of 26 of *Spitzer*-selected transitional disks in the Ophiuchus molecular cloud ($d\sim 125$ pc, Loinard et al. 2008). Their basic properties are listed in Table \[properties\]. The mass accretion rates for our targets, inferred from the H$\alpha$ emission line widths (Natta et al. 2004), range from 1.3$\times$10$^{-10}$ to 5$\times$10$^{-8}$ M$_\odot/$yr.
[lcccccccccccc]{} & [*Spitzer* ID]{} & [SpT]{} & [$T_{\rm eff}$]{} & [$L_\star$]{} & [$R_\star$]{}& [$M_\star^b$]{} & [Age]{} & [A$_V$]{} & [$\log\dot{M}_{\rm acc}^c$]{} & [$\lambda_{\rm turn-off}$]{} & [$\alpha_{\rm excess}$]{}& [$M_{\rm disk}^d$]{}\
& [SSTc2d\_]{} & & [(K)]{} & [($L_{\odot}$)]{} & [($R_{\odot}$)]{}& [($M_{\odot}$)]{} & [(Myr)]{} & [(mag)]{} & [($M_{\odot}$/yr)]{} & [($\mu$m)]{} & & [($M_{\rm JUP}$)]{}\
Tran 11 & J162506.9-235050 & M3 & 3470 & 0.24 & 1.25 & 0.3 & 2.1 & 3.8 & -8.8 & 8.0 & 0.65 & $<$1.5\
Tran 21 & J162854.1-244744 & M2 & 3580 & 0.51 & 1.74 & 0.4 & 1.4 & 5.4 & -9.3 & 8.0 & 0.69 & 1.3\
Tran 31 & J163205.5-250236 & M2 & 3580 & 0.19 & 1.08 & 0.4 & 4.1 & 5.0 & -7.3 & 8.0 & 0.30 & $<$1.3\
Tran 32 & J163355.6-244205 & K7 & 4000 & 0.78 & 1.70 & 0.7 & 2.0 & 5.0 & -9.9 & 8.0 & 0.72 & 11.1\
\[properties\]
For all of them we have at hand the photometric fluxes in the $R$-band from the USNO-B1 catalog, the $J$, $H$, and $K_S$ bands, from the Two Micron All Sky Survey (2MASS), as well as fluxes at 3.6, 4.5, 5.8, 8.0, 24, and 70 $\mu$m obtained by *Spitzer*[^1]. We have assumed errors that are within observational standards ($\la 20 \%$) . In addition, all of our targets have been observed with the Submillimeter Array (SMA). For two of them (Tran 21 and 32) we obtained the fluxes at 1.3 mm, while for Tran 11 and 31 we could only derive upper limits (see Paper I for details on SED data). For Tran 32, $F_{850\,\mu{\rm m}}$ was given in Nutter et al. (2006). One of our targets, Tran 11, was recently included in the study of [*cold*]{} disks (disks with large inner dust holes) performed by Merín et al (2010), i.e. their source \# 24. It was modeled including a positive detection from MIPS photometry: $F_{70 \mu{\rm m}}= 537\pm78.8$ mJy, and the measured *Spitzer*-IRS spectrum, but no constraints of the flux at (sub)-millimeter wavelengths. For the sake of completeness, we have redone the fit to the SED of this source with our set of free parameters and including the 1.3 mm flux upper limit.
Radiation transfer models
=========================
The modeling that we have applied is similar to those performed by Andrews et al. (2009, 2010a) and Brown et al. (2009), who have confirmed their physical estimates from SED modeling through direct imaging. A 2–D structure model for flared disks is combined with the Monte Carlo continuum radiative transfer package RADMC v3.1 (Dullemond & Dominik, 2004), modified to include a density reduction as an inner cavity. The code computes a temperature structure consistent with the given density profile, and in equilibrium with the irradiation by the central star. The disk is presumed to be passive, an assumption that is supported by the low disk to stellar luminosity ratios of our sample, $\la$0.005 according to estimates in Paper I.\
We consider a surface density profile characterized by a power-law, $\Sigma \propto R^{-\gamma}$, with an exponential taper at larger radii ($\propto e^{-(R/R_c)^{2-\gamma}}$), where $R_c$ is the characteristic radius. This is physically motivated by the success of similarity solutions of viscous disks to reproduce the observed gradual density decay at large radii (Hughes et al. 2008). $\Sigma$ is normalized to obtain the total mass of the disk, $M_d$, when integrated. The radial index was fixed to be $\gamma=1$ which is a typical value within the range $\gamma = 0.4 - 1.1$ established by Andrews et al (2010a). Our option could be questioned in the light of results by Isella et al. (2009) who have independently inferred slopes from steep to quite shallow in their sample of spatially resolved disks.\
Resolved images are therefore mandatory to obtain more accurate estimates of $\gamma$ for our particular targets. We set the characteristic radius $R_c= 100$ AU. However, there is no spatially resolved information in the SEDs alone, and the data can be reproduced equally well with a wide range of outer disk values (Andrews et al. 2010b). The value we choose is representative of the disks with resolved interferometric visibilities, that are $R_c=14-198$ AU (Andrews et al. 2009, 2010b) in Ophiuchus, and $R_c\simeq 30-230$ AU for Taurus-Auriga (Isella, Carpenter & Sargent, 2009). Note that larger outer radii (100 - 1100 AU) have been obtained with different fitting techniques (sharply truncated power law fits to CO observations) and are not directly comparable with the $R_c$ values (see Williams & Cieza 2011). Aside from the extreme case of a nearly edge-on viewing angle, the disk inclination cannot be determined from unresolved observations. Scattered light images have proben useful in this sense (Pinte et al. 2007). We have set an intermediate representative inclination $i=30^\circ$ in our modeling.\
There is strong observational evidence that circumstellar dust can present some degree of settling to the midplane (Furlan et al. 2006, McClure et al. 2010), which is in agreement with theoretical predictions (e.g., Dominik et al. 2007 and references). Indeed, the growth of the dust grains, which proceeds in a complex fashion, can be accelerated by dust settling, and it is expected that the larger particles are aggregated close to the equatorial plane. The dust can therefore be distributed differently than the gas, which is expected to remain in hydrostatic equilibrium (Chiang & Goldreich, 1997). This is described by a vertical Gaussian density profile with scale-height $H$ and radial index $\psi$, i.e. $H\propto R^{1+\psi}$, anchored at $R=100$ AU by $H_{100}$. Vertically extended disks (with larger $H$) are heated more efficiently and thus reemit more radiation at mid- and far-IR wavelengths. Here we have fixed $\psi=0.2$ in order to reduce the number of free parameters of the models.\
Following Andrews et al. (2009), we consider a radius, $R_{\rm cav}$, such that the modified surface density is $\Sigma^{\prime} =
\delta_{\rm cav} \Sigma$ when $R \le R_{\rm cav}$, and $\delta_{\rm cav} < 1$ artificially reduces the density inside the cavity, in order to mimic the deficit in the disk emission. The whole disk extends down to the distance, $R_{in}$, where the temperature is enough to sublimate the dust grains ($\sim 1500$ K). Our 4 targets present a small or negligible NIR excess above the photospheric value (note the $\lambda_{\rm
turn-off}$ values in Table \[properties\]), suggesting cavities that are largely evacuated of dust, therefore we have varied $\log(\delta_{\rm cav})$ as a sensitive parameter to be established by the data.\
We have adopted the opacity spectrum given by Andrews et al. (2009) who considered the silicate and graphite abundances determined for the ISM, and updated optical properties. The grain size distribution, $n(a) \propto a^{-3.5}$, extends in diameters from $a_{{\rm min}} = 0.005$$\mu$m up to $a_{\rm max}$=1 mm, and the opacities were calculated with a simple code for Mie scattering.
The stellar properties given in Table \[properties\] are in each case fixed inputs for the models, and Kurucz spectra are used as models of the central stars. As the disk SED is not influenced by the $\lambda\la 3\, \mu$m fluxes, they were not included in the final fit, avoiding a systematic error term (shift in the $\chi^2$). For each disk, we have explored the parameter space, $\{ M_{\rm disk}, \, H_{100}, \, \, R_{\rm cav}, \, \log \delta_{\rm cav}\}$, by generation of Monte Carlo Markov Chains, which is a parameter space exploration technique designed to provide the best fitting values and their uncertainties. This technique attains a better performance than classical data fitting methods (e.g., Press et al. 1992; Gregory 2005) for problems with large numbers of parameters, which is our case. In the fitting procedure we have followed Ford (2005) and Gregory (2005). The marginal probability distribution for each parameter where derived from the MCMC chains. The 90% percentile of such distributions[^2] simmetrically centered in the mode when possible (or using the larger side if not) was used to define the error estimates within a conservative approach.
Results
=======
The estimated parameters that best reproduce the data for our planet forming disk candidates are listed in Table \[resul\], while the corresponding SEDs are shown in Figure \[own\]. These fits have reduced $\bar{\chi}^2 = 0.9 - 10$, i.e. the $\chi^2$ considering only the observational data point errors, and divided by the number of degrees of freedom. The disk masses obtained are within a factor $\sim$2 than the rough estimates from Paper I, which were based on a single (sub)millimeter flux. We find $H_{100}$ $\lesssim$ 2 – 6 AU in all four cases, with estimated uncertainties i.e. $\la 1.5$ AU. The comparison between the dust scale heights to the scale heights of the gas, which is in hydrostatic equilibrium, give a settling ratio 0.13–0.25. Specifically for Tran 32 we get a rather flat geometry, with the smallest $H_{100}$ in our sample. This disk presents also the largest cavity and is bright enough to be detected by SMA extended configuration. A simultaneous SED + image fit will be presented in a future paper.
To test our results against an alternative model, we have used the precomputed grid of SEDs by Robitaille et al. (2007). A cavity is also inferred in all 4 cases, with $R_{\rm cav}\sim 4-38$ AU, as well as some dust settling leading to flat geometries ($H_{100}\la 4$ AU). However, the on-line fitting tool fails to accurately reproduce the SED for 2 of our targets, Tran 11 and 32. A discussion on the preferential use of RADMC to model transitional disks and comments on alternative observational characterization of them can be found in Merín et al (2010).
[ccccc]{} & [$M_{disk} ^{a}$]{} & [$H_{100}^{b}$]{}& [$R_{\rm cav}$]{} &[$\log \delta_{\rm cav}$]{}\
& [($M_{JUP})$]{} & [(AU)]{}& [(AU)]{}& [(dex)]{}\
11& $<\!$ 2.0 $\pm$ 0.7 & $<\!$ 4.2 $\pm$ 1.3 & 4.8 $\pm$ 2.5 & $\la$ -6.2\
21& 0.6 $\pm$ 0.2 & 3.1 $\pm$ 0.8 & 1.9 $\pm$ 0.3 & -4.9 $\pm$ 0.8\
31& $<\!$ 1.7 $\pm$ 0.1 & $<\!$ 1.9 $\pm$ 0.1 & 1.5 $\pm$ 0.4 & -5.2 $\pm$ 0.5\
32& 17.0 $\pm$ 5.3 & 2.0 $\pm$ 0.5 & 7.9 $\pm$ 2.3 & $\la$ -6.3\
\[resul\]
![Best fit SEDs (solid lines) obtained using RADMC, in the context of a flared disk model with an inner cavity. The dashed line shows the input stellar model atmosphere. The filled circles are extinction corrected values, while the arrows are upper limits.[]{data-label="own"}](f1.ps){width="0.75\linewidth"}
Discussion
==========
Extensive work in non-linear model fitting have been applied in many areas, and recently to disk modeling by Isella et al. (2009). They have used the two layer approximation of Chiang & Goldreich (1997), whereas we compute the thermal structures with significant detail using the RADMC code. In this study we have followed the literature by Andrews et al. (same parametric model and opacities) who have performed refined grid searches over 8 parameters. Their acute comments on the technical obstacles (Andrews et al. 2011) and non-uniqueness of the fit, also apply for our approach.
We have estimated some of the physical parameters describing the dust structures of 4 transition disk systems that are excellent candidates for ongoing giant planet formation. The cavities sizes inferred for our targets, $R_{\rm cav}=2-8$ AU, are in agreement with the distribution of semi-major axis of exoplanets, which has a bi-modal behavior, peaking at $\sim 0.05$ and $2$ AU and extending up to $\sim 10$ AU (exoplanets database, [http://exoplanets.org]{}). The mass of the disks and the actual size of their inner holes depend on the assumed opacities. The small scale height of the dust in all our targets suggest this could be a characteristic property of planet forming disks. This is an intriguing results that deserved to be investigated with further modeling and follow up observations.
An effort was done here ($\sim 10^4$ model runs for disk) to provide the set of best fitting parameters and their confidence regions through the application of Bayesian methods, but the real weight of the results could be yet unclear given the degeneracy of the model into some of the parameters we have fixed. An illustration is the case of Tran 11, for which a cavity radius $R_{\rm cav}=3\pm 2$ AU, and a settling parameter that translates into $\psi=0.07$ have been found by Merín et al. (2010) when setting $R_c=200$ AU. We used $R_c=100$ AU, $\psi=0.2$ and obtained $R_{\rm cav}=4.8\pm 2.5$ AU.
The disk masses obtained here cover a rather wide range of masses, from $\sim 0.6 M_{\rm JUP}$ to $\sim 17 M_{\rm JUP}$ that extends to lower values the mass range obtained by Andrews et al., i.e. $5-42 M_{\rm JUP}$ for systems that have been modeled in the same way, i.e. with same parameterizations and opacity tables. However, Andrews et al. (2009, 2010a) fitted their systems with larger values of the depletion factor, $\delta_{\rm cav}\sim 10^{-4}-10^{-2}$, and obtain larger cavities, $R_{\rm cav}=20-40$AU. We note that in Andrews et al. (2011) where the model include a more complex surface density profile (i.e. with an inner disk inside the cavity) the range of masses estimated for 12 disks with resolved images is $\sim 8 - 128 M_{\rm JUP}$ but comparisons could be misleading in this case. Some differences are probably the result of a selection effect. While Andrews et al. selected their targets based on large (sub)millimeter fluxes, our selection criterion is based on the slope of IR SEDs, and the presence of accretion.
At a southern declination of $\sim$25 deg and a distance of $\sim$125 pc, our targets are excellent targets for follow up studies with *Herschel* and the Atacama Large Millimeter Array (ALMA) to investigate their properties in more detail, and in particular, the small holed transition disks will require its better spatial resolution. *Herschel* far-IR photometry would bridge the gap between the mid-IR and the (sub)mm wavelengths accessible from the ground and help to better constrain the scale heights and the flaring angles of the disks. *Herschel* spectroscopy of fine structure lines, such as the 63.2 $\mu$m \[O I\] line, could help to probe their gas content and hence the gas to dust mass ratio. Similarly, resolved images with ALMA will break some of the degeneracies of the models and will allow to better understand the physical properties of these fascinating disks, thereby helping to elucidate the conditions in which planets are formed.
We acknowledge very useful discussions with C. Dullemond and S. Andrews. We thank Hervé Bouy for explanations. M.O. was supported by ALMA-CONICYT (31070021) and ANPCyT PICT 2007 00848/Prestamo BID. MRS acknowledges support from Millennium Science Initiative, Chilean ministry of Economy: Nucleus P10-022-F.
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[^1]: two of the 70 $\mu$m measurements given in Paper I are upper limits.
[^2]: Note that there is no guarantee that the marginal distribution is Gaussian so the dispersion of the sample can only be a rough indicator of the probable error as in Isella et al (2009).
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abstract: 'In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.'
author:
- |
Yiping Zhang[^1]\
Academy of Mathematics and Systems Science, CAS;\
University of Chinese Academy of Sciences;\
Beijing 100190, P.R. China.
bibliography:
- 'propagation.bib'
title: 'Approximate Two-Sphere One-Cylinder Inequality in Parabolic Periodic Homogenization'
---
Introduction
============
Quantitative propagation of smallness is one of the central issues in the quantitative study of solutions of elliptic and parabolic equations. It can be stated as follows: a solution $u$ of a PDE $Lu=0$ on a domain $X$ can be made arbitrarily small on any given compact subset of $X$ by making it sufficiently small on an arbitrary given subdomain $Y$. There are many important applications in quantitative propagation of smallness, such as the stability estimates for the Cauchy problem [@alessandrini2009stability] and the Hausdorff measure estimates of nodal sets of eigenfunctions [@lin1991nodal], [@logunov2018nodal].
For solutions of second order parabolic equations $$\left(\partial_t-\mathcal{L}\right)u=\partial_tu-\partial_i\left(a_{ij}(x,t)\partial_j u\right)+b_i\partial_i u+cu=0,$$(the summation convention is used throughout the paper). There exists a large literature on the three cylinder inequality for solutions to parabolic equations. If $A(x,t)=\left(a_{ij}(x,t)\right)$ is three times continuously differentiable with respect to $x$ and one time continuously differentiable with respect to $t$, and $b$ and $c$ are bounded, then a not optimal three-cylinder inequality has been obtained in [@glagoleva1967some] and [@varin1992three-cylinder]. In [@glagoleva1967some], the three-cylinder inequality is derived by the Carleman estimates proved in [@Li1963An]. In 2003, Vessella[@vessella2003carleman] has obtained the following optimal three-cylinder inequality:
$$\left\|u
\right\|_{L^{2}(Q_R^{T/2})} \leqslant C\left( ||u||_{L^2(Q_\rho^{T})}\right)^{\kappa_\rho}\left( ||u||_{L^2(Q_{R_0}^{T})}\right)^{\kappa_\rho},$$
for every $0<\rho<R<R_0$, under the assumptions that the derivatives $\partial A/\partial t$, $\partial A/\partial x^i$, $\partial^2 A/(\partial t\partial
x^j)$, $\partial^2 A/(\partial x^i\partial x^j)$, for every $i,j\in \{1,\cdots,d\}$, $b=(b_1,\cdots,b_d)$ and $c$ are bounded, where $Q_r^t=B_r\times (-t,t)$ with $B_r$ the $d$-dimensional open ball centered at $0$ of radius $r$ and $\kappa_\rho\sim |\log \rho|^{-1}$ as $\rho\rightarrow 0$. The optimality of Equation $(1.2)$ consists in the growth rate of the exponent $\kappa_\rho$. Later on, Escauriaza and Vessella [@escauriaza2003optimal] have obtained the inequality $(1.2)$ under the assumptions that $A\in C^{1,1}(\mathbb{R}^{n+1})$ , $b$ and $c$ are bounded. Moreover, Vessella [@vessella2008quantitative] has obtained the following two-sphere one-cylinder inequality: $$\left\|u\left(., t_{0}\right)\right\|_{L^{2}\left(B_{\rho}\left(x_{0}\right)\right)} \leqslant C\left\|u\left(., t_{0}\right)\right\|_{L^{2}\left(B_{r}\left(x_{0}\right)\right)}^{\theta}\|u\|_{L^{2}\left(B_{R}\left(x_{0}\right) \times\left(t_{0}-R^{2}, t_{0}\right)\right)}^{1-\theta},$$under the assumptions that $A$ satisfies the Lipschitz continuity: $|A(y,s)-A(x,t)|\leq C\left(|x-y|+|t-s|^{1/2}\right)$, $b=0$ and $c=0$, where $\theta=\left(C\log {\frac{R}{Cr}}\right)^{-1}$, $0<r<\rho<R$ and $C$ depends neither on $u$ nor on $r$ but may depend on $\rho$ and $R$, see also [@escauriaza2006doubling], where the two-sphere one-cylinder inequality $(1.3)$ for time-dependent parabolic operators was first established. And the estimate $(1.3)$ has first been obtained by Landis and Oleinik [@landis1974generalized], when $A$ does not depend on $t$.
In general, the Carleman estimates are tools often used to obtain a three-cylinder inequality and the unique continuation properties for solutions. The Carleman estimates are weighted integral inequalities with suitable weight functions satisfying some convexity properties. The three-cylinder inequality is obtained by applying the Carleman estimates by choosing a suitable function. For Carleman estimates and the unique continuation properties for the parabolic operators, we refer readers to [@escauriaza2000carleman; @escauriaza2004unique; @escauriaza2001carleman; @fernandez2003unique; @vessella2008quantitative; @koch2009carleman] and their references therein for more results.
Recently, Guadie and Malinnikova [@guadie2014on] developed the three-ball inequality with the help of Poisson kernel for harmonic functions. This method also has been used in [@kenig2019propagation] to obtain an approximate three-ball inequality in elliptic periodic homogenization.
In this paper, we intend to develop an approximate two-sphere one-cylinder inequality for the $L^\infty$-norm in parabolic periodic homogenization equation, parallel to the inequality $(1.3)$, with the different exponent $\theta$.
We consider a family of second-order parabolic equation in divergence form with rapidly oscillating and time-dependent periodic coefficients, $$\partial_t u_\varepsilon-\operatorname{div}\left(A(x/\varepsilon,t/\varepsilon^2)\nabla u_\varepsilon\right)=0,$$where $1>\varepsilon>0$ and $A(y,s)=(a_{ij}(y,s))$ is a symmetric $d\times d$ matrix-valued function in $\mathbb{R}^d\times \mathbb{R}$ for $d\geq 2$. Assume that $A(y,s)$ satisfies the following assumptions:
\(i) Ellipticity: For some $0<\mu<1$ and all $(y,s)\in \mathbb{R}^d\times \mathbb{R}$, $\xi\in \mathbb{R}^d$, it holds that $$\mu |\xi|^2\leq A(y,s)\xi\cdot\xi\leq \mu^{-1}|\xi|^2.$$
\(ii) Periodicity: $$A(y+z,s+t)=A(y,s) \quad \text{for } (y,s)\in \mathbb{R}^d\times \mathbb{R} \text{ and }(z,t)\in\mathbb{Z}^d\times\mathbb{Z}.$$
(iii)Hölder continuity: There exist constants $\tau>0$ and $0<\lambda<1$ such that $$|A(x,t)-A(y,s)|\leq \tau \left(|x-y|+|t-s|^{1/2}\right)^\lambda$$ for any $(x,t),(y,s)\in \mathbb{R}^d\times \mathbb{R}$.
We are able to establish the following approximate two-sphere one-cylinder inequality in ellipsoids. The definition of ellipsoids $E_r$ depending on the coefficients $A(y,s)$ is given in Section 2.
(Interior two-sphere one-cylinder inequality) Let $u_\varepsilon$ be a solution of $(1.4)$ in $B_{{R}}\times (-T,T)$. For $0<r_1<r_2<{r_3}/4<R/8$, then there holds $$\sup_{E_{r_2}}|u_\varepsilon(\cdot,t_0)|\leq C\left\{\frac{2 r_{2}}{{r_3}} (\sup_{E_{r_1}}|u_\varepsilon(\cdot,t_0)|)^{\alpha}(\sup_{\tilde{\Omega}_{{r_3},t_0}}|u_\varepsilon|)^{1-\alpha}+\frac{r_3^{2}}{r_{1}^{2}}\left[\frac{\varepsilon}{{r_3}}\log (2+\frac{\varepsilon}{{r_3}})\right]^{\alpha}\sup_{\tilde{\Omega}_{{r_3},t_0}}|u_\varepsilon|\right\},$$ where $\alpha=\frac{\log \frac{{r_3}}{2r_2}}{\log \frac{{r_3}}{r_1}}$, $C$ depends only on $d$, $\mu$ and $(\tau,\lambda)$, and $\tilde{\Omega}_{{r_3},t_0}=E_{r_3} \times (t_0-{r_3}^2,t_0)$ is a subdomain of $B_{R}\times (-T,T)$ with $R$ and $T$ fixed.
A direct consequence of Theorem 1.1 is the following approximate two-sphere one-cylinder inequality in balls.
Let Let $u_\varepsilon$ be a solution of $(1.4)$ in $B_{R}\times (-T,T)$. For $0<r_1<r_2<\mu {r_3}/4<\mu R/8$, then there holds $$\sup_{B_{r_2}}|u_\varepsilon(\cdot,t_0)|\leq C\left\{\frac{2 r_{2}}{{r_3}} (\sup_{B_{r_1}}|u_\varepsilon(\cdot,t_0)|)^{\alpha}(\sup_{{\Omega}_{{r_3},t_0}}|u_\varepsilon|)^{1-\alpha}+\frac{r_3^{2}}{r_{1}^{2}}\left[\frac{\varepsilon}{{r_3}}\log (2+\frac{\varepsilon}{{r_3}})\right]^{\alpha}\sup_{{\Omega}_{{r_3},t_0}}|u_\varepsilon|\right\},$$ where $\alpha=\frac{\log \frac{C_1 {r_3}}{r_2}}{\log \frac{{r_3}}{r_1}}$, $C$ depends only on $d$, $\mu$ and $(\tau,\lambda)$ and $C_1$ depends only on $\mu$ and $d$, and ${\Omega}_{{r_3},t_0}=B_{r_3} \times (t_0-r_3^2,t_0)$ is a subdomain of $B_{R}\times (-T,T)$ with $R$ and $T$ fixed.
Compared with the Lipschitz regularity needed to obtain the inequality $(1.3)$, only Hölder continuity is imposed to obtain the inequality $(1.9)$, with the different exponent $\theta$.
Preliminaries
=============
Let $\mathcal{L}_\varepsilon =-\operatorname{div}\left(A_\varepsilon(x,t)\nabla \right)$, where $A_\varepsilon(x,t)=A(x/\varepsilon,t/\varepsilon^2)$. Assume that $A(y,s)$ is 1-periodic in $(y,s)$ and satisfies the ellipticity condition. For $1\leq j\leq d$, the corrector $\chi_j=\chi_j(y,s)$ is defined as the weak solution to the following cell problem: $$\left\{\begin{array}{l}
\left(\partial_{s}+\mathcal{L}_{1}\right)\left(\chi_{j}\right)=-\mathcal{L}_{1}\left(y_j\right) \quad \text { in } Y, \\
\chi_{j}=\chi_{j}^{\beta}(y, s) \text { is } 1 \text { -periodic in }(y, s), \\
\int_{Y} \chi_{j}^{\beta}=0,
\end{array}\right.$$where $Y=[0,1)^{d+1}.$ Note that
$$\left(\partial_{s}+\mathcal{L}_{1}\right)\left(\chi_{j}+y_j\right)=0\ \text{in }\mathbb{R}^{d+1}.$$
By the rescaling property of $\partial_t+\mathcal{L}_\varepsilon$, we obtain that
$$\left(\partial_{s}+\mathcal{L}_{1}\right)\left(\varepsilon\chi_{j}(x/\varepsilon,t/\varepsilon^2)+y_j\right)=0\ \text{in }\mathbb{R}^{d+1}.$$
Moreover, if $A=A(y,s)$ is Hölder continuous in $(y,s)$, then by standard regularity for $\partial_s+\mathcal{L}_1$, $\nabla \chi_j(y,s)$ is Hölder continuous in $(y,s)$, thus $\nabla \chi_j(y,s)$ is bounded.\
Let $\widehat{A}=(\widehat{a}_{ij})$, where $1\leq i,j\leq d$, and $$\widehat{a}_{ij}=\fint_Y\left(a_{ij}+a_{ik}\frac{\partial \chi_j}{\partial y_k}\right)dyds.$$ It is known that the constant matrix $\widehat{A}$ satisfies the ellipticity condition, $$\mu |\xi|^2\leq \widehat{a}_{ij}\xi_i\xi_j,\quad\quad \text{for any }\xi\in\mathbb{R}^d,$$and $|\widehat{a}_{ij}|\leq \mu_1$, where $\mu_1$ depends only on $d$ and $\mu$ [@bensoussan2011asymptotic]. Denote $$\mathcal{L}_0=-\operatorname{div}(\widehat{A}\nabla).$$
Then $\partial_t+\mathcal{L}_0$ is the homogenized operator for the family of parabolic operators $\partial_t+\mathcal{L}_\varepsilon$, $\varepsilon>0$. Since $\widehat{A}$ is symmetric and positive definite, there exists a $d\times d$ matrix $S$ such that $S\widehat{A}S^T=I_{d\times d}$. Note that $\widehat{A}^{-1}=S^TS$ and $$\langle \widehat{A}^{-1}x,x\rangle=|Sx|^2.$$ We introduce a family of ellipsoids as $$E_r(\widehat{A})=\{x\in\mathbb{R}^n:\widehat{A}^{-1}x,x\rangle< r^2\}.$$ It is easy to see that $$B_{\sqrt{\mu}r}(0)\subset E_r(0)\subset B_{\sqrt{\mu_1}r}(0).$$We will write $E_r(\widehat{A})$ as $E_r$ if the context is understood.
To move forward, let $\Gamma_\varepsilon(x,t;y,s)$ and $\Gamma_0(x,t;y,s)$ denote the fundamental solutions for the parabolic operators $\partial_t+\mathcal{L}_\varepsilon$, $\varepsilon>0$ and the homogenized operator $\partial_t+\mathcal{L}_0$, respectively. Moreover, it is easy to see that $$\Gamma_0(x,t;y,s)=\frac{1}{\left(2\sqrt{\pi}\right)^d}\left(t-s\right)^{-d/2}|S|\exp\left\{-\frac{|Sx-Sy|^2}{4(t-s)}\right\},$$ for any $x,y\in\mathbb{R}^d$ and $-\infty<s<t<\infty$ with the matrix $S$ defined in $(2.5)$.\
The following lemmas state the asymptotic behaviors of $\Gamma_\varepsilon(x,t;y,s)$ with $\varepsilon>0$, whose proof could be found in [@geng2020asymptotic].
Suppose that the coefficient matrix $A$ satisfies the assumptions $(1.5)$ and $(1.6)$, then $$\left|\Gamma_{\varepsilon}(x, t ; y, s)-\Gamma_{0}(x, t ; y, s)\right| \leq \frac{C \varepsilon}{(t-s)^{\frac{d+1}{2}}} \exp \left\{-\frac{\kappa|x-y|^{2}}{t-s}\right\}$$for any $x,y\in\mathbb{R}^d$ and $-\infty<s<t<\infty$, where $\kappa>0$ depends only on $\mu$. The constant C depends only on $d$ and $\mu$.
The next lemma states the asymptotic behaviors of $\nabla_x\Gamma_\varepsilon(x,t;y,s)$ and $\nabla_y\Gamma_\varepsilon(x,t;y,s)$.
Suppose that the coefficient matrix $A$ satisfies the assumptions $(1.5)$, $(1.6)$ and $(1.7)$, then $$\begin{aligned}
\left|\nabla_{x} \Gamma_{\varepsilon}(x, t ; y, s)-\left(1+\nabla \chi\left(x / \varepsilon, t / \varepsilon^{2}\right)\right) \nabla_{x} \Gamma_{0}(x, t ; y, s)\right| \\
\quad \leq \frac{C \varepsilon}{(t-s)^{\frac{d+2}{2}}} \log \left(2+\varepsilon^{-1}|t-s|^{1 / 2}\right) \exp \left\{-\frac{\kappa|x-y|^{2}}{t-s}\right\}
\end{aligned}$$ for any $x,y\in\mathbb{R}^d$ and $-\infty<s<t<\infty$, where $\kappa>0$ depends only on $\mu$. The constant C depends only on $d$, $\mu$ and $(\tau,\lambda)$ in $(1.7)$. Similarly, there holds $$\begin{aligned}
\left|\nabla_{y} \Gamma_{\varepsilon}(x, t ; y, s)-\left(I+\nabla \widetilde{\chi}\left(y / \varepsilon,-s / \varepsilon^{2}\right)\right) \nabla_{y} \Gamma_{0}(x, t ; y, s)\right| \\
\quad \leq \frac{C \varepsilon}{(t-s)^{\frac{d+2}{2}}} \log \left(2+\varepsilon^{-1}|t-s|^{1 / 2}\right) \exp \left\{-\frac{\kappa|x-y|^{2}}{t-s}\right\},
\end{aligned}$$where $\tilde{\chi}(y,s)$ denote the correctors for $\partial_t+\tilde{\mathcal{L}}_\varepsilon$ with $\tilde{\mathcal{L}}_\varepsilon=-\operatorname{div}\left(A(x/\varepsilon,-t/\varepsilon^2)\nabla\right)$.
With the summation convention this means that for $1\leq i,j\leq d$, $$\left|\frac{\partial\Gamma_{\varepsilon}(x, t ; y, s)}{\partial x_i}-\frac{\partial\Gamma_{0}(x, t ; y, s)}{\partial x_i}-\frac{\partial\chi_j\left(x / \varepsilon, t / \varepsilon^{2}\right)}{\partial x_i}\frac{ \partial \Gamma_{0}(x, t ; y, s)}{\partial x_j}\right|$$ is bounded by the RHS of $(2.10)$. And the similar result holds for $\nabla_{y} \Gamma_{\varepsilon}(x, t ; y, s)$.
The next lemma will be used in the proof of Theorem 1.1.
Let $u_\varepsilon$ be a week solution of $\partial_t u_\varepsilon+\mathcal{L}_\varepsilon u_\varepsilon=0$ in $B_{R}\times (-T,T)$, then $$\int_{E_{4{r_3}/5}\backslash E_{3{r_3}/4}}|\nabla u_\varepsilon|^2(x,t)dx\leq Cr_3^d||u_\varepsilon||^2_{L^\infty\left(E_{r_3}\times (t-r_3^2,t)\right)},$$ where $C$ depends only on $\mu$, and $E_{r_3}\times (t-r_3^2,t)$ is a subdomain of $B_{R}\times (-T,T)$.
The proof is standard. Choosing a cut-off function $\varphi\in [0,1]$ such that $\varphi=\varphi(x)=1$ if $x\in E_{3{r_3}/4}$, and $\varphi(x)=0$ if $x\notin E_{4{r_3}/5}$ together with $|\nabla \varphi|\leq C/{r_3}$, then multiplying the equation $\partial_t u_\varepsilon+\mathcal{L}_\varepsilon u_\varepsilon=0$ by $\varphi^2u_\varepsilon$ and integrating the resulting equation over $\mathbb{R}^d\times (t-r_3^2,t)$ leads to
$$\begin{aligned}
&\int_{\mathbb{R}^d}\varphi^2u_\varepsilon^2(x,t)dx+\int_{t-r_3^2}^t\int_{\mathbb{R}^d}\varphi^2|\nabla u_\varepsilon|^2(x,s)dxds\\
\leq& C\int_{\mathbb{R}^d}\varphi^2u_\varepsilon^2(x,t-{r_3}^2)dx+C\int_{t-{r_3}^2}^t\int_{\mathbb{R}^d}|\nabla \varphi|^2 u_\varepsilon^2(x,s)dxds\\
\leq & Cr_3^d||u_\varepsilon||^2_{L^\infty\left(E_{r_3}\times (t-r_3^2,t)\right)}.
\end{aligned}$$
Thus we have completed the proof of $(2.13)$ after noting the choice of $\varphi$.
Approximate two-sphere one-cylinder inequality
==============================================
Following [@guadie2014on], we are going to apply the Lagrange interpolation method to obtain the approximate two-sphere one-cylinder inequality. Actually, the similar method in [@guadie2014on] has been used by the author in [@kenig2019propagation] to obtain the approximate three-ball inequality in elliptic periodic homogenization. Let us briefly review the standard Lagrange interpolation method in numerical analysis. Set $$\Phi_m(z)=(z-p_1)(z-p_2)\cdots(z-p_m)$$for $z,p_j\in \mathcal{C}$ with $j=1,\cdots,m$. Let $\mathcal{D}$ ba a simply connected open domain in the complex plane $\mathcal{C}$ that contains the nodes $\tilde{p},p_1,\cdots,p_m$. Assume that $f$ is an analytic function without poles in the closure of $\mathcal{D}$. By well-known calculations, it holds that
$$\frac{1}{z-\tilde{p}}=\sum_{j=1}^{m} \frac{\Phi_{j-1}(\tilde{p})}{\Phi_{j}(z)}+\frac{\Phi_{m}(\tilde{p})}{(z-\tilde{p}) \Phi_{m}(z)}.$$
Multiplying the last identify by $\frac{1}{2\pi i}f(z)$ and integrating along the boundary of $\mathcal{D}$ leads to $$\frac{1}{2 \pi i} \int_{\partial \mathcal{D}} \frac{f(z)}{z-\tilde{p}} d z=\sum_{j=1}^{m} \frac{\Phi_{j-1}(\tilde{p})}{2 \pi i} \int_{\partial \mathcal{D}} \frac{f(z)}{\Phi_{j}(z)} d z+\left(R_{m} f\right)(\tilde{p}),$$ where
$$\left(R_{m} f\right)(\tilde{p})=\frac{1}{2 \pi i} \int_{\partial \mathcal{D}} \frac{\Phi_{m}(\tilde{p}) f(z)}{(z-\tilde{p}) \Phi_{m}(z)} d z.$$
By the residue theorem, we obtain that $$\begin{aligned}
\left({R}_{m} f\right)(\tilde{p}) &=\sum_{j=1}^{m} \frac{\Phi_{m}(\tilde{p})}{\left(p_{j}-\tilde{p}\right) \Phi_{m}^{\prime}\left(p_{j}\right)} f\left(p_{j}\right)+f(\tilde{p}) \\
&=-\sum_{j=1}^{m} \prod_{i \neq j}^{m} \frac{\tilde{p}-p_{i}}{p_{j}-p_{i}} f\left(p_{j}\right)+f(\tilde{p}),
\end{aligned}$$ where $\left(R_{m} f\right)(\tilde{p})$ is called the interpolation error. See chapter 4 in [@Bjorck2008Numerical] for more information.
In order to obtain the approximate two-sphere one-cylinder inequality for the solution in $(1.4)$, we consider the Lagrange interpolation for $f(h)=\Gamma_0(hx_0\frac{r_1}{r_2},t_0;y,s)$, where $0<r_1<r_2<{{r_3}}/{4}<R/8$ and $(x_0,t_0)$ is a fixed point such that $\sqrt{\langle\widehat{A}^{-1}x_0,x_0\rangle}=|Sx_0|< r_2$. In view of $(3.5)$, we need to estimate the error term $(R_m\Gamma_0)(x_0,t_0;y,s)$ of the approximation. Following the idea in [@kenig2019propagation], we choose points $x_i=h_ix_0\frac{r_1}{r_2}$ on the segment $[0,x_0\frac{r_1}{r_2}]$ with $h_i\in (0,1)$, then $x_i\in E_{r_1}$. Select $p_i=h_i$ in the definition of $\Phi_m$ in $(3.1)$ and $\tilde{p}=r_2/r_1$. Define $$c_i=\prod_{j\neq i}^{m}\frac{r_2r_1^{-1}-h_i}{h_j-h_i}.$$ Since $0<h_i<1$, direct computation shows that $$|c_i|\leq \frac{\left(r_2r_1^{-1}\right)^m}{|\Phi_m'(h_i)|}.$$ To estimate $|c_i|$, we choose $h_i$ to be the Chebyshev nodes, which means, $t_i=\cos\left(\frac{(2i-1)\pi}{2m}\right)$. Then we can write $$\Phi_m(h)=2^{1-m}T_m(h),$$ where $T_m$ is the Chebyshev polynomial of the first kind. There also holds that $$\Phi_m'(h)=m2^{1-m}U_{m-1}(t),$$ where $U_{m-1}$ is the Chebyshev polynomial of the second kind. See e.g. section 3.2.3 in [@Bjorck2008Numerical]. At each $h_i$, there hold
$$U_{m-1}(h_i)=U_{m-1}\left(\cos\left(\frac{(2i-1)\pi}{2m}\right)\right)=\frac{\sin{\frac{(2i-1)\pi}{2}}}{\sin{\frac{(2i-1)\pi}{2m}}}
=\frac{(-1)^{i-1}}{\sin\frac{(2i-1)\pi}{2m}}.$$
According to $(3.8)$ and $(3.9)$, there holds
$$|\Phi_m'(h_i)|\geq m 2^{1-m}.$$
Therefore, by $(3.7)$, we have $$|c_i|\leq (2m)^{-1}\left(\frac{2r_2}{r_1}\right)^m.$$
To estimate the error term $(R_m\Gamma_0)(x_0,t_0;y,s)$, we do an analytic extension of the function $f(h)=\Gamma_0(hx_0\frac{r_1}{r_2},t_0;y,s)$ to the disc of radius $\frac{{r_3}}{2r_1}$ centered at the origin in the complex plane $\mathcal{C}$. According to $(2.8)$, we have $$f(z)=\frac{1}{\left(2\sqrt{\pi}\right)^d}\left(t-s\right)^{-d/2}|S|\exp\left\{-\frac{|z\frac{r_1}{r_2}Sx_0-Sy|^2}{4(t-s)}\right\}.$$ Note that $|z\frac{r_1}{r_2}Sx_0|\leq \frac{{r_3}}{2}$ in the disk, then if ${r_3}$ is fixed, there holds
$$|f(z)|\leq \tilde{C}\left(t-s\right)^{-d/2}\exp\left\{-\frac{Cr_3^2}{t-s}\right\}\quad \text{for } y\in E_{4{r_3}/5}\backslash E_{3{r_3}/4},$$
where $C$ and $\tilde{C}$ are universal constants.\
Similarly, with the notations above, consider the Lagrange interpolation for $g(h)=\nabla_y\Gamma_0(hx_0\frac{r_1}{r_2},t_0;y,s)$, and we do an analytic extension of the $g(h)$ to the disc of radius $\frac{{r_3}}{2r_1}$ centered at the origin in the complex plane $\mathcal{C}$. Then according to $(2.8)$ again, there holds
$$g(z)=\frac{S\left(z\frac{r_1}{r_2}Sx_0-Sy\right)}{2\left(2\sqrt{\pi}\right)^d}\left(t-s\right)^{-d/2-1}|S|\exp\left\{-\frac{|z\frac{r_1}{r_2}Sx_0-Sy|^2}{4(t-s)}\right\}.$$
Note that $|z\frac{r_1}{r_2}Sx_0|\leq \frac{{r_3}}{2}$ in the disc, then if ${r_3}$ is fixed, we have
$$|g(z)|\leq \tilde{C}{r_3}\left(t-s\right)^{-d/2-1}\exp\left\{-\frac{Cr_3^2}{t-s}\right\}\quad \text{for } y\in E_{4{r_3}/5}\backslash E_{3{r_3}/4},$$
where $C$ and $\tilde{C}$ are some universal constants.\
The following lemma gives the interpolation error terms $(R_m(\nabla_y \Gamma_0))(x,t;y,s)$ and $(R_m\Gamma_0)(x,t;y,s)$ for $\nabla_y\Gamma_0(x,t;y,s)$ and $\Gamma_0(x,t;y,s)$, respectively.
If $x_0\in E_{r_2}$ and $y\in E_{4{r_3}/5}\backslash E_{3{r_3}/4} $ with $0<r_1<r_2<{r_3}/4<R/8$ and $-\infty<s<t<\infty$, then there hold $$|(R_m\Gamma_0)(x_0,t;y,s)|\leq \frac{\tilde{C}2^mr_2^m}{{r_3}^m}(t-s)^{-d/2}\exp\left\{-\frac{Cr_3^2}{t-s}\right\},$$and
$$|(R_m(\nabla_y\Gamma_0))(x_0,t;y,s)|\leq \frac{\tilde{C}2^mr_2^m}{{r_3}^{m-1}}(t-s)^{-d/2-1}\exp\left\{-\frac{Cr_3^2}{t-s}\right\},$$
where $\tilde{C}$ and $C$ are some universal constants.
First, to see $(3.16)$. According to $(3.1)$, it is easy to see that $$|\Phi_m(z)|\geq \left(\left(\frac{r_3}{2r_2}\right)-1\right)^m \text{ on the circle } |z|=\frac{r_3}{2r_1}$$and $$|\Phi_m(r_2/r_1)|\leq (r_2/r_1)^m.$$ In view of $(3.4)$-$(3.6)$ and $(3.18)$-$(3.19)$, we have $$\begin{aligned}
\left|(R_m\Gamma_0)(x_0,t;y,s)\right| &=|\Gamma_0(x_0,t;y,s)-\sum_{i=1}^{m} c_{i} \Gamma_0(x_i,t;y,s)| \\
&=|f\left(r_2/r_1\right)-\sum_{i=1}^{m} c_{i} f\left(h_{i}\right)| \\
&=|\frac{1}{2 \pi i} \int_{|z|=\frac{{r_3}}{2 r_{1}}} \frac{\Phi_{m}\left(r_{2} r_{1}^{-1}\right) f(z)}{\left(z-r_{2} r_{1}^{-1}\right) \Phi_{m}(z)} d z| \\
& \leq C \frac{(r_2/r_1)^m(2r_1)^m}{({r_3}-2r_2)({r_3}-2r_1)^m}\cdot {r_3}(t-s)^{-d/2}\exp\left\{-\frac{Cr_3^2}{t-s}\right\}\\
& \leq \frac{\tilde{C}2^mr_2^m}{{r_3}^m}(t-s)^{-d/2}\exp\left\{-\frac{Cr_3^2}{t-s}\right\},
\end{aligned}$$where we have used estimate $(3.13)$, the assumption that $0<r_1<r_2<{r_3}/4<R/8$ in the last inequality, and the constants $\tilde{C}$ and $C$ in the last inequality do not depend on $m$ and $R$.\
Similarly, for $(R_m(\nabla_y) \Gamma_0(x,t;y,s)$, there holds $$\begin{aligned}
\left|(R_m(\nabla_y\Gamma_0))(x_0,t;y,s)\right| &=|\nabla_y\Gamma_0(x_0,t;y,s)-\sum_{i=1}^{m} c_{i} \nabla_y\Gamma_0(x_i,t;y,s)| \\
&=|g\left(r_2/r_1\right)-\sum_{i=1}^{m} c_{i} g\left(h_{i}\right)| \\
&=|\frac{1}{2 \pi i} \int_{|z|=\frac{{r_3}}{2 r_{1}}} \frac{\Phi_{m}\left(r_{2} r_{1}^{-1}\right) g(z)}{\left(z-r_{2} r_{1}^{-1}\right) \Phi_{m}(z)} d z| \\
& \leq C \frac{(r_2/r_1)^m(2r_1)^m}{({r_3}-2r_2)({r_3}-2r_1)^m}\cdot {r_3}^2(t-s)^{-d/2-1}\exp\left\{-\frac{Cr_3^2}{t-s}\right\}\\
& \leq \frac{\tilde{C}2^mr_2^m}{{r_3}^{m-1}}(t-s)^{-d/2-1}\exp\left\{-\frac{Cr_3^2}{t-s}\right\},
\end{aligned}$$where we have used estimate $(3.15)$ instead of $(3.13)$, compared to $(3.20)$, and the assumption that $0<r_1<r_2<{r_3}/4<R/8$ in the last inequality, and the constants $\tilde{C}$ and $C$ in the last inequality do not depend on $m$ and $R$. Thus we have completed the proof of Lemma 3.1.
To continue the proof of Theorem 1.1, since $u_\varepsilon$ satisfies $$\partial_t u_\varepsilon-\operatorname{div}\left(A_\varepsilon \nabla u_\varepsilon\right)=0 \quad \text{in }B_{{R}}\times (-T,T),$$then simple calculation shows that $$\begin{aligned}
&\partial_t \left(\eta u_\varepsilon\right)-\operatorname{div}\left(A_\varepsilon \nabla \left(\eta u_\varepsilon\right)\right)\\
&=-\operatorname{div}\left[\left( A_\varepsilon\cdot \nabla \eta\right) u_\varepsilon \right]-A_\varepsilon \nabla u_\varepsilon \nabla \eta +u_\varepsilon \partial_t \eta\\
&=:\tilde{f}(x,t),
\end{aligned}$$where $\eta\in[0,1]$ is a cut-off function such that $\eta=\eta(x,t)=1$ if $(x,t)\in E_{3{r_3}/4} \times (t_0-r_3^2/2,t_0)$, and $\eta=0$ if $(x,t)\notin E_{4{r_3}/5} \times (t_0-2r_3^2/3,t_0)$ for some fixed $t_0$ with $|\nabla \eta|\leq C/{r_3}$ and $|\partial_t \eta|\leq C/r_3^2$. Then $$\begin{aligned}
\left(\eta u_\varepsilon\right)(x_0,t_0)&=\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\Gamma_\varepsilon(x_0,t_0;y,s)\tilde{f}(y,s)dyds\\
&=\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\nabla_y\Gamma_\varepsilon(x_0,t_0;y,s)\left( A_\varepsilon\cdot \nabla \eta\right) u_\varepsilon dyds \\
&\quad+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\Gamma_\varepsilon(x_0,t_0;y,s)\left(u_\varepsilon \partial_t \eta- A_\varepsilon \nabla u_\varepsilon \nabla \eta\right)dyds\\
&=:I_1+I_2,
\end{aligned}$$where $x_0\in E_{r_2} $ is a fixed point.
It is easy to see that $$\begin{aligned}
I_1=&\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}c_i\nabla_y\Gamma_\varepsilon(x_i,t_0;y,s)A_\varepsilon \nabla \eta u_\varepsilon dyds\\
&+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left[\nabla_y\Gamma_\varepsilon(x_0,t_0;y,s)-\left(I+\nabla
\widetilde{\chi}_\varepsilon\right)\nabla_y\Gamma_0(x_0,t_0;y,s)\right]\left( A_\varepsilon\cdot \nabla \eta\right) u_\varepsilon dyds\\
&+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left(I+\nabla\widetilde{\chi}_\varepsilon\right)\left[\nabla_y\Gamma_0(x_0,t_0;y,s)
-c_i\nabla_y\Gamma_0(x_i,t_0;y,s)\right]\left( A_\varepsilon\cdot \nabla \eta\right) u_\varepsilon dyds\\
&+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}c_i\left[\left(I+\nabla\widetilde{\chi}_\varepsilon\right)\nabla_y\Gamma_0(x_i,t_0;y,s)
-\nabla_y\Gamma_\varepsilon(x_i,t_0;y,s)\right]\left( A_\varepsilon\cdot \nabla \eta\right) u_\varepsilon dyds\\
=&:M_0+M_1+M_2+M_3,
\end{aligned}$$ where $c_i$ are defined in $(3.6)$ and $x_i=h_ix_0\frac{r_1}{r_2}$ on the segment $[0,x_0\frac{r_1}{r_2}]$ with $h_i\in (0,1)$, $i=1,\cdots,m$ and $\nabla \widetilde{\chi}_\varepsilon=\nabla \widetilde{\chi}\left(y / \varepsilon,-s / \varepsilon^{2}\right)$. It is easy to see that $x_i\in E_{r_1} $, $i=1,\cdots,m$. Similarly, we have $$\begin{aligned}
I_2=&\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}c_i\Gamma_\varepsilon(x_i,t_0;y,s)\left(u_\varepsilon \partial_t \eta- A_\varepsilon \nabla u_\varepsilon \nabla \eta\right)dyds\\
&+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left[\Gamma_\varepsilon(x_0,t_0;y,s)-\Gamma_0(x_0,t_0;y,s)\right]\left(u_\varepsilon \partial_t \eta- A_\varepsilon \nabla u_\varepsilon \nabla \eta\right) dyds\\
&+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left[\Gamma_0(x_0,t_0;y,s)-c_i\Gamma_0(x_i,t_0;y,s)\right]\left(u_\varepsilon \partial_t \eta- A_\varepsilon \nabla u_\varepsilon \nabla \eta\right) dyds\\
&+\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}c_i\left[\Gamma_0(x_i,t_0;y,s)-\Gamma_\varepsilon(x_i,t_0;y,s)\right]\left(u_\varepsilon \partial_t \eta- A_\varepsilon \nabla u_\varepsilon \nabla \eta\right) dyds\\
=&\tilde{M}_0+\sum_{i=4}^{i=9}M_i,
\end{aligned}$$ with the same $c_i$, $x_i$ and $x_0$ as in $(3.25)$. Clearly, it follows from the representation formula $(3.22)$ that $$M_0+\tilde{M}_0=c_i\left(\eta u_\varepsilon\right)(x_i,t_0).$$Before we continue, we give some notations first. Denote $\tilde{E}$ and $\tilde{\Omega}_{r_3,t_0}$ by $E_{4r_3/5} \setminus E_{3r_3/4}$ and $E_{r_3} \times (t_0-r_3^2,t_0)$, respectively. Next, we need to estimate $M_1$-$M_9$ term by term. In view of $(3.25)$, we have
$$\begin{aligned}
|M_1|&\leq C\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left|\nabla_y\Gamma_\varepsilon(x_0,t_0;y,s)
-\left(I+\nabla\widetilde{\chi}_\varepsilon\right)\nabla_y\Gamma_0(x_0,t_0;y,s)\right||\nabla \eta|| u_\varepsilon| dyds\\
&\leq C\varepsilon\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}} \exp \left\{-\frac{\kappa|x_0-y|^{2}}{t_0-s}\right\}|\nabla \eta|| u_\varepsilon| dyds\\
&\leq C\varepsilon\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}} \exp \left\{-\frac{C|Sx_0-Sy|^{2}}{t_0-s}\right\}|\nabla \eta|| u_\varepsilon| dyds\\
&\leq \frac{C\varepsilon}{r_3}\int_{t_0-r_3^2}^{t_0}\int_{\tilde{E}}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}} \exp \left\{-\frac{C|Sx_0-Sy|^{2}}{t_0-s}\right\}| u_\varepsilon| dyds\\
&\leq C\varepsilon r_3^{d-1}\int_{t_0-r_3^2}^{t_0}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}} \exp \left\{-\frac{Cr_3^2}{t_0-s}\right\}ds\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq C\varepsilon r_3^{-1}\int_1^\infty\log (2+\varepsilon^{-1}\tilde{s}^{-1/2}r_3){\tilde{s}}^{d/2-1}\exp \{-C\tilde{s}\}d\tilde{s}\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq C\varepsilon r_3^{-1}\log(2+\varepsilon^{-1}r_3)||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})},
\end{aligned}$$
where we have used $(2.11)$ in Lemma 2.2 in the above inequality.
To estimate $M_2$, we first note that $\nabla \tilde{\chi}_\varepsilon$ is bounded, then according to Lemma 3.1, there holds $$\begin{aligned}
|M_2|&\leq\frac{C(2r_2)^m}{r_3^{m}}\int_{t_0-r_3^2}^{t_0}\int_{\tilde{E}}(t_0-s)^{-d/2-1}\exp\left\{-\frac{Cr_3^2}{t_0-s}\right\}dyds\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C(2r_2)^m}{r_3^{m-d}}\int_{t_0-r_3^2}^{t_0}(t_0-s)^{-d/2-1}\exp\left\{-\frac{Cr_3^2}{t_0-s}\right\}ds\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C(2r_2)^m}{r_3^{m}}\int_{1}^{\infty}{\tilde{s}}^{d/2-1}\exp\left\{-C\tilde{s}\right\}ds\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C(2r_2)^m}{r_3^{m}}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.
\end{aligned}$$
As for $M_3$, noting that $x_i\in E_{r_1}$ with $i=1,\cdots,m$, then the estimate $(2.11)$ yields that $$\begin{aligned}
|M_3|&\leq \frac{C\varepsilon}{r_3}\int_{t_0-r_3^2}^{t_0}\int_{\tilde{E}}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}}|c_i| \exp \left\{-\frac{\kappa|x_i-y|^{2}}{t_0-s}\right\}dyds\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C\varepsilon}{r_3}\int_{t_0-r_3^2}^{t_0}\int_{\tilde{E}}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}}|c_i| \exp \left\{-\frac{C|Sx_i-Sy|^{2}}{t_0-s}\right\}dyds\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq C\varepsilon r_3^{d-1}\sum_{i}|c_i|\int_{t_0-r_3^2}^{t_0}\frac{\log \left(2+\varepsilon^{-1}|t_0-s|^{1 / 2}\right)}{(t_0-s)^{\frac{d+2}{2}}}\exp \left\{-\frac{Cr_3^2}{t_0-s}\right\}dyds\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C\varepsilon(2r_2)^m}{r_1^mr_3}\int_1^\infty\log (2+\varepsilon^{-1}\tilde{s}^{-1/2}r_3){\tilde{s}}^{d/2-1}\exp \{-C\tilde{s}\}d\tilde{s}\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C(2r_2)^m}{r_1^m}\varepsilon {r_3}^{-1}\log (2+\varepsilon^{-1}r_3)||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.
\end{aligned}$$
Next, we give the estimate of $I_2$ term by term in $(3.26)$. In view of $(2.9)$, then we have $$\begin{aligned}
|M_4|&\leq C\varepsilon r_3^{d-2}\int_{t_0-r_3^2}^{t_0}(t_0-s)^{-\frac{d+1}{2}} \exp \left\{-\frac{Cr_3^2}{t_0-s}\right\}ds\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq C\varepsilon r_3^{d-2} \int_1^\infty {\tilde{s}}^{\frac{d-3}{2}}\exp (-C\tilde{s}) r_3^{-d+1}d\tilde{s}\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq C\varepsilon r_3^{-1}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.
\end{aligned}$$
As for $M_5$, we have $$\begin{aligned}
|M_5|&\leq C\left(\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left|\Gamma_\varepsilon(x_0,t_0;y,s)-\Gamma_0(x_0,t_0;y,s)\right|^2|\nabla \eta|dyds\right)^{1/2}\\
&\quad \times\left(\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left| \nabla u_\varepsilon \right|^2|\nabla \eta| dyds\right)^{1/2}\\
&\leq C\varepsilon r_3^{d/2-1}\left(\int_{t_0-r_3^2}^{t_0} (t_0-s)^{-d-1}\exp \left\{-\frac{Cr_3^2}{t_0-s}\right\}ds\right)^{1/2}\left(\int_{t_0-r_3^2}^{t_0}\int_{\tilde{E}} |\nabla u_\varepsilon|^2dyds\right)^{1/2}\\
&\leq C\varepsilon r_3^{-1}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})},
\end{aligned}$$where we have used $(2.9)$ in the second inequality and $(2.13)$ in the third inequality.
Due to $(3.16)$ and similar to the estimate of $M_2$, we have $$\begin{aligned}
|M_6|&\leq\frac{\tilde{C}(2r_2)^m}{r_3^{m-d+2}}\int_{t_0-r_3^2}^{t_0}(t_0-s)^{-d/2}\exp\left\{-\frac{Cr_3^2}{t_0-s}\right\}ds
\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C(2r_2)^m}{r_3^{m}}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.
\end{aligned}$$According to $(3.16)$ and $(2.13)$, then there holds
$$\begin{aligned}
|M_7|&\leq C\left(\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left|\Gamma_0(x_0,t_0;y,s)-c_i\Gamma_0(x_i,t_0;y,s)\right|^2|\nabla \eta|dyds\right)^{1/2}\\
&\quad \times\left(\int_{t_0-r_3^2}^{t_0}\int_{\mathbb{R}^d}\left| \nabla u_\varepsilon \right|^2|\nabla \eta| dyds\right)^{1/2}\\
&\leq \frac{\tilde{C}(2r_2)^m}{r_3^{m+1-d}}\left(\int_{t_0-r_3^2}^{t_0}(t_0-s)^{-d}\exp\left\{-\frac{Cr_3^2}{t_0-s}\right\}ds\right)^{1/2}\cdot ||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq\frac{C(2r_2)^m}{r_3^{m}}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.
\end{aligned}$$
Similarly, in view of $(2.9)$, we have $$\begin{aligned}
|M_8|&\leq C\varepsilon r_3^{d-2}\sum|c_i|\int_{t_0-r_3^2}^{t_0}(t_0-s)^{-\frac{d+1}{2}} \exp \left\{-\frac{Cr_3^2}{t_0-s}\right\}ds
\cdot||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&\leq \frac{C\varepsilon (2r_2)^m}{r_3r_1^m}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.
\end{aligned}$$ Moreover, similar to the proof of $M_5$ and $(3.11)$, we have $$|M_9|\leq \frac{C\varepsilon (2r_2)^m}{r_3r_1^m}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}.$$
Consequently, noting that $\sum_{i}|c_i|\leq (2r_2)^m/{r_1}^m$, then combining $(3.24)$-$(3.36)$ yields that $$\begin{aligned}
|u_\varepsilon(x_0,t_0)|\leq& \frac{(2r_2)^m}{{r_1}^m}\sup_{E_{r_1}}|u_\varepsilon(\cdot,t_0)|+\frac{C(2r_2)^m}{r_3^{m}}||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})}\\
&+\frac{C(2r_2)^m}{r_1^m}\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)||u_\varepsilon||_{L^\infty(\tilde{\Omega}_{r_3,t_0})},
\end{aligned}$$where $C$ does not depend on $m$, $r_1$, $r_2$ or $r_3$. Since $x_0\in E_{r_2}$ is an arbitrary point, then it follows that
$$\begin{aligned}
\sup_{E_{r_2}}|u_\varepsilon(\cdot,t_0)|\leq& C\left\{ \frac{(2r_2)^m}{{r_1}^m}\sup_{E_{r_1}}|u_\varepsilon(\cdot,t_0)|+\frac{(2r_2)^m}{r_3^{m}}\sup_{\tilde{\Omega}_{r_3,t_0}}|u_\varepsilon|\right.\\
&\quad\quad\left.+\frac{(2r_2)^m}{r_1^m}\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\sup_{\tilde{\Omega}_{r_3,t_0}}|u_\varepsilon|\right\}.
\end{aligned}$$
Now we need to minimize the summation of the terms in the RHS of $(3.38)$ by choosing the integer value $m$. Actually, the similar proof can be found in [@kenig2019propagation], we give it just for completeness. For simplicity, let $$\sup_{E_{r_1}}|u_\varepsilon(\cdot,t_0)|=\delta,\quad \sup_{\tilde{\Omega}_{r_3,t_0}}|u_\varepsilon|=N.$$First, choose $m$ such that $$\delta\left(\frac{2r_2}{r_1}\right)^m=N\left(\frac{2r_2}{r_3}\right)^m,$$which gives $$m=\frac{\log {M/\delta}}{\log {r_3/r_1}}.$$ Consequently, define $$m_0\leq \lfloor\frac{\log {M/\delta}}{\log {r_3/r_1}}\rfloor+1,$$where $\lfloor\cdot\rfloor$ denotes its integer part. We minimize the above terms by considering two cases.\
**Case 1**. $\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\left(\frac{2r_2}{r_1}\right)^{m_0}\leq \left(\frac{2r_2}{r_3}\right)^{m_0}$.
In this case, let $m=m_0$ in $(3.38)$. Then the third term can be absorbed into the second one in the right hand side of $(3.38)$. Consequently, it follows that $$\begin{aligned}
\sup _{E_{r_{2}}}\left|u_{\epsilon}\right| & \leq C\left\{\delta\left(\frac{2 r_{2}}{r_{1}}\right)^{m_{0}}+M\left(\frac{2 r_{2}}{r_3}\right)^{m_{0}} \right\} \\
& \leq C\frac{2 r_{2}}{r_3} M^{1-\alpha} \delta^{\alpha},
\end{aligned}$$where $$\alpha=\frac{\log {\frac{r_3}{2r_2}}}{\log\frac{r_3}{r_1}}.$$
**Case 2.** $\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\left(\frac{2r_2}{r_1}\right)^{m_0}> \left(\frac{2r_2}{r_3}\right)^{m_0}$.
In this case, from the definition of $m_0$, there holds that $$\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)>\frac{\delta r_1}{NR}.$$That is,
$$\sup_{E_{r_1}}|u_\varepsilon(\cdot,t_0)|\leq \varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\frac{r_3}{r_1}\sup_{\tilde{\Omega}_{r_3,t_0}}|u_\varepsilon|.$$
Then, we choose $\widehat{m}$ such that
$$\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\left(\frac{2r_2}{r_1}\right)^{\widehat{m}}=\left(\frac{2r_2}{r_3}\right)^{\widehat{m}},$$
which gives $$\widehat{m}=\frac{\log{\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)}}{\log \frac{r_1}{r_3}}.$$ Therefore, we can choose $$m_1=\lfloor\frac{\log{\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)}}{\log \frac{r_1}{r_3}}\rfloor+1.$$Taking $m=m_1$ in $(3.38)$, then the second term can be absorbed into the third term in the RHS of $(3.38)$. In view of $(3.45)$, then we have
$$\begin{aligned}
\sup_{E_{r_2}}|u_\varepsilon(x,t_0)| & \leq C\left\{\left(\frac{2 r_{2}}{r_{1}}\right)^{m_{1}} \varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3) \frac{r_3}{r_{1}}N+\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\left(\frac{2 r_{2}}{r_{1}}\right)^{m_{1}} N\right\} \\
& \leq C \frac{2 r_{2} r_3}{r_{1}^{2}} \exp \left\{\frac{\log \frac{2 r_{2}}{r_{1}} \log \left[\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\right]}{\log \frac{r_{1}}{r_3}}\right\} \varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3) N \\
& \leq C \frac{r_3^{2}}{r_{1}^{2}}\left[\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\right]^{\frac{\log\frac{r_3}{2r_2}}{\log{r_3}{r_1}}}N \\
&=C \frac{r_3^{2}}{r_{1}^{2}}\left[\varepsilon r_3^{-1}\log (2+\varepsilon^{-1}r_3)\right]^{\alpha} N,
\end{aligned}$$
where $$\alpha=\frac{\log {\frac{r_3}{2r_2}}}{\log\frac{r_3}{r_1}}.$$Notice that $0<\alpha<1$ according to the assumption of $r_1,r_2$ and $r_3$. Consequently, combining the two cases above yields the result of Theorem 1.1. And Corollary 1.2 directly follows from Theorem 1.1 and $(2.7)$.
[**Acknowledgements**]{}
The author thanks Prof. Luis Escauriaza for helpful discussions.
[^1]: Email:zhangyiping161@mails.ucas.ac.cn
|
---
abstract: 'We consider gauge vortices in symmetry breaking models with a non-canonical kinetic term. This work extends our previous study on global topological $k$-defects (hep-th/0608071), including a gauge field. The model consists of a scalar field with a non-canonical kinetic term, while for the gauge field the standard form of its kinetic term is preserved. Topological defects arising in such models, $k$-vortices, may have quite different properties as compared to “standard” vortices. This happens because an additional dimensional parameter enters the Lagrangian for the considered model — a “kinetic” mass. We briefly discuss possible consequences for cosmology, in particular, the formation of cosmic strings during phase transitions in the early universe and their properties.'
author:
- 'E. Babichev'
title: 'Gauge $k$-vortices'
---
ø ł
Introduction {#SIntro}
============
Vortices are a class of topological defects which may form as a result of symmetry-breaking phase transitions. In condensed matter physics linear defects arise rather commonly. Well-known examples are flux tubes in superconductors [@Abr] and vortices in superfluid helium-4. In cosmology topological defects attract much interest because they might appear in a rather natural way during phase transitions in the early universe. The breaking of discrete symmetries leads to the appearance of domain walls, while the breaking of a global or a local $U(1)$-symmetry is associated with global [@glstr] and local [@lostr] cosmic strings, respectively. Localized defects or monopoles may form in gauge models possessing a $SO(3)$ symmetry which is spontaneously broken to $U(1)$ [@Pol; @tHo].
Many properties of topological defects arising in symmetry-breaking models with a canonical kinetic term are well-known, see, e.g. [@Vilenkin; @book; @Rubakov]. Adding non-linear terms to the kinetic part of the Lagrangian has interesting consequences for topological defects. For example, defects can exist without a symmetry-breaking potential term [@Sky]. Non-standard kinetic terms in the form of some non-linear function of the canonical term may arise in string theory, due to the presence of higher-order corrections to the effective action for the scalar field. Non-canonical kinetic structures appear also commonly in effective field theories.
During the last years Lagrangians with non-canonical fields were intensively studied in the cosmological context. So-called $k$-fields were first introduced in the context of inflation [@k-inflation] and then $k$-essence models were suggested as solution to the cosmic coincidence problem [@k-essence; @kessence; @others]. Tachyon matter [@tachyon] and ghost condensates [@ghost] are other examples of non-canonical fields in cosmology. An interesting application of $k$-fields is the explanation of dark matter as a self-gravitating coherent state of $k$-field matter [@Halo]. The production of strong gravitational waves in models of inflation with nontrivial kinetic term was considered in [@MukVik]. The effects of scalar fields with non-canonical kinetic terms in the neighborhood of a black hole were investigated in [@BH].
Recently, symmetry-breaking models with $k$-essence-like terms have been discussed in literature. General properties of global topological defects appearing in such models were studied in [@kdefect]. It was shown that the properties of such defects (dubbed $k$-defects) are quite different from “standard” global domain walls, vortices and monopoles. In particular, depending on the concrete form of the kinetic term, the typical size of such a defect can be either much larger or much smaller than the size of a standard defect with the same potential term. A detailed study of global defect solutions for one space dimension was carried out in [@Bazeia]. A self-gravitating $k$-monopole was considered in [@Jin]. In [@Adam] the authors argued that a special type of $k$-defects may be viewed as “compactons”, i.e. solutions with a compact support. Global strings with a Dirac-Born-Infeld (DBI) term were considered in [@DBIstring].
In this paper we study properties of gauge vortices arising in a model with a $k$-essence-like kinetic term and a symmetry breaking potential. We dubb such defects “gauge $k$-vortices”, in analogy to global $k$-defects. We extend our previous investigation on $k$-defects [@kdefect] including a gauge field into the model. As for the global $k$-defects, the scalar field has a non-canonical structure of the kinetic term, while for the gauge field we keep the canonical form of the kinetic term. The existence of non-trivial configurations is ensured by the symmetry-breaking potential term. The generic feature of the model with a non-canonical kinetic term is the appearance of a new scale — the kinetic “mass”. The presence of a new mass scale in the model radically changes basic properties of vortices.
We show that generally the size of the scalar core of the gauge vortex solution is almost independent on the presence of the gauge field. Its value can be approximated by the core’ size of the global $k$-vortex with the same kinetic structure [@kdefect]. With an additional, natural assumption we find that the vector core has roughly the same size as a standard vortex. A particularly interesting result is that the mass of a vortex radically vary depending on the choice of kinetic term. As the concrete examples we study numerically the vortex solutions for the models with DBI and with a power-law kinetic terms.
The paper is organized as follows. In Sec. \[SModel\], we describe our model and derive its equations of motion and the energy functional for a vortex solution. General properties of $k$-vortices are studied in Sec. \[SGeneral\]. In Sec. \[SConstraints\] we find constraints on the parameters of the model. Numerical solutions for particular choices of the non-canonical kinetic term are presented in Sec. \[Sec Numerics\]. We summarize and discuss results and cosmological applications in the concluding Sec. \[SDiscussion\].
Model {#SModel}
=====
We consider the action $$\label{act}
S=\int {\rm d}^4 x\left[M^4 K(X/M^4) - U(f) - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right],$$ with $$\label{X}
X=(D_\mu\phi)(D^\mu\phi)^{*},\quad
F_{\mu\nu}=\pd_\mu A_\nu-\pd_\nu A_\mu \,,$$ and $$\label{D_mu}
D_\mu\equiv \pd_\mu - ieA_\mu \,.$$ The potential term which provides the symmetry breaking is given by $$\label{U}
U(\phi)=\frac{\lambda}{4}(\mid\phi\mid^2-\eta^2)^{2} \,,$$ where $\eta$ has dimension of a mass, while $\lambda$ is a dimensionless constant. Note that throughout this paper we use a metric with signature $(+,-,-,-)$. The kinetic term $K(X)$ in (\[act\]) is in general some non-linear function of $X$. The action (\[act\]) contains three mass scales: The “usual” scalar and vector masses, $\sqrt\lambda\eta$ and $e\eta$ correspondingly, and the “kinetic” mass $M$. It is worth to note that a kinetic term that is non-linear in $X$ unavoidably leads to a new scale in the action. In the standard case $K=X/M^4$ and the kinetic mass $M$ drops out from the action. For non-trivial choices of the kinetic term, the kinetic mass enters the action and changes the properties of the resulting topological defects.
In what follows it is convenient to make the following redefinition of variables to dimensionless units, $$\label{newv}
x\to \frac{x}{M},\, \phi\to M\phi,\, A_\mu\to M A_\mu \,.$$ In terms of the new variables the energy density $\e$ is also dimensionless: $\e\to M^4 \e$. It is easy to see that $D_\mu~\to~MD_\mu$, $X~\to~M^4 X$ and the action (\[act\]) becomes $$\label{S}
S=\int{\rm d}^4 x
\left[K(X)-V(\phi)-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} \right],$$ where $$\label{V}
V(\phi)=\frac{\lambda}{4}(\mid\phi\mid^2-v^2)^{2},$$ with $v\equiv\eta/M$ being a dimensionless quantity. One can calculate the energy-momentum tensor from the action (\[S\]), $$\begin{aligned}
\label{emt}
T_{\mu\nu} &=& 2 K_X\mid D_\mu\phi\mid^2 - g_{\mu\nu}\left[K(X) - V(f)\right]\nonumber\\
&-& F_{\mu\alpha}F_\nu^{\,\,\,\alpha}+
\frac 14 g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\nonumber,\end{aligned}$$ where we denoted $K_X\equiv dK/dX$, $K_{XX}\equiv d^2K/dX^2$ etc. In the gauge $A_0=0$, the energy density for a static configuration, $\dot\phi=0$, $\pd A_i=0$, is $$\label{E}
T_0^0=- K(X) + V(\phi)+\frac 14 F_{ij}^2.$$ Note also that for static configurations $X=-D_i\phi (D_i\phi)^*$. The mass per unit length of a vortex, $E$, can be expressed as: $$\label{Ef}
E=\int\left[-K\left(-|D_i\phi|^2\right)+V(\phi)+\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right]
{\rm d}^2x.$$ From the variation of the action (\[S\]) with respect to $\phi^*$ and $A^\mu$ we obtain as equations of motion (EoM) $$\begin{aligned}
\label{eom phi}
K_X D_\mu D^\mu\phi + K_{XX} X_{,\mu}D^\mu\phi + \frac{dV}{d\phi^*} &=& 0,\\
\label{eom A}
\pd_\mu F^{\mu\nu} &=& e j^\nu \,,\end{aligned}$$ where the current $j_\mu$ is given by $$\label{current}
j_\mu = -iK_X\left[\phi^*D_\mu\phi-\phi\left(D_\mu\phi\right)^*\right].$$ (Note the additional $K_X$ in the above expression as compared to the standard case.) One can check that the current $j_\mu$ is conserved, $$\pd_\mu j^\mu=0,$$ similar to the standard case. To obtain the solution describing a vortex we use the following ansatz, $$\begin{aligned}
\label{ansatz}
\phi(x) &=& e^{i\theta} f(r),\\
A_i(x) &=& -\frac{1}{e r^2}\epsilon_{ij} r_j\alpha(r)\nonumber \,,\end{aligned}$$ where $r=\left(x^2+y^2\right)^{1/2}$. It is worth to note that we use the same ansatz (\[ansatz\]) as in the standard case. Substituting (\[ansatz\]) into (\[eom phi\]) and (\[eom A\]) we obtain the EoM for the functions $f(r)$ and $\alpha(r)$, $$\begin{aligned}
-K_X\left[\frac1r \frac{d\,(rf')}{dr}-\frac{f(1-\alpha)^2}{r^2}\right]\quad\quad\quad\quad
&& \nonumber\\
-K_{XX}X'f'+\frac{\lambda}{2}\left(f^2-v^2\right)f&=&0,\label{EOM}\\
-\frac{d}{d r}\left(\frac{\alpha'}{r}\right)
- \frac{2 e^2 f^2}{r}(1-\alpha)K_X &=& 0. \label{EOMa}\end{aligned}$$ One can check that in the standard case, $K(X)=X$, Eqs. (\[EOM\]) and (\[EOMa\]) take the familiar form of EoMs for “usual” vortices.
We assume that the kinetic term $K(X)$ has the standard asymptotic behavior at small $X$. This means that in the perturbative regime in trivial backgrounds the dynamics of the considered system is the same as with a canonical kinetic term. This requirement is introduced to avoid troubles at $X=0$: In the case of $X^\delta$ with $\delta<1$ there is a singularity at $X=0$, while for $\delta>1$ the system becomes non-dynamical at $X=0$ [@kdefect]. One can understand this better in terms of an “emergent geometry”: Because of the non-linearity of the EoMs, small fluctuations of the scalar field feel an “effective” metric, which in general differs from the gravitational one (in our case — from the Minkowski metric). As it was shown in [@causality], in the case when a kinetic term does not coincide with the canonical one (probably, up to some constant) in the limit of small $X$, the effective metric for small perturbations diverges as $X\to 0$. This means that such models are physically meaningless.
In the opposite limit, $X \gg 1$, we restrict our attention to the modifications of the kinetic term having the following asymptotic, $$\label{pow}
K(X)=-\left(-X\right)^n.$$ Note that for a static configuration $X<0$ and a minus sign in the expression above for $K(X)$ provides a positive contribution to the energy density (\[E\]). Below we find the criteria for the Lagrangians to have the desired asymptotic $X\gg 1$ in the core of a vortex. Summarizing, we will consider kinetic terms with the following asymptotic behavior, $$\begin{aligned}
\label{model}
K(X)=\left\{
\begin{array}{lcl}
X,&& X\ll 1,\\
-\left(-X\right)^n, && X\gg 1.
\end{array}
\right.\end{aligned}$$ Assuming $X\gg 1$, one can easily obtain from (\[model\]), (\[EOM\]) and (\[EOMa\]) EoMs in this regime, $$\begin{aligned}
\left[\frac1r \frac{d\,(rf')}{dr}-\frac{f(1-\alpha)^2}{r^2}\right]
+(n-1)\left(\ln X\right)'f'&& \nonumber\\
-\left(-X\right)^{1-n}\frac{\lambda}{2n}\left(f^2-v^2\right)f \!\!&=&\!\! 0,\label{eomnf}\\
\frac{d}{d r}\left(\frac{\alpha'}{r}\right)
+ \frac{2 n e^2 f^2}{r}(1-\alpha)\left(-X\right)^{n-1}\!\! &=&\!\! 0.\label{eomna}\end{aligned}$$ As a particular example we choose a DBI-like kinetic term for the scalar field, $$\label{BI}
K(X)=1-\sqrt{1-2X}.$$ It is easy to see that for this choice in the limit $X\gg 1$ the kinetic term is of form (\[model\]) with $n=1/2$. Another particular example we will study is a power-law form of the Lagrangian: $$\label{X3}
K(X)=X+X^3.$$
General properties {#SGeneral}
==================
The EoMs (\[eomnf\]), (\[eomna\]) for arbitrary $K(X)$ are highly non-linear and cannot be solved analytically. We restrict our attention to the study of vortices arising from Lagrangians having the asymptotic behavior (\[model\]) for the kinetic term. Although the EoMs can not be integrated even in this case, some general features can be extracted without the knowledge of explicit solutions.
At some point of our estimations we will use an additional simplifying assumption, which makes the understanding of the results more transparent. We will assume that the parameters of the Lagrangian satisfy the following natural relation, $$\label{1mass}
e\sim\sqrt\lambda,$$ which means that in the linear regime for the kinetic term, $K(X)=X$, the “scalar” and “vector” masses are of the same order. Thus the assumption (\[1mass\]) reduces the number of different scales in the model from $3$ to $2$: one is the usual “scalar” or “vector” mass, $e\eta\sim \sqrt\lambda\eta$, and the other is a new kinetic mass $M$.
The region $r\to 0$
-------------------
We start our study from the region close to the center of a vortex, $r\to 0$. As we assume that in the core of a defect the kinetic term can be approximated by (\[model\]), we must require that for models (\[BI\]) and (\[X3\]), $X\gg 1$. In the opposite case, $X\lesssim 1$, we end up with a solution which does not deviate much from the standard one. For $r\to 0$ we search a solution in the following form, $$\begin{aligned}
\label{r0}
f(r) &=& A_f r+B_f r^3+ O(r^5),\nonumber\\
\alpha(r) &=& A_\alpha r^2 + B_\alpha r^4+ O(r^6) \nonumber\end{aligned}$$ with unknown constants $A_f$, $B_f$, $A_\alpha$, $B_\alpha$. Substituting the above expressions into (\[eomnf\]) and (\[eomna\]) we find that $A_f$ and $A_\alpha$ are arbitrary, while the others are $$\begin{aligned}
\label{appr}
B_\alpha &=& -2^{n-3}e^2 n A_f^{2n},\nonumber\\
B_f &=& -\frac{2^{n-5}}{n^2}\lambda v^2 A_f^{2n-1}
+\frac{n(n-2)}{4}A_f A_\alpha.\nonumber\end{aligned}$$ The standard asymptotics for $K(X)=X$ are recovered from the above expression by setting $n=1$. Note that the constants $A_f$ and $A_\alpha$ are left undetermined, which means that the size of a defect and its mass are undetermined too. It is possible, however, to estimate these quantities without solving explicitly EoMs, as we will see in \[subs structure\] and \[subs mass\].
Structure of a vortex {#subs structure}
---------------------
The model (\[act\]) contains a complex scalar field and a gauge vector field. In accordance to this, there are two distinct cores for a vortex solutions: One is associated with the scalar field and the other with the vector field. The typical sizes of the cores depend on the parameters of the Lagrangian. In this subsection we will estimate the typical sizes of cores without explicitly solving EoMs. In what follows it will be helpful to use an additional rescaling: $$\label{rescaling}
f\to vf,\quad r\to r\,L_H,$$ where $$\label{LH}
L_H=v\;\e^{-1/2n},$$ and $\e$ was defined as $$\label{eps}
\e\equiv \lambda v^4,$$ in analogy with global defects [@kdefect]. Later we will see that the quantity $\e$ (\[eps\]) corresponds to the energy density inside the scalar core of a vortex. The rescaling (\[rescaling\]) brings the EoMs (\[eomnf\]), (\[eomna\]) to the following form, $$\begin{aligned}
\left[\frac1r \frac{d\,(rf')}{dr}-\frac{f(1-\alpha)^2}{r^2}\right]
+(n-1)\left(\ln X\right)'f'&&\quad\quad \nonumber \\
-\frac{1}{2n}\left(-X\right)^{1-n}\left(f^2-1\right)f &=& 0, \label{eomHf} \\
\frac{d}{d r}\left(\frac{\alpha'}{r}\right)
+ \frac{2n}{\gamma}\left(-X\right)^{n-1}\frac{f^2(1-\alpha)}{r} &=& 0, \label{eomHa}\end{aligned}$$ where $$\label{gamma}
\gamma %=\left(2\beta\right)^{-n/(n-2)}
= \frac{\lambda}{e^2}\;\e^{-2(n-1)/n}.$$ Notice that in the standard case, $n=1$, the parameter $\gamma$ coincides with the usual parameter, defined as the ratio of the scalar and vector masses. The EoM for $f(r)$, Eq. (\[eomHf\]), contains parameters of order of $1$ as well as the function $\alpha$, going from $0$ to $1$. Therefore one can guess that the typical scale on which the function $f(r)$ varies is of the order $L_H$. Thus the typical size of the scalar core, $l_H$, is given by $$\label{lH}
l_H \sim L_H,%=\left(\frac{v^{n-2}}{\sqrt{\lambda}}\right)^{1/n}=
% \frac{v}{\left(\lambda v^4\right)^{1/(2n)}},$$ and is almost independent on $\gamma$ and $e$. In fact, the value $l_H$ (\[lH\]) coincides (up to an irrelevant numerical factor of order of $1$) with the size of the core in the case of a global $k$-string, found in [@kdefect]. Very roughly speaking, the scalar core remains unaffected by the gauge field. Intuitively it can be understood as follows. Topological defects exist due to the presence of a potential, which provides symmetry breaking for the scalar field $\phi$; the gauge field is in a sense merely an auxiliary component. Therefore, the presence of the gauge field should not radically change the size of the scalar core.
To estimate the size of the vector core is a more tricky task. First of all we note that when $\gamma\sim 1$, i.e. $$\label{valideq}
\frac{e}{\sqrt\lambda}\;\e^{(n-1)/n}%\sqrt{2n}
\sim 1,$$ the size of the vector core, $l_V$, is of order of the size of the scalar core, $$\label{lVeq}
l_V\sim l_H,$$ since in this case the EoMs (\[eomHf\]), (\[eomHa\]) do not contain any large or small parameters.
To consider other cases, when the vector core is much larger/smaller than the scalar core, let us turn back to Eq. (\[eomna\]). The inflection point for the function $\alpha(r)$ is at the point $r\sim l_V$. Then, taking also into account that $\alpha'(r)\sim 1/l_V$ at $r\sim l_V$, we obtain from (\[eomna\]), $$\label{eq lV}
\frac{1}{l_V^2} \sim e^2 f^2 \left(-X\right)^{n-1}.$$ Let us now assume that the vector core is much smaller than the scalar one, $l_V\ll l_H$. From (\[eq lV\]) using the estimates, $f\sim v\,l_V/l_H$ and $(-X)^n\sim\lambda v^4$ at $r\sim l_V$, we find $$\label{lVless}
l_V \sim \frac{1}{\left(e^2\e\right)^{1/4}}.$$ Note, that the above result is valid if $l_V\ll l_H$, which can be recasted as follows, using (\[lVless\]): $$\label{validVH}
\frac{e}{\sqrt\lambda}\;\e^{(n-1)/n}\gg 1.$$ In the opposite case, $l_V\gg l_H$, the non-linearity in $X$ is negligible, i.e. one has to set $n=1$ in (\[eq lV\]). Then we immediately find the size of the vector core as $$\label{lVgreater}
l_V \sim \frac{1}{ev}.$$ Eq. (\[lVgreater\]) is valid for $l_V \gg l_H$, or $$\label{validHV}
\frac{e}{\sqrt\lambda}\;\e^{(n-1)/2n}\ll 1.$$ For us the most interesting case is $\e\gg 1$, as we will see later, this corresponds to the regime when the non-linearity in $X$ become important. Taking into account our assumption (\[1mass\]) we may summarize our results for $\e\gg 1$ as follows. The size of the scalar core (with the restored physical units) is given by $$\label{lH1}
l_H= \frac{\eta}{M^2}\left[\lambda\left(\eta/M\right)^4\right]^{-1/2n},$$ and the size of the vector core is $$\label{lV1}
l_V\sim \frac{1}{e\eta}.$$ It is worth to note that the vector core in our model is roughly as large as in the standard case. This is what one can naively expect from the action (\[act\]): The kinetic term for the vector field is unchanged as compared with the standard Lagrangian, so it is unlikely that the vector core varies much.
Vortex’ mass {#subs mass}
------------
Another way to see how the parameter $\gamma$ (\[gamma\]) appears in the model is to make the change of variables in the action (\[S\]) as follows: $$\label{changeV}
\phi=v f,\quad x=L_H\,y, \quad A_\mu = \frac{B_\mu}{eL_H},$$ with $L_H$ given by (\[LH\]). Substituting the rescaling (\[changeV\]) into (\[S\]) we immediately find the functional of the energy density (with the restored physical units): $$\label{Efunc}
E= \eta^2 \left[\lambda\left(\eta/M\right)^4\right]^{(n-1)/n} \mathfrak{F}(n,\gamma),$$ where $$\label{F}
\mathfrak{F}(n,\gamma)=\int{\rm d}^2y \left[ \frac{\gamma}{4} F_{\mu\nu}F^{\mu\nu}
+ (D_i f)^{2n} +\left(\mid f \mid^2-1\right)^2 \right].\nonumber$$ Is it important to note that the above expression is only valid in the non-linear regime in $X$, i.e. when $X\gg1$ inside the scalar core. In addition, we have to require that the vector core is not larger than the scalar one, $l_V\ll l_H$; otherwise the vector core is partly outside the scalar one and Eq. (\[Efunc\]) is inapplicable.
To find the energy density of the vortex for particular parameters of the Lagrangian, one needs to calculate the functional of the energy density (\[Efunc\]) applied to a solution. An alternative way is to minimize this functional. All these methods require numerical methods to involve. It is possible, however, to roughly estimate the energy density of a vortex, based on the results of the previous subsection \[subs structure\].
There are three different contributions to the energy density of the vortex, each associated with different terms in the action (\[S\]): The kinetic energy of the scalar, $$\label{epsK}
\e_s\equiv -K(X),$$ the potential energy, $$\label{epsP}
\e_{\rm pot}\equiv V(\phi),$$ and the kinetic energy of the gauge field, $$\label{epsV}
\e_V\equiv \frac 14 F_{ij}^2.$$ First we note that the energy density inside the scalar core associated with the kinetic term $K(X)$ is approximately equal to the potential energy: $$\label{eps1}
\e_s\sim \epsilon_{\rm pot}\sim \e,$$ while the energy density of the gauge field is given by $$\label{eps2}
\e_V\sim F_{ij}^2\sim \frac{1}{e^2 l_V^4}.$$ Using (\[eps1\]), (\[eps2\]) and taking into account (\[lH\]), (\[lVless\]) and (\[lVgreater\]) it is easy to estimate the energy density of the vortex for different forms of the kinetic term $K(X)$: $$\label{Eest}
E\sim \left\{\begin{array}{lcl}
\eta^2, & & n\leq 1, \\
\eta^2 \left(\lambda v^4\right)^{1-1/n}, & & n>1.\\
\end{array} \right. \,$$ Notice that Eq. (\[Eest\]) is in agreement with Eq. (\[Efunc\]). For $n>1$ the non-linearity in $X$ is important for both the scalar and the vector fields, thus (\[Efunc\]) is directly applicable. For $n<1$ the vector core spreads wider than the scalar one, so in the region $r\gtrsim l_H$ the kinetic term takes the standard form, therefore we set $n=1$ in (\[Efunc\]) and arrive at Eq. (\[Eest\]).
An important consequence of Eq. (\[Eest\]) is that for a particular choice of the non-canonical kinetic term (namely, $n>1$), the energy per unit length of a vortex can be (much) larger than that for the standard vortex. The opposite is impossible: There is no Lagrangian that leads to vortices with small energy per unit length. One can understand this as follows: Although the scalar core can be adjusted to have a small size, (exactly as in the case of global defects [@kdefect]), the vector core nevertheless spreads widely, with the configuration close to the standard case. Thus the contribution of the vector field to the energy is roughly the same as for an usual vortex, as Eq. (\[Eest\]) shows.
Constraints on the parameters of the action {#SConstraints}
===========================================
Let us now discuss constraints on the parameters of the model. In this section we will closely follow the similar consideration for the case of global $k$-defects [@kdefect] with necessary adjustments.
First of all we must satisfy the hyperbolicity condition. Physically it means that small perturbations on the background solution do not grow exponentially. As applied to our problem, we have to check that the perturbed Eqs. (\[eom phi\]), (\[eom A\]) give hyperbolic EoMs for the propagation of small perturbations. Note that for small enough wavelengths the gauge derivative $D_\mu$ is replaced by the partial derivative $\partial_\mu$. The Eq. (\[eom phi\]) for high wave-numbers becomes the EoM for a global scalar $k$-field. The hyperbolicity condition for perturbations for $k$-essence field reads [@Halo; @Rendall] $$\label{hyperbolicity}
\frac{K_{,X}(X)}{2X K_{,XX}(X)+K_{,X}(X)}>0.$$ It is easy to check that for the Born-Infeld-like kinetic term (\[BI\]) the hyperbolicity condition (\[hyperbolicity\]) is met for $X<1/2$, while for the second example we consider, Eq. (\[X3\]), inequality Eq. (\[hyperbolicity\]) is always true.
Meantime the EoM for the gauge field (\[eom A\]) in the limit of small wavelengths coincides with the standard EoM for the normal electromagnetic field, since the r.h.s of (\[eom A\]) can be neglected in this limit.
Thus we have proved that the system of equations (\[eom phi\]), (\[eom A\]) is hyperbolic provided that the inequality (\[hyperbolicity\]) holds, and therefore there are no instabilities for small wavelengths. It is worth to note that with the above argumentation we have not proved the stability of the system for long wavelengths. This problem, however, deserves a separate investigation and is not addressed in this paper.
\[l\][$v$]{} \[l\][$\ln \lambda v^4$]{} \[l\][standard]{} \[l\][vortex]{} \[l\][quantum]{} \[l\][k-vortex]{} ![\[constraint\] Constraints on the parameters $\lambda$ and $v$ for the model (\[model\]) are shown. There are three regions in the plane of parameters: i) standard vortex solution, when the non-linearity in $X$ inside the core is small and the standard solution is restored; ii) quantum defect, when the classical picture is not valid; iii) $k$-vortex, when the non-linearity in $X$ is large and the properties of a vortex are considerably different from the standard case.](constr.eps "fig:"){width="40.00000%"}
As the second constraint on the parameters of the model we demand that the nonlinear part of $K(X)$ dominates inside the core of the defect. Otherwise we end up with a “standard” solution arising in the model with the canonical kinetic term. Thus we require $ X\gtrsim 1$, which can be brought to $$\label{Xc}
\lambda \left(\frac{\eta}{M}\right)^4\gtrsim 1.$$ Finally, the third restriction comes from the validity of the classical description. We consider vortices as classical objects, neglecting quantum effects. This picture is valid if the Compton wave length of the cube with the edge $l_H$ is smaller than the size of a scalar core $l_H$, and similar must be true for the vector core. This gives $$\label{quantum}
l_H^4 \e_s \gtrsim 1,$$ and $$\label{quantum2}
l_V^4 \e_V \gtrsim 1.$$ Eq. (\[quantum\]) can be rewritten as follows: $$\label{quant}
\l \lesssim \e^{2-2/n},$$ while (\[quantum2\]) gives simply $$\label{quant2}
e \lesssim 1.$$ It is interesting to note that the only additional constraint, as compared to the global $k$-vortices [@kdefect], is a natural inequality Eq. (\[quant2\]). We summarize the requirements (\[Xc\]) and (\[quantum\]) in Fig. \[constraint\].
Numerical solutions {#Sec Numerics}
===================
\[l\][$r$]{} \[l\] \[l\][$\alpha(r)$]{} \[l\][canonical]{} \[l\][DBI]{} \[l\][power-law]{} ![\[Fig num\] The numerical solutions for the field profiles $f(r)/v$ (solid), $\alpha(r)$ (dashed) are shown for different choice of the kinetic term $K(X)$. From the up to bottom: the standard case, $K(X)=X$; DBI term, $K(X)=1-\sqrt{1-2X}$; power-law term, $K(X)=X+X^3$. The parameters of the model are chosen such that the non-linearity in $X$ inside the core of a vortex is large, $\lambda=e=1/4$, $v=5$, The field profile for the vector part $\alpha$, is roughly the same for the different kinetic terms, in accordance with (\[lV1\]). While one can notice a strong dependence of the scalar field profile $f(r)$ on the choice of $K(X)$. The size of scalar core is in a good agreement with our estimations (\[lH1\]).](num.eps "fig:"){width="48.00000%"}
In this section we present the numerical solutions for the vortices in the model (\[act\]) \[or, equivalently, (\[S\])\] with different choices of the non-canonical kinetic term $K(X)$. We compare the obtained solutions to the standard ones. With the help of these explicit solutions we verify our general results on the properties of the gauge $k$-vortices, presented in Sec. \[SGeneral\].
We solve numerically the system of ordinary differential equation (\[EOM\]), (\[EOMa\]) for the model (\[S\]) with the following kinetic terms, $K(X)$:
- canonical term, $K(X)=X$;
- DBI-like term, $K(X)=1-\sqrt{1-2X}$;
- the power-law kinetic term, $K(X)=X+X^3$.
In Fig. \[Fig num\] the functions $f(r)$ and $\alpha(r)$ are shown for the vortex solution in the case of canonical, DBI and power-law kinetic terms. We have chosen the parameters of the Lagrangian as $\lambda=e=1/4$ and $v=5$, thus providing the non-linear in $X$ regime for the model with non-canonical terms (\[BI\]) and (\[X3\]), since for these parameters $X\sim 10^2$. One can see that the results of our general consideration, Sec. \[SGeneral\], are in a perfect agreement with the numerical results \[compare Eqs. (\[lH1\]), (\[lV1\]) with the numerical values for the sizes of the scalar and vector cores\]. The properties of $k$-vortices are indeed quite different from those for a standard vortex in the regime when the non-linearity in $K(X)$ is important. We also have found the functions $f(r)$, $\alpha(r)$ for such parameters of the model, that $X\leq 1$ inside the core of the defect. As it was expected on general grounds, the obtained solutions do not deviate much from the standard vortex solutions, since in this regime the kinetic terms (\[BI\]), (\[X3\]) have almost the canonical form.
Summary and Discussion {#SDiscussion}
======================
We have studied topological linear gauge defects (gauge vortices), in the model with a non-canonical kinetic term. The action for the model (\[act\]) contains kinetic terms for the scalar and gauge vector fields and a symmetry-breaking potential. The principal difference of the studied model from the standard one is the presence of a non-standard kinetic part for the scalar field. The term $K(X)$ in the action is in general some non-linear function of canonical kinetic term $X$.
A remarkable feature of the model is that the non-linearity of the kinetic term inevitably leads to the appearance of a new scale in the action, a “kinetic” mass, in addition to the “usual” scalar and vector masses. The presence of another mass scale in the model changes radically the basic properties of a vortex: the size of the scalar core and the energy of a vortex per unit length vary considerably as compared to the standard case.
We have investigated general properties of $k$-vortices and found restrictions on the parameters of the model, having in mind a rather general form of a kinetic term with the asymptotic behavior $K(X)\sim X^n$. Also, for the sake of simplicity and clarity of results we assumed that the scalar and vector masses are of the same order, $e\eta \sim \sqrt\lambda\eta$. We can summarize our general estimations as follows. The size of the scalar core, $l_H$, depends on the coupling $\lambda$, and mass scales $\eta$ and $M$, Eq. (\[lH1\]). A remarkable point is that $l_H$ roughly coincides with the characteristic size of the core in the case of a global $k$-vortex [@kdefect]. The size of the vector core $l_V$ does not depend on the kinetic mass $M$ and is roughly the same as in the standard case, Eq. (\[lV1\]).
Having the values for the core’ sizes one can estimate the energy of $k$-vortex per unit length, see Eq. (\[Eest\]). An important result is that the mass of a vortex radically vary depending on the choice of kinetic term. In the case $n>1$ and the limit $\lambda v^4\gg 1$, we have found a simple exact expression for the energy functional of $k$-vortex, Eqs. (\[Efunc\]), (\[F\]).
As particular examples, we studied numerically two concrete models having non-canonical kinetic terms: A DBI-like term, Eq. (\[BI\]), and a power-law term, Eq. (\[X3\]). The field profiles of domain walls for different choices of $K(X)$ are shown in Fig. \[Fig num\]: The numerical solutions are in agreement with our general estimations.
As we already discussed in our previous work [@kdefect], interesting properties of $k$-defects may have important consequences for cosmological applications. Standard cosmic strings which might have been formed during phase transitions in the early universe have a mass scale directly connected to the temperature of a phase transition $T_c$, $\mu\sim \eta^2 \sim T_c^2$. By contrast, the mass scale of a resulting $k$-string depends both on $T_c$ and the kinetic mass $M$. This means that the tension of $k$-strings may not be close to $T_c^2$, thus helping to avoid constrains on cosmic strings $G\mu\lesssim 10^{-7}$ [@csc1; @csc3] (or even stronger, $G\mu<3\times 10^{-8}$, see [@csc2]). Meanwhile theoretical predictions give $G\mu\sim 10^{-6}-10^{-7}$ for GUT strings. If, however, physics at the GUT scale involves non-standard kinetic terms, then the GUT phase transition may have lead to the formation of cosmic strings with smaller tension, $G\mu \ll 10^{-6}$, thereby evading conflicts with the present observations.
It is a pleasure to thank M. Kachelriess for critical reading of the manuscript. This work was supported by an INFN fellowship grant.
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---
abstract: 'Optical dipole traps and fractional Talbot optical lattices based on the interference between multiple co-propagating laser beams are proposed. The variation of relative amplitudes and phases of the interfering light beams of these traps makes it possible to manipulate the spatial position of trapped atoms. Examples of spatial translation and splitting of atoms between a set of the interference traps are considered. The prospect of constructing all-light atom chips based on the proposed technique is presented.'
author:
- 'Yuri B. Ovchinnikov'
title: 'Coherent manipulation of atoms by co-propagating laser beams'
---
Introduction
============
There are three general types of conservative atom traps: magnetic, electrostatic and optical dipole traps. This paper concentrates on a special kind of all-light atom chip traps. First we review the progress in magnetic surface traps, which have been the basis for atom chips to date.
Magnetic atom chips [@Folman] are becoming more and more popular experimental technique to produce and manipulate Bose-Einstein condensed (BEC) coherent ensembles of ultra-cold atoms [@BEC] . The main advantage of magnetic chips is their compactness, relatively low energy consumption and low cost, which is related to the proximity of atoms to the sources of magnetic field. Most of the chips use current caring wires to create strong non-uniform magnetic fields. Initially magnetic chips were used only to trap [@Hansch] and guide [@Prentiss] cold thermal atoms, now the more complicated versions of them are used to get BEC [@Hansch_2] and even to implement integrated atom interferometers [@Schumm]. In spite the fact that the current consumed by each of the elements of the magnetic chip is essentially smaller compared to the macroscopic magnetic traps, it is far above the currents used in traditional integrated electronic elements. Further complication of magnetic atom chips will sooner or later have to face up to the problem of removing dissipated heat. The further miniaturization of the magnetic chip elements is restricted by the fluctuations of the magnetic field caused by thermally excited currents in the metal wires and some other effects \[8\], which rapidly increase with decreasing distance between the trapped atoms and the surface of the chip.
All-light atom chips involve manipulation of atomic motion with light forces only. To provide coherent manipulation of atoms in these chips far-detuned dipole traps [@GWO] have to be used. There are multiple advantages of these chips over the magnetic chips. The all-light chips are free of the problems of heat dissipation and the absorption of light in their substrates is rather small. The Van-der-Waals interaction of atoms with a surface of a dielectric substrate becomes important only at sub-micron distances [@Aspect]. Another advantage of the light traps is that they can confine atoms in all magnetic energy states including the magnetically in-sensitive ones. This last property of the all-light chips makes them very advantageous for precise measurements. Finally, the cold ensembles of molecules can be formed and stored in these traps [@Grimm].
The development of all-light atom chips lags behind that of magnetic chips. One approach to the manipulation of atoms near the surface of a dielectric substrate is based on evanescent light waves [@Dowling]. In fact the earliest proposal of a surface trap for neutral atoms was a light dipole trap based on a superposition of two evanescent light waves of different frequencies [@Ovchinnikov]. The first observation of atoms cooled in an evanescent light trap was made in 1997 [@Ovchinnikov_2] after the cooling mechanism of atoms in evanescent waves became available [@Soeding]. The evanescent light wave traps are still under development and a recent result is that BEC Cs atoms have been produced in such a trap [@Rychtarik], this was not possible in magnetic traps due to specific properties of collisions between cold Cs atoms. Recently also the surface waveguides [@Prentiss_2] and two-dimensional surface optical lattices [@Christandl] based on the bichromatic evanescent waves have been proposed.
Another approach to the surface light traps, which was developed in parallel to evanescent wave traps, is based on standing light waves [@Mlynek]. Recently an integrated atom-optical circuit, which combines cooling of atoms in a surface magneto-optical trap and their subsequent transportation along the waveguide formed by a standing light wave, has been demonstrated [@Mlynek_2]. Finally, in [@Ertmer_1; @Ertmer_2] a microscopic array of independent dipole light traps formed by an array of microlenses was realized.
These light and magnetic chips technologies complement each other nicely, as demonstrated by the new generation of combined atom chips [@Wang; @Eriksson].
This paper introduces another approach to the construction of near-surface all-light atom optics elements based on the interference between multiple co-propagating laser beams. It also demonstrates how to coherently transfer and split atoms between arrays of such traps, which opens the prospects of building new interferometers, quantum registers and other missing elements of all-light chips.
Following this introduction the second section considers basic multi-beam traps, and concentrates on a two-beam trap, because it is the simplest configuration interference trap. In the third section the basic manipulations of the spatial position of the trapped atoms are considered. The fourth section considers optical lattices based on a fractional Talbot effect. In the outlook section the prospects of building all-light atom chips are briefly discussed. Finally, in the conclusion the results of this investigation are summarized.
Multi-beam interference dipole traps
====================================
The simplest trap based on the interference of co-propagating laser beams is a two-beam dipole trap. Although the two-beam trap is not the most ideal trap of this kind, it is the best to explain the general principle. The trap is formed by the two Gaussian laser beams, which are co-propagating in z direction (see Fig.1). The waists of the beams locate in the X0Y plane. Our interest is the region A, where the interference of the two beams forms the fringes, which can be used for three-dimensional conservative trapping of atoms by dipole light forces [@GWO]. The amplitude of the light field can be written as
$$\begin{split}
E(x,y,z)=\sum_{j=1}^n\sqrt{\frac{2}{\pi}} \frac{1}{w^2(z)} \exp \left[-\frac{(x-x_j)^2+(y-y_j)^2}{w^2(z)}\right]\times\\
\exp\left[-i \left( kz-\tan^{-1}(z/z_R)+\frac{k((x-x_j)^2+(y-y_j)^2)}{2(z+z_R^2/z)} \right) \right],
\end{split}$$
where for a two-beam trap $n=2$, $x_1=-d/2$, $x_2=d/2$, $y_{1,2}=0$, $d$ is the distance between the beams, $w(z)=w_0
\sqrt{1+(z/z_R)^2}$ is the radius of each of the two beams, $w_0$ is their waist radiuses, $k=2 \pi/\lambda$ is the wave vector and $z_R=\pi w_0^2/\lambda$ is the Rayleigh range. The amplitude of each of the two beams is normalized to unit power. We suppose here that both beams are of the same linear polarization directed along the y-axis. The spatial distribution of the light intensity resulted from the interference between the two beams is described by the formula
$$\begin{split}
I(x,y,z)=\frac{2}{\pi w^2(z)}\times \exp\left(-\frac{d^2+4(x^2+y^2)}{2 w^2(z)}\right)\times\\
\left[\exp\left(-\frac{2 d x}{w^2(z)}\right)+
\exp\left(\frac{2 d x}{w^2(z)}\right)+
2\cos\left(\frac{2 \pi d x z}{\lambda (z^2+z_R^2)}\right)\right],
\end{split}$$
The interference between the two beams forms intensity fringes along the x-axis, which are described by the cosine term inside the square brackets. These fringes are very well known from the Young’s double slit interferometer. The first exponential term of equation (2) shows that the intensity of the central interference fringes reaches maximum at $z \neq 0$. For the central fringe, when $x=0$ and $y=0$, the maximum of intensity is reached at
$$z=z_{max}=z_R \sqrt{\frac{d^2}{2 w_0^2}-1}.$$
When the waist radii, $w_0$ of the beams and the distance, $d$ between them are comparable to each other, the interference pattern consists mostly of a single central fringe. Fig.2 shows the intensity distribution of the two-beam interference pattern in the X0Z plane for $z>0$, $w_0=\lambda$ and $d=3 \lambda$. One can see that the central interference fringe is partially separated from the waist regions of the two source beams. The cross sections of the spatial intensity distribution of the central interference fringe along the x, y and z coordinate axis are shown on Fig.3a,b,c. The maximum of the light intensity in this fringe is reached at $z=5.66\,\lambda$ in a full agreement with the formula (3). Fig.3d shows the intensity distribution along the line $z=z_{max} \pm 2 x z_{max}/d$, which connects the centre of the fringe to the centres of the waists of the source beams. These two directions define the potential depth of the corresponding red-detuned dipole trap formed by the central interference fringe. It is expected that the escape of atoms from the trap due to their heating or tunnelling will happen mostly along these directions.
According to the formula (2) the interference trap is scalable. If the sizes of the interfering beams $w_0$ and the distance between them $d$ are increased proportionally by the same factor $a$, the transverse size of the trap along the x and y axes also increases by factor $a$, while the longitudinal size along the z-axis increases by factor $a^2$. This scalability law is exactly the same as for a single Gaussian light beam. The situation when the sizes of the light beams stays fixed and only the distance between them is increased is very different. In that case there additional intensity fringes along the x-axis of the interference trap become visible.
To increase the degree of localization of the central interference fringe, a larger number of the interfering light beams must be used. Let us skip the three-beam case and consider the trap, which consists of four equidistant co-propagating light beams with waist radiuses of $w_0=\lambda$, which are located at the corners of a square with side length $d=3\,\lambda$, as it is shown in Fig.4. The amplitude of the interference field produced by the interference between the four beams is described by the formula (1), where the coordinates of the centres of the four Gaussian beams $(x_j,\,y_j)$ are (-d/2,d/2), (-d/2,-d/2), (d/2,d/2) and (d/2,-d/2). The spatial distribution of the light intensity in the r0Z plane, which is crossing the square trap along its diagonal r (see Fig.4), is shown in Fig.5. The maximum of the intensity in the central interference fringe is localized at $z=8.885 \lambda$. Fig.6a,b,c,d shows the spatial intensity distribution along four different directions around the maximum of the fringe. The last graph shows the intensity distribution along the line which joins the waist centre of one of the source beams and the centre of the interference fringe. One can see that the separation of the fringe is nearly complete.
To estimate the potential depth of the red-detuned trap formed by the interference of the four co-propagating laser beams we will take an example of Rb$^{87}$ atom trapped by the coherent light of a Nd:YAG laser with wavelength of $\lambda=1.064\, \mu$m. We will use a two-level model of the atom, the wavelength of which $\lambda_0=0.780\, \mu$m corresponds to the strongest $5S^{1/2} \rightarrow 5P^{3/2}$ transition of Rb$^{87}$. Such a frequency of the trapping light is chosen to minimize the probability of the spontaneous scattering of photons by an atom and therefore to preserve the coherence of the light-atom interaction. The dipole potential formed by the light can be estimated from the formula [@GWO]
$$U_{dip}(x,y,z)=-\frac{3 \pi c^2}{2 \omega_0^3}\left(\frac{\Gamma}{\omega_0-\omega}+\frac{\Gamma}{\omega_0+\omega}\right) I(x,y,z),$$
where $\omega_0=2 \pi c/ \lambda_0$ is the frequency of the atomic transition, $\omega$ is the frequency of the light field, $\Gamma$ is the linewidth of the transition and $I(x,y,z)$ is the spatial distribution of the light intensity. We neglect here the fine structure of the transition between the ground 5S state of Rb$^{87}$ and its first 5P excited state, which causes just a small correction to the magnitude of the potential (4) for the chosen frequency of the trapping light. For the laser power 1.6mW in each beam of the four-beam trap this formula gives the total depth of the central interference fringe of 82$\mu$K. The corresponding oscillation frequencies of the Rb atoms around the minimum of the trap are equal to $f_x=f_y=18.7$kHz and $f_z=2.6$kHz, which gives the mean frequency of the trap of $f_0=16.6$kHz. The rate of spontaneous scattering of the trapped atoms can be estimated from the formula [@GWO]
$$\begin{split}
\Gamma_{scat}(x,y,z)= -\frac{3 \pi^2 c^2}{2 \hbar \omega_0^3}
\left( \frac{\omega}{\omega_0}\right)^3\\
\left(\frac{\Gamma}{\omega_0-\omega}+\frac{\Gamma}{\omega_0+\omega}\right)^2
I(x,y,z).
\end{split}$$
For the parameters of the light field given above the minimum time between the spontaneous scattering events $1/\Gamma_{scat}$ near the bottom of the trap is about 1 second. To get longer coherence times without scarifying the potential depth, CO$_2$ lasers with a wavelength of 10$\mu$m can be used [@THG].
Additional heating of the trapped atoms can be caused by technical noise of the laser beams. Periodic change of the depth of the dipole trap causes transition of the atoms to the higher vibrational states of the trap potential. Such parametric heating of the atoms in the multi-beam interference trap due to fluctuations of the light intensity is expected to be the same as in other dipole traps and is relatively small [@Savard; @Alt] even for free-running industrial lasers.
The other source of parametric heating is a motion of the trapping dipole potential caused by fluctuations of the relative phases of the interfering laser beams. The corresponding transition rate of trapped atoms from the ground to the first vibration state of the trap is
$$R_{0 \rightarrow 1}=\frac{2 \pi^3 M f_x^3}{\hbar} S_x(f_x),$$
where $f_x$ is the transverse frequency of the trap, $M$ is the mass of atom and $S_x(f_x)$ is the spectral density of the position fluctuations in the trap centre along the $x$ axis [@Gehm]. According to this equation, if an atom is confined in a trap with an oscillation frequency $f_x=18.7\,$kHz, achievement of the transition rate of $R_{0 \rightarrow 1}=1\,s^{-1}$ requires a position stability of $\sqrt{S_x(f_x)}=2.4\times10^{-6}\,\mu$m/$\sqrt{\textnormal{Hz}}$. The change of the position of the interference trap along the $x$ axis $\epsilon=\alpha \Delta\phi$ is a linear function of the phase difference between the interfering laser beams $\Delta\phi$, where the proportionality coefficient $\alpha=D/2\pi$ is determined by the period of the interference fringes $D \approx
z_{max} \lambda/d$ along the $x$ axis at $z=z_{max}$. For the given above example of a four beam interference trap the period of the interference fringes along $x$ axis at $z=z_{max}=9.45\,\mu$m is $D \approx 3\,\mu$m. Therefore, to achieve $R_{0 \rightarrow
1}=1\,s^{-1}$, the required relative phase stability of the laser beams is $\sqrt{S_{\phi}(f_x)}=5\times10^{-6}\,\textnormal{rad}/\sqrt{\textnormal{Hz}}$. According to [@Ye] the phase spectral density of a phase locked Nd:YAG laser at frequency $f=1$kHz is better than $\sqrt{S_{\phi}(f)}=3\times10^{-6}\,\textnormal{rad}/\sqrt{\textnormal{Hz}}$. It means that even independent lasers can provide the phase stability required. If all the trapping beams are generated from the same laser, their relative phase fluctuations should be essentially smaller than the absolute phase noise of the laser light. In fact in standard optical lattices the demands on the phase stability of the laser beams are stricter than in the multibeam copropagating interference dipole traps for two reasons. First, the typical vibration frequency of the optical lattice potential is higher ($\ge$100kHz) and according to the equation (6) the parametric heating rate of atoms grows rapidly with the frequency. Second, all the beams of the copropagating interference trap can be formed from different parts of a single laser beam. It means the relative phases of these beams will be as stable as the phase across a single laser beam, which is not the case of standard optical lattices, where the laser beams have to propagate along different passes before they interfere with each other.
Manipulation of atoms
=====================
A coherent dipole light trap is a very useful element of atom optics. However many important applications such as atom interferometry [@Berman] or quantum computing [@QC] require the ability to manipulate atoms between or within these traps. In this paper only the manipulation of a spatial position of atoms between the interference traps will be described. There are two basic types of such manipulation, the translation of atoms and their splitting between two or more dipole traps. Let us consider a spatial translation of atoms first. Certainly the simplest way to translate trapped atoms in space is by adiabatic change of the spatial position of the trapping light beams, but that is not possible for all-light atom chips, in which the position of the light beams emitted from the surface of the chip is fixed and determined by the design of the chip. The only parameters, which can be changed are the intensities, phases, polarizations or frequencies of the individual light beams. Multi-beam interference traps offer the possibility of translating atoms in all directions, but only the translation of atoms across the light beams is considered here.
One way to move atoms across the interfering light beams is to change their relative phases. For a two-beam interference trap the change of the phase of one of the beams will lead to the transverse displacement of the interference fringes. The problem of the two-beam trap is that the shifted central interference fringe eventually becomes merged with one of the source beams, which leads to transfer of the trapped atoms into the waist region of this beam. For an atom chip, where the waists of the source beams are located in the plane of the surface of the chip, that means atom reaching the surface of the chip, thermalize, and escape from the trap.
As shown in the previous section the four-beam trap allows atoms to be confined and well separated from the source light beams. The translation of the trapped atoms along the x- and y-axis of this trap preserves that separation of the central interference fringe from the source beams. Fig.7 shows the intensity distribution of the light field in the X0Z plane while the phases of the beams 3 and 4 of the four-beam trap are changed by the same amount relative to the beams 1 and 2 (see Fig.4). The parameters of this four-beam trap are the same as ones used in the previous section. One can see that the centre of the trap is smoothly moving along the x-axis while the phase of the beams 3 and 4 is changed. During the shift of the central fringe the second interference fringe from the side opposite to the direction of motion of the central fringe is developed. Nevertheless these two fringes stay well separated from each other by a distance of about 3$\mu$m. By choosing the proper intensities of the laser beams and the rate of the spatial translation of the fringe it is possible to keep the tunnelling probability of the translated atoms to the other interference fringe negligibly small. The procedure described provides a unidirectional shift of the trapped atoms along the x-axis. Fig.8 shows the evolution of the intensity distribution of the trap along the x-axis taken at z=8.885$\lambda$, where the maximum of the fringe is located, as a function of the phase difference between the two pairs of the light beams. The dashed lines show the borders of the trap, which correspond to the x-coordinates of the centres of the four trapping light beams. From the last graph of Fig.8 it follows that at $\phi=\pi$ the position of the shifted interference fringe reaches the border of the trap. Let us suppose now that there are two four-beam traps are set up next to each other in such a way that they have a common border with two pairs of their light beams are superimposed as shown in fig.9. The adiabatic transfer of atoms between these two traps can be organized as follows. First the second trap is completely off. By variation of the phase difference between the light beams of the first trap form 0 to $\pi$ atoms are moved towards the common border. Then the intensity of the first trap is switched off while the intensity of the second trap is simultaneously switched on. The phases of the beams in the second trap are set in such a way to pick up the atoms at the common border of the traps. Finally the adiabatic change of the phases of the light beams of the second trap moves atoms to its centre and the process of transfer of atoms between the two traps is completed. This elementary process can be expanded to a large number of neighboring traps so that an all-light conveyor belt similar to the magnetic one [@Hansel] can be realized.
The next important type of atomic manipulation is the coherent splitting of the atomic ensemble into two identical parts. Such a coherent beam splitter is a key element of any interferometer. In the last few years the interferometers based on conservative traps, which can be adiabatically transformed into double-well traps, were actively investigated [@Hansel_2; @Shin_1; @Shin_2; @Schumm]. In [@Hansel_2] it is proposed to split the atomic ensemble by adiabatic transportation of a single well potential into a double well trap and then to recombine the two wells back into a single well trap. Some possible phase difference between the atoms in two “arms” of the interferometer would lead to variation of the population of the excited oscillation states of that final single well trap. A simpler approach is used in [@Shin_1; @Shin_2; @Schumm], where it is proposed to split atoms coherently in a double well trap and then to release atoms by switching off the trap potential. As a result interference fringes like a classical Young’s double slit interferometer were observed. The experiments [@Shin_2; @Schumm] relied on magnetic microtraps. A very different approach was demonstrated in [@Shin_1] where an optical double well trap was used. This optical trap consists of two Gaussian beam traps of different frequencies, which do not interfere with each other in a sense that they do not produce a stationary interference pattern. A smooth spatial translation of these traps with respect to each other allows both merger into one trap and separation to form a double well potential.
The splitter based on the interference dipole traps can also be made by placing two non-interfering traps next to each other. The difference is that the transformation of the dipole potential is achieved now, not by moving of the trapping light beams, but by changing their relative phases. The corresponding configuration of the trap, which consists of six co-propagating laser beams is shown in Fig.9. This figure shows the intensity distribution of light in the X0Y plane, where the waists of all the beams are located. The distance between the adjacent beams along the x- and y-axes is d=3$\lambda$ and their waist radii are $w_0=\lambda$. The amplitude of the light field in the beams 3 and 4 is by factor $\sqrt{2}$ larger compared to the beams 1, 2, 5 and 6. The arrows shows the direction of the polarization of light in each of the beams. This configuration corresponds to the two four-beam traps with crossed polarizations, which have a common side located at x=0. We suppose that initially the phases of the outer beams 1, 2, 5 and 6 $\phi_1=\phi_2=\phi_5=\phi_6=\pi$ are in anti-phase with the phases of the central beams 3 and 4 $\phi_3=\phi_4=0$. In these conditions the interference of the beams forms a fringe, the maximum of which is located at x=y=0, z=8.43$\lambda$. The intensity distribution of light in that fringe along the x-axis taken at z=8.43$\lambda$ is shown by the highest curve in fig.10. The other curves in this figure shows the stages of the transformation of this single well potential into a double well trap while the phases of the outer beams are changed gradually from $\pi$ to 0. Finally at $\phi_1=\phi_2=\phi_5=\phi_6=0$ the two maximums of the double well trap are located exactly at the centres of the two adjacent four-beam traps at x=$\pm 1.5 \, \lambda$.
It is possible to perform a coherent splitting of atoms between the two four-beam traps formed by six beams of the same polarization by changing their relative phases. In that case an additional interference between the outer most beams of the traps leads to more complicated evolution of the trap potential.
Talbot optical lattices
=======================
The idea of trapping atoms in the dipole potential of a standing light wave has been put forward in 1968 [@Letokhov] long before the laser cooling of atoms was achieved. The potential formed by a standing light wave is unique, because it has an ideally sinusoidal form of periodicity, determined by the wavelength of light. Standing light waves were used in a huge number of experiments on diffraction and interference of thermal atomic beams [@Berman; @Metcalf]. A new interest to the standing light waves appeared after Bose-Einstein condensates of cold atoms became available. The BEC loaded into three-dimensional lattices has allowed one for the first time to get filling factor more than 1 and to observe such an effect as a superfluid - Mott insulator transition [@Bloch]. The interest of the standing light waves might be explained also by the fact that an essential part of proposals for quantum computing rely on optical lattices [@QC]. The advantage of the optical lattices in the quantum computing applications is the the possibility of performing the same quantum logic operations in parallel over a large number of identical sites of the lattice. However there are still some unsolved problems with addressing and selective manipulation of atoms between several distinct sites of optical lattices based on standing light waves.
In [@Ertmer_1; @Ertmer_2] a two-dimensional lattice of about 80 dipole traps, separated from each other by a macroscopic distance of 125$\mu$m has been demonstrated. The lattice was made by focussing a laser beam with a microfabricated array of microlenses. Addressing of a single site of the lattice and modification of the trapping potential with an additional inclined laser beam was demonstrated experimentally.
An alternative approach is to trap of atoms using fractional Talbot optical lattices, which are based on the interference of an array of multiple co-propagating equidistant laser beams. Compared to a case of an array of independent dipole traps [@Ertmer_1; @Ertmer_2], the Talbot optical lattice permits a much smaller period of the lattice and control of the form of all or part of the interference potential by changing the phases and amplitudes of the individual source beams.
The Talbot effect [@Talbot] consists of reproducing the transverse spatial distribution of light coming through a transmission diffraction grating at the distances from the grating, which are multiples of a Talbot period $T=2 d^2/\lambda$, where $d$ is a period of the diffraction grating. For a grating with infinite number of slits, the transverse spatial distribution of light intensity at Talbot distances is exactly the same as the intensity distribution of light in the plane of the grating. This is valid for both a one-dimension and two-dimension gratings of a definite period.
At fractional distances $T \times M/N$, where $M$ and $N$ are coprime integers, the fractional Talbot images are observed [@FT]. The transverse period of the intensity distribution of light at fractional Talbot distances is $2 d/N$ for even $N$ and $d/N$ for odd values of $N$. The fractional Talbot optical images of different orders are separated from each other in space, because they are located at different distances from the grating and also shifted with respect to each other transversally.
These periodic spatial distributions of light formed at fractional Talbot distances by the interference of a periodic array of light beams can be used as atom traps.
We will skip the case of a one-dimensional Talbot grating and describe the most interesting case of the square two-dimension grating, which is formed by 5$\times$ 5 square array of 25 co-propagating Gaussian laser beams. The waists of the beams are located in the X0Y plane with coordinates of their centres $x_j=n d$ and $y_j=m d$, where $n,m=-2,\,-1,\,0,\,1,\,2$ and $d=10\,\lambda$ is the period of the array. The waist radiuses of each of the beams are $w_0=\lambda$. The corresponding Talbot period of the lattice is $T=200\,\lambda$. Fig.11a shows the spatial intensity distribution of the corresponding interference pattern in the XZ plane at y=$d/2$. That specific plane was taken to see the intensity maximums of the fractional Talbot lattice at $z=T/2=100\,\lambda$, where the interference fringes are shifted by half of a period with respect to the position of the source beams in the X0Y plane. In full accordance with theory the transverse period of the $T/2$-lattice is equal to $d$. The other fractional Talbot lattice of nearly same intensity and of twice smaller period $d/2$ is located at $z=T/4=50\, \lambda$. Fig11b and 11c show the transverse distributions of the light intensity in the XY plane, taken at $z=T/2$ and $z=T/4$ correspondingly. To get a better idea on the intensities of the different fractional Talbot lattices, the distribution of the light intensity along the $z$-axis at $x=y=d/2$ is presented in fig.12a. One can see that the intensity of the $T/4$-lattice is comparable to the intensity of the $T/2$-lattice. Calculations show that the intensity of the interference fringes at the full Talbot distance $z=T=200\,\lambda$, the locations of which in the XY plane are the same as the positions of the source beams, is about four times smaller compared to the fringes of the $T/2$-lattice. Finally, fig.12b shows the transverse distribution of the light intensity in the $T/2$-lattice taken along the x-axis at $z=T/2$ and $y=d/2$.
To estimate the potential depth of the Talbot lattices we use again an example of Rb$^{87}$ atoms and the wavelength of light of a Nd:YAG laser. For the laser power of 4mW in each of the 25 source beams of the lattice the potential depth of the central sites of the $T/2$-lattice is about 91$\mu$K. The oscillation frequencies of the atoms near the centres of the sites of the lattice are $f_z=1.74$kHz and $f_x=f_y=16.25$kHz.
Due to the large number of the source beams involved in the Talbot lattices, they offer a huge variety of ways to manipulate the trapped atoms by changing the phases and amplitudes of different groups of the source beams. The methods described in the previous section for the four-beam interference traps can be easily extended to the Talbot lattices. For example, a change of the relative phases and amplitudes of different rows of the source beams can provide a unidirectional transfer of atoms between the rows of the Talbot lattice. The shift of the phases of the even source beams with respect to the odd ones can change the period of the Talbot lattice potential by factor of two [@Ovchinnikov_3].
Another important difference of the Talbot lattice compared to the standing light wave lattices is that the form of its potential can essentially differ from a sinusoidal form. The general rule is that the intensity distribution of light in the Talbot lattice reproduces the corresponding distribution of the light beams in the source plane X0Y. Therefore, the different sites of the Talbot lattice can be well isolated from each other, which is important for applications where the tunnelling of atoms between different sites of the lattice is undesirable.
Outlook
=======
In this paper the interference traps formed by co-propagating Gaussian laser beams have been considered. The same principles are also valid for the light beams formed by diffraction of light at round holes. The construction of an all-light atom chip might be as follows. The desired matrix of holes is made in a non-transparent screen, which is mounted on top of some optical substrate. Micro-electro-optical elements are positioned right above or below each of the holes to control the amplitudes and phases of the individual source light beams. This construction would allow one to address each of the beams with electrical signals. It was demonstrated that interference traps are scalable, which means all sizes of the traps and their distance from the waists of the source beams depend on the sizes of the beams and the distances between them. Therefore using holes of different diameter in the chip would allow manipulation of the spatial position of atoms not only along but also across the surface of the chip.
An alternative chip construction might use an array of single mode polarization mantaining fibres, which could each be addressed individually.
The negative sign of the frequency detuning of the trapping light of the interference traps and Talbot lattices provides an opportunity to use these traps for trapping such atoms as Sr, Yb, Ca and some other atoms in a regime when the optical Stark shift of the intercombination transition of these atoms is cancelled [@Katori]. Thus fractional Talbot lattices may provide an alternative option for the construction of portable optical frequency standards.
Conclusion
==========
The dipole traps based on the interference between multiple co-propagating laser beams can be configured to enable atom manipulation by changing the phases, polarizations and intensities of the individual light beams. Such basic manipulation of cold atoms in these traps as their spatial displacement and coherent splitting in two identical parts have been outlined. A Talbot optical lattice dipole trap based on the interference of arrays of co-propagating laser beams is proposed. It is shown that single sites of the Talbot lattice can be well isolated from each other. In addition the Talbot lattice provides a number of ways to transfer atoms between lattice sites. The variable period of the considered Talbot optical lattices enables one to choose the right compromise between the addressability of the individual sites of the lattice and the interaction of the sites with each other.
It is shown that with rather moderate power (1-4mW) laser beams with wavelength 1$\mu$m, the depth of interference traps for Rb$^{87}$ atoms can be $\sim$100$\mu$K, while the coherence time is of the order of 1 second.
This technique offers the opportunity of building a new generation of all-light atom chips based on interference traps, which might find applications in the design of integrated elements of quantum logic, atom interferometers and other atom optics components.
The examples of the interference traps and their manipulation presented in this paper do not pretend to be the most optimal ones, but are intended only to demonstrate the main principles. The most optimal and efficient schemes for interference traps and methods of atom manipulation still have to be developed.
Acknowledgments
===============
Many thanks to Hugh Klein for his valuable comments to the paper. This work was funded by the UK National Measurement System Directorate of the Department of Trade and Industry via the Strategic Research Programme of NPL.
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Fig.1 Schematics of the two-beam interference trap.
Fig.2 Spatial intensity distribution of light in the XOZ-plane of a two-beam interference trap.
Fig.3 Cross sections of the intensity distribution of light in the central interference fringe of the two-beam interference trap, which are taken along four different directions.
Fig.4 Intensity distribution in the four-beam trap in the XOY-plane, where the waists of the beams are located.
Fig.5 Spatial intensity distribution of light in the XOZ-plane of the four-beam interference trap.
Fig.6 Cross sections of the intensity distribution of light in the central interference fringe of the four-beam interference trap, which are taken along four different directions.
Fig.7 Intensity distribution in the four-beam trap in the XOZ-plane as a function of the phase difference between the two pairs of source laser beams.
Fig.8 Cross sections of the intensity distribution of light in the moving central interference fringe of the four-beam interference trap, which are taken along the x-axis at $z=8.885\,\lambda$ for several different values of the phase shift between the two pairs of source light beams.
Fig.9 Intensity distribution in the two superimposed four-beam traps in the XOY-plane, where the waists of the beams are located. The arrows shows the polarizations of the six light beams.
Fig.10 Evolution of the intensity distribution of light along the x-axis in a superposition of two four-beam interference traps, which is taken at $z=8.43\, \lambda$ for five different values of the phases of the outer laser beams 1, 2, 5 and 6.
Fig.11 a) Intensity distribution of light in the XZ-plane at y=d/2 produced by 5x5 two-dimensional array of Gaussian light beams. b) Intensity distribution across the fractional Talbot optical lattice taken at $z=T/2=100\, \lambda$. c) Intensity distribution in a higher order lattice at $z=T/4=50\, \lambda$.
Fig.12 a) Cross section of the intensity distribution in the fractional Talbot lattices along the z-axis at x=y=d/2. b) Cross section of the transverse intensity distribution along the x-axis at $z=T/2=100\, \lambda$ and y=d/2.
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abstract: 'We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical expectations for two distinct probability measures describing the statistics of the length vectors when the number of links tends to infinity. In the proof we use a combination of geometric and analytic tools. The average Betti numbers are expressed in terms of volumes of intersections of a simplex with certain half-spaces.'
author:
- |
Michael Farber\
Department of Mathematical Sciences\
University of Durham, UK
- |
Thomas Kappeler\
Institute of Mathematics\
University of Zurich, Switzerland
date: 'November 19, 2006'
title: Betti numbers of random manifolds
---
Introduction
============
In various fields of applications, such as topological robotics, configuration spaces of mechanical systems depend on a large number of parameters, which typically are only partially known and often can be considered as random variables. Since these parameters determine the topology of the configuration space, the latter can be viewed in such a case as a [*random topological space*]{} or a [*random manifold*]{}. To control such a system one has to understand geometry, topology and control theory of random manifolds.
One of the most natural notion to investigate is the mathematical expectation of the Betti numbers of random manifolds. Clearly, these average Betti numbers encode valuable information for engineering applications; for instance they provide an average lower bound for the number of critical points of a Morse function (i.e. observable) on such manifolds.
In this paper we consider a specific instance of this general problem. We study closed planar $n$-gons whose sides have fixed lengths $l_1, \dots, l_n$ where $l_{i}>0$ for $1 \le i \le n$. The [*polygon*]{} space $$\begin{aligned}
M_\ell = \{(u_1, \dots, u_n)\, \in \, S^1\times \dots\times S^1;
\, \sum_{i=1}^n l_iu_i \, =\, 0\in {{\mathbb C}}\}\, /\, {\rm {SO}}(2)\end{aligned}$$ parametrizes the variety of all possible shapes of such planar $n$-gons with sides of length $l_1, \dots, l_n$. The unit vector $u_i\in {{\mathbb C}}$ indicates the direction of the $i$-th side of the polygon. The condition $\sum l_iu_i=0$ expresses the property of the polygon being closed. The rotation group ${\rm {SO}}(2)$ acts on the set of side directions $(u_1, \dots, u_n)$ diagonally.
The polygon space $M_\ell$ emerges in topological robotics as the configuration space of the planar linkage, a simple mechanism consisting of $n$ bars of length $l_1, \dots, l_n$ connected by revolving joints forming a closed planar polygonal chain. The positions of two adjacent vertices are fixed but the other vertices are free to move and the angles between the bars are allowed to change.
The spaces $M_\ell$ also appear in molecular biology where they describe the space of shapes of closed molecular chains.
Statistical shape theory is another subject where the spaces $M_\ell$ play a role: they describe the space of shapes having certain geometric properties with respect to the central point, see [@Kendall].
The configuration space $M_\ell$ depends on the length vector $$\begin{aligned}
\ell=(l_1, \dots,l_n)\in {{\mathbb R}}^n_+ \end{aligned}$$ in an essential way. Here ${{\mathbb R}}^n_+$ denotes the set of vectors in ${{\mathbb R}}^n$ having nonnegative coordinates. Clearly, $M_\ell = M_{t\ell}$ for any $t>0.$
The length vector $\ell$ is called [*generic*]{} if $\sum\limits_{i=1}^n l_i\epsilon_i\not=0$ for any choice $\epsilon_i=\pm 1$. It is known that for a generic length vector $\ell$ the space $M_\ell$ is a closed smooth manifold of dimension $n-3$. If the length vector $\ell$ is not generic then $M_\ell$ is a compact $(n-3)$-dimensional manifold having finitely many singular points.
The moduli spaces $M_\ell$ of planar polygonal linkages were studied extensively by W. Thurston and J. Weeks [@TW], K. Walker [@Wa], A. A. Klyachko [@Kl], M. Kapovich and J. Millson [@KM1], J.-Cl. Hausmann and A. Knutson [@HK] and others. The Betti numbers of $M_\ell$ as functions of the length vector $\ell$ are described in [@FS]; we recall the result of [@FS] later in §\[sec1\]. A. A. Klyachko [@Kl] found the Betti numbers of spacial polygon spaces.
In this paper we view the length vector $\ell\in {{\mathbb R}}^n_+$ as a random variable whose statistical behavior is described by a probability measure $\mu$ on ${{\mathbb R}}^n_+$. The $p$-dimensional Betti number $b_p(M_\ell)$ is then a function $$\begin{aligned}
b_p: {{\mathbb R}}^n_+\to {{\mathbb Z}}, \quad \ell \, \mapsto \, b_p(M_\ell)\in
{{\mathbb Z}}.\end{aligned}$$ Our goal is to find the asymptotics for $n$ large of the [*average Betti numbers*]{}, defined by $$\begin{aligned}
\label{barbi} b_p(n,\mu) =
\int_{{{\mathbb R}}^n_+} b_p(M_\ell)d\mu.\end{aligned}$$ The choice of measure $\mu$ reflects apriori information about the problem. We study in detail two special choices for $\mu$:
**
1. $\mu=\mu_a$ is the probability measure on ${{\mathbb R}}^n_+$ supported on the unit simplex $\Delta^{n-1}$ such that $\mu|_{\Delta^{n-1}}$ coincides with the Lebesgue measure on $\Delta^{n-1}\subset {{\mathbb R}}^n_+$ normalized so that $\mu(\Delta^{n-1})=1$. Recall that $\Delta^{n-1}\subset {{\mathbb R}}^n_+$ is described by the inequalities $l_i\geq 0$, $\sum l_i=1$.
2. $\mu=\mu_b$ is the probability measure on ${{\mathbb R}}^n_+$ supported on the unit cube $\square^n\subset {{\mathbb R}}^n_+$ such that $\mu|_{\square^n}$ is the Lebesgue measure. Here $\square^n$ is given by the inequalities $0\leq l_i\leq 1$, $i=1, \dots, n$ and $\mu(\square^n)=1$.
The main result of this paper states that in cases (a) and (b) for any fixed $p \ge 0$ and large $n$ the following asymptotic formula holds $$\begin{aligned}
\label{asym}
b_p(n, \mu) \sim \left(\begin{array}{c} n-1\\ p\end{array}
\right).\end{aligned}$$ This can be expressed by saying that for large $n$ the random manifold $M_\ell$ is connected, its first Betti number is $n-1$, the second is $(n-1)(n-2)/2$ and so on. More precisely, our main result is the following:
\[thm1\] Let $\mu$ be either of the measures $\mu_a$ or $\mu_b$ described above. Then for any $p \ge 0$ there exist constants $C
>0$ and $0<a<1$ such that for any $n$ the average Betti number $b_p(n,\mu)$, given by (\[barbi\]), satisfies $$\begin{aligned}
\label{ineqthm1}
\left| b_p(n,\mu) - \left(\begin{array}{c} n-1\\ p\end{array}
\right)\right| \, < Ca^n.\end{aligned}$$
This result appears quite surprising for two reasons. Firstly, it states that the asymptotic values of the average Betti number $b_p(n, \mu)$ as $n\to \infty$ are equal for the measures $\mu_a$ and $\mu_b$ and raises the intriguing question about the universality of the obtained asymptotic values.
Secondly, the binomial coefficient which appears in estimate (\[ineqthm1\]) equals the $p$-dimensional Betti number $b_p(M_{\ell^{*}})$ of the configuration space of the equilateral linkage $\ell^{*} = (1, \dots , 1)$ for any $p$ satisfying $2p<n-3$, see [@FS], Examples 3 & 4. Hence the Betti numbers of the moduli space $M_{\ell^{*}}$ approximate the average Betti numbers $b_p(n, \mu)$ for $n$ large. On the other hand, by Theorem 2 of [@FS] the total Betti number, $\sum_{p=0}^{n}
b_p(M_{\ell}),$ viewed as a function of the length vector $\ell\in
{{\mathbb R}}^n_+$, is maximal for the equilateral linkage $\ell=\ell^{*}$. This comparison shows that at least for some values of $p$ the maximum of the individual Betti numbers $b_p(M_\ell)$ viewed as a function of $\ell$ must be higher than $b_p(M_{\ell^\ast})$.
Indeed, this is the case since for the length vector $\ell_\epsilon=(1,1,1, \epsilon, \dots, \epsilon)$, where $\epsilon$ is a small positive number which appears $n-3$ times, the Betti number $b_p(M_{\ell_\epsilon})$ equals $$2\cdot \left(\begin{array}{c}n-3\\ p\end{array}
\right) = 2\cdot \frac{(n-2-p)(n-1-p)}{(n-1)(n-2)}\cdot
\left(\begin{array}{c}
n-1\\ p\end{array}\right)$$ (see [@FS], Example 2) which for large $n$ is nearly twice the average Betti number $b_p(n,\mu)$. However if $p\sim n/2$ then $$b_p(M_{\ell_\epsilon})\sim \frac{1}{2}\left(\begin{array}{c} n-1\\
p\end{array}\right).$$
In a subsequent work we shall describe a generalization of Theorem \[thm1\] which allows the dimension $p$ to grow with $n$.
Reduction of the problem {#sec1}
========================
In this section we express the average Betti numbers in terms of certain volumes.
To state this result we need to introduce some more notations. For a subset $J\subset \{1, \dots, n\}$ we denote by $\phi_J: {{\mathbb R}}^n \to
{{\mathbb R}}$ the linear functional given by $$\begin{aligned}
\label{functional}
\phi_J(l_1, \dots, l_n) = \sum_{i\in J} l_i - \sum_{i\notin J} l_i\end{aligned}$$ and by $H_J$ the half-space $$\begin{aligned}
\label{hj}
H_J = \{\ell\in {{\mathbb R}}^n; \phi_J(\ell)<0\}.\end{aligned}$$ Further, $C^n$ denotes the cone $$\begin{aligned}
C^n \, =\, \{(l_1, \dots, l_n); \, l_1\geq l_2\geq \dots l_n\geq
0\}\subset {{\mathbb R}}^n_+.\end{aligned}$$
The main result of this section provides a reduction of the problem of computing the average Betti numbers.
\[prop2\] Let $\mu$ be a probability measure on ${{\mathbb R}}^n_+$ having the following two properties:
- $\mu$ is invariant with respect to the action of the symmetric group $\Sigma_n$ on ${{\mathbb R}}^n_+$ permuting coordinates;
- $\mu(L\cap {{\mathbb R}}^n_+)=0$ for any proper linear subspace $L\subset
{{\mathbb R}}^n$.
Then the average Betti number $b_p(n,\mu)$ equals $$\begin{aligned}
\label{propstatement}
b_p(n,\mu) \, =\, n!\cdot \sum_J \mu(H_J\cap C^n),\end{aligned}$$ where $J$ runs over all subsets $J\subset \{1, 2, \dots, n\}$ containing $1$ and having cardinality either $|J|=p+1$ or $|J|=n-2-p$.
First, recall the result of [@FS] stating that the Betti numbers $b_p(M_\ell)$, as functions of the length vector $\ell=(l_1, \dots, l_n)$, can be computed by counting certain subsets of the index set $\{1, \dots, n\}$. A subset $J\subset
\{1, \dots, n\}$ is called [*short*]{} if $$\begin{aligned}
\sum_{i\in J}l_i \, < \, \sum_{i\notin J}l_i.\end{aligned}$$ A subset is called [*long*]{} if its complement is short. A subset $J\subset \{1, \dots, n\}$ is called [*median*]{} if $$\begin{aligned}
\sum_{i\in J} l_i=\sum_{i\notin J} l_i.\end{aligned}$$
Fix an index $1\leq i\leq n$ such that $l_i$ is maximal among $l_1, \dots, l_n$. Denote by $a_p(\ell)$ the number of short subsets $J\subset \{1, \dots, n\}$ of cardinality $|J|=1+p$ containing $i$. Denote by $\tilde a_p(\ell)$ the number of median subsets $J\subset \{1, \dots, n\}$ containing $i$ and such that $|J|=1+p$. It was proven in [@FS] that for $p=0, 1, \dots,
n-3$ one has $$\begin{aligned}
\label{fs}
b_p(M_\ell) = a_p(\ell) + \tilde a_p(\ell) + a_{n-3-p}(\ell).\end{aligned}$$
It is easy to see that the manifolds $M_{\ell_1}$ and $M_{\ell_2}$ are diffeomorphic if the length vector $\ell_1\in {{\mathbb R}}^n_+$ is obtained from $\ell_2\in {{\mathbb R}}^n_+$ by permuting the components. In other words, the order of the coordinates $l_1, \dots, l_n$ in the length vector $\ell=(l_1, \dots, l_n)$ is irrelevant.
Let $\mu$ be a probability measure on ${{\mathbb R}}^n_+$ having properties (I) and (II). Property (II) implies that $$\begin{aligned}
\int_{{{\mathbb R}}^n_+} \tilde a_p(\ell)d\mu \, =\, 0\end{aligned}$$ since the function $\ell\mapsto\tilde a_p(\ell)$ is zero on the complement of a union of finitely many linear hyperplanes. Integrating (\[fs\]) we have $$\begin{aligned}
\label{bpmu}
b_p(n, \mu)= a_p(n, \mu) + a_{n-3-p}(n,\mu).\end{aligned}$$ where $$\begin{aligned}
\label{apmu}
a_p(n, \mu) \, =\, \int_{{{\mathbb R}}^n_+} a_p(\ell)d\mu \, =\,
n!\cdot \int_{C^n} a_p(\ell) d\mu.
\end{aligned}$$ The second equality follows from property (I) of $\mu$. The function $a_p|_{C^n}$ is quite simple. Denote by $\sigma$ the step function $$\begin{aligned}
\sigma(x) = \left\{
\begin{array}{lll}
1, &\mbox{if} & x<0,\\
0, &\mbox{if} & x\geq 0,
\end{array}\right.\end{aligned}$$ where $x\in {{\mathbb R}}$. Then we may write $$\begin{aligned}
a_p(\ell) = \sum_J \sigma((\phi_J(\ell)))\end{aligned}$$ where $J$ runs over all subsets of $\{1, \dots, n\}$ of cardinality $p+1$ containing $1$. Integrating we find $$\begin{aligned}
\label{apell}
a_p(n,\mu) = n! \cdot \int_{C^n} a_p(\ell)d\mu \nonumber\\
= n! \cdot \sum_J
\int_{C^n} \sigma(\phi_J(\ell))d\mu.\end{aligned}$$ In formula (\[apell\]) $J$ runs over all subsets $J\subset \{1,
\dots, n\}$ satisfying $1\in J$, $|J|=p+1$. Obviously $$\begin{aligned}
\label{vj}
\int_{C^n}\sigma(\phi_J(\ell))d\mu \, =\, \mu(H_J\cap
C^n)\end{aligned}$$ where $H_J$ is given by (\[hj\]). Our statement (\[propstatement\]) now follows by combining (\[bpmu\]), (\[apmu\]), (\[apell\]) and (\[vj\]).
Simplices and volumes
=====================
In this section we give a geometric interpretation of the quantities appearing on the RHS of formula (\[propstatement\]).
Recall that we denote by $\mu_a$ the measure on ${{\mathbb R}}^n_+$ with support on the unit simplex $\Delta^{n-1}\subset {{\mathbb R}}^n_+$ such that $\mu_a|_{\Delta^{n-1}}$ coincides with the normalized Lebesgue measure, $\mu_a(\Delta^{n-1}) =1$. Similarly, $\mu_b$ denotes the probability measure on ${{\mathbb R}}^n_+$ supported on the unit cube $\square^n\subset {{\mathbb R}}^n_+$ such that $\mu_b|_{\square^n}$ coincides with the Lebesgue measure.
Denote by $A\subset {{\mathbb R}}^n_+$ the simplex of dimension $n$ having the vertices $$\begin{aligned}
c_0&=&(0, \dots, 0),\nonumber\\
c_1&=&(1, 0, \dots, 0),\nonumber\\
c_2&=& \frac{1}{2}(1,1,0, \dots, 0),\label{verticesA}\\
\dots\nonumber\\
c_n &=& \frac{1}{n}(1,1,\dots, 1).\nonumber\end{aligned}$$ Similarly, denote by $B\subset {{\mathbb R}}^n_+$ the simplex of dimension $n$ having the vertices $$\begin{aligned}
\label{verticesB}
c'_i =(\underbrace{1, \dots, 1}_{i \, \, \mbox{times}}, 0\dots,0)=
i\cdot c_i,\end{aligned}$$ where $i=0, 1, \dots, n$.
\[prop3\] For any subset $J\subset \{1, \dots, n\}$ one has $$\begin{aligned}
n!\cdot \mu_a(H_J\cap C^n) = \frac{{{\rm {vol}}}(H_J\cap A)}{{{\rm {vol}}}(A)},
\quad n!\cdot \mu_b(H_J\cap C^n) = \frac{{{\rm {vol}}}(H_J\cap
B)}{{{\rm {vol}}}(B)}.\end{aligned}$$ In other words, the quantities appearing in Proposition \[prop2\] can be interpreted as ratios of volumes of certain simplices and their parts cut off by a half-space.
One checks that the intersection $\Delta^{n-1}\cap C^n$ coincides with the $(n-1)$-dimensional simplex $A'\subset A$ with vertices $c_1,
\dots, c_n$ and the intersection $\square^n\cap C^n$ equals $B$. Clearly $$\mu_a(A') = \frac{{{\rm {vol}}}_{n-1}(A')}{{{\rm {vol}}}_{n-1}(A)}=(n!)^{-1}, \quad \mu_b(B)= {{\rm {vol}}}(B) =
(n!)^{-1}.$$ Here ${{\rm {vol}}}_{n-1}$ denotes the $(n-1)$-dimensional Euclidean area. Hence we find that $$\begin{aligned}
n!\cdot \mu_a(H_J\cap C^n) = \frac{{{\rm {vol}}}_{n-1}(H_J\cap
A')}{{{\rm {vol}}}_{n-1}(A')} = \frac{{{\rm {vol}}}(H_J\cap A)}{{{\rm {vol}}}(A)}\end{aligned}$$ and $$\begin{aligned}
n!\cdot \mu_b(H_J\cap C^n) = \frac{{{\rm {vol}}}(H_J\cap B)}{{{\rm {vol}}}(B)}.\end{aligned}$$ This completes the proof.
We are led to consider the following simple geometric problem.
\[cutvol\] Given $n+1$ points $v_0, \dots, v_n\in {{\mathbb R}}^n$ in general position and a linear functional $\phi: {{\mathbb R}}^n\to {{\mathbb R}}$, consider the simplex $\Sigma$ spanned by $v_0, \dots, v_n$ and its intersection with the half space $H=\{v\in {{\mathbb R}}^n; \phi(v)<0\}$. Denote by $$\begin{aligned}
q_i=\phi(v_i), \quad i=0, \dots, n\end{aligned}$$ the values of the functional $\phi$ on the vertices of $\Sigma$. Assume that the numbers $q_0, q_1, \dots, q_n$ are all distinct and the vertices are labelled so that $q_i<0$ for all $i=0,\dots,
m$ and $q_i\geq 0$ for $i=m+1, \dots, n$. Then the ratio of the volumes $$\begin{aligned}
\label{ratio2} r= \frac{{{{\rm {vol}}}}(\Sigma\cap
H)}{{{{\rm {vol}}}}(\Sigma)}\end{aligned}$$ equals $$\begin{aligned}
\label{slice}
r = \sum_{i=0}^m \, \prod\limits_{\begin{array}{c}0\leq j\leq n\\
j\not= i\end{array}}\, \frac{q_i}{q_i- q_j}.\end{aligned}$$
Proposition \[cutvol\] is well known. Different expressions for the volume cut off a simplex by a half space were obtained in [@CS] (see Theorem 2 of [@CS]) and more recently in [@A], [@Ge], [@Va]. Proposition \[cutvol\] can be easily obtained from the results mentioned above. For convenience of the reader we briefly sketch the proof.
Consider for $x\in {{\mathbb R}}$ the half space $H_x=\{v\in {{\mathbb R}}^n; \phi(v)<x\}$ and the real valued function $$\begin{aligned}
r(x) = \frac{{{{\rm {vol}}}}(H_x\cap B )}{{{{\rm {vol}}}}(B)}.\end{aligned}$$ Without loss of generality we will assume that $q_0<q_1<
\dots<q_n$; for convenience we set $q_{-1}=-\infty$ and $q_{n+1}=+\infty$.
The function $r(x)$ has the following properties:
1. when restricted on each subinterval $[q_{i-1}, q_i]$, the function $r(x)$ is a polynomial $p_i(x)$ of degree $n$ with real coefficients where $i=0, \dots, n+1$.
2. $p_0(x)\equiv 0$ and $p_{n+1}(x)\equiv 1$.
3. $r(x)$ has continuous derivatives up to order $ n-1$.
Properties (i) and (iii) (proven in [@CS]) imply that for any $0 \le i \le n$ there exists $\beta_{i} \in {{\mathbb R}}$ such that $$\begin{aligned}
p_{i+1}(x) - p_i(x) = \beta_i(x-q_i)^n.\end{aligned}$$ Hence $$\begin{aligned}
\label{pii}
p_{i+1}(x) = \sum_{k=0}^i \beta_k(x-q_k)^n\end{aligned}$$ where the coefficients $\beta_i$ can be found from the polynomial identity $$\begin{aligned}
\label{system}
\sum_{k=0}^n \beta_k(x-q_k)^n\equiv 1.\end{aligned}$$ Equation (\[system\]) has a unique solution given by $$\begin{aligned}
\label{alphai}
\beta_k = (-1)^n \cdot\prod_{\begin{array}{c} j=0\\
j\not=k\end{array}}^n (q_k-q_j)^{-1}, \quad k=0, 1, \dots, n.\end{aligned}$$ Indeed, by comparison of coefficients in (\[system\]) we obtain the linear system $$\begin{aligned}
\sum_{k=0}^n \beta_k q_k^{n-i} = \left\{
\begin{array}{lll}
(-1)^n & \mbox{if} & i=0,\\ 0 & \mbox{if} & i=1, \dots, n.
\end{array} \right.\end{aligned}$$ which can be written in the matrix form $$\begin{aligned}
\left[
\begin{array}{cccc}
1 & 1& \dots & 1\\
q_0 & q_1& \dots & q_n\\
\dots& \dots&\dots &\dots\\
q_0^{n-1}& q_1^{n-1} & \dots & q_n^{n-1}\\
q_0^n& q_1^n & \dots & q_n^n
\end{array}
\right] \left[
\begin{array}{c}
\beta_0\\
\beta_1\\
\dots\\
\beta_{n-1}\\
\beta_n\end{array} \right] = \left[\begin{array}{c} 0\\ 0\\
\dots\\0\\ (-1)^n\end{array} \right]\end{aligned}$$ To solve it one applies Cramer’s rule. The determinant of the system is the Vandermonde determinant $\prod_{j>l} (q_j-q_l)$ and the numerator of the fraction expressing $\beta_k$ can be computed to be the Vandermonde determinant $$(-1)^k\, \cdot
\prod_{\begin{array}{c} j>l\\ j, l\not= k\end{array}} (q_j-q_l)$$ so that we obtain $$\beta_k= (-1)^k \prod_{k>l}(q_k - q_l)^{-1}\cdot
\prod_{j>k}(q_j-q_k)^{-1} = (-1)^{n}\prod_{j \not=
k}(q_k-q_j)^{-1}$$ which coincides with (\[alphai\]). Substituting the obtained value into (\[pii\]) we find that for $x\in [q_{i-1},q_i]$ one has $$\begin{aligned}
\label{slice1}
r(x) \, = \, p_i(x) = \sum_{k=0}^{i-1}
\prod\limits_{\begin{array}{c} 0\leq j\leq n\\
j\not=k\end{array}} \, \frac{q_k-x}{q_k-q_j}.\end{aligned}$$ Formula (\[slice\]) is obtained from (\[slice1\]) by setting $x=0$.
For our applications we need to have a more general formula for the ratio (\[ratio2\]) covering the case when some values $q_i=\phi(c_i)$ coincide. Formula (\[slice\]) is not well defined in this case since some of the denominators might vanish. Our setting is as follows. Let $v_0, \dots, v_n\in {{\mathbb R}}^n$ be points in general position spanning a simplex $B$. Let $\phi: {{\mathbb R}}^n\to {{\mathbb R}}$ be a linear functional. We consider the ratio of the volumes (\[slice\]) where $H$ is the half-space $H=\{v\in {{\mathbb R}}^n; \phi(v)
< 0\}$. Denote $q_i=\phi(v_i)\in {{\mathbb R}}$, where $i=0, \dots, n$. Suppose that there is a decomposition of the set of indices into disjoint subsets $$\{0, 1, \dots, n\} = \bigsqcup_{l=0}^s I_l$$ such that $q_i=Q_l$ for all $i\in I_l$ and the numbers $Q_0,
\dots, Q_s\in {{\mathbb R}}$ are pairwise distinct. We denote by $k_l$ the number $|I_l|-1$. Thus, the multiplicity of $Q_l$ is $k_l+1$, where $l=0,1, \dots, s$ and one has $$\begin{aligned}
\label{sum5}
k_0+k_1+\dots+k_s =n-s.\end{aligned}$$ Given two nonnegative integers $s$ and $a$, we denote by $P(s,a)$ the set of all functions $$\delta: \{0,1, \dots, s\}\to {{\mathbb Z}}_{\ge
0}, \quad i\mapsto \delta_i$$ satisfying $\sum_{j=0}^s\delta_j
=a.$ The set $P(s,a)$ labels partitions of $a$ in $s+1$ summands.
\[ratio3\] Assume that $Q_i<0$ for all $i=0,\dots, m$ and $Q_i\geq 0$ for all $i=m+1, \dots, s$. Then the ratio of volumes (\[ratio2\]) equals $$\begin{aligned}
\label{ratiogeneral}
r= \sum_{i=0}^m \,\left[ F_i\cdot \prod\limits_{\begin{array}{c} 0\leq j\leq s\\
j\not= i\end{array}}\left(
\frac{Q_i}{Q_i-Q_j}\right)^{k_j+1}\right]\end{aligned}$$ where $F_i$ equals $$\begin{aligned}
\label{fi}
\sum\limits_{\delta\in P(s, k_i)}
\left(
\begin{array}{c} n\\ \delta_i\end{array}
\right)\cdot (-Q_i)^{k_i-\delta_i}\cdot
\prod\limits_{\begin{array}{c} 0\leq j\leq s\\ j\not=i\end{array}}
\left(\begin{array}{c} k_j+\delta_j\\
\delta_j\end{array}\right)\cdot (Q_i-Q_j)^{-\delta_j}.\end{aligned}$$
Note that if $k_i=0$, then $P(s,k_{i})= \{ \delta \equiv 0\}$ and hence $F_i=1$. In particular, if $s = n$ and therefore $k_{i} = 0$ for any $0 \le i \le n$ formula (\[fi\]) coincides with formula (\[slice\]). Moreover, we point out that the coefficients $F_{i}$ are homogenous of degree $0$ in the variables $Q_{0},
\dots , Q_{s}.$
Let $\psi: {{\mathbb R}}^n \to {{\mathbb R}}$ be a linear functional, a small perturbation of $\phi: {{\mathbb R}}^n\to {{\mathbb R}}$, such that the values $p_i=\psi(v_i)$ are all distinct. Without loss of generality we may assume that $p_i<0$ for $0\leq i \leq m'$ and $p_i>0$ for $m'
<i \leq n$ where $m'=m +1+\sum_{i=0}^m k_i$. Denote $H'=\{v\in
{{\mathbb R}}^n; \psi(v)<0\}$. The perturbed ratio $$\begin{aligned}
\label{ratioperturb}
r'= \frac{{{\rm {vol}}}(H'\cap \Sigma)}{{{\rm {vol}}}(\Sigma)}\end{aligned}$$ tends to $r$ when $\psi$ tends to $\phi$. By Proposition \[cutvol\] it can be written in the form $$\begin{aligned}
\label{ratioper}
r' = \sum_{i=0}^{m'} \left[ p_i^n \cdot \prod\limits_{\begin{array}{c} j\not= i\\
0\leq j \leq n\end{array}}\left(p_i - p_j\right)^{-1}\right].\end{aligned}$$ To find its limit we will use the following general formula, see [@MT], $$\begin{aligned}
\label{formula}
\sum_{i=0}^k f(x_i)\cdot \prod_{\begin{array}{c} 0\leq j\leq k\\
j\not= i\end{array}} (x_i-x_j)^{-1} = \frac{1}{k!} f^{(k)}(\xi)\end{aligned}$$ where $f(x)$ is a real valued smooth function, $x_0, \dots, x_k$ are distinct real numbers and $\xi$ is a number lying in the smallest interval containing these points. Applying (\[formula\]) $(m+1)$ times to (\[ratioper\]) and passing to the limit as $\{p_0, \dots, p_n\} \to \{Q_0, \dots, Q_s\}$ we obtain $$\begin{aligned}
\label{multiple}
r = \sum_{i=0}^m \frac{1}{k_i!}\left[x^n\cdot
\prod_{\begin{array}{c} 0\leq l\leq s\\ l\not=i\end{array}}
(x-Q_l)^{-k_l-1}\right]^{(k_i)}_{x=Q_i}.\end{aligned}$$ Recall the following formula for higher derivatives of products $$(f_0f_1\cdots f_s)^{(k)} = \sum_{\delta\in P(s,k)}
\frac{k!}{\delta_0!\cdots \delta_s!}\, \, f_0^{(\delta_0)}\cdots
f_s^{(\delta_s)}.$$ Applying it to (\[multiple\]) and using that $n=k_i+\sum_{j\not= i} (k_j+1)$ one obtains (\[ratiogeneral\]) after certain elementary transformations.
Sequences of densities
======================
By Proposition \[prop2\] combined with Proposition \[prop3\], to compute the average Betti numbers $b_p(n,\mu)$ one has to know the volumes cut off a simplex by certain half-spaces. The result of Propositions \[cutvol\] and \[ratio3\] show that to find these volumes it is enough to know the values of the functionals determining the half-spaces on the vertices of the simplices $A$ and $B$. We investigate these values in this section.
Let $J\subset \{1, \dots, n\}$ be an arbitrary subset and $\phi_J:
{{\mathbb R}}^n\to {{\mathbb R}}$ be the linear functional (\[functional\]). The values of the functional $\phi_J$ on the vertices $c_i$ of the simplex $A$ (see (\[verticesA\])) equal $\phi_J(c_0) =0$ and for $i\geq 1$ $$\begin{aligned}
\label{phi}
\phi_J(c_i) = \frac{1}{i} \left|J\cap \{1, \dots, i\}\right|\,-\,
\frac{1}{i}\left|\bar J\cap \{1, \dots, i\}\right|\end{aligned}$$ where $\bar J$ denotes the complement of $J$ in $\{1, \dots ,
n\}$. Let $$\begin{aligned}
\label{dens}
\alpha_i(J) = \frac{1}{i}\left|J\cap \{1, \dots, i\}\right|, \quad
i=1, \dots, n\end{aligned}$$ denote the [*density*]{} of the set $J$ in the interval $\{1,
\dots, i\}$. Clearly $0 \, \leq \, \alpha_i(J)\, \leq\, 1$.
\[lm1\] Let $J\subset \{1, \dots, n\}$ be a subset of cardinality $|J|=p\geq 1$. Then the following estimates hold:
1. For $2p \le i \le n$ one has $$\begin{aligned}
\label{half}
\alpha_i(J) \leq \frac{1}{2} \end{aligned}$$ and equality in (\[half\]) may only hold for $i=2p$.
2. If $1\leq i,
j\leq 2p$, then either $\alpha_i(J)=\alpha_j(J)$ or $$\left|\alpha_i(J) - \alpha_j(J)\right| \geq \frac{1}{(2p)^2}.$$
3. If $\alpha_i(J) <\frac{1}{2}$ for some $1 \le i \le
n$, then $$\alpha_i(J) \leq \frac{1}{2} - \frac{1}{2(2p+1)}.$$
4. For any $8p^3 \le i\leq n$ one has $$\begin{aligned}
0\leq \alpha_i(J) \leq \frac{1}{8p^2}.\end{aligned}$$
\(a) Let $k_i$ denote $|J\cap \{1, 2, \dots, i\}|$. Then for $i\geq
2p$ one has $$\alpha_i(J) = \frac{k_i}{i} \leq \frac{p}{i} \leq \frac{p}{2p} =1/2.$$ If $\alpha_i(J) = 1/2,$ then the above inequalities imply that $i = 2p.$
\(b) Suppose now that $1\leq i, j\leq 2p$. Then $$\begin{aligned}
\label{dif}
\left|\alpha_i(J)-\alpha_j(J)\right| = \left|\frac{k_i}{i} -
\frac{k_j}{j}\right|=
\frac{\left|jk_i-ik_j\right|}{ij}.\end{aligned}$$ We obtain that either (\[dif\]) vanishes or $\left|jk_i - ik_j\right|\geq 1$ and hence (\[dif\]) is greater or equal to $$\frac{1}{ij}\geq \frac{1}{(2p)^2}$$ proving (b).
\(c) Note that for $i\geq 2p+2$ one has $$\alpha_i(J) =\frac{k_i}{i} \leq \frac{p}{2p+2} = \frac{1}{2} -
\frac{1}{2(p+1)}.$$ Now consider the case $i\leq 2p+1.$ By assumption $\alpha_i(J) <1/2$, i.e. $2k_i<i$ or $2k_i\leq i-1.$ It implies that $$\alpha_i(J) =\frac{k_i}{i} \leq \frac{i-1}{2i} = \frac{1}{2} -
\frac{1}{2i} \leq \frac{1}{2} - \frac{1}{2(2p+1)}.$$
\(d) Note that if $i \ge 8p^3$ and $p \ge 1,$ then $$\alpha_i(J) =\frac{k_i}{i}\leq
\frac{p}{i}\leq \frac{1}{8p^2}.$$
\[lm3\] Let $J \subset \{1, \dots , n\}$ with $|J|=p\geq 1$. A nonzero number may appear in the sequence of densities $\alpha_1(J),
\dots, \alpha_n(J)$ at most $p$ times.
The densities $\alpha_i(J)$ satisfy the following recurrent relation: $$\begin{aligned}
\alpha_{i+1}(J) = \left\{
\begin{array}{lll}
\frac{i}{i+1} \alpha_i(J),& \mbox{if} & i+1\notin
J,\\ \\
\frac{i}{i+1} \alpha_i(J) +\frac{1}{i+1},& \mbox{if} & i+1\in J.\\
\end{array}
\right.\end{aligned}$$ It follows that $\alpha_{i+1}(J)<\alpha_i(J)$ if $\alpha_i(J)>0$ and $i+1\notin J$. On the other hand, $\alpha_{i+1}(J)\geq
\alpha_i(J)$ if $i+1\in J$. Hence, for $i<j$, the equality $\alpha_i(J)=\alpha_j(J)>0$ implies that at least one of the intermediate indices $i+1, \dots, j$ belongs to $J$.
Assume now that for $i_1<i_2<\dots < i_k$ one has $\alpha_{i_1}(J)
= \dots = \alpha_{i_{k}}(J)> 0$. The set $J$ divides $\{1, \dots,
n\}$ into $p+1$ subintervals, i.e. subsets of consecutive integers in $\{1, \dots, n\}$ all of which are not in $J$ except the first one. The leftmost interval contains no elements of $J$ and might be empty. As explained above, each of the subintervals may contain at most one of the integers $i_1, \dots, i_k$. The density $\alpha_i(J)$ vanishes iff $i$ lies in the leftmost subinterval. This shows that $k\leq p$ as claimed.
The following examples show that Lemma \[lm3\] cannot be improved.
\(i) Consider $J = \{ 1,2,\dots, p\}$. Then $\alpha_i(J)=1$ if and only if $1 \le i \le p$, i.e. the multiplicity of the value $1$ is $p$.
\(ii) Let $J=\{1, 3, 5, \dots, 2p-1\}$. Then $\alpha_i(J)$ equals $1/2$ exactly $p$ times.
\(iii) Suppose that $J=\{n-p+1, n-p+2, \dots, n\}$. Then $\alpha_i(J)=0$ for $i=1, 2, \dots, n-p$, i.e. the multiplicity of the value zero is $n-p$. This shows that the bound of the multiplicity of Lemma \[lm3\] does not hold for the value zero.
The values $$\begin{aligned}
\label{values}
q_{i}=\phi_J(c_i)\end{aligned}$$ of the functional $\phi_J$ on the vertices of the simplex $A$ (see (\[verticesA\])) will play an important role in the sequel. Clearly $q_0=0$ and $$\begin{aligned}
q_i = 2\alpha_i(J) -1\, \, \in [-1, 1], \quad i=1, \dots,
n,\end{aligned}$$ see (\[phi\]). Let us restate Lemmas \[lm1\] and \[lm3\] in terms of the $q_i$’s:
\[lm2\] Let $J\subset \{1, \dots, n\}$ be a subset of cardinality $|J|=p\geq 1$. Then:
1. For $2p \le i \le n$ one has $q_i\leq 0$ with equality possible only for $i=2p$.
2. If $1\leq i, j\leq
2p$ then either $q_i=q_j$ or $$\left|q_i - q_j\right| \geq \frac{1}{2p^2}.$$
3. If $q_i < 0 $ for some $1 \le i \le n$ then $$q_i \leq - \frac{1}{2p+1}.$$
4. For $8p^3 \leq i\leq n$ one has $$\begin{aligned}
-1\leq q_i \leq -1+ \frac{1}{4p^2}.\end{aligned}$$
5. A number distinct from $-1$ may appear in the sequence $q_0, \dots, q_n$ at most $p$ times.
Next we consider the values $q'_i=\phi_J(c'_i)$ of the functional $\phi_J$ on the vertices of simplex $B$ (see (\[verticesB\])).
\[lmqprime\] Let $J\subset \{1, \dots, n\}$ be a subset of cardinality $|J|=p\geq 1$. Then:
1. The numbers $q'_i$ satisfy $-i\le
q'_i\le i$;
2. For $2p \le i \le n$ one has $q'_i\leq
0$ with equality possible only for $i=2p$.
3. If $q'_i\not=q'_j$, then $\left|q'_i - q'_j\right| \geq 1.$
4. If $q'_i < 0 $ for some $1 \le i \le n$, then $q'_i
\leq - 1.$
5. A number may appear in the sequence $q'_0,
\dots, q'_n$ at most $p+1$ times.
Since $q'_i=iq_i$, statements (a) - (d) follow from Lemma \[lm2\] and from the observation that $q'_i$ is an integer. To prove (e) we note that the numbers $q'_i$ satisfy the following recurrent relation: $$q'_{i+1} = \left\{
\begin{array}{ll}
q'_i -1 & \mbox{if}\, \, i+1\notin J,\\ \\
q'_i +1 & \mbox{if}\, \,
i+1\in J.
\end{array}
\right.$$ Hence the sequence $q'_0, q'_1, \dots, q'_n$ has exactly $p$ jumps up and decays between the jumps. This proves (e).
[Consider the sequence $J=\{1, 3, \dots, 2p-1\}$. Then (i) $q'_i=0$ for $i=0, 2, 4, \dots, 2p$, (ii) $q'_i=1$ for $i=1, 3,
\dots 2p-1$ and (iii) $q'_i<0$ for $i>2p$. We see that in this case zero appears in the sequence $q'_0, \dots, q'_n$ exactly $p+1$ times.]{}
Proof of Theorem \[thm1\] for $\mu=\mu_a$
=========================================
In this section we prove Theorem \[thm1\] for the measure $\mu=\mu_a$ described before the statement of Theorem \[thm1\].
Fix a subset $J\subset\{1, 2, \dots, n\}$ of cardinality $p\geq 1$ where we think of $p$ as being fixed and of $n$ as being large. Let $A$ be the simplex with vertices (\[verticesA\]). Our first goal is to estimate the ratios of the form $$\begin{aligned}
r_J= \frac{{{\rm {vol}}}(H_J\cap A)}{{{\rm {vol}}}(A)}\end{aligned}$$ where $H_J\subset {{\mathbb R}}^n$ is the half-space $H_J=\{v\in {{\mathbb R}}^n;
\phi_J(v)<0\}$. The average Betti numbers are sums of ratios of this kind, see Propositions \[prop2\] and \[prop3\]. Recall that $\phi_J$ denotes the linear functional $\phi_J: {{\mathbb R}}^n\to {{\mathbb R}}$ given by (\[functional\]).
The values of the functional $\phi_J$ on the vertices $c_i$ of $A$ are described in Lemma \[lm2\]. In particular, by statement (a) of Lemma \[lm2\], for large $n$ the majority of the values $q_i$ are negative, more precisely, at most $2p$ of them are positive. Hence, one may expect that the volume of the section $H_J\cap A$ approximates ${{\rm {vol}}}(A)$ for large $n$. This is indeed the case.
To estimate the difference $1-r_J$ from above we will apply Propositions \[cutvol\] and \[ratio3\]; our aim is to show that it is exponentially small.
\[estim2\] Given an integer $p\geq 1$, there exist constants $C>0$ and $0<a<1$ such that for any $n\geq p$ and any subset $J\subset \{1,
2, \dots, n\}$ of cardinality $p$ one has $$\begin{aligned}
\label{estim1}
1-r_J < C\cdot a^n.\end{aligned}$$
It will be apparent from the proof that one can take for $a$ an arbitrary number satisfying $(2p+1)/(2p+2)<a<1$.
Consider the values $q_i=\phi_J(c_i)$, where $i=0,
1, \dots, n$. They may have multiplicities, i.e. the same value may appear several times. We will denote by $Q_0> Q_1 >, \dots,
> Q_s\in [-1, 1]$ the different values of the sequence $q_0, \dots, q_n$. Then there is a surjective mapping $\tau: \{0,1, \dots, n\}\to
\{0,1,\dots, s\}$ such that $q_i=Q_{\tau(i)}$. For $0\leq i\leq s$ we denote by $k_i+1$ the cardinality of the preimage $\tau^{-1}(i)$. Clearly, $$\sum_{i=0}^s k_i = n-s.$$ If $Q_i\leq 0$ for all $0\leq i\leq s$ then $H_J\cap A= A$, hence $r_J=1$ and therefore (\[estim1\]) trivially holds. Thus without loss of generality we may assume that for some $0\leq m\leq s$ one has $Q_i>0$ for $i=0, \dots, m$ and $Q_i\leq 0$ for $i= m+1,
\dots, s$. By statement (a) of Lemma \[lm2\] we have $$\begin{aligned}
\label{mki}
\sum_{i=0}^m (k_i+1)\, \leq 2p.\end{aligned}$$ We also have $$\begin{aligned}
\label{ineq3}
k_i\leq p, \quad i=0, 1, \dots, s-1,\end{aligned}$$ see (e) of Lemma \[lm2\]. Inequality (\[ineq3\]) also applies to the multiplicity $k_s$ if $Q_s$ is distinct from $-1$.
Let $b=1+(2p)^{-2}.$ From statements (a), (b), and (d) of Lemma \[lm2\] we obtain that for any $0\leq i\leq m$ and $j\not=i$ one has $$\begin{aligned}
\label{lowerbound}
|Q_i-Q_j|\, \geq\, \left\{
\begin{array}{ll}
(2p^2+1)^{-1},& \mbox{for}\, \,0 \leq j\leq 8p^3,\\ \\
(2p^2)^{-1} - (-1 +\frac{1}{4p^2})=b, & \mbox{for}\, \, 8p^3 <
j\leq n.
\end{array}
\right.\end{aligned}$$ Let us explain the first line of this inequality. If $j$ satisfies $0\leq j\leq 2p$ then the statement follows from (a) of Lemma \[lm2\]. If $j>2p$ then $Q_j<0$ and hence $Q_i-Q_j\geq
(2p+1)^{-1}\geq (2p^2+1)^{-1}$ by (c) of Lemma \[lm2\].
By applying Proposition \[ratio3\] to $-\phi_{J}$ we have
$$\begin{aligned}
\label{vol-formula}
1-r_J= \sum_{i=0}^m \, \, \left[F_i\cdot\prod\limits_{\begin{array}{c} 0\leq j\leq s\\
j\not= i\end{array}} \left(
\frac{Q_i}{Q_i-Q_j}\right)^{k_j+1}\right],\end{aligned}$$
where $F_i$ is given by $$\begin{aligned}
\label{fi1}
\sum\limits_{\delta\in P(s, k_i)}\left(
\begin{array}{c} n\\ \delta_i\end{array}
\right)\cdot(-Q_i)^{k_i-\delta_i}\cdot
\prod\limits_{\begin{array}{c} 0\leq j\leq s\\ j\not=i\end{array}}
\left(\begin{array}{c} k_j+\delta_j\\
\delta_j\end{array}\right)\cdot (Q_i-Q_j)^{-\delta_j}.\end{aligned}$$
We claim that for any $0 \le i \le m,$ $$\begin{aligned}
\label{estim}
|F_i| \leq (2p^2+1)^p n^{2p}.\end{aligned}$$ Indeed, observe that by (\[lowerbound\]), $$\left| Q_i-Q_j \right| ^{-\delta_j} \le (2p^{2}+1)^{\delta_{j}}.$$ To estimate the binomial coefficient $\left(\begin{array}{c} k_j+\delta_j\\
\delta_j\end{array}\right)$ note that $\delta_j\le k_i$ and thus $k_j+\delta_j\leq k_j+k_i\leq n$ and therefore $$\begin{aligned}
\label{binom1}
\left(\begin{array}{c} k_j+\delta_j\\
\delta_j\end{array}\right) \le n^{\delta_{j}}\quad \mbox{and}\quad
\left(\begin{array}{c} n\\
\delta_i\end{array}\right) \le n^{\delta_{i}} .\end{aligned}$$ As $|Q_{i} | \le 1,$ each term in the sum (\[fi1\]) can be estimated as follows $$\begin{aligned}
\left(
\begin{array}{c} n\\ \delta_i\end{array}
\right)\cdot(Q_i)^{k_i-\delta_i}\cdot
\prod\limits_{\begin{array}{c} 0\leq j\leq s\\ j\not=i\end{array}}
\left(\begin{array}{c} k_j+\delta_j\\
\delta_j\end{array}\right)\cdot (Q_i-Q_j)^{-\delta_j} \\
\le (n \cdot (2p^{2}+1))^{\sum_{0}^{s}\delta_{j}}
\le (2p^{2}+1)^{p} \cdot n^{p}\end{aligned}$$ where we have used that $\sum_{j=0}^{s}\delta_{j} = k_{i} \le p$. The total number of terms in the sum (\[fi1\]) is $$\begin{aligned}
\label{partitions}
|P(s,k_i)| = \left(\begin{array}{c} s+k_i\\
k_i\end{array}\right) \leq n^{p}\end{aligned}$$ since $s+ k_{i} \le n$ and $k_i\leq p$. This proves (\[estim\]).
Now consider the fractions $\frac{Q_i}{Q_i-Q_j}$ which appear in (\[vol-formula\]). For $0 \le j\leq m$, $j\not=i$ we have by (\[lowerbound\]) $$\begin{aligned}
\label{factors3} \left|\frac{Q_i}{Q_i-Q_j}\right|\leq
\frac{1}{\left| Q_i-Q_j\right|} \leq 2p^2+1.\end{aligned}$$
Note that $Q_j< Q_{m+1}\leq 0$ for any $m+2\leq j\leq s$. Using statement (c) of Lemma \[lm2\] we have for $m+2\leq j \le s$ $$\begin{aligned}
\label{factors}
0 < \frac{Q_i}{Q_i-Q_j}
\leq \frac{Q_i}{Q_i+ (2p+1)^{-1}}\leq \frac{1}{1+
(2p+1)^{-1}}=\frac{2p+1}{2p+2}.\end{aligned}$$ If $Q_{m+1}<0$ then estimate (\[factors\]) continues to hold whereas if $Q_{m+1}=0$ then the corresponding factor equals $1$ and by Lemma \[lm2\], (e), using $k_{m+1}+1\leq p$, we get $$\begin{aligned}
\label{factors1}
\prod\limits_{\begin{array}{c} m < j\leq s \end{array}} \left|
\frac {Q_i}{Q_i-Q_j} \right| ^{k_j+1} \le a_0^{(\sum_{j = m+1}^{s}
(k_{j}+1)-p)} \le C' \cdot a_0^{n}\end{aligned}$$ where the constant $C'$ depends on $p$ only. The number of summands in formula (\[vol-formula\]) equals $m+1$; by (\[mki\]) it is bounded above by $2p$.
Combining inequalities (\[estim\]), (\[factors3\]) and (\[factors1\]) we obtain an estimate of the form $1-r_J<
Cn^\gamma a_0^n$ where the constants $C$ and $\gamma$ depend on $p$ but are independent of $n$. This clearly gives (\[estim1\]).
The following statement is equivalent to Proposition \[estim2\]; we will need it in the proof of Theorem \[thm1\].
\[estim3\] Given an integer $p\geq 1$, there exist constants $C>0$ and $0<a<1$ such that for any $n\geq
p$ and any subset $J\subset \{1, 2, \dots, n\}$ of cardinality $n-p$ one has $$\begin{aligned}
\label{estim4}
r_J < C\cdot a^n.\end{aligned}$$
The claimed statement follows from Proposition \[estim2\] by observing that $r_J = 1-r_{\bar J}$ where $\bar J$ denotes the complement of $J$ in $\{1, \dots, n\}$.
By Propositions \[prop2\] and \[prop3\] we have $$b_p(n,\mu) = \sum_J r_J$$ where $J\subset \{1, \dots, n\}$ runs over all subsets containing $1$ and being of cardinality either $p+1$ or $n-2-p$. By Proposition \[estim2\] each $r_J$ with $|J|=p+1$ contributes to $b_p(n,\mu)$ a quantity exponentially close to $1$ and by Proposition \[estim3\] each term $r_J$ with $|J|=n-2-p$ is exponentially small. Adding up all these contributions we arrive at the desired inequality (\[ineqthm1\]).
Proof of Theorem \[thm1\] for $\mu=\mu_b$
=========================================
The proof of Theorem \[thm1\] in the case $\mu=\mu_b$ is quite similar. Propositions \[estim2\] and \[estim3\] remain true but their proofs are slightly different. The difference between the two cases stems only from different simplices involved: for $\mu=\mu_a$ we consider the simplex $A$ with vertices $c_0, \dots,
c_n$ and for $\mu=\mu_b$ we have to consider instead the simplex $B$ with vertices $c'_0, \dots, c'_n$, see (\[verticesB\]).
Let us examine the arguments of the proof of Proposition \[estim2\] when $c_i$ is replaced by $c'_i$. Inequality (\[mki\]) follows from (b) of Lemma \[lmqprime\] and inequality (\[ineq3\]) follows from statement (e) of Lemma \[lmqprime\]. One introduces the points $Q'_0>Q'_1>\dots >Q'_s$ as the distinct values appearing in the sequence $q'_0, q'_1,
\dots, q'_n$. Instead of (\[lowerbound\]) we have a simpler inequality $|Q'_i-Q'_j|\geq 1$ where $i\not=j$ which is a consequence of Lemma \[lmqprime\], (c).
Let us assume that $Q'_i>0$ for $i=0, \dots, m$ and $Q'_i\leq 0$ for $i=m+1, \dots, s$. We claim that for any $0\leq i\leq m$ the quantity $F_i$ given by (\[fi1\]) satisfies inequality (\[estim\]). Indeed, $|Q'_i|\leq 2p$ for $0\leq i\leq m$ (see (a) and (b) of Lemma \[lmqprime\]) and hence $$|F_i|\leq \sum_{\delta\in P(s, k_i)} (2p)^{k_i-\delta_i}\cdot
n^{\sum \delta_j} \leq (2p)^p\cdot n^{2p} \leq (2p^2+1)^pn^{2p}.$$ Here we have used inequalities (\[binom1\]) and (\[partitions\]). To estimate (\[vol-formula\]) from above we note that for $0\leq i\leq m$ and $j\not=i$ one has $$\left| \frac{Q'_i}{Q'_i-Q'_j}\right| \, \leq\, \left\{
\begin{array}{ll}
|Q'_i|\leq 2p & \mbox{for}\, \, j\leq m,\\ \\
1 & \mbox{for}\, \, j= m+1,\\ \\
\frac{Q'_i}{Q'_i+1} \leq \frac{2p}{2p+1} & \mbox{for} \, \, j \geq
m+2.
\end{array} \right.$$ Here we have used statement (d) of Lemma \[lmqprime\]. Combining the obtained inequalities we obtain that the statement analogous to Proposition \[estim2\] holds for $\mu=\mu_b$. The remaining arguments of the proof of Theorem \[thm1\] for $\mu=\mu_b$ are very similar to those described in the case $\mu=\mu_a$.
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|
---
author:
- 'Stephane Perrin [^1] '
- Paul Montgomery
title: 'Fourier optics: basic concepts'
---
**Based on diffraction theory and the propagation of the light, Fourier optics is a powerful tool allowing the estimation of a visible-range imaging system to transfer the spatial frequency components of an object. The analyses of the imaging systems can thus be performed and the the performance retrieved. For a better understanding of the optical study, I present a short introduction of the Fourier optics and I review the mathematical treatment depending on the illumination conditions of the imaging system. Furthermore, resolution criteria based on Fourier optics are detailed. Also, the impact of aberrations on the imaging quality are discussed.**
[2]{}
INTRODUCTION {#introduction .unnumbered}
------------
Performance of a linear optical system can be evaluated using Fourier optics [@Malacara03]. Indeed, the ability of a single lens or a more complex system to reproduce an image of an (1D or 2D) object can be quantify by decomposing the object in Fourier series. In Fourier optics, the object is thus considered as a sum of spatial sinusoidal waves at specific frequencies by analogy with electronics [@Goodman96]. Despite the resolution limit of a microscope by the diffraction of the light was explicitly mentioned by E. Verdet in 1869 [@Verdet69], the concept of Fourier series decomposition appeared firstly in 1873 with the works of E. Abbe [@Abbe74]. He considered the objects as periodic diffraction elements and he showed that at least two orders of a diffraction grating should be collected by the objective lens of a microscope in order to barely resolve the features of the grating. He describes in words the first definition of the spatial resolution $\delta_{x,y}$ which can be mathematically expressed as: $$\delta_{x,y}=\frac{\lambda}{n~\sin\left(\theta\right)}=\frac{\lambda}{NA}
\label{EqAbbe1}$$ Where $\lambda$ is the wavelength of the monochromatic light source and $\theta$, the diffraction angle of the periodic object. According to the Abbe theory, only diffracted components generated by the object having a spatial period higher than $\delta_{x,y}$ are intercepted by the finite pupil of the lens. Thus, the features with higher spatial frequencies are not resolved. In addition, the term of numerical aperture (NA) were firstly introduced as the sine of the diffraction angle $\theta$ [@Abbe81]. Moreover, E. Abbe reported that the resolution $\delta_{x,y}$ can be improved using an focusing (or oblique) illumination of the object. This assumption allows determining the total NA of a microscope as the sum of the NAs of each optical component of the microscope. If the NA of the illumination part $NA_{i}$ equals the NA of the collection part $NA_{c}$, Eq. \[EqAbbe1\] can hence be written: $$\delta_{x,y}=\frac{\lambda}{NA_{i} + NA_{c}}=\frac{\lambda}{2~NA}
\label{EqAbbe2}$$ It should be noted that a direct focusing illumination through a Abbe condenser lens has however the disadvantage of imaging the lamp filament on the sample. A. Köhler took an interest in illumination conditions for microscopy and developed thus a method for homogenising the intensity of the incident beam using an arrangement of lenses and diaphragms [@Kohler93].\
In 1876, H. von Helmholtz confirmed the Abbe theory through a mathematical demonstration of the resolution limit [@Helmholtz76]. At the same time, he exposed the impact of the coherence of the light for avoiding the phase relations [@Lauterbach12] and he showed that Eq. \[EqAbbe1\] also requires a factor $1/2$ when the illumination light source is incoherent ($\lambda$ becomes thus $\lambda_{0}$, the central wavelength of the broadband light source). Furthermore, H. von Helmholtz discussed a new definition of the spatial resolution, reported previously in 1874 by Lord Rayleigh [@Rayleigh74].\
J. W. Rayleigh did not treat microscopic objects as grating elements illuminated with plane waves but rather as as a sum of white-light-emitting point sources. Using the Airy works who calculated the diffraction pattern of a bright point source [@Airy35], J. W. Rayleigh defined the angular separation of an imaging system as the distance between the intensity maximum and the first intensity minimum of the diffraction pattern of an on-axis point source[^2]. Based on Fourier’s theorem, this definition leads to a new resolution expression: $$\delta_{x,y}\approx0.61\frac{f~\lambda}{R}
\label{Rayleigh}$$ Where $f$ is the focal length of the lens and $R$ is the radius of the finite pupil of the lens. Some years later, in 1879, Lord Rayleigh presented a view equivalent to regarding diffraction effects as resulting from the exit pupil [@Rayleigh79]. In 1896, Lord Rayleigh extended his investigations to different objects (points, lines and gratings) and aperture shapes [@Rayleigh96].\
Twenty years later, based on the limits of photodetectors, C. M. Sparrow defined the smallest recognizable inter-space as the distance for which the irradiance pattern of two incoherent point sources has no curvature in the center [@Sparrow16], giving the expression: $$\delta_{x,y}=0.47\frac{\lambda}{NA}
\label{Sparrow}$$ These criteria assume an aberrations-free imaging system. Nevertheless, in practice, evaluating the resolution is more complex due to the noise and the aberrations of optical components. Furthermore, the nature of the light source, i.e. coherent, partially coherent, partially incoherent or incoherent, should be considered. In 1927, W. V. Houston proposes thus to use the full-width at half-maximum (FWHM) of the diffraction pattern of a point source (being the PSF of the imaging system), to quantify the lateral resolution [@Houston27] because more useful in practice and also applicable to diffraction patterns that do not fall off to zero, i.e. a Gaussian PSF or a Lorentzian PSF.\
In 1946, P.M. Duffieux introduced Fourier optics for evaluating the spatial frequency transfers through the optical system using sinusoidal test patterns [@Duffieux46]. And, latter, H.H. Hopkins led the way in the use of transfer function methods for the assessment of the quality of optical imaging systems, making the analogy with analog electronic systems. In 1960, V. Ronchi highlighted the importance of considering both the sensitivity of the sensor and the illumination conditions, to determine the resolving power of an imaging system [@Ronchi61].\
Nowadays, Fourier optics is often used for the design of new optical components or the analysis of imaging system. And the mathematical treatment of Fourier optics was demonstrated and is supposed to be known [@Saleh91]. However, the nature of the light dependence is often neglected and not considered in the literature. Thus, this manuscript reviews the influence of the coherence of the light source on the transfer function of a visible-range (or infra-red) imaging system. The resolution criteria based on Fourier optics are explained. Furthermore, the effects of aberrations on the imaging quality are discussed.
IMAGING SYSTEMS {#imaging-systems .unnumbered}
---------------
In Fourier optics, the imaging systems are supposed time invariant and linear. Figure. \[model\] represents a generalized scheme of an optical system, i.e. an assembly of optical elements, collecting the electric field from a point source placed in the object plane and propagating it in the image plane. The object plane is placed at a distance $z_{o}$ of the entrance pupil, i.e. the working distance, and the imaging plane at a distance $z_{i}$ of the exit pupil. The exit pupil (or simply the finite circular pupil function $P(x,y)$) of the imaging system is considered in the following diffraction equations. Indeed, only the output pupil plays a role in the diffraction of the light. The pupil function $P(x,y)$ has a diameter $D$ and is unity inside and zero outside the projected aperture. Gauss approximations are considered, meaning that the incident angles are small and the point light sources are close to the optical axis[^3].
![**Model of an imaging system and representation of its pupil function.** The electric field from the emitting point source in the object plane is transmitted by the imaging system in the image plane. The imaging system has a normalized pupil function $P(x,y)$ with a diameter $D$. $\xi$ and $\eta$ are the Cartesian coordinates in the object plane. $x$ and $y$ are the Cartesian coordinates in the principal planes of the imaging system. $u$ and $v$ are the Cartesian coordinates in the image plane. $z$ is the optical axis.[]{data-label="model"}](model.png){width="8cm"}
Furthermore, the imaging system is first assumed to be free of aberration, i.e. a diffraction-limited optical system. An imaging system is said to be diffraction-limited if a diverging spherical wave, emanating from a point-source object, is converted by the system into a new wave, again perfectly spherical, that converges towards an ideal point in the image plane, where the location of that ideal image point is related to the location of the original object point through a simple scaling factor, i.e. the magnification $M$. The magnification factor must be the same for all points in the lateral field of view of the imaging system [@Goodman96].\
In the image plane, the 2D complex amplitude distribution of the image of a point light source is represented by a superposition integral, $$U_{i}\left(u,v\right) = \displaystyle \iint_{\infty}^{\infty} h\left(u,v;\xi,\eta\right) \times U_{o}\left(\xi,\eta\right) \, \mathrm{d}\xi \mathrm{d}\eta,
\label{eq1}$$ with $U_{o}\left(\xi,\eta\right)$, the complex electric field of the emitting point source. The coordinates $u$ and $v$ are given by $u=M\xi$ and $v=M\eta$ where the magnification $M$ of the imaging system could be negative or positive. Using the convolution theorem [@conv], Eq. \[eq1\] can be rewritten: $$U_{i}\left(u,v\right) = h\left(u,v;\xi,\eta\right) \circledast U_{o}\left(\xi,\eta\right)
\label{eq2}$$ The amplitude response to a point-source object $h\left(u,v;\xi,\eta\right)$ of the imaging system (also called amplitude point spread function) at a position $\left(\xi,\eta\right)$ is defined as the Fourier transform of its pupil function $P(x,y)$. $$\begin{aligned}
h\left(u,v;\xi,\eta\right) = \frac{A}{\lambda z_{i}} \iint_{\infty}^{\infty} P\left(x,y\right) \nonumber \times \\
\exp\left\lbrace -j \frac{2\pi}{\lambda z_{i}} \left[ \left(u-M\xi\right)x+\left(v-M\eta\right)y \right]\right\rbrace \, \mathrm{d}x \mathrm{d}y
\label{eq3}\end{aligned}$$ In order to determine the irradiance $I_{i}\left(u,v\right)$ recorded by a photo-detector placed in the image plane, the square of the image amplitude is time-averaged. $$I_{i}\left(u,v\right) = \left\langle |U_{i}\left(u,v\right)|^{2}\right\rangle
\label{eq4}$$
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[2]{}
COHERENT IMAGING {#coherent-imaging .unnumbered}
----------------
The coherent illumination of an object yields an imaging system linear in complex amplitude. Indeed, by defining a time-invariant phasor amplitude $U_{i}\left(u,v\right)$ in the image space, the imaging system is found to be described by an amplitude convolution equation. By combining the Eq.\[eq2\] and Eq.\[eq4\], the expression of the image intensity can thus defined as the square of the convolution product of the object signal and the amplitude spread function $h\left(u,v;\xi,\eta\right)$. $$I_{i}\left(u,v\right) = \left\langle\big| h\left(u,v;\xi,\eta\right) \circledast U_{o}\left(\xi,\eta\right) \big|^{2} \right\rangle
\label{eq5}$$ Figure \[asf\] shows the image formation of an imaging system with coherent illumination in one dimensions. Assuming a rectangular-aperture amplitude distribution $P(x)$, the amplitude spread function $h\left(v;\eta\right)$, represented in blue, is a cardinal sinus function, i.e. a zero-order spherical Bessel function. Two objects (two continuous periodical waves) having a low and a high spatial frequencies are illustrated in green and in red, respectively. The imaging system transmits the low frequency object pattern keeping the same contrast (Fig.\[asf\](a)). Whereas, when the frequency of the entrance signal is higher (Fig.\[asf\](b)), the contrast of the image equals zero, i.e. the imaging system cannot resolve such frequency. Through a frequency analysis, a better visualization of the coherent imaging process is provided. Using convolution theorem and Eq. \[eq2\], the frequency-domain electric field in the image plane can be expressed by: $$\mathscr{F} \left\lbrace U_{i}\left(u,v\right) \right\rbrace = \mathscr{F} \left\lbrace h\left(u,v;\xi,\eta\right) \right\rbrace \times \mathscr{F} \left\lbrace U_{o}\left(\xi,\eta\right) \right\rbrace,
\label{eq6}$$ and, from Eq. \[eq3\], $$\mathscr{F} \left\lbrace U_{i}\left(u,v\right) \right\rbrace \propto P(x,y) \times \mathscr{F} \left\lbrace U_{o}\left(\xi,\eta\right) \right\rbrace,
\label{eq7}$$ Then, the frequency-domain irradiance is defined by: $$I_{i}\left(f_{y},f_{x}\right) \propto \big| P(x,y) \times \mathscr{F} \left\lbrace U_{o}\left(\xi,\eta\right) \right\rbrace\big|^{2}
\label{eq8}$$ These equations show that the term of normalized amplitude transfer function $H\left(f_{x},f_{y}\right)$ of the imaging system is its pupil function $P(x,y)$. $$\begin{aligned}
H\left(f_{x},f_{y}\right) &= \mathscr{F} \left\lbrace h\left(u,v;\xi,\eta\right) \right\rbrace \nonumber \\
&\propto \mathscr{F} \left\lbrace \mathscr{F} \left\lbrace P(x,y) \right\rbrace \right\rbrace \propto P \left( \lambda z_{i} f_{x}, \lambda z_{i} f_{y} \right)
\label{eq9}\end{aligned}$$ Figure \[atf\] illustrates the image formation in frequency domain of a finite-aperture imaging system using a coherent illumination. The object consists of two sinusoidal signals having a high and a low frequency equivalent to Fig. \[asf\]. When the frequency of the entrance object signal is below the cut-off frequency $f_{c}$ of the imaging system, e.g. green signal, the resulting contrast of its image stays unchanged. However, for higher frequency signal, e.g. red signal, the contrast drop to zero. The frequencies of the object being higher than the cut-off frequency of transfer function of the coherent imaging system are thus not resolved. The cut-off frequency $f_{c}$ of a perfectly-coherent imaging system is given by: $$f_{c} = \frac{D}{\lambda z_{i}} = \frac{NA}{\lambda}
\label{eq10}$$ This formula remind us the Abbe theory with Eq. \[EqAbbe1\]. However, this assumption of strictly monochromatic illumination is overly restrictive. The illumination generated by real optical sources, including the LASER, are never perfectly monochromatic. The value of the cut-off frequency could hence be slightly reduced.
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[2]{}
INCOHERENT IMAGING {#incoherent-imaging .unnumbered}
------------------
Using an incoherent illumination, the image formation by the optical system is linear in intensity. The resulting intensity in the image plane is defined as the square of the amplitude spread function $h\left(u,v;\xi,\eta\right)$ convoluted at the object irradiance point source. Equation \[eq5\] can thus be expressed: $$I_{i}\left(u,v\right) = \big| h\left(u,v;\xi,\eta\right) \big|^{2} \circledast \big| U_{o}\left(\xi,\eta\right) \big|^{2}
\label{eq11}$$ or, $$I_{i}\left(u,v\right) = \big| h\left(u,v;\xi,\eta\right) \big|^{2} \circledast I_{o}\left(\xi,\eta\right)
\label{eq12}$$ The imaging process of an aberration-free optical system using an incoherent light source is schemed in Fig.\[isf\] in one dimension. The object having two distinct frequencies, as shown in Fig.\[asf\], are illustrated in green and in red. Here, the imaging system transmits the low-frequency object signal keeping the same contrast (Fig.\[isf\](a)). However, when the frequency of the entrance signal is higher (Fig.\[isf\](b)), the contrast of the image is decreased but still resolved by the imaging system. The incoherent spread function, also called intensity spread function or intensity point spread function (IPSF), is the square of the amplitude spread function $h\left(u,v;\xi,\eta\right)$ of the imaging system. This function is called Airy disk [@Airy35] where the full width at half maximum (FWHM) equals 1.029 and the first zero position is at $\eta = 1.219$. J. W. Rayleigh reported the use of the first zero of the Airy pattern in order to determine the distance $\delta_{x}$ between two resolved point sources [@Rayleigh74]. $$\eta = 1.219 = \delta_{x}\frac{2~NA}{\lambda}$$ In frequency domain, Eq. \[eq12\] can be retrieved using the convolution theorem. $$\mathscr{F} \left\lbrace I_{i}\left(u,v\right) \right\rbrace = \mathscr{F} \left\lbrace \big| h\left(u,v;\xi,\eta\right) \big|^{2} \right\rbrace \times \mathscr{F} \left\lbrace I_{o}\left(\xi,\eta\right) \right\rbrace,
\label{eq13}$$ or $$\mathscr{F} \left\lbrace I_{i}\left(u,v\right) \right\rbrace \propto \mathscr{F} \left\lbrace \big| \mathscr{F} \left\lbrace P(x,y) \right\rbrace \big|^{2} \right\rbrace \times \mathscr{F} \left\lbrace I_{o}\left(\xi,\eta\right) \right\rbrace
\label{eq14}$$ In this equations, the term of optical transfer function $OTF\left(f_{x},f_{y}\right)$ is revealed, being the normalized autocorrelation function of the amplitude transfer function. $$\begin{aligned}
OTF\left(f_{x},f_{y}\right) &= \mathscr{F} \left\lbrace \big| h\left(u,v;\xi,\eta\right) \big|^{2} \right\rbrace \nonumber \\
&\propto \mathscr{F} \left\lbrace \big| \mathscr{F} \left\lbrace P(x,y) \right\rbrace \big|^{2} \right\rbrace
\label{eq15}\end{aligned}$$ From the Fourier transform operation, a real and an imaginary parts result, leading to the modulation transfer function (MTF) and the phase transfer function (PTF). $$MTF = \big| OTF \big|
\label{eq16}$$ $$PTF = angle \left\lbrace OTF \right\rbrace
\label{eq17}$$ The PTF is liable for the transversal shift of the image. Whereas, the MTF represents the contrast (or the visibility) distribution ratio between the image $M_{image}$ with the object $M_{object}$ at a given spatial frequency $f_{x}$,$f_{y}$. $$MTF\left(f_{x},f_{y}\right) = \frac{M_{image}}{M_{object}}\left(f_{x},f_{y}\right)$$ Where the contrast $M\left(f_{x},f_{y}\right)$ is related to the intensity modulation depth of the sinusoid signal. $$M\left(f_{x},f_{y}\right) = \frac{I_{max} - I_{max}}{I_{max} + I_{max}}\left(f_{x},f_{y}\right)$$ Figure \[mtf\] illustrates the transfer function of an incoherent imaging system with a rectangular pupil function $P(x,y)$. The object signals are sinusoidal functions with different frequencies (see Fig. \[isf\]). An entrance signal with a frequency below the cut-off frequency $f_{c}$ of the transfer function of the imaging system can be transmitted without reducing of the contrast. However, unlike in coherent imaging, the contrast of higher frequency signal than $f_{c}$ can be decreased, e.g. the red higher frequency signal, and still resolved. The cut-off frequency $f_{c}$ of the transfer function in incoherent imaging is defined by: $$f_{c} = 2\frac{NA}{\lambda_{0}},
\label{eq18}$$ confirming the results reported previously by H. von Helmholtz [@Helmholtz76]. The long wavelengths tend thus to decrease the cut-off frequency value. While, shorter wavelengths increase $f_{c}$. For polychromatic illumination, the MTF is determined by averaging the sum of $MTF\left(f_{x},f_{y}\right)$ of each wavelength, i.e. additioning the MTFs over the bandwidth of the light source [@telescope]. In experiments, the resulting polychromatic transfer function is assumed similar to the one at the central wavelength $\lambda_{0}$ of the spectrum.\
Mathematically, the MTF function of an aberration-free imaging system, having a circular pupil, can be expressed as: $$MTF\left(f_{x},f_{y}\right) = \frac{2}{\pi}\left\lbrace\phi\left(f_{x},f_{y}\right) -\cos\left[\phi\left(f_{x},f_{y}\right)\right]\sin\left[\phi\left(f_{x},f_{y}\right)\right]\right\rbrace$$ where, in one dimension, $$\cos\left[\phi\left(f_{x}\right)\right] = \frac{f_{x}}{f_{c}}~~~~\text{and}~~~~\cos\left[\phi\left(f_{y}\right)\right] = \frac{f_{y}}{f_{c}}$$
INFLUENCE OF THE ABERRATIONS {#influence-of-the-aberrations .unnumbered}
----------------------------
Up to now, the manuscript relates the performance Fourier analysis of an aberration-free imaging system, i.e. a diffraction-limited system. In reality, the imaging systems provide often optical aberrations especially high-numerical-aperture systems. These aberrations decrease dramatically the performance of imaging systems through the deformation of the transfer function in both spatial and frequency domain. Indeed, in spacial domain, the point spread function can be widen, lowing the lateral resolution, or can be flattened, reducing the contrast of the image. Therefore, criteria exist for estimating the performance of an aberrated imaging system.\
The Strehl ratio (SR) allows a quantification of the contrast as a function of a perfect optical component, having the same numerical aperture [@Strehl95; @Baranski14], through a mathematical operation based on the ratio of intensity peaks. $$SR = \frac{max \left(IPSF_{aberrated}\right)}{max \left(IPSF_{perfect}\right)}
\label{eq19}$$ where $max \left(IPSF_{aberrated}\right)$ is the IPSF peak intensity of the system to be characterized and $max \left(IPSF_{perfect}\right)$ is the IPSF peak intensity of an aberration-free system. An system system is assumed diffraction limited only when the SR is higher than 0.8. Other criteria subsist and are listed in Tab. \[citerionPSF\].
[Strehl ratio]{} RMS (wave) Criterion
------------------ ------------------ -------------------
0.96 $\lambda / 32$ $ $
0.82 $\lambda / 14$ Maréchal
0.8 $\lambda / 13.4$ Diffraction limit
: \[citerionPSF\]Relation between SR and RMS of the wavefront.
According to Maréchal’s Strehl approximation, the root mean square (RMS) of the wavefront can then be deducted from SR when the aberrations are low (SR > 0.8). $$SR = \exp\left[-\left(RMS_{Phi}\right)^{2}\right]
\label{eq20}$$ In frequency domain, optical aberrations decrease the contrast of the MTF at some frequencies $f_{x,y}$ or yield to a linear lateral shift of the image pattern, i.e. PTF derivation. Figure \[aberration\] illustrates the impact of the commonly optical aberrations on the MTF distribution. The cut-off frequency, depending on the wavelength of the light source and the NA of the imaging system, stays unchanged. However, the MTF of strongly-aberrated imaging system could fall off to zero at lower frequencies, leading perhaps, in practice, to wrong interpretation of the results.
![**Modulation transfer function of an incoherent imaging system** being aberration-free or having defocus, spherical or astigmatism aberration.[]{data-label="aberration"}](aberration.png){width="8cm"}
H. H. Hopkins defined thus a criterion based on a drop-off contrast of the MTF equaling 20%, by analogy to the Strehl ratio for IPSF [@telescope; @Hopkins57].
DISCUSSIONS {#discussions .unnumbered}
-----------
First, periodic white and black bars, i.e. Ronchi rulings or USAF 1951 target, are widely preferred in experiments for example for the characterization of imaging system because more convenient to fabricate than sinusoidal grey-level pattern. The response of an alternating black and white lines is called contrast transfer function (CTF). By decomposing a one-dimension square-wave object in Fourier series, the CTF is related to the MTF by: $$MTF\left(f_{x}\right) = \frac{\pi}{4} \sum_{n=0}^{N} \frac{1}{2n+1} CTF \left(\left(2n+1\right) f_{x}\right)$$ However, the difference of the evolution curves between the MTF and the CTF is low [@telescope] and, in practice, is often not considered.\
Finally, in optical metrology, Fourier optics made it possible to measure the wavefront generated by an optical system using an iteration process of the propagation and the back-propagation of the light in several image planes [@Saxton72]. This characterization technique is called phase retrieval [@Fienup82; @Perrin15].
CONCLUSION {#conclusion .unnumbered}
----------
This manuscript reviews the concept of Fourier optics for optical imaging. Therefore, the transfer function of finite-aperture imaging systems is detailed in both the spatial and the frequency domains considering the nature of the light. Indeed, the response of a coherent imaging system is linear in complex amplitude. Whereas, for incoherent imaging, the transfer function is related to the irradiance and the intensity. Also, the effect of optical aberrations on the imaging quality is discussed. Furthermore, resolution criteria are mathematically defined, assuming the sensor satisfies at least the Nyquist-based sampling.\
Recently, several imaging techniques have been developed in order to overcome the physical barrier of the diffraction of the light. Indeed, confocal microscopy \[US patent **US3013467** (1961)\], stimulated-emission-depletion fluorescence microscopy \[Opt. Lett. **19**, 780 (1994)\], scanning photonic jet scanning microscopy \[Opt. Express **15**, 17334 (2007)\] or microsphere-assisted microscopy in 2D \[Nat. Commun. **2**, 218 (2011)\] and in 3D \[Appl. Opt. **56**, 7249 (2017)\] made the resolution possible to reach $\lambda$/7.
[1]{} Z. Malacara, “Handbook of Optical Design, Second Edition” (CRC Press, 2003). J.W. Goodman “Introduction to Fourier optics, Second Edition” (McGraw-Hill companies, 1996). E. Verdet, “Leçons d’optique physique,” (Victor Masson et fils, 1869). E. Abbe, “A contribution to the theory of the microscope and the nature of microscopic vision,” Proceedings of the Bristol Naturalists Society **1** 200-261 (1874). E. Abbe, “On the estimation of aperture in the microscope,” Journal of the Royal Microscopical Society **1**, 388-423 (1881). A. Köhler, “Ein neues Beleuchtungsverfahren für mikrophotographische Zwecke,” Zeitschrift für wissenschaftliche Mikroskopie und für Mikroskopische Technik **10**, 433-440 (1893). H. von Helmholtz with a preface by H. E. Fripp, “On the limits of the optical capacity of the microscope,” The Monthly Microscopical Journal **16**, 15-39 (1876). M. A. Lauterbach, “Finding, defining and breaking the diffraction barrier in microscopy – a historical perspective,” Optical Nanoscopy **1**, 8 (2012). J. W. Rayleigh, “On the manufacture and theory of diffraction-gratings,” Philosophical Magazine and Journal of Science 4 **47**, 81-93 (1874). G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Transactions of the Cambridge Philosophical Society **5**, 283-291 (1835). J. W. Rayleigh, “Investigations in optics, with special reference to the spectroscope,” Philosophical Magazine and Journal of Science 4 **5**, 261-274 (1879). J. W. Rayleigh, “On the theory of optical images, with special reference to the microscope,” Philosophical Magazine and Journal of Science 5 **42**, 167-195 (1896). C. M. Sparrow, “On spectroscopic resolving power,” Astrophysical Journal **44**, 76-86 (1916). W. V. Houston, “A Compound Interferometer for Fine Structure Work,” Physical Review **29**, 478-484 (1927). P.M. Duffieux, “L’intégrale de Fourier et ses applications a l’optique” (Faculté des Sciences de Besançon, 1946). V. Ronchi, “Resolving Power of Calculated and Detected Images,” Journal of Optical Society of America **51**, 458-460 (1961). B.E.A. Saleh and M. Carl Teich “*Ch. 4: Fourier optics,*” in “Fundamentals of Photonics” (John Wiley and Sons, 1991). R. Dror, “[Fourier transforms and convolution](https://web.stanford.edu/class/cs279/lectures/lecture8.pdf),” \[Online; accessed Nov. 1, 2016\]. V. Sacek, “[Amateur Telescope Optics](http://www.telescope-optics.net/index.htm),” \[Online; accessed 05-July-2017\]. K. Strehl, “Aplanatische und fehlerhafte Abbildung im Fernrohr,” Zeitschrift fur Instrumentenkunde **15** 362-370 (1895). M. Baranski, S. Perrin, N. Passilly, L. Froehly, J. Albero, S. Bargiel, and C. Gorecki, “A simple method for quality evaluation of micro-optical components based on 3D IPSF measurement,” Optics Express **22**, 13202-13212 (2014). H. H. Hopkins, “The Aberration Permissible in Optical Systems,” Proceedings of the Physical Society B **70** 449-470 (1957). R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik **35**, 237 (1972). J. R. Fienup “Phase retrieval algorithms: a comparison” Applied Optics **21**, 2758-2769 (1982). S. Perrin, M. Baranski, L. Froehly, J. Albero, N. Passilly, and C. Gorecki, “Simple method based on intensity measurements for characterization of aberrations from micro-optical components,” Applied Optics **54**, 9060-9064 (2015).
**Coherence effect on the irradiance**\
\
The coherence of the light can be divided in three cases: the spatial and the temporal coherence and the polarization. Here, only the two first cases are considered. The polarization of the light is neglected. An optical system behaves differently if illuminated by a temporally or spatially coherent or incoherent light. Indeed, a temporally incoherent illumination, typically a broadband light source, provides chromatic aberration which is typical evidence of temporal incoherence. The degree of spatial coherence alters the description of an optical system as a linear system. A coherent illumination is obtained whenever light appears to originate from a single point. The most common example of a source of such light is a laser, although more conventional sources, e.g. zirconium arc lamps, can yield coherent light if their output is first passed through a pinhole. Whereas, spatial incoherent light is obtained from diffuse or extended sources. We digress temporarily to consider the very important effects of polychromaticity.\
\
In coherent imaging, the two wave sources have a constant initial phase difference and the same frequency. Thus, the intensity distribution $I$ resulting on the interference pattern from a wavefront division (temporally coherence) or an amplitude division (spatially coherence) can be written: $$\begin{aligned}
I &= \big| a_{1}\exp \left( j\phi_{1} \right) + a_{2}\exp \left( j\phi_{2} \right) \big|^{2} \nonumber \\
&= I_{0} \left( 1+m \cos\left( \phi_{2} - \phi_{1} \right) \right)\end{aligned}$$ with $j$ is the imaginary unit. The average intensity is given by $I_{0}=a_{1}^{2}+a_{1}^{2}=I_{1}+I_{2}$ and the contrast by $m=2\sqrt{I_{1}I_{2}}/\left(I_{1}+I_{2}\right)$.\
\
The phase delay $\Phi=\phi_{2}-\phi_{1}$ incurred in the paths of the two arms of the interferometer is:\
- $\phi_{2}-\phi_{1} = \frac{2\pi}{\lambda}\left(x_{2}-x_{1}\right)$ for an amplitude division interferometer,\
- $\phi_{2}-\phi_{1} = \frac{2\pi}{\lambda}\left( t_{2}-t_{1}\right)$ for a wavefront division interferometer.\
$\lambda$ is the wavelength of the light source.\
\
In incoherent imaging, the object illumination has the opposite property that the phasor amplitudes at all points on the object are varying in totally uncorrelated way. In this case, the phase delay $\Phi$ changes rapidly over the time ($\sum_{i} \Phi \left(\lambda_{i}\right) = 0$), leading to an intensity averaging out to zero: $$I = \left\langle I_{0}\left( 1+m \cos\left( \phi_{2} - \phi_{1} \right) \right) \right\rangle$$ $$I = I_{0}\left( 1+m \left\langle \cos\left( \phi_{2} - \phi_{1} \right) \right\rangle \right)$$ $$I = I_{0} = a_{1}^{2}+a_{1}^{2}$$\
However, in the perfectly coherent case, the intensity is computed as the modulus-squared of the sum of the phasors of the electric fields. The image intensity is thus linear in amplitude. Whereas, in the perfectly incoherent case, the intensity is computed as the sum of the modulus-squared of the phasors of the electric fields, leading to an intensity linearity of the image distribution.\
\
In conclusion, if the illumination is coherent, the output field is described as the convolution of the input field with the amplitude spread function. While, if the illumination is incoherent, the output intensity is described as the convolution of the input intensity with the intensity spread function.
**Two dimensional representations of image formation using Fourier optics**.\
\
![Image formation of a low (in green) and a high (in red) frequency modulation pattern using the concept of Fourier optics in spatial domain. The illumination is incoherent and the imaging system, aberration-free. The imaging system resolves the two sinusoidal object patterns. However, the contrast of the resulting image of the higher-frequency pattern is reduced.[]{data-label="2dfrequency"}](2dfrequency.png)
![Image formation of a low frequency modulation pattern using the concept of Fourier optics in spatial domain and in frequency domain. The illumination is incoherent and the imaging system, aberration-free. The imaging system resolves the sinusoidal object pattern. However, the contrast of the resulting image is reduced by the point spread function (in spatial domain) or the modulation transfer function (in frequency domain).[]{data-label="2ddomain"}](2ddomain.png)
![Image formation of a high frequency modulation pattern using the concept of Fourier optics in frequency domain. The illumination is incoherent (top of the figure) and coherent (bottom of the figure) and the imaging system, aberration-free. The incoherent imaging system resolves the sinusoidal object pattern but the contrast of the resulting image is reduced by the modulation transfer function, whereas the coherent imaging system is not able to transmit the features of the object pattern.[]{data-label="2dcoherence"}](2dcoherence.png)
Some figures were extracted from the Kurt Thorn’s presentation ([Microscopy optics II](http://nic.ucsf.edu/dokuwiki/doku.php?id=presentations)" \[Online: accessed 2018\]).
[^1]: Corresponding author: <stephane.perrin@unistra.fr>
[^2]: The diffraction image of an emitting incoherent point source through an optical system is the point spread function (PSF) of the system.
[^3]: Formerly, a Gaussian image was an image formed from these approximations
|
---
abstract: |
The correlations between properties of hydrogenated diamond like carbon films and their Raman spectra have been investigated. The films are prepared by plasma deposition technique, keeping different hydrogen to methane ratio during the growth process. The hydrogen concentration, sp$^3$ content, hardness and optical Tauc gap of the materials have been estimated from a detail analysis of their Raman spectra. We have also measured the same parameters of the films by using other commonly used techniques, like sp$^3$ content in films by x-ray photoelectron spectroscopy, their Tauc gap by ellipsometric measurements and hardness by micro-hardness testing. The reasons for the mismatch between the characteristics of the films, as obtained by Raman measurements and by the above mentioned techniques, have been discussed. We emphasize on the importance of the visible Raman spectroscopy in reliably predicting the above key properties of DLC films.
Keywords : Diamond like carbon, Raman scattering.
author:
- 'Achintya Singha$^1$, Aditi Ghosh$^1$, Nihar Ranjan Ray$^{2}$'
- 'Anushree Roy$^{1}$'
title: Quantitative Analysis of Hydrogenated DLC Films by Visible Raman Spectroscopy
---
introduction
============
In recent years, hard diamond like carbon (DLC) films have attracted a great deal of research interest as they are useful materials for protective optical and tribological coating [@geis:1993; @koidl:1989; @angus:1988; @robertson:1992; @tsai:1987]. These films can be of greater economic importance in the long run, if their properties can be understood to the extent we know the bulk characteristics of diamond today. The characteristics of DLC films depend considerably on the atomic structure of the films. For example, opaque samples with hardness one-fifth of that for diamond and the transparent ones nearly as dense and hard as diamond [@tamor:1994], have been considered as DLC films, for which specific properties have been investigated. This indicates that over a wide range of atomic structures one can achieve diamond-like behavior.
DLC films can be defined as composites of nanocrystalline diamond and/or amorphous carbon with/without hydrogen (required to passivate the dangling bonds of carbon). The films with 20 - 50 $\%$ of hydrogen content is commonly known as hydrogenated DLC (HDLC) films. DLC films are usually composed of amorphous carbon with a significant amount of sp$^3$ bonds [@ferrari:2000]. The films with higher (more than 85%) sp$^3$ content are known as tetrahedral amorphous carbon (ta-C) rather than DLC. Such films along with a considerable amount of hydrogen content are known as ta-C:H. In general, the amorphous carbon can have sp$^3$ (diamond-like), sp$^2$ (graphite-like) and even sp$^1$ sites. The properties of any carbon material depend on the ratio of the amount of sp$^2$ to that of the sp$^3$ bonds. In addition to the sp$^3$ content, the clustering of sp$^2$ phase also plays an important role in determining different properties of these materials, especially, their optical, electrical and mechanical characteristics. Moreover, the presence of nanocrystalline (NC) diamond in DLC/HDLC films results in a special characteristic of these materials compared to graphite or diamond. It has been shown by Badziag *et al.* that the nanodiamond clusters of less than 5 nm in diameter are more stable than the graphite clusters of same size [@badziag:1990].
A variety of analytic techniques have been used to characterize the DLC/HDLC films. Common tools to measure sp$^2$ and sp$^3$ contents in the carbon films are nuclear magnetic resonance (NMR) [@kaplan:1985; @tamor:1991] or electron energy loss spectroscopy (EELS) [@fink:1983; @fallon:1993]. X-ray photoelectron (XP) spectroscopy has also been used to estimate sp$^3$ content in DLC films [@yan:2004]. Ellipsometric measurement is in general used to measure the Tauc gap of the material [@franta:2002; @vinnichenko:2004]. The standard techniques to measure hardness of DLC films are nano-indentation or micro-Vicker hardness testing [@marchon:1997; @wazumi:2005]. All the above techniques have their own limitations when used for DLC films. For example, NMR and EELS are time-consuming and destructive techniques. The aim of this article is to show that Raman spectroscopy, a non-destructive probe, can be employed as a single technique, to predict all the above critical properties of DLC/HDLC films from a single measurement with better accuracy.
Raman modes
-----------
The Raman spectrum of diamond consists of T$_{2g}$ mode at 1332 cm$^{-1}$ (sp$^3$ mode). The phonon confinement in NC diamond results in a downshift in the Raman spectrum of the diamond line. The amount of shift depends on the grain size. For grains of less than 1 nm in size, the maximum of the vibrational density of states appears at 1260 cm$^{-1}$ [@ferrari:2001]. On the other hand, the Raman spectrum of graphite usually shows two modes, zone center E$_{2g}$ phonon mode at around 1580-1600 cm$^{-1}$ (commonly known as G-peak, sp$^2$ mode) and K-point phonons at around 1350 cm$^{-1}$ (commonly known as D-peak : disordered allowed zone edge mode of graphite). The sp$^1$ mode is present in the HDLC film in a negligible amount. In HDLC films, other than NC diamond, the G- and D- peaks, features near 1150 cm$^{-1}$ and its companion mode 1450 cm$^{-1}$ appear due to sum and difference in combinations of C=C chain stretching and CH wagging modes \[v$_1$ and v$_3$ modes of transpolyacetylene (Trans-PA)\] lying in the grain boundary [@ferrari:2001]. Trans-PA is an alternate chain of sp$^2$ carbon atoms, with a single hydrogen bonded to each carbon atom. In addition, recent work has shown that a particularly stable defect in diamond is the dumbbell defect, which consists of $<$100$>$ split interstitials [@prawer:2000] in HDLC films. This localized defect consists of an isolated sp$^2$ bonded pair. In vibrational density of states calculation of a largely four folded amorphous carbon network there is a strong evidence for paired three-fold coordinated defects which appear as a sharp localized mode at about 1600 cm$^{-1}$ [@drabold:1994].
Three Stage model
-----------------
At this point it is to be noted that there is an inherent problem in applying visible Raman spectroscopy directly to estimate the ratio of sp$^3$ and sp$^2$ fraction in carbon materials by measuring the intensities of corresponding Raman modes : the visible Raman scattering cross-section is 50-230 times more sensitive to sp$^2$ sites compared to that of sp$^3$ sites, as visible photons preferentially excite the $\pi$-states of sp$^2$ sites. It was believed that visible Raman spectroscopy has a limited use to characterize DLC/HDLC films, especially to estimate the content of sp$^3$ and sp$^2$ fraction in these materials. Recently, based on atomic and electronic structure of disordered carbon, Ferrari and Robertson [@ferrari:2000] have proposed a *three-stage model* and have shown that disordered, amorphous and diamond like carbon phases in amorphous C-H films can be characterized by measuring the position and width of G-peak and intensity ratio of G- and D-peaks in Raman spectra, rather than by directly measuring their intensities. The changes in line shape of the Raman spectrum for carbon material, when its phase changes from graphite to NC carbon (stage one) to amorphous carbon (stage two) and then to ta-C carbon with about 85-90 $\%$ sp$^3$ bonding (stage three), have been explicitly shown in their article. During the first stage, with an increase in sp$^3$ content in the material, the ratio of the intensity of D peak (I$_D$) to that of G peak (I$_G$) increases from 0.0 to 2.0 and simultaneously, the G peak position ($\omega_{G}$) increases from 1580 cm$^{-1}$ to 1600 cm$^{-1}$. However, in the second stage, a reverse trend is observed for both parameters with an increase in sp$^3$ content : the ratio I$_D$/I$_G$ decreases from 2.0 to 0.25, whereas, the value of $\omega_G$ decreases from 1600 to 1510 cm$^{-1}$. For a phase transition from amorphous carbon to ta-C phase, $\omega_G$ increases (from 1520 to 1560 cm$^{-1}$) with an increase in sp$^3$ content and the intensity ratio I$_D$/I$_G$ drops down further (from 0.25 to 0) from what was observed in stage two. For each stage, relations between G-peak position and sp$^3$ content as well as Tauc gap of the carbon material have been discussed by authors.
In this article, we show how visible Raman spectra can be used as a fingerprint to characterize the DLC/HDLC -films grown by plasma deposition technique. Our emphasis will be on four important characteristics of HDLC films : sp$^3$ content, hydrogen concentration, optical gap and hardness. Section II covers the sample preparation technique, which we have followed and other details regarding the instruments, which we have used for various measurements. In Section III, we have analyzed our experimental Raman spectra using the *three-stage model* proposed by Ferrari and Robertson, mentioned above. The analysis provides an insight into the chemical composition (eg. hydrogen content, sp$^3$ content) and hardness of these films. The estimated sp$^3$ content and hardness of the films, measured from Raman data analysis, have been compared with the results obtained from the XP spectroscopic and nano-indentation measurements, respectively. We could also make an approximate judgement of the optical Tauc gap in these films from the shift in $\omega_G$ in their Raman spectra. Later on, the measured gap energies have been compared with the values we get from ellipsometric measurements. Finally, in section IV, we have discussed our results with a few concluding remarks.
Here, we would like to mention that in the article [@tamor:1994], Tamor and Vassel described a systematic study of the Raman spectra of amorphous carbon films. Based on experimental data, this phenomenological report revealed a clear correlation between the variation in the G-peak position and the change in optical gap and hardness of films. In this present report, based on the theoretical understanding proposed by Ferrari and Robertson [@ferrari:2000], we have shown how Raman measurements can be used to study HDLC films in more detail. We have also shown the presence of NC diamond in our films. Furthermore, in this report, we have pointed out drawbacks of other commonly used probes to study this system.
Experiments
===========
Carbon thin films are deposited on mirror-polished Si(100) substrate at room temperature using asymmetric capacitively coupled RF (13.56 MHz) plasma system. The depositions are made systematically as follows: a pretreatment of the bare mirror polished Si (100) substrate has been done for 15 minutes using pure hydrogen plasma at pressure 0.2 mbar and dc self-bias of -200 volts. The deposition has been made for 30 minutes at pressure 0.7 mbar keeping the flow rate of helium (He) at 1500 sccm, hydrogen (H$_2$) at 500 sccm, and varying the flow rate of methane (CH$_4$). Five samples (for which the CH$_4$ flow rates are 50 sccm, 30 sccm, 20 sccm, 18 sccm and 15 sccm), thus grown with varying H$_2$ to CH$_4$ ratio during deposition, will be represented as Sample A to Sample E, in the rest of the article.
Raman spectra are measured in back-scattering geometry using a 488 nm Argon ion laser as an excitation source. The spectrometer is equipped with 1200 g$/$mm holographic grating, a holographic super-notch filter, and a Peltier cooled CCD detector. With 100 $\mu$m slit-width of the spectrometer the resolution of our Raman measurement is 1 cm$^{-1}$. XP spectra are taken using PHI-5702 X-ray Photoelectron Spectroscope operating with monochromated Al K$\alpha$ irradiation (photon energy 1476.6 eV) as an excitation source at a pass energy of 29.4 eV. The chamber pressure is maintained at 10$^{-8}$ Pa.
Ellipsometric measurements of the five DLC films (Sample A - Sample E) are performed for the spectral range from 370 to 990 nm (1.25–3.35 eV) using J.A. Wollam ellipsometer( make USA) in the reflection mode. The angle of incidence is 70$^\circ$. From the ellipsometric measurements, the thickness of the films are estimated to be $\sim$ 120 $\pm$ 20 nm.
Hardness of the films are measured by Leica micro- hardness tester operated using Vickers diamond indenter by applying a load of 50 gf at a minimum of three places for each sample. Diagonals of the indentations are measured to eliminate the asymmetry of the diamond pyramid.
Samples for Transmission Electron Microscopy (TEM) are deposited on 400 mesh copper TEM grids coated with carbon films. The suspended film is obtained by sonicating the substrate with film in Acetone; which is then directly added drop-wise on the grid. The excess acetone is allowed to evaporate in air. The grids are examined in Hitachi H600 microscope operated at 75 kV.
Characterization of DLC films by Raman measurements and by other techniques
===========================================================================
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Fig. 1 (a) shows Raman spectra of Sample A to Sample E for the range from 950 to 2500 cm$^{-1}$. The increasing PL background of each spectra is a typical signature of hydrogenated DLC (HDLC) films [@ferrari:2000; @marchon:1997]. Keeping in mind the possibility of the presence of different phases of carbon, mentioned in section I, we have deconvoluted each spectrum with six Lorentzian functions (shown by dashed lines in Fig. 1) keeping intensities, width and peak positions as free fitting parameters. Table I provides all parameters, which we have obtained, in this analysis. It is to be noted that the asymmetric Breit-Wigner-Fano (commonly known as BWF) line shape is usually used to fit the G-peak which appears due to asymmetry of the vibrational density of states of graphite towards lower wavenumber. The tail of BWF line takes into account the Raman modes at 1100 and 1400 cm$^{-1}$, mentioned in the previous section, without giving rise to extra peaks. In Fig. 1 (and Table I) we note that the intensities of the D-peak and other low frequency peaks are quite less than the intensity of the G-peak. Thus, if we deconvolute each spectrum with all low frequency components to obtain the information of individual modes, there is an inherent difficulty in using BWF line shape for the G-peak. For this reason, we have used Lorentzian line shapes for all modes to analyze our spectra. For clarity, in Fig 1 (b) we show the experimental data, deconvoluted lines and net fitted spectrum for Sample A. Below we discuss the properties of our DLC films obtained from above Raman data analysis :
Sample Sample E Sample D Sample C Sample B Sample A Remarks
--------------------------- ---------- ---------- ---------- ---------- ---------- ----------------
Peak position (cm$^{-1})$ 1591 1589 1590 1589 1590 Dumbbell
Width (cm$^{-1}$) 38 45 39 39 39 defect
Intensity 1708 3082 3045 2283 2331 in NC
Peak position (cm$^{-1})$ 1539 1524 1518 1515 1517
Width (cm$^{-1}$) 54 54 50 50 50 G-peak
Intensity 10943 9885 7059 6182 6132
Peak position (cm$^{-1})$ 1451 1451 1451 1451 1451
Width (cm$^{-1}$) 67 64 66 64 65 Trans-PA
Intensity 4304 2919 4225 2919 900 ($\nu_3$ mode)
Peak position (cm$^{-1})$ 1341 1341 1341 1341 1341
Width (cm$^{-1}$) 90 105 105 105 105 D-peak
Intensity 2813 2800 2332 1416 2146
Peak position (cm$^{-1})$ 1258 1258 1258 1258 1258
Width (cm$^{-1}$) 73 91 110 148 177 NC
Intensity 2214 2894 4000 4858 5363 diamond
Peak position (cm$^{-1})$ 1150 1146 1146 1146 1146
Width (cm$^{-1}$) 51 50 52 50 50 Trans-PA
Intensity 84 70 73 58 74 ($\nu_1$ mode)
: Assignment of vibrational bands for DLC Raman spectra
=3.5in
\(a) *Hydrogen content*: The main effect of hydrogen in HDLC films is to modify its C-C network. Instead of increasing the fraction of C-C bond, hydrogen saturates the C=C bonds as $\equiv$ CH$_x$ groups to increase the sp$^3$ content in the film [@ferrari:2000]. In visible Raman spectra, C-H stretching mode lies above 3000 cm$^{-1}$, whereas, C-H bending mode (1290-1400 cm$^{-1}$) is masked by the D-peak region [@ristein:1998]. Moreover, these modes are not resonantly enhanced in visible Raman spectroscopy. Thus, it is difficult to estimate hydrogen content in a HDLC film directly by using this technique. However, as a result of the recombination of electron-hole pairs within sp$^2$ bonded clusters in HDLC films, the hydrogen content in films gives rise to a strong PL background for first order Raman spectra [@marchon:1997]. The ratio between the slope $m$ of the fitted linear background of the Raman spectrum (shown by dashed-dotted line in Fig. 1 (a)) due to photoluminescence and the intensity of G-peak (I$_G$), $m/I_G$, can be used as a measure of the bonded H content in the film. The slope parameter is described in micrometer unit [@marchon:1997]. Using this analysis, the decrease in intensity of bonded hydrogen content from Sample A to Sample E is shown in Fig. 2. It is to be noted that the observed decrease in H$_2$ content is expected to increase the sp$^2$ cluster size and results in a decrease in band gap from Sample A to Sample E [@ferrari:2000].
=6.5in
\(b) *sp$^3$ fractions in the films* : The variation in $\omega_G$ from Sample A to Sample E, as obtained from their Raman spectra in Fig.1 (a), is shown in Fig. 3(a). The frequency of the Raman shift for $\omega_G$ and the presence of NC diamond in the films (which will be discussed latter), indicate that the characteristics of the films fall under the second stage of the three stage model. If one carefully notes the dispersion in Raman G-peak position, $\omega_G$, in amorphous HDLC films and the sp$^3$ content in the same samples obtained by other measurements (NMR and EELS), a certain correlation between these two parameters can be clearly observed. Ferrari and Robertson in Fig. 14 of ref [@ferrari:2000] summarized experimental data from different sources and have shown that sp$^3$ content in the HDLC film is related to $\omega_G$ in the films. Similar characteristic has also been shown in Ref. [@tamor:1991] by Tamor et al. We have taken average of all data presented in these references in Fig. 3(b) and obtained an empirical relation by fitting the data points between 1525 cm$^{-1}$ and 1580 cm$^{-1}$ by a polynomial equation
$$\mbox {sp}^{3} \mbox{content}= 0.24 - 48.9 (\omega_{G} - 0.1580)$$
In this equation, $\omega_G$ has been taken in unit of inverse of micrometer unit. Eqn. 1 also demonstrates that for $\omega_G$ at 1580 cm$^{-1}$, the sp$^3$ fraction in the film is $\sim$ 0.24. Using the shift in G peak position in the above equation, the changes in sp$^3$ content in the films are shown in Fig. 3(c) (they are also tabulated in Table II). If one looks into the ternary phase diagram for HDLC film [@robertson:1991], as shown in Fig. 4, it is clear that with a decrease in hydrogen content the C-C sp$^3$ bonding in the film should also decrease. Thus, the observed decrease in hydrogen content (shown in Fig. 2) supports the decrease in sp$^3$ content in films, shown in Fig. 3(c).
=3.0in
=4.5in
Next, we have used XP Spectroscopy on these samples to measure their sp$^3$ content. The XP spectra for Sample A to Sample E are shown in Fig. 5. After subtracting the background signal, C1s spectra of HDLC films are deconvoluted into four components around 283.6 $\pm$0.2 eV, 284.6 $\pm$0.2 eV, 285.4 $\pm$0.2 eV, and 286.9 $\pm$0.3 eV (Fig. 5). The peaks at around 284.3 eV and 285.3 eV correspond to sp$^2$ carbon atoms [@cardona:1978] and sp$^3$ C–C bond [@zhao:2000], respectively. The feature at 286.6 eV is assigned to C–O contamination formed on the film surface due to air exposure [@yan:2004; @zhao:2000]. The presence of lower energy peak at around 283 eV indicates that some carbon atoms in HDLC films are bonded to silicon substrate (carbon peak of carbide) [@yan:2004]. The full width at half maxima for both peaks at 284.3 eV and 285.3 eV are kept at 1.4$\pm$0.05 eV. The sp$^3$ content in the films, as obtained from the ratio of the corresponding sp$^3$ peak area over the total C1s peak area, have been tabulated in Table II along with the same obtained from visible Raman data analysis.
Sample
---------- ------------ ----------
from Raman from XPS
Sample A 55 44
Sample B 56 43
Sample C 54 42
Sample D 51 43
Sample E 44 31
: Comparison of sp$^3$ content in HDLC films as obtained from Raman and XP spectroscopic measurements.
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(c)*Tauc gap* : For quantitative analysis of the Tauc gap in DLC films, as done in reference [@ferrari:2000], the authors have shown a correlation between $\omega_G$ with the optical Tauc gap of the material. Similar correlation between the same parameters has been reported by Tamor et al in Ref. [@tamor:1994]. We make a summary of all experimental data from the above references in Fig. 6(a) and fit them with a polynomial equation
$$\mbox{Tauc Gap} = 0.92 + 28410 (\omega_{G} - 0.1580)^2$$
As in Eqn. 1, $\omega_G$ has been taken in unit of inverse of micrometer unit. Eqn. 2 also demonstrates that for $\omega_G$ at 1580 cm$^{-1}$, the Tauc gap of the film is $\sim$ 0.92 eV. From the variation in $\omega_G$ as shown in Fig. 3(b) and using Eqn. 2, we have estimated the variation in Tauc gap of the samples, which has been shown in Fig. 6(b) and also listed in Table III. Here, we would like to recall that a decrease in Tauc gap was expected due to the decreasing hydrogen content from Sample A to Sample E (see Fig. 2).
Sample
---------- ------------ -------------------
from Raman from Ellipsometry
Sample A 2.05 1.59
Sample B 2.12 1.53
Sample C 2.01 1.49
Sample D 1.98 1.48
Sample E 1.40 1.41
: Comparison of Tauc gap of the HDLC films as obtained from Raman and Ellipsometric measurements.
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The optical gap obtained from the analysis of Raman data, has been compared with the same measured directly from Ellipsometry. Both layer and substrate are characterized by spectral dependencies of the refractive indices (n and n$^\star$) and the extinction coefficients (k and k$^\star$).The spectral dependency of n$^\star$ and k$^\star$ of the silicon substrate are taken from ref. [@ohlidal:1999]. The spectral dependencies of the optical constants (n and k) of the DLC films for a wide spectral range can be interpreted by a semi-classical dispersion model with Lorentz oscillators corresponding to $\pi \rightarrow \pi*$ (originating from sp$^2$ sites) and $\sigma \rightarrow \sigma*$ (originating from sp$^3$ sites) interband transitions [@zhang:1998]. We have used this model to obtain n and k from the ellipsometric data. The shape of the imaginary part of the complex dielectric function $\zeta [\varepsilon(E)]$ ( = 2nk) above the band gap is given by the Tauc equation $$\zeta [\varepsilon(E)]=\frac{\alpha (E-E_{g})^2}{E^2}$$ $\alpha$ is the absorbance of the film. The intersection of the plot $\sqrt{\zeta[\varepsilon(E)]}E$ vs. $E$ (Tauc plot) with the $E$ axis, measures the energy of the band gap, $E_g$. Tauc plots for Sample A to Sample E as obtained from the ellipsometric measurements are shown in Fig. 7. It is to be noted that plots for our DLC films are not strictly straight lines as indicated by Eqn. 3. This can be due to the existence of the Urbach tail (the transitions between localized states inside the band gap and extended states inside valence or conduction bands, which causes an exponential broadening of the absorption edge). From the intercept of the tangent of Tauc plots on $E$ axis we have obtained the optical band gap of HDLC films. The variation in optical band gap from Sample A to Sample E, as observed by ellipsometric measurements has been compared with that obtained from Raman measurements in Table IV.
(d)*Hardness*: In reference [@marchon:1997] we find a comment on variation in hardness of the DLC films with the percentage of Hydrogen in the film. Using this reference, we propose an empirical relation by fitting the data points by a linear equation
$$\mbox {Hardness in GPa}= 44.195 -0.93 \times \mbox {(\% of
hydrogen content)}$$
Using Eqn. 4 and the variation in hydrogen contents in our samples, we have estimated the hardness of the films. The hardness of each film, as obtained indirectly from Raman data analysis has been tabulated in Table IV along with the same obtained from micro-hardness testing.
Sample
---------- ------------ --------------------------
from Raman from micro-hardness test
Sample A 11.35 13.1
Sample B 15.37 12.7
Sample C 17.22 13.2
Sample D 17.60 11.4
Sample E 18.15 13.3
: Comparison of hardness of HDLC films as obtained from Raman and micro-hardness measurements.
From Fig. 2, Fig. 3(c) and the above hardness analysis we conclude (a) a decrease in sp$^3$ bonding from Sample A to Sample E does not degrade the hardness and scratch resistance of the deposit and (b) the low hydrogen content in the deposit strengthens the inter-molecular structure and, hence, leads to improved mechanical properties. Similar observation has been reported for Si implanted DLC films, where it has been shown from XP spectroscopic measurements that low hydrogen content in the DLC films results in a better hardness in these films [@zhao:2000].
=5.5in
\(e) *Nanocrystalline diamond* : The decrease in intensity of the peak at 1260 cm$^{-1}$ \[shown in Fig. 8 (a)\] indicates a decrease in NC diamond phase from Sample A to Sample E. From our above analysis we summarize that the NC component in these films decreases along with (i) an increase in G-peak position from 1517 cm$^{-1}$ to 1539 cm$^{-1}$ \[Fig. 3(b)\] and (ii) a decrease in sp$^3$ content \[Fig. 3(c)\]. The TEM image of one of our films is shown in Fig. 8(b). Presence of NC diamond along with larger particles is clear in the micrograph. This, in a way, supports our Raman data analysis by deconvolution.
Discussion
==========
In this article we have tried to establish a workable phenomenological picture for characterizing DLC films using visible Raman spectroscopy. We are able to estimate sp$^3$ and hydrogen content, optical gap and hardness of films, prepared by plasma deposition technique, by visible Raman measurements. We have summarized a number of data, available in the literature and then utilized the ‘*three stage model*’, which critically assesses the position of G peak in the Raman spectra of the films, to propose empirical relations between G-peak position and the values of sp$^3$ content and Tauc gap in HDLC films.
We have also measured the sp$^3$ content of the film by XP spectroscopy and optical band gap by ellipsometry. In addition, the hardness of the films are estimated from micro-hardness testing. In Table II, III and IV, we have compared the characteristics of DLC films as obtained from Raman data analysis and directly by the above mentioned measurements.
We now discuss the limitations of techniques while analyzing the characteristics of HDLC films on Si substrate. XP spectroscopy is a surface sensitive probe. Thus, this technique underestimates the value of sp$^3$ fraction (obtained by fitting the XP spectra) if the thickness of the film is more than the electron mean free path ($\sim$ 50 Å). Indeed, in Table II, we find that though both Raman and XP spectroscopic analysis exhibit decrease in sp$^3$ content from Sample A to Sample E, the absolute values of the same obtained from XP spectroscopic measurements are less than the values obtained from Raman data analysis. Here, we would like to point out that rich sp$^3$ phase of DLC films is often located below (depends on C energy) the surface layer. The surface layers are much more graphitic [@Lifshitz:1999]. XP spectroscopic measurement after etching the surface of the film under ultra high vacuum can be a solution to get the information of the inner layers; however, this experiment will damage the sample.
Furthermore, ellipsometry does not measure optical constants of materials directly. In this technique, we measure spectral dependency of the complex quantity $\rho$, which is defined as $\rho = \frac{r_p}{r_s}$. $r_s$ and $r_p$ represent the Fresnel coefficients of the sample. To get the spectral dependencies of the optical constants in the entire spectral region, one needs to understand the nature of dispersion in HDLC films. We have used ‘single sample method’ to analyze the ellipsometric data. We interpreted the spectral dependencies of the optical constants using Lorentz oscillator model. However, this model does not take into account the hydrogenated amorphous phase of the HDLC layer in a true sense. A more detailed nontrivial numerical analysis on modified Lorentz oscillator model is required to obtain the correct band gap of DLC films using ellipsometric measurements [@franta:2002]. may explain the mismatch in the Tauc gap of the samples obtained from Raman and ellipsometry measurements (shown in Table III); though both exhibit decrease in Tauc gap from Sample A to Sample E
Regarding the hardness measurements the source of discrepancy lies in the fact that our films are grown on Si substrate. The hardness of crystalline Si is $\sim$ 13 GPa, which is of the same order of magnitude as that of DLC. Thus, by indentation it is difficult to obtain the correct hardness of HDLC films subtracting the effect of the substrate. Indentation with less load does not measure the hardness of the films correctly. We attribute the discrepancy in hardness of the films, obtained from Raman and micro-hardness test to the fact that the latter is not a fool-proof technique to obtain the hardness of relatively thin DLC films on Si substrate.
Our Raman data analysis, based on a few empirical relations, may not be very accurate. However, by this, we overcome some of the above-mentioned shortcomings of the other techniques. The skin depth of the 488 nm excitation source, which we have used in Raman measurements is $\sim$ 6 $\mu$. Thus, using this technique we get information for the whole material rather than only for the surface as was the case in XP spectroscopic measurements. Unlike ellipsometric measurements, Raman analysis takes into account all phases of carbon. The hardness of the films, obtained from Raman data analysis, does not require an understanding of the effect of the substrate separately. By correct data analysis, we have also shown the presence of NC diamond in our films.
In addition to $\omega_G$, the intensity ratio of the D- and G peaks, can also be used to estimate the above parameters of HDLC films. We did not follow this procedure because of the following reasons : i) the intensity of the D peak is quite low compared to that of G peak in our samples (so the numerical value of I$_D$/I$_{G}$ can be erroneous) and ii) in the low frequency regime of the Raman spectra the weak D-peak coexists with Raman components of other phases of carbon; hence the measured intensity of the D-peak from the curve fitting is not unique, it varies strongly with the choice of width and intensity of other nearby peaks.
Table V summarizes the physical properties of our HDLC films, as obtained by Raman data analysis.
Sample $\%$ of sp$^3$ content Tauc gap (eV) hardness (GPa)
---------- ------------------------ --------------- ----------------
Sample A 57 2.03 11.35
Sample B 56 2.11 15.37
Sample C 55 1.91 17.22
Sample D 54 1.96 17.60
Sample E 43 1.39 18.15
: Properties of HDLC films as obtained from Raman measurements
Acknowledgement
===============
AR thanks Department of Science and Technology, India, for financial assistance. Authors thank Dr. S. Varma, Institute of Physics, Bhubaneswar, for their help on XPS measurements, Prof. K.K. Ray, Indian Institute of Technology, Kharagapur for Micro -Vicker’s testing, Mr. P. Roy, Saha Institute of Nuclear Physics for TEM measurements and Dr. Jens Raacke, University of Wuppertal, Germany for Ellipsometric measurements.
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abstract: 'This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left nonsingular left CS and right Utumi. A nonsingular Utumi ring is right max (resp. right min, right max-min) CS if and only if it is left min (resp. left max, left max-min) CS. In addition, a semiprime nonsingular ring is right max-min CS with finite right uniform dimension if and only if it is left max-min CS with finite left uniform dimension.'
address:
- '$^{(1)}$Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam'
- '$^{(2)}$Department of Science and Technology, Nguyen Tat Thanh University, 300A Nguyen Tat Thanh St, District 4, Ho Chi Minh City, Vietnam'
- 'International Cooperation Office, Hong Duc University, 565 Quang Trung St, Dong Ve ward, Thanh Hoa city, Vietnam'
- 'Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam'
author:
- Thuat Do
- Hai Dinh Hoang
- 'Truong Dinh Tu$^\dag$'
title: Symmetry of extending properties in nonsingular Utumi rings
---
[^1]
Introduction
============
Right-left symmetry of extending properties in associative (generally not commutative) rings is extensively studied by many authors. DV. Huynh et al. [@GCS] showed that a prime ring is right Goldie right CS with finite right uniform dimension at least two if and only if it is left Goldie left CS with finite left uniform dimension at least two, and a semiprime ring is right Goldie left CS if and only if it is left Goldie, right CS. Later, DV. Huynh [@CS1] investigated the symmetry of the CS condition on one-sided ideals in prime rings. SK. Jain et al. [@m-mCS] proved the right-left symmetry of the max-min CS property and nonsingularity on prime rings. In more general setting, DV. Thuat et al. [@thuat] studied the CS and Goldie conditions in prime and semiprime modules and their endomorphism rings. It is proved that a finite generated, quasi-projective self-generator $M$ is a prime, Goldie and CS module with uniform dimension at least two if and only if its endomorphism ring $S$ is a prime, left Goldie and left CS ring with left uniform dimension at least two; and $S$ is left Goldie and $M$ is CS if and only if $M$ is Goldie and $S$ is left CS. In the mentioned papers, primeness plays an important role to obtain the symmetric properties. We ask here:
[*“If primeness is omitted, can we find some classes of rings in which CS, max CS, min CS and max-min CS properties are right-left symmetric?”*]{}
Firstly, we provide some preliminaries in Section 2. The answer which involves our main results is presented in Section 3. There, the right-left symmetry of the extending properties (we mean the CS, max CS, min CS and max-min CS properties) is proved for the case of associative rings without primeness and even without having finite uniform dimension (see Theorem $\ref{th3.5}$ and Theorem $\ref{th3.7}$). The symmetry of the CS condition on one-sided ideals generated by idempotents is studied in Theorem $\ref{th3.12}$. In addition, the right-left symmetry of the CS, max CS, min CS, max-min CS conditions and finiteness of uniform dimension on nonsingular semiprime rings is shown in Theorem $\ref{thpr}$. Then, we apply the results to the class of nonsingular retractable modules and their endomorphism rings (see Theorem $\ref{th3.10}$, Proposition $\ref{p2}$ and Corollaries $\ref{c1}$, $\ref{c3}$ and $\ref{c4}$). Finally, some examples are discussed to guarantee that our results make sense.
Preliminaries
=============
Throughout this paper, $R$ is an associative (generally not commutative) ring with identity, $M$ is a unitary right $R-$module with the endomorphism ring $S={\operatorname {End}}(M_R).$ We denote ${\mathbf {r}}_X(Y)$ and ${\mathbf {l}}_X(Y)$ for the right annihilator and the left annihilator of $Y$ in $X,$ respectively. If there is no chance for misunderstanding of the space $X,$ then we simply write ${\mathbf {r}}(Y), {\mathbf {l}}(Y)$.
We write $X\hookrightarrow M$ (resp. $X \stackrel{*}{\hookrightarrow}M$) for a submodule (resp. an essential submodule) $X$ of $M.$ A submodule $X$ of $M$ is called a *closed submodule* if $X \stackrel{*}{\hookrightarrow} Y \Rightarrow X=Y,$ for any submodule $Y$ of $M.$ A module $M$ has *finite uniform dimension* if it contains no direct sum of infinitely many nonzero submodules. An $M-$annihilator $X$ of $M_R$ is a submodule provided $X={\mathbf {r}}_M(T)$ for some subset $T$ of $S$. If $M=R,$ then $M-$annihilators are exactly right annihilators of $R$ as usual. A *Goldie module* $M$ is provided that $M$ has finite uniform dimension and $M$ satisfies the ACC (i.e. ascending chain condition) on $M-$annihilators. A *right (left) Goldie* ring $R$ is provided that $R$ has finite right (left) uniform dimension and $R$ satisfies the ACC on right (left) annihilators. We denote the uniform dimension of a module $M_R$ by ${\operatorname {u-dim}}(M_R).$\
A [*CS*]{} (resp. [*uniform extending*]{}) module is provided that every closed (resp. closed and uniform) submodule is a direct summand. $M$ is called a *max CS module* if every maximal closed submodule with nonzero left annihilator in $S$ is a direct summand. $M$ is called a *min CS module* if every minimal closed submodule is a direct summand. $M$ is called a *max-min CS module* if it is both max CS and min CS. $R$ is called a *right max CS* (resp. *right min CS, right max-min CS*) ring if $R_R$ is a max CS (resp. min CS, max-min CS) module. Left max CS, left min CS and left max-min CS ring are defined analogously. It is clear that min CS modules are exactly uniform extending modules. If $M$ has finite uniform dimension, then $M$ is CS if and only if it is min CS. The original notion of right and left max-min CS rings may be seen in [@m-mCS].
The concepts of *nonsingular modules* and *nonsingular rings* are understood as usual. According to [@khuri3], $M$ is a nonsingular module if and only if for any $X\hookrightarrow M,$ ${\mathbf {r}}_R(X)\stackrel{*}{\hookrightarrow} R_R$ implies $X=0.$ $M$ is said to be *cononsingular* if for any $X\hookrightarrow M,$ ${\mathbf {l}}_S(X)\stackrel{*}{\hookrightarrow}_SS$ implies $X=0.$ It is equivalent to say that $R$ is right (left) nonsingular if and only if every essential right (left) ideal of $R$ has zero left (right) annihilator. Therefore, $R$ is right (left) nonsingular if and only if $R_R$ is a nonsingular (cononsingular) module. The following proposition is clear.
\[p0\] The following statements hold for the module $M.$
\(1) If $M$ is a nonsingular module, then for any $f\in S, {\operatorname {Ker}}(f) \stackrel{*}{\hookrightarrow} M$ implies $f=0.$ Furthermore, any essential submodule of $M$ has zero left annihilator in $S.$
\(2) If $M$ is a cononsingular module, then for any left ideal $K\hookrightarrow S, K \stackrel{*}{\hookrightarrow} _SS$ implies ${\mathbf {r}}_M(K)=0.$
Now, we consider the converse statements of Proposition $\ref{p0}$. According to [@khuri3], a nonsingular module $M$ is called a *Utumi module* if every submodule $X$ of $M$ with zero left annihilator in $S$ is essential in $M,$ i.e. ${\mathbf {l}}_S(X)=0 \Rightarrow X\stackrel{*}{\hookrightarrow}M.$ A cononsingular module $M$ is called a *co-Utumi module* if every left ideal $K$ of $S$ with zero right annihilator in $M$ is essential in $S,$ that is ${\mathbf {r}}_M(K)=0 \Rightarrow K\stackrel{*}{\hookrightarrow} _SS.$ $R$ is called a *right (left) Utumi ring* if $R_R$ is a Utumi (co-Utumi) module. By a nonsingular (Utumi) ring we mean that it is right and left nonsingular (Utumi). The two following lemmas are easy.
\[lm3.3\]
If $M$ is a CS module, then $M$ is Utumi. In particular, a right CS ring is right Utumi.
\[lm3.4\]
If $M$ is a nonsingular CS module, then $M$ is cononsingular. In particular, a right nonsingular right CS ring is left nonsingular.
For a submodule $X$ of $M,$ we write $I_X:=\{f\in S|f(M)\subseteq X\}.$ For a subset $K$ of $S,$ we write $KM=K(M):=\sum\limits_{f\in K}f(M).$ It is clear that $I_X$ is a right ideal of $S$ and $KM$ is a submodule of $M.$ The two following conditions are introduced and investigated in [@khuri-nd; @khuri4].
- \(I) For submodules $X, Y$ of $M, X \stackrel{*}{\hookrightarrow} Y$ if and only if $I_X\stackrel{*}{\hookrightarrow}I_Y.$
- \(II) For right ideals $K, L$ of $S, K \stackrel{*}{\hookrightarrow} L$ if and only if $KM\stackrel{*}{\hookrightarrow}LM.$
We observe that every finitely generated, quasi-projective self-generator is retractable and it possesses (I) and (II) (see [@thuat Lemma 2.2]). The same assertion holds for nondegenerate modules (see [@khuri-nd]). In the following lemma, we sum up [@khuri4 Theorem 2.2] and [@khuri4 Theorem 2.5] to make a tool to prove our main results in the subsequent section.
(see [@khuri4]) \[lm3.9\]
\(1) If $M$ is a nonsingular and retractable module, then (I) holds.
\(2) Given the condition (I), then the condition (II) holds if and only if $K\stackrel{*}{\hookrightarrow}I_{KM}$ for every right ideal $K\hookrightarrow S_S.$
\(3) Given the condition (II), then the condition (I) holds if and only if $I_X(M)\stackrel{*}{\hookrightarrow}X$ for every submodule $X\hookrightarrow M.$
The main results
================
We agree an abbreviation that MRQR and MLQR indicate maximal right quotient ring and maximal left quotient ring, successively. According to [@khuri4], $M$ is a [*retractable module*]{} if and only if ${\operatorname {Hom}}(M, X)\neq 0$ for every $0\neq X\hookrightarrow M.$ We denote the injective hull (or the envelope) of $M$ by $E(M),$ and the endomorphism ring of $E(M)$ by $T={\operatorname {End}}_R(E(M)).$ The following lemma plays an important role in our investigation.
\[lm3.2\] [@khuri3 Theorem 2]
Let $M$ be a retractable, nonsingular and cononsingular module. Then, $T={\operatorname {End}}_R(E(M))$ is both the MRQR and the MLQR of $S={\operatorname {End}}(M_R)$ if and only if $M$ is both a Utumi and co-Utumi module. In particular, for a nonsingular ring $R,$ the MRQR and the MLQR of $R$ coincide if and only if $R$ is right and left Utumi.
Note that in the case of Lemma $\ref{lm3.2}$, if $Q$ is the MRQR and the MLQR of $R$, then $Q$ is also the injective hull of $R_R$ and $_RR$. Therefore, $Q$ is von Neumann regular, right and left self-injective. Moreover, by [@utumi Lemma 1.4], $Q$ can be regarded as the ring consisting of element $x$ such that the set of $y\in R$ with $xy\in R$ forms an essential right ideal of $R.$ This notation will serve us in proof of subsequent theorems. Under the aid of the conditions (I) and (II), we derive the following results.
\[lm3.13\] Let $M$ be a retractable module which possesses (I) and (II). Then, ${\operatorname {u-dim}}(M_R)=n$ if and only if ${\operatorname {u-dim}}(S_S)=n$, where $n\geq 0$ is an integer.
We give some observations before mutually converting finiteness of uniform dimension between $M_R$ and $S_S.$ For nonzero right ideals $K, H$ of $S,$ we have $K(M)\neq 0$ and $H(M)\neq 0$. Moreover, we claim that $K\cap H=0$ if and only if $K(M)\cap H(M)=0$. We assume that $K\cap H=0$ and $K(M)\cap H(M)=Y\neq 0$. Then, by retractability of $M,$ there exists $0\neq s\in {\operatorname {Hom}}(M, Y)$, whence $s\in I_{KM}\cap I_{HM}$. By Lemma $\ref{lm3.9}$, we have $K\overset{*}{\hookrightarrow}I_{KM}$ and $H\overset{*}{\hookrightarrow}I_{HM}$. Therefore, there exists $f\in S$ such that $0\neq sf\in K$ and $sfS\cap H\neq 0.$ This means $K\cap H\neq 0,$ a contradiction. Thus, we must have $K(M)\cap H(M)=0.$ The converse is proved similarly.
For nonzero submodules $A$ and $B$ of $M,$ we see that $I_A\neq 0, I_B\neq 0$. Moreover, $A\cap B=0$ if and only if $I_A\cap I_B=0$ because of retractability of $M$. If we have $A\oplus B$, then we also get $I_A\oplus I_B$. It is obvious that $(I_A\oplus I_B)(M)\subseteq I_A(M)\oplus I_B(M)$. For $z=a+b\in I_A(M)\oplus I_B(M),$ where $a\in I_A(M), b\in I_B(M),$ there are $f_A\in I_A, f_B\in I_B$ and $x, y\in M$ such that $a=f_A(x), b=f_B(y).$ We see that $a\in f_A(M)\subset (I_A\oplus I_B)(M)$ and $b\in f_B(M)\subset (I_A\oplus I_B)(M)$. This implies $a+b\in (I_A\oplus I_B)(M)$, whence $I_A(M)\oplus I_B(M)\subseteq (I_A\oplus I_B)(M)$. Therefore, we get $(I_A\oplus I_B)(M)=I_A(M)\oplus I_B(M)$.
Now, let $A$ and $B$ be submodules of $M$ with $A\oplus B\overset{*}{\hookrightarrow}M$. Then, we have $(I_A\oplus I_B)(M)=I_A(M)\oplus I_B(M)$. On the other hand, by Lemma $\ref{lm3.9}, I_A(M)\oplus I_B(M)\overset{*}{\hookrightarrow}A\oplus B$ and $I_A\oplus I_B\overset{*}{\hookrightarrow}I_{A\oplus B}\overset{*}{\hookrightarrow}S_S$. Similarly, for right ideals $K, H$ of $S,$ if $K\oplus H\overset{*}{\hookrightarrow} S_S,$ then $(K\oplus H)(M)\overset{*}{\hookrightarrow} M_R$ and hence $K(M)\oplus H(M)\overset{*}{\hookrightarrow} M_R$. By these arguments, we inductively induce that for any integer $n\geq 0, {\operatorname {u-dim}}(M_R)=n$ if and only if ${\operatorname {u-dim}}(S_S)=n$.
\[p1\] Let $M$ be a nonsingular and co-nonsingular, Utumi and co-Utumi retractable module which possesses (II). Then, we have ${\operatorname {u-dim}}(M_R)=n$ if and only if ${\operatorname {u-dim}}(S_S)=n$ if and only if ${\operatorname {u-dim}}(_SS)=n$, where $n\geq 0$ is an integer. In this case, $M_R, S_S$ and $_SS$ are Goldie modules.
In particular, let $R$ be a nonsingular Utumi ring. Then, ${\operatorname {u-dim}}(R_R)=n$ if and only if ${\operatorname {u-dim}}(_RR)=n$. In this case, $R$ is right and left Goldie.
Since $M$ is nonsingular and retractable, Lemma $\ref{lm3.9}$ asserts that $M$ possesses (I). By Lemma $\ref{lm3.13}$, we have ${\operatorname {u-dim}}(M_R)=n={\operatorname {u-dim}}(S_S).$ By [@khuri Theorem 3.1], nonsingularity of $M$ implies right nonsingularity of $S$. Since $M$ is co-nonsingular, by [@khuri3 Proposition 1], $S$ is left nonsingular, so it is nonsingular. By Lemma $\ref{lm3.2}$, $T$ is both the MRQR and the MLQR of $S.$ Thus, we have $n={\operatorname {u-dim}}(S_S)={\operatorname {u-dim}}(T_S)={\operatorname {u-dim}}(_ST)={\operatorname {u-dim}}(_SS).$
Since $M$ is nonsingular module with finite uniform dimension, $M$ satisfies the ACC on $M-$annihilators. Thus, $M$ is a Goldie module. Since $S$ is nonsingular with finite right and left uniform dimensions, $S$ satisfies the ACC on right and left annihilators. Thus, $S$ is right and left Goldie.
\[th3.5\] The following statements are equivalent for a ring $R.$
\(1) $R$ is a right nonsingular, right CS and left Utumi ring;
\(2) $R$ is a left nonsingular, left CS and right Utumi ring.
In this case, if either $R_R$ or $_RR$ has finite uniform dimension, then $R$ is a right and left Goldie ring.
We assume that $R$ is a right nonsingular, right CS and left Utumi ring. By Lemma $\ref{lm3.3}$, $R$ is right Utumi, so it is Utumi. By Lemma $\ref{lm3.4}$, $R$ is left nonsingular, so it is nonsingular. Since $R$ is a nonsingular, Utumi ring, the MRQR and the MLQR of $R$ coincide by Lemma $\ref{lm3.2}$, and denoted by $Q.$
Now, we prove that $R$ is left CS. For any closed left ideal $I$ of $R,$ by the lattice isomorphism [@john Corollary 2.6], we have $I=J\cap R$ for some closed left ideal $J$ of $Q.$ Then, $J$ is a direct summand of $Q,$ writing $J=Qe$ for some idempotent $e\in Q.$ We easily see that ${\mathbf {r}}_Q(e)=(1-e)Q$ is a closed right ideal of $Q,$ thus $(1-e)Q\cap R$ is a closed right ideal of $R.$ Since $R$ is right CS, we get $(1-e)Q\cap R=fR$ for some $f=f^2\in R.$ We set $K=\{k\in R|(1-e)k\in R\}.$ Then, $K$ is an essential right ideal of $R.$ We have $R(1-f)={\mathbf {l}}_R(fR)={\mathbf {l}}_R[(1-e)Q\cap R]={\mathbf {l}}_R[(1-e)K]=\{x\in R|x(1-e)K=0\}=\{x\in R|x(1-e)=0\}={\mathbf {l}}_Q(1-e)\cap R=Qe\cap R=I.$ Thus, $I$ is a direct summand of $R.$ This implies that $R$ is left CS.
The converse is right-left symmetric.
The last statement is referred to Proposition $\ref{p1}.$
A right nonsingular, right CS and left Utumi ring is directly finite.
It follows from Theorem $\ref{th3.5}$ and the fact that a right and left CS ring is directly finite.
\[c1\] If $M$ is a nonsingular, retractable module, then the following statements are equivalent:
\(1) $M$ is a co-Utumi, CS module;
\(2) $S$ is a left Utumi, right CS ring;
\(3) $S$ is a right Utumi, left CS ring.
In addition, if $M$ has finite uniform dimension, then $M_R, S_S$ and $_SS$ are Goldie modules, and ${\operatorname {u-dim}}(M_R)={\operatorname {u-dim}}(S_S)={\operatorname {u-dim}}(_SS).$
We observe that $S$ is right nonsingular by [@khuri Theorem 3.1].
[*(1) $\Leftrightarrow$ (2)*]{} Since $M$ is nonsingular and retractable, by [@khuri4 Theorem 3.2], $M$ is CS if and only if $S$ if right CS. Since $M$ is nonsingular and CS, $M$ is a Utumi and co-nonsingular module by Lemma $\ref{lm3.3}$ and Lemma $\ref{lm3.4}$, respectively. Therefore, $S$ is left Utumi if and only if $M$ is co-Utumi by [@khuri3 Lemma 4].
[*(2) $\Leftrightarrow$ (3)*]{} It follows from Theorem $\ref{th3.5}$.
See Proposition $\ref{p1}$ for the last statement.
\[th3.7\] The following statements hold for every nonsingular Utumi ring $R.$
\(1) $R$ is right min CS if and only if $R$ is left max CS.
\(2) $R$ is right max CS if and only if $R$ is left min CS.
\(3) $R$ is right max-min CS if and only if $R$ is left max-min CS.
Since $R$ is nonsingular Utumi, the MRQR and the MLQR of $R$ coincide by Lemma $\ref{lm3.2}$, and denoted by $Q.$
*(1)* We assume that $R$ is right min CS. For any maximal closed left ideal $I$ of $R$ with ${\mathbf {r}}_R(I)\neq 0,$ by the lattice isomorphism [@john Corollary 2.6], we have $I=J\cap R$ for some closed left ideal $J$ of $Q.$ If $J$ is contained in some closed left ideal $K$ of $Q,$ then $K\cap R$ is a closed left ideal of $R$ and $I\subseteq K\cap R.$ Since $I$ is maximal closed, $I=K\cap R=J\cap R$ so $K=J.$ This shows that $J$ is a maximal closed left ideal of $Q.$ It is clear that $J$ is a direct summand of $Q,$ so $J=Qe$ for some idempotent $e\in Q.$ We easily see that ${\mathbf {r}}_Q(Qe)={\mathbf {r}}_Q(e)=(1-e)Q$ is a closed right ideal of $Q,$ thus $(1-e)Q\cap R$ is a closed right ideal of $R.$ We will show that $(1-e)Q$ is minimal closed in $Q.$ Suppose that $H=tQ, t=t^2\in Q,$ is a closed right ideal of $Q$ such that $H\subseteq (1-e)Q.$ Then, ${\mathbf {l}}_Q(H)=Q(1-t)\supseteq {\mathbf {l}}_Q[(1-e)Q]=Qe.$ Since $Qe=J$ is maximal closed in $Q,$ $Qe=Q(1-t)$ and hence $(1-e)Q={\mathbf {r}}_Q(Qe)={\mathbf {r}}_Q[Q(1-t)]=tQ=H.$ This implies that $(1-e)Q$ is minimal closed in $Q$. Thus $(1-e)Q\cap R$ is a minimal closed right ideal of $R.$ Since $R$ is right min CS, $(1-e)Q\cap R=fR$ for some idempotent $f\in R.$ We set $F=\{k\in R|(1-e)k\in R\}.$ Then, $F$ is an essential right ideal of $R.$ We have $R(1-f)={\mathbf {l}}_R(fR)={\mathbf {l}}_R[(1-e)Q\cap R]={\mathbf {l}}_R[(1-e)F]=\{x\in R|x(1-e)F=0\}=\{x\in R|x(1-e)=0\}={\mathbf {l}}_Q(1-e)\cap R=Qe\cap R=I.$ Thus, $I$ is a direct summand of $R.$ This shows that $R$ is left max CS.
Conversely, let $R$ be a left max CS ring. For any properly minimal closed right ideal $I$ of $R,$ by the lattice isomorphism [@john Corollary 2.6], we have $I=J\cap R$ for some closed right ideal $J$ of $Q.$ If $J$ contains a closed right ideal $K$ of $Q,$ then $K\cap R$ is a closed right ideal of $R$ and $(K\cap R)\subseteq (J\cap R)=I.$ Since $I$ is minimal closed, $I=K\cap R=J\cap R$ so $K=J.$ This shows that $J$ is a minimal closed right ideal of $Q.$ We write $J=eQ$ for some idempotent $0\neq e\in Q.$ We observe that ${\mathbf {l}}_Q(eQ)={\mathbf {l}}_Q(e)=Q(1-e)$ is a closed left ideal of $Q,$ thus $Q(1-e)\cap R$ is a closed left ideal of $R.$ We will prove that $Q(1-e)$ is maximal closed in $Q.$ Suppose that $H=Qt, t=t^2\in Q,$ is a closed left ideal of $Q$ such that $H\supseteq Q(1-e).$ Then, ${\mathbf {r}}_Q(H)=(1-t)Q\subseteq {\mathbf {r}}_Q[Q(1-e)]=eQ.$ Since $eQ=J$ is minimal closed in $Q,$ $eQ=(1-t)Q$ and hence $Q(1-e)={\mathbf {l}}_Q(eQ)={\mathbf {l}}_Q[(1-t)Q]=Qt=H.$ This implies that $Q(1-e)$ is maximal closed in $Q,$ thus $Q(1-e)\cap R$ is a maximal closed left ideal of $R.$
Because of $e\neq0,$ we have $0\neq eQ\cap R\subset {\mathbf {r}}_R[Q(1-e)\cap R]$. Since $R$ is left max CS, $Q(1-e)\cap R=Rf$ for some idempotent $f\in R.$ We set $F=\{k\in R|k(1-e)\in R\}.$ Then, $F$ is an essential left ideal of $R.$ We have $(1-f)R={\mathbf {r}}_R(Rf)={\mathbf {r}}_R[Q(1-e)\cap R]={\mathbf {r}}_R[F(1-e)]=\{x\in R|F(1-e)x=0\}=\{x\in R|(1-e)x=0\}={\mathbf {r}}_Q(1-e)\cap R=eQ\cap R=I.$ Thus, $I$ is a direct summand of $R.$ This shows that $R$ is right min CS.
*(2)* It is dual to the proof of *(1)*.
*(3)* It is induced from *(1)* and *(2)*.
By [@khuri4 Theorem 3.2], a nonsingular retractable module is CS if and only if its endomorphism ring is right CS. We wish to find an analogue for the max-min CS property. With the aid of (I) and (II), we will transfer the max CS, min CS and max-min CS properties of a module to its endomorphism in the next theorem.
\[th3.10\] Let $M$ be a nonsingular and retractable module which possesses the condition (II). Then, the following statements hold.
\(1) $M$ is min CS if and only if $S$ is right min CS.
\(2) $M$ is max CS if and only if $S$ is right max CS.
\(3) $M$ is max-min CS if and only if $S$ is right max-min CS.
It is clear that [*(3)*]{} follows from [*(1)*]{} and [*(2)*]{}. Note that since $M$ is a nonsingular module, every submodule $X$ has a unique closure (i.e. there is a unique closed submodule of $M$ that essentially contains $X$).
[*(1)*]{} Let $M$ be a min CS module. For a minimal closed (or uniform closed) right ideal $K$ of $S,$ we have $K\stackrel{*}{\hookrightarrow}I_{KM}$ by Lemma $\ref{lm3.9}$ so $K=I_{KM}$. For nonzero submodules $U, V$ of $KM,$ since $M$ is retractable, $I_U$ and $I_V$ are nonzero. It is clear that $I_U$ and $I_V$ are contained in $I_{KM}=K,$ thus $I_U \cap I_V\neq 0.$ Then, there exists $0\neq s \in I_U \cap I_V,$ whence $0\neq s(M)\subset U\cap V.$ Therefore, $KM$ is uniform. Since $M$ is min CS, $KM$ is essential in a direct summand of $M,$ namely $X.$ We have $X=e(M)$ for some $e=e^2\in S$ and by the condition (I), $K=I_{KM}\stackrel{*}{\hookrightarrow}I_{e(M)}=eS.$ Closeness of $K$ implies that $K=eS$, and hence $K$ is a direct summand of $S.$ Consequently, $S$ is right min CS.
Conversely, let $S$ be a right min CS ring. For a uniform closed submodule $X$ of $M,$ $I_X$ is a right ideal of $S.$ If arbitrary nonzero right ideals $K, L$ are contained in $I_X,$ then $KM, LM$ are nonzero submodules contained in $X.$ Since $X$ is uniform, $KM\cap LM=Y\neq 0$. By retractability of $M,$ there exists $0\neq s\in S$ such that $s(M)\subset Y.$ Therefore, we get $s\in I_{KM}\cap I_{LM}.$ On the other hand, $K \stackrel{*}{\hookrightarrow}I_{KM}$ and $L\stackrel{*}{\hookrightarrow}I_{LM}$ follows from Lemma $\ref{lm3.9}.$ Thus, there exists $f\in S$ such that $0\neq sf\in K,$ and $sfS\cap L\neq 0.$ This implies that $K\cap L\neq 0$ so $I_X$ is uniform. Since $S$ is right min CS, $I_X$ is essential in a direct summand $J=eS$ for some $e=e^2\in S.$ Then, by the condition (II), $I_X(M)$ is essential in $eS(M)=e(M)$, a direct summand of $M.$ By Lemma $\ref{lm3.9},$ $I_X(M)$ is essential in $X.$ But $I_X(M)$ has one closure only. Therefore, we must have $X=e(M).$ This shows that $M$ is min CS.
[*(2)*]{} Let $M$ be a max CS module. For a maximal closed right ideal $K$ of $S$ with ${\mathbf {l}}_S(K)\neq 0,$ we have $K=I_{KM}$ as arguing in [*(1)*]{}. It is induced from the condition (II) that $KM$ is not essential in $M$, since $K$ is not essential in $S$. Thus, there exists a maximal closed submodule $X\hookrightarrow M$ containing $KM$ and $X\neq M$. We have $K=I_{KM}\subset I_X$ so $K=I_X$ by maximality of $K$. This implies that $KM=I_X(M)\stackrel{*}{\hookrightarrow}X$. Because ${\mathbf {l}}_S(I_X)={\mathbf {l}}_S(K)\neq 0$, there is some $f\in S$ so that $f(KM)=0.$ By Proposition $\ref{p0}$, we also have $f(X)=0$ so ${\mathbf {l}}_S(X)\neq 0$. Since $M$ is max CS, $X$ is a direct summand of $M,$ writing $X=e(M)$ for some $e=e^2\in S.$ Then, we have $K\subset I_{e(M)}=eS$. Since $K$ is maximal closed, $K=eS$ holds true. Thus, $S$ is right max CS.
Conversely, let $S$ be a right max CS ring. For a maximal closed submodule $X$ of $M$ with nonzero left annihilator in $S,$ we have $I_X(M)\stackrel{*}{\hookrightarrow}X$ and $0\neq{\mathbf {l}}_S(X)\subset{\mathbf {l}}_S(I_X)$. By the condition (I), $I_X$ is not essential in $S$, since $X$ is not essential in $M$. Thus, there exists a maximal closed right ideal $K\hookrightarrow S$ containing $I_X$ and $K\neq S.$ We observe that $I_X(M)\subset KM.$ On the other hand, $I_X(M)$ has a unique maximal essential extension, so $KM\subset X$ because of maximality of $X$. This shows that $K=I_X$ and hence $I_X=eS$ for some $e=e^2\in S,$ since $S$ is right max CS. Therefore, we get $I_X(M)\hookrightarrow e(M),$ whence $X=eM,$ a direct summand of $M.$ This proves that $M$ is max CS.
By Theorem $\ref{th3.7}$ and Theorem $\ref{th3.10}$, we do have.
\[c3\]
Let $M$ be a retractable, nonsingular and co-nonsingular, Utumi and co-Utumi $R-$module which possesses the condition (II). Then, the following statements hold.
\(1) $M$ is min CS if and only if $S$ is right min CS if and only if $S$ is left max CS.
\(2) $M$ is max CS if and only if $S$ is right max CS if and only if $S$ is left min CS.
\(3) $M$ is max-min CS if and only if $S$ is right max-min CS if and only if $S$ is left max-min CS.
Motivated by [@CS1 Theorem 3], we study the symmetry of the CS property on one sided-ideals in the following theorem.
\[th3.12\] Let $R$ be a nonsingular Utumi ring. Then, the following conditions are equivalent for every $e=e^2\in R.$
\(1) $eR_R$ is CS with finite uniform dimension;
\(2) $_RRe$ is CS with finite uniform dimension;
\(3) $eRe$ is right CS with finite right uniform dimension;
\(4) $eRe$ is left CS with finite left uniform dimension.
In this case, $eR_R$ and $_RRe$ are Goldie modules, $eRe$ is a right and left Goldie ring, and ${\operatorname {u-dim}}(eR_R)={\operatorname {u-dim}}(_RRe)={\operatorname {u-dim}}(eRe_{eRe})={\operatorname {u-dim}}(_{eRe}eRe)$.
Since $R$ is nonsingular Utumi, the MRQR and MLQR of $R$ coincide by Lemma $\ref{lm3.2}$, denoted by $Q$.
*(1)$\Leftrightarrow$(3)* Let $eR_R$ be a CS module with finite uniform dimension. Then, $eQ$, the injective hall of $eR,$ is a semisimple artinian right ideal of $Q.$ Furthermore, because ${\operatorname {End}}(eQ)\cong eQe$, we see that $eQe$ is a semisimle artinian ring which is the MRQR of $eRe$. Thus, $eRe$ has finite right uniform dimension. In order to show that $eRe$ is right CS, it is sufficient to prove that every uniform closed right ideal $V$ of $eRe$ is a direct summand of $eRe.$ Clearly, $V=f(eQe)\cap eRe$ for some $f=f^2\in eQe.$ We observe that $f=ef=fe$ and $fQ\hookrightarrow eQ.$ Thus, $fQ\cap R$ is closed in $R$ and contained in $eR.$ Therefore, $fQ\cap R$ is closed in $eR$. Since $eR_R$ is a CS module, $fQ\cap R$ is a direct summand of $eR$ so of $R$. This means $fQ\cap R=gR$ for some idempotent $g\in R.$ We have $ge=ege=(ege)^2.$ Hence, $ege(eRe)=geRe$ is a direct summand of $eRe$ which is also contained in $V.$ Since $V$ is minimal closed, $V=geRe$ is a direct summand of $eRe.$ This implies that $eRe$ is a right CS ring.
Conversely, let $eRe$ be a right CS ring with finite right uniform dimension. Then, $eQe$ is a semisimple artinian ring and $eQ_R$, the injective hall of $eR_R$, is a semisimple artinian and noetherian module. Thus, $eR_R$ has finite uniform dimension. Let $V$ be a minimal closed submodule of $eR.$ We have $V=fQ\cap R$, where $f=f^2\in Q$ and $fQ\hookrightarrow eQ.$ We observe that $fQ$ of a simple component of $eQ.$ Thus, $fQe$ is a simple component of $eQe$. Therefore, $fQe\cap eRe$ is a minimal closed right ideal of $eRe,$ hence $fQe\cap eRe=g(eRe)$, where $g=g^2\in eRe.$ We see that $gR$ is minimal closed. Because of $Ve\overset{*}{\hookrightarrow}fQ, Ve\cap gRe\neq 0$ and $V\cap gR\neq 0, V$ and $gR$ are both the unique closure of $V\cap gR$, whence $V=gR$. This shows that $V$ is a direct summand of $eR$ and hence $eR_R$ is CS.
*(2)$\Leftrightarrow$(4)* We argue similarly to *(1)$\Leftrightarrow$(3)*.
*(3)$\Leftrightarrow$(4)* Let $eRe$ be a right CS ring with finite right uniform dimension. Then, $eQe$, the MRQR of $eRe,$ is semisimple artinian and is also the MLQR of $eRe.$ Therefore, $eRe$ has finite left uniform dimension and is left CS (see proof of Theorem \[th3.5\]). The converse is symmetric.
The last statement is referred to Proposition $\ref{p1}.$
\[p2\]
Let $M$ be a nonsingular retractable $R-$module which possesses (II). Then, the following statements hold for any $e=e^2\in S.$
\(1) $e(M)$ is CS if and only if $eS_S$ is CS.
\(2) $e(M)$ is min CS if and only if $eS_S$ is min CS.
\(3) $e(M)$ is max CS if and only if $eS_S$ is max CS.
\(4) $e(M)$ is max-min CS if and only if $eS_S$ is max-min CS.
We argue similarly to the proof of Theorem $\ref{th3.10}$. Note that if $K$ is a closed right ideal of $S$ contained in $eS$, then $KM$ is contained in $e(M).$ Conversely, if $Y$ is a closed submodule of $M$ contained in $e(M),$ then $I_Y$ is a right ideal of $S$ contained in $I_{e(M)}=eS.$
By Theorem $\ref{th3.12}$ and Proposition $\ref{p2}$, we have.
\[c4\] Let $M$ be a retractable $R-$module which possesses (II). If $M$ is nonsingular and co-nonsingular, Utumi and co-Utumi, then the following conditions are equivalent for every $e=e^2\in S.$
\(1) $eM$ is CS with finite uniform dimension;
\(2) $eS_S$ is CS with finite uniform dimension;
\(3) $_SSe$ is CS with finite uniform dimension;
\(4) $eSe$ is right CS with finite right uniform dimension;
\(5) $eSe$ is left CS with finite left uniform dimension.
As we mentioned in the introduction, this paper mainly consider rings without primeness. However, the following theorem give us an additional symmetry of the extending properties and finiteness of uniform dimension on nonsingular semiprime rings. This is not investigated in [@GCS; @CS1; @m-mCS].
\[thpr\] Let $R$ be a semiprime ring.
\(1) $R$ is right CS, right nonsingular with finite right uniform dimension if and only if $R$ is left CS, left nonsingular with finite left uniform dimension.
\(2) If $R$ is nonsingular, then the following statements hold true.
- (2.1) $R_R$ is max CS with finite uniform dimension if and only if $_RR$ is min CS with finite uniform dimension.
- (2.2) $R_R$ is min CS with finite uniform dimension if and only if $_RR$ is max CS with finite uniform dimension.
- (2.3) $R_R$ is max-min CS with finite uniform dimension if and only if $_RR$ is max-min CS with finite uniform dimension.
In all the cases above, $R$ is right and left Goldie with ${\operatorname {u-dim}}(R_R)={\operatorname {u-dim}}(_RR)=n$ for some integer $n\geq 0.$
For the case of *(1)*, Lemma $\ref{lm3.4}$ implies that a right CS right nonsingular ring is left nonsingular, and a left CS left nonsingular ring is right nonsingular. Thus, $R$ is right and left nonsingular for both cases *(1)* and *(2)*.
Since $R$ is nonsingular, $R$ has a maximal two-sided quotient ring $Q$ by [@utumi Lemma 1.4]. Since $R_R$ has finite uniform dimension, $Q$ is semisimple. Therefore, ${\operatorname {u-dim}}(_RQ)$ is finite so is ${\operatorname {u-dim}}(_RR)$. Since $R$ is nonsingular with finite right and left uniform dimension, $R$ is a right and left Goldie ring by [@Go Corollary 3.32]. Therefore, $Q$ is a classical right and left quotient ring of $R$ as well as a maximal right and left quotient ring of $R$ by [@Go Theorem 3.37]. We argue similarly when $_RR$ has finite uniform dimension.
Now, the proof about equivalence of the extending conditions on the right and left sides of $R$ is similar to Theorem $\ref{th3.5}$ and Theorem $\ref{th3.7}$.
[**Examples.**]{} It is easy to find examples of right and left max-min CS rings. In particular, one of such a ring is $R=\left(
\begin{array}{cc}
F & F \\
0 & F \\
\end{array}
\right),$ where $F$ is a field.
There is a module which is neither max CS nor min CS. Let $Z$ be the set of all integers. Consider $Z-$module $M=(Z/Z2)\oplus (Z/Z8).$ We observe that $A=((1+Z2)\oplus (2+Z8))Z$ is a minimal closed submodule of $M$ but not a direct summand. It is easy to verify that $A$ is also a maximal closed submodule with non-zero left annihilator in the endomorphism ring of $M$. Thus, $M$ is neither max CS nor min CS, although $M$ has finite uniform dimension.
There exists a ring which is a right Ore domain but not a left Ore domain. Such a ring is mentioned (namely $R$) in [@Go Exercise 1, page 101]. It is not difficult to see that $R$ is right max-min CS but not left min CS. If $R$ is min CS, then $R$ must be left uniform, so is left Ore, a contradiction.
[10]{}
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R. E. Johnson. Quotient rings with zero singular ideal. [*Pacific J. Math*]{}, [**11**]{}(1961), 1358-1392.
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S. M. Khuri. Correspondence theorems for modules and their endomorphism rings. [*J. Algebra*]{}, [**122**]{}(1989), 380-396.
S. M. Khuri. Nonsingular retractable modules and their endomorphism rings. [*Bull. Austral. Math. Soc.*]{}, [**43**]{}(1991), 63-71.
D. V. Thuat, H. D. Hai and N. V. Sanh. On Goldie prime CS-modules. [*East-West J. Math.*]{}, [**16:2**]{}(2014), 131-140.
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[^1]: [**2010 MSC**]{}: 16D70, 16S50\
$^\dag$ The corresponding author, email: truongdinhtu@tdtu.edu.vn
|
---
abstract: 'We study a class of nonlinear schrödinger system with external sources terms as perturbations in order to obtain existence of multiple solutions, this system arises from Bose-Einstein condensates etc.. As these external sources terms are positive functions and small in some sense, we use Nehari manifold to get the existence of a positive ground state solution and a positive bound state solution.'
author:
- |
Zexin Qi$^a$, Zhitao Zhang$^{b,}$[^1]\
[$^a$ College of Mathematics and Information Science,]{}\
[Henan Normal University, Xinxiang 453007, China]{}\
[E-mail: qizedong@126.com]{}\
[$^b$ Academy of Mathematics and Systems Science,]{}\
[the Chinese Academy of Sciences, Beijing 100190, China]{}\
[E-mail: zzt@math.ac.cn]{}
title: '**Existence of multiple solutions to a class of nonlinear Schrödinger system with external sources terms** '
---
[*Keywords:*]{} nonlinear schrödinger system; multiple solutions; ground state; Nehari manifold 0.2cm
[*AMS Subject Classification (2010):*]{} 35B20, 35J47, 35J50.
Introduction
============
We are concerned with the following nonlinear schrödinger system with external sources terms $$\label{QZ1.1}
\left\{
\begin{array}{ll}
-\Delta u+\lambda_{1}u=\mu_{1}u^{3}+\beta uv^{2}+f(x), & x\in \Omega, \\
-\Delta v+\lambda_{2}v=\mu_{2}v^{3}+\beta u^{2}v+g(x), & x\in \Omega, \\
u=v=0, & x\in\partial \Omega,
\end{array}
\right.$$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}, N\leq3$, $\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}$ and $\beta$ are positive constants, $f(x),g(x)$ are external sources terms.
(1.1) is a perturbed version of the following system $$\label{1.2}
\left\{
\begin{array}{ll}
-\Delta u+\lambda_{1}u=\mu_{1}u^{3}+\beta uv^{2}, & x\in \Omega, \\
-\Delta v+\lambda_{2}v=\mu_{2}v^{3}+\beta u^{2}v, & x\in \Omega, \\
u=v=0, & x\in\partial \Omega.
\end{array}
\right.$$
System (1.2) arises from many physical problems, especially in describing some phenomenon in nonlinear optics ([@2],[@9]). It also models the Hartee-Fock theory for a double condensate, i.e., a binary mixture of Bose-Einstein condensates in two different hyperfine states $|1\rangle$ and $|2\rangle$ ([@15]). We refer the reader to [@3; @1; @10; @ChenZ; @CTV2; @CTV3; @41; @42; @43; @LW; @18; @NTTV1; @NTTV2; @34], and the references therein for interesting existence of solutions or properties of solutions. The parameters $\mu_i$ and $\beta$ are the intraspecies and interspecies scattering lengths respectively. The sign of the scattering length $\beta$ determines whether the interaction of states $|1\rangle$ and $|2\rangle$ are repulsive or attractive. When $\beta<0$, the interactions of states $|1\rangle$ and $|2\rangle$ are repulsive. In contrast, when $\beta>0$ the interactions of states $|1\rangle$ and $|2\rangle$ are attractive. For atoms of the single state $|j\rangle$, when $\mu_j>0$, the interactions of the single state $|j\rangle$ are attractive.
Naturally, people concern nontrivial solutions(solutions with both components nonzero) of the system. In recent years, many interesting works have been devoted to searching ground states and bound states for this system, see [@3; @1; @10; @ChenZ; @LW; @18; @34] etc. and the references therein.
***A positive ground state solution*** we mean a solution of a schrödinger system which has the least energy among all nonzero solutions, and both of its components are positive. Note here we call a function positive if it is nonnegative and nonzero. A ***bound state solution*** refers to limited-energy solution. As for system (1.2), well-known results indicate the existence of positive ground state is closely related to the parameters, see [@1],[@34] etc. and a remark for the bounded-domain case in [@17].
In this paper, multiplicity result is established when the perturbations are sufficiently small. If $f(x)$ and $g(x)$ are both positive, we can find a positive ground state.
To be precise, let $S_{4}$ be the best Sobolev constant of the embedding: $ H_{0}^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$, then we have\
\
**Theorem 1.1** *Assume that* $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, *both nonzero. Then there exists a positive constant* $\Lambda =\Lambda(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}, \beta,S_{4})$, *such that whenever $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda $, system* (1.1) *has two nontrivial solutions. Furthermore, if $f$ and $g$ are both positive, system* (1.1) *has one positive ground state solution and one positive bound state solution*.\
Let $H:=H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)$ with the norm
$\|(u,v)\|:=\left(\displaystyle\int_{\Omega}(|\nabla u|^{2}+\lambda_{1}u^{2})+\displaystyle\int_{\Omega}(|\nabla v|^{2}+\lambda_{2}v^{2})\right)^{\frac{1}{2}}.$
An element ${(}u,v{)}\in H$ is called a weak solution of (1.1), if the equality $$\displaystyle\int_{\Omega}(
\nabla u \nabla \varphi+\nabla v\nabla \psi-\lambda_{1}u\varphi-\lambda_{2}v\psi-\mu_{1}u^{3}\varphi-\mu_{2}v^{3}\psi$$ $$-\beta uv^{2}\varphi-\beta u^{2}v\psi-f\varphi-g\psi)dx=0~~~~~~~~~~~~$$ holds for all $(\varphi,\psi)\in H$. A weak solution of (1.1) corresponds to a critical point of the following $C^{1}$-functional $$J(u,v)=\frac{1}{2}\|(u,v)\|^{2}-\frac{1}{4}(\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\int_{\Omega}u^{2}v^{2})-\int_\Omega(fu+gv).$$
Denote the Nehari manifold associated with the functional by
$\mathcal{N}:=\{(u,v)\in H : \langle J^{'}(u,v),(u,v)\rangle=0\}.$
It is well-known that all critical points lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed $(u,v)\in H \setminus\{(0,0)\}$, denote $$\label{}
\phi(t)=\phi_{(u,v)}(t):=J(tu,tv),~t>0$$ the so called fibering map in the direction $(u,v)$. Such maps are often used to investigate Nehari manifolds for various semi-linear problems([@27; @Poh; @32; @40]). Our method is, roughly speaking, to figure out two non-degenerate parts of the Nehari manifold, and then consider minimization problems in the two parts respectively to obtain two nontrivial solutions, especially to obtain two positive solutions. Under some other assumptions, there may exist more solutions, after we submitted this manuscript, we thank the referee to tell us that existence of infinitely many nontrivial solutions of systems related to are recently obtained in [@YueZ] by different methods under different conditions, they assume the coefficient matrix is either positive definite when $N=1,2$ or anti-symmetric when $N=3$. But here for , the coefficient matrix $\left(\begin{array}{lll}
\mu_1~ \beta\\
\beta~ \mu_2\end{array}
\right)$ usually doesn’t satisfy those assumptions, especially the anti-symmetric assumption implies that $\beta=0$ when $N=3$, then becomes two independent differential equations without any couplings. Our assumption here on $\beta$ can be $\beta>-\sqrt{\mu_1\mu_2}$(notice the remark at the end of the paper).
Our paper is organized as follows. In section 2, we use fibering maps to divide the Nehari manifold into three parts. A basic work related can be found in [@Poh],[@32]. The number $\Lambda$ in Theorem 1.1 is determined to ensure a satisfactory partition. In Section 3 we set up a critical point lemma to consider two minimization problems. In the last section, we give the proof of Theorem 1.1.
Partition of the Nehari manifold
================================
Denote $\Phi(u,v):= \langle J^{'}(u,v),(u,v)\rangle$. By (1.3) one has
$\langle J^{'}(u,v),(u,v)\rangle=\|(u,v)\|^{2}-(\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\int_{\Omega}u^{2}v^{2})-\int_\Omega(fu+gv),$
$\langle \Phi^{'}(u,v),(u,v)\rangle=2\|(u,v)\|^{2}-4(\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\int_{\Omega}u^{2}v^{2})-\int_\Omega(fu+gv).$
Now we divide the Nehari manifold into three parts:
$ \mathcal{N}^{+}:=\{(u,v)\in \mathcal{N} : \langle \Phi^{'}(u,v),(u,v)\rangle>0\};$
$ \mathcal{N}^{0}:=\{(u,v)\in \mathcal{N} : \langle \Phi^{'}(u,v),(u,v)\rangle=0\};$
$ \mathcal{N}^{-}:=\{(u,v)\in \mathcal{N} : \langle \Phi^{'}(u,v),(u,v)\rangle<0\}.$
Obviously, only $\mathcal{N}^{0}$ contains the element $(0,0)$, and it is easy to see $\mathcal{N}^{+}\cup \mathcal{N}^{0}$ and $\mathcal{N}^{-}\cup \mathcal{N}^{0}$ are both closed subsets of $H$.
In order to make an explanation of such partition, for $(u,v)\in H\setminus\{(0,0)\}$, let us consider the fibering map defined in (1.4). Since
$\phi'(t)=\dfrac{1}{t}\langle J'(tu,tv),(tu,tv)\rangle,$
we know $(u,v)\in \mathcal{N}$ if and only if $\phi'(1)=0$. Moreover, $(tu,tv)\in \mathcal{N}$ with $t>0$ if and only if $\phi'(t)=0$.
Thus for a fixed direction $(u,v)\in H \setminus\{(0,0)\}$, we can obtain all elements on the Nehari manifold which lie in this direction if we can find all stationary points of the fibering map. As a result, one would obtain all nonzero elements of the Nehari manifold if one could find stationary points of fibering maps in all directions. We remark that the number of roots of the equation $\phi'(t)=0$ does not depend on the norm of $(u,v)$, once this direction is fixed. Indeed, for $\delta>0$ we have
$\phi_{(\delta u,\delta v)}^{'}\left(\dfrac{t}{\delta}\right)=\delta\phi_{(u,v)}^{'}(t)$.
Thus
$\phi_{(u,v)}^{'}(t)=0,~t>0\Longleftrightarrow\phi_{(\delta u,\delta v)}^{'}\left(\dfrac{t}{\delta}\right)=0,~t>0.$
Furthermore,
$\phi_{(\delta u,\delta v)}^{''}\left(\dfrac{t}{\delta}\right)=\delta^{2}\phi_{(u,v)}^{''}(t)$,
which means it is also irrelevant with the norm of $(u,v)$ if we consider the sign of second derivative of the fibering map at its stationary points. Moreover, stationary points can be classified into three types, namely local minimum, local maximum and turning point, according to the sign of second derivative of the fibering map at these points.
We now disclose the relationship between such classification and the former partition of the Nehari manifold. By direct calculation, we get
$\phi^{'}(t)=\dfrac{1}{t} \langle J^{'}(tu,tv),(tu,tv)\rangle=\dfrac{1}{t}\Phi(tu,tv),$
$\phi^{''}(t)=\dfrac{1}{t^{2}}[\langle\Phi^{'}(tu,tv),(tu,tv)\rangle-\Phi(tu,tv)].$
Thus if $\phi^{'}(t)=0$, then $(tu,tv)\in \mathcal{N}$, and $\Phi(tu,tv)=0$, which yields
$\phi^{''}(t)=\dfrac{1}{t^{2}} \langle\Phi^{'}(tu,tv),(tu,tv)\rangle.$
Now it is easy to check:
$t(u,v)\in \mathcal{N}^{+},~t>0 \Longleftrightarrow \phi^{'}(t)=0, \phi^{''}(t)>0;$
$t(u,v)\in \mathcal{N}^{0},~t>0 \Longleftrightarrow \phi^{'}(t)=0, \phi^{''}(t)=0;$
$t(u,v)\in \mathcal{N}^{-},~t>0 \Longleftrightarrow \phi^{'}(t)=0, \phi^{''}(t)<0.$
We can explore the Nehari manifold through fibering maps. In fact, we give the following important lemma to show when the degenerate part of the Nehari manifold is clear and simple.
To simplify the calculation, let us introduce some notations that will be used repeatedly in the rest. For $(u,v)\in H$, define $$\label{}
A=A(u,v):=\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\int_{\Omega}u^{2}v^{2},$$ $$\label{}
B=B(u,v):=\int_\Omega(fu+gv).$$ **Lemma 2.1** *Suppose that $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, both nonzero, then there exists a positive constant $\Lambda =\Lambda(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2}, \beta,S_{4})$ such that $\mathcal{N}^{0}=\{(0,0)\}$ when $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$.\
*\
**Proof.** By the analysis above, we only need to prove for each $(u,v)\in H $ with $\|(u,v)\|=1$, the fibering map $\phi(t)=\phi_{(u,v)}(t)$ has no stationary point that is a turning point. By the notations (2.1) and (2.2), we can write $$\begin{aligned}
\label{}
\phi(t)&=J(tu,tv)\notag\\
&=\dfrac{t^{2}}{2}\|(u,v)\|^{2}-\dfrac{t^{4}}{4}(\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\displaystyle\int_{\Omega}u^{2}v^{2})-t\int_\Omega(fu+gv)\notag\\
&=\dfrac{t^{2}}{2}-\dfrac{At^{4}}{4}-Bt.\notag\end{aligned}$$
Thus
$\phi'(t)=t-At^{3}-B$,$\phi''(t)=1-3At^{2}.$
Notice $A=A(u,v)=\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\int_{\Omega}u^{2}v^{2}>0$. Define $\psi(t)=t-At^{3},~t>0$, then $\phi'(t)=0\Longleftrightarrow \psi(t)=B$.
Let us consider the graph of $\psi(t)$: $\psi''(t)=-6At<0$, so $\psi(t)$ is strictly concave; $\psi'(t)=0\Longleftrightarrow t=\sqrt{\frac{1}{3A}}$, thus $\psi(t)$ takes its maximum at $t=\sqrt{\frac{1}{3A}}$, the value of which is $\frac{2}{3}\sqrt{\frac{1}{3A}}$. Also, $\lim_{t\rightarrow0^{+}} \psi(t)=0$, $ \psi(\infty)=-\infty$. All the above implies if $0<B<\frac{2}{3}\sqrt{\frac{1}{3A}}$, the equation $\phi'(t)=0$ has exactly two roots $t_{1}$, $t_{2}$ satisfying $0<t_{1}<\sqrt{\frac{1}{3A}}<t_{2}$; if $B\leq0$ the equation $\phi'(t)=0$ has exactly one root denoted $t_{2}^{'}$, which satisfies $\sqrt{\frac{1}{3A}}<t_{2}^{'}$. Since $\phi''(t)=1-3At^{2}$, considering the above two cases we have $\phi''(t_{1})>0$, $\phi''(t_{2})<0$, and $\phi''(t_{2}^{'})<0$. So when $0<B<\frac{2}{3}\sqrt{\frac{1}{3A}}$, one has $t_{1}(u,v)\in \mathcal{N}^{+}$, $t_{2}(u,v)\in \mathcal{N}^{-}$; when $B\leq0$ one has $t_{2}^{'}(u,v)\in \mathcal{N}^{-}$. Since $f(x),g(x)$ are both nonzero, it is easy to check that the sets $\{(u,v)\in H :\|(u,v)\|=1,0<B<\frac{2}{3}\sqrt{\frac{1}{3A}}\}\not=\emptyset$ and $\{(u,v)\in H :\|(u,v)\|=1,B\leq0 \}\not=\emptyset$, so $\mathcal{N}^{+}\not=\emptyset$ and $\mathcal{N}^{-}\not=\emptyset$. To finish the proof, we are in a position to determine a number $\Lambda$ such that whenever $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$ we have $$\label{}
B<\frac{2}{3}\sqrt{\frac{1}{3A}}.$$
In fact, since $(u,v)$ lies on the unit sphere of $H$, by Sobolev inequality and Hölder inequality one obtains an upper bound for $A$, which yields the existence of a positive constant $\alpha=\alpha(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2},\beta,S_{4})$ such that $$\label{}
0<\alpha\leq\frac{2}{3}\sqrt{\frac{1}{3\sup\limits_{\|(u,v)\|=1}A(u,v)}}.$$
On the other hand, by Sobolev inequality and Hölder inequality, we obtain that
$B=\displaystyle\int_\Omega(fu+gv)
\leq \|f\|_{\frac{4}{3}}\|u\|_{4}+\|g\|_{\frac{4}{3}}\|v\|_{4}
\leq \sqrt{2}S_{4}\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}\|(u,v)\|.$
That is, $$\label{}
B\leq \sqrt{2}S_{4}\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}.$$
Now take $\Lambda:=\Lambda(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2},\beta,S_{4}) =\frac{\alpha}{\sqrt{2}S_{4}}$, by (2.4),(2.5) we know when $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$, (2.3) holds.
Reduction to minimization problems
==================================
Here we reduce our discussion into two minimization problems through a critical point lemma. As to constraint minimization problems, we refer the reader to [@8]. Let $X$,$Y$ be real Banach spaces, $U\subset X$ be an open set. Suppose that $f:U\rightarrow
\mathbb{R}^{1}$, $g:U\rightarrow Y$ are $C^{1}$ mappings in this section. Let
$ M=\{x\in U:g(x)=\theta\}.$
For the following minimization problem $$\label{}
\min_{x\in M}f(x),$$ we have\
\
**Lemma 3.1** ([@8], Theorem 4.1.1) *Suppose that $x_{0}\in M$ solves* (3.1), \
Let us introduce two minimization problems $$\label{}
\theta^{+}= \inf_{(u,v)\in \mathcal{N}^{+}}J(u,v);$$ $$\label{}
\theta^{-}= \inf_{(u,v)\in \mathcal{N}^{-}}J(u,v).$$ **Lemma 3.2** *If $(u_{1},v_{1})$ solves* (3.2), *then $(u_{1},v_{1})$ is a nontrivial weak solution of* (1.1); *if $(u_{2},v_{2})$ solves* (3.3), *then $(u_{2},v_{2})$ is a nontrivial weak solution of* (1.1).\
\
**Proof.** We prove the first assertion, since the second is similar.
Recall $\Phi(u,v)= \langle J^{'}(u,v),(u,v)\rangle$ and
$ \mathcal{N}^{+}=\{(u,v)\in \mathcal{N} : \langle \Phi^{'}(u,v),(u,v)\rangle>0\}.$
Let $U=\{(u,v)\in H :\langle \Phi^{'}(u,v),(u,v)\rangle>0\}$, and rewrite
$ \mathcal{N}^{+}=\{(u,v)\in U:\Phi(u,v)=0 \}.$
Now we use Lemma 3.1 to consider problem (3.2) by taking $X=H,~Y=\mathbb{R}^{1},~U=\{(u,v)\in H :\langle
\Phi^{'}(u,v),(u,v)\rangle>0\},~f=J,~g=\Phi,~M=\mathcal{N}^{+}$. Then there exists a real pair $(\lambda_{1},\lambda_{2})\neq
(0,0)$, satisfying
$\lambda_{1}J'(u_{1},v_{1})=\lambda_{2}\Phi'(u_{1},v_{1}).$
Since $\langle \Phi^{'}(u_{1},v_{1}),(u_{1},v_{1})\rangle>0$, we have $\Phi^{'}(u_{1},v_{1})\neq 0$, again by Lemma 3.1, one may assume $\lambda_{1}\neq0$. In other words, there exists a real number $\mu$ such that $$\label{}
J'(u_{1},v_{1})=\mu\Phi'(u_{1},v_{1}).$$
Since
$\mu \langle \Phi^{'}(u_{1},v_{1}),(u_{1},v_{1})\rangle=\langle J^{'}(u_{1},v_{1}),(u_{1},v_{1})\rangle=0$
and $\langle \Phi^{'}(u_{1},v_{1}),(u_{1},v_{1})\rangle>0$, we obtain $\mu =0$. By (3.4) we get $J'(u_{1},v_{1})=0$. Obviously $(u_{1},v_{1})\neq (0,0)$, noticing that (1.1) doesn’t admit semi-trivial (nonzero but one component being trivial) solutions, we know that $(u_{1},v_{1})$ is a nontrivial weak solution of (1.1).
Proof of the main result
========================
When $\mathcal{N}^{0}=\{(0,0)\}$, we show the solvability of the two problems (3.2) and (3.3), and give the proof of Theorem 1.1. The following lemma implies that $\theta^{+}$ and $\theta^{-}$ are both finite, and any minimizing sequence for (3.2) or (3.3) are bounded.\
\
**Lemma 4.1** *Assume that $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, then $J$ is coercive and bounded from below on $\mathcal{N}$ *(*thus on $\mathcal{N}^{+}$ and $\mathcal{N}^{-}$*).\
\
**Proof.** Let $(u,v)\in \mathcal{N}$, from the definition of the Nehari manifold we have
$
\|(u,v)\|^{2}-\displaystyle\int_\Omega(fu+gv)=(\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\displaystyle\int_{\Omega}u^{2}v^{2})$,
this equality together with (1.3) yield $$J(u,v)=\frac{1}{4}\|(u,v)\|^{2}-\frac{3}{4}\int_\Omega(fu+gv).$$
Note that $$\label{}
\displaystyle\int_\Omega(fu+gv)
\leq \|f\|_{\frac{4}{3}}\|u\|_{4}+\|g\|_{\frac{4}{3}}\|v\|_{4}
\leq \sqrt{2}S_{4}\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}\|(u,v)\|,$$ combining (4.1) and (4.2) we obtain $$J(u,v)\geq \frac{1}{4}\|(u,v)\|^{2}-\frac{3\sqrt{2}}{4}S_{4}\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}\|(u,v)\|.$$
Since the right hand side of the inequality (4.3) is a quadratic function of $\|(u,v)\|$, it is easy to know that $J$ is coercive and bounded from below on $\mathcal{N}$.\
From now on, $\Lambda$ refers to the number in Lemma 2.1. Although the lemma below is valid under weaker conditions, we would assume $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, both nonzero, and $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$ in order to ensure $\mathcal{N}^{0}=\{(0,0)\}$ by Lemma 2.1, because our discussion bases on a good partition of the Nehari manifold. We will often use the proof of Lemma 2.1 to analyze some properties of the manifold as well as associated fibering maps.
For the first minimization problem, we establish:\
\
**Lemma 4.2** *Assume that $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, both nonzero, and that $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$. Then $\theta^{+} < 0$*.\
\
**Proof.** By the proof of Lemma 2.1, we may take one $(u,v)\in H $, $\|(u,v)\|=1$ such that $0<B<\frac{2}{3}\sqrt{\frac{1}{3A}}$. Now the derivative $\phi'(t)$ of the fibering map in this direction has exactly two positive zeros: $t_{1}$, $t_{2}$, satisfying $0<t_{1}<\sqrt{\frac{1}{3A}}<t_{2}$, and $t_{1}\cdot(u,v)\in \mathcal{N}^{+}$. By $\phi'(t)=t-At^{3}-B$ we have $\lim_{t\rightarrow0^{+}}
\phi'(t)=-B<0$, and $\phi''(t)>0$, $\forall
t\in(0,\sqrt{\frac{1}{3A}})$. Since $\phi'(t_{1})=0$, we know that $\phi(t_{1})<\lim_{t\rightarrow0^{+}} \phi(t)=0$. Since $\phi(t_{1})=J(t_{1}u,t_{1}v)\geq\theta^{+}$, we obtain that $\theta^{+} < 0$.\
For the second minimization problem we need the following lemma, from which we know $\mathcal{N}^{-}$ stays away from the origin.\
\
**Lemma 4.3** *Assume that $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, both nonzero, and that $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$. Then $\mathcal{N}^{-}$ is closed*.\
\
**Proof.** Under the assumptions one has
$cl~({{\mathcal{N}^{-}}})\subset\mathcal{N}^{-}\cup\mathcal{N}^{0}=\mathcal{N}^{-}\cup \{(0,0)\},$
where we denote $cl~({{\mathcal{N}^{-}}})$ the closure of $\mathcal{N}^{-}$. Thus we only need to prove $(0,0) \not\in
cl~({{\mathcal{N}^{-}}})$, which is equivalent to prove that
$dist((0,0),\mathcal{N}^{-})>0.$
Take $(u,v)\in\mathcal{N}^{-}$, and denote
$(\widetilde{u},\widetilde{v})=\dfrac{(u,v)}{\|(u,v)\|},$
then $\|(\widetilde{u},\widetilde{v})\|=1$. Under the assumptions, by the proof of Lemma 2.1, we obtain that $A:=A(\widetilde{u},\widetilde{v}),~B:=B(\widetilde{u},\widetilde{v})$ satisfy $B<\frac{2}{3}\sqrt{\frac{1}{3A}}$. Furthermore, if $0<B<\frac{2}{3}\sqrt{\frac{1}{3A}}$, the equation $\phi'_{(\widetilde{u},\widetilde{v})}(t)=0$ has exactly two roots also denoted by $t_{1}$, $t_{2}$, which satisfy $t_{2}\cdot(\widetilde{u},\widetilde{v})\in
\mathcal{N}^{-}$, $t_{1}\cdot(\widetilde{u},\widetilde{v})\in
\mathcal{N}^{+}$, we have $t_{2}\cdot(\widetilde{u},\widetilde{v})=
(u,v)$, so $t_{2}= \|(u,v)\|$. If $B\leq 0$, then the equation $\phi'_{(\widetilde{u},\widetilde{v})}(t)=0$ has exactly one root still denoted by $\widetilde{t}_{2}$, thus we get $\widetilde{t}_{2}\cdot(\widetilde{u},\widetilde{v})= (u,v)$, then $\widetilde{t}_{2}= \|(u,v)\|$. Since ${t_{2}}>\sqrt{\frac{1}{3A}}$, $\widetilde{t}_{2}>\sqrt{\frac{1}{3A}}$ in the proof of Lemma 2.1, so no matter which of the above two cases happens, we always obtain $\|(u,v)\|>\sqrt{\frac{1}{3A}}$. Noticing that $A$ is bounded from above, we know that there exists $\tau>0$ such that
$\|(u,v)\|>\tau.$
We obtain
$dist((0,0),\mathcal{N}^{-})=\inf\limits_{(u,v)\in\mathcal{N}^{-}}\|(u,v)\|\geq\tau>0,$
which completes the proof.\
In order to abstract a $(PS)_{\theta^{+}}$ sequence from the minimizing sequence for problem (3.2), we use the idea of [@32] to obtain the following lemma.\
\
**Lemma 4.4** *Assume that $f(x),g(x)\in
L^{\frac{4}{3}}(\Omega)$, both nonzero. Then for $(u,v)\in
\mathcal{N}^{+}$, there exists $\epsilon=\epsilon(u,v)>0$ and a differentiable function $\xi^{+}:B_{\epsilon}(0,0)\rightarrow\mathbb{R}_{+}:=(0,+\infty)$ such that*
- $\xi^{+}(0,0)=1;$
- $\xi^{+}(w,z)(u-w,v-z)\in \mathcal{N}^{+},\forall(w,z)\in B_{\epsilon}(0,0);$
- $ \langle (\xi^{+})'(0,0),(w,z)\rangle=[\|(u,v)\|^{2}-3A(u,v)]^{-1}[2\displaystyle\int_{\Omega}(\nabla u\nabla w+\lambda_{1}uw+\nabla v\nabla z+\lambda_{2}vz)-4\displaystyle\int_{\Omega}(\mu_{1}u^{3}w+\mu_{2}v^{3}z+\beta uv^{2}w+\beta u^{2}vz)-\displaystyle\int_{\Omega}(fw+gz)].$
**Proof.** Define a $C^{1}$-mapping $F:\mathbb{R}_{+}\times H \rightarrow \mathbb{R}$ as follows:
$F(t,(w,z))=t\|(u-w,v-z)\|^{2}-t^{3}A(u-w,v-z)-\displaystyle\int_\Omega[f\cdot(u-w)+g\cdot(v-z)].$
By (2.1), we know that $A(u-w,v-z)=\mu_{1}\|u-w\|_{4}^{4}+\mu_{2}\|v-z\|_{4}^{4}+2\beta\displaystyle\int_{\Omega}(u-w)^{2}(v-z)^{2}$. Since $(u,v)\in \mathcal{N}^{+}$ we have
$F(1,(0,0))=0.$
Consider the fibering map $\phi(t)=\phi_{(u,v)}(t):=J(tu,tv)$, since
$F(t,(0,0))=t\|(u,v)\|^{2}-t^{3}(\mu_{1}\|u\|_{4}^{4}+\mu_{2}\|v\|_{4}^{4}+2\beta\displaystyle\int_{\Omega}u^{2}v^{2})-\displaystyle\int_\Omega(fu+gv),$
we have
$F(t,(0,0))=\phi'(t).$
Since $(u,v)\in \mathcal{N}^{+}$, we get $\phi''(1)>0$, thus
$\dfrac{\partial F}{\partial t}(1,(0,0))=\phi''(1)>0.$
We apply the implicit function theorem at point (1,(0,0)) to obtain the existence of $\epsilon=\epsilon(u,v)>0$ and differentiable function $\xi^{+}(u,v)(i.e.,~t(u,v)):B_{\epsilon}(0,0)\rightarrow\mathbb{R}_{+}$ such that
- $\xi^{+}(0,0)=1;$
- $\xi^{+}(w,z)\cdot(u-w,v-z)\in \mathcal{N},~\forall(w,z)\in B_{\epsilon}(0,0).$
Besides, we obtain the third conclusion of this lemma by calculation. To finish the second one, we only need to choose $\epsilon=\epsilon(u,v)>0$ small, such that $\xi^{+}(w,z)(u-w,v-z)\in \mathcal{N^{+}},~\forall(w,z)\in B_{\epsilon}(0,0)$. Indeed, since $\mathcal{N}^{-}\cup \mathcal{N}^{0}$ is closed, $dist((u,v),\mathcal{N}^{-}\cup \mathcal{N}^{0})>0$. Since $\xi^{+}(w,z)\cdot(u-w,v-z)$ is continuous with respect to $(w,z)$, when $\epsilon=\epsilon(u,v)>0$ is small enough, we know
$ \|\xi^{+}(w,z)\cdot(u-w,v-z)-(u,v)\| < \dfrac{1}{2}dist((u,v),\mathcal{N}^{-}\cup \mathcal{N}^{0}),\forall(w,z)\in B_{\epsilon}(0,0).$
That is, $\xi^{+}(w,z)\cdot(u-w,v-z)$ does not belong to $
\mathcal{N}^{-}\cup \mathcal{N}^{0}$. Thus $\xi^{+}(w,z)\cdot(u-w,v-z)\in \mathcal{N^{+}}$ and our proof is completed.\
Similarly, we can establish the following lemma, which will be used to abstract a $(PS)_{\theta^{-}}$ sequence from the minimizing sequence for problem (3.3).\
\
**Lemma 4.5** *Assume that $f(x),g(x)\in
L^{\frac{4}{3}}(\Omega)$, both nonzero. Then for $(u,v)\in
\mathcal{N}^{-}$, there exists $\epsilon=\epsilon(u,v)>0$ and a differentiable function $\xi^{-}:B_{\epsilon}(0,0)\rightarrow\mathbb{R}_{+}$ such that*
- $\xi^{-}(0,0)=1;$
- $\xi^{-}(w,z)(u-w,v-z)\in \mathcal{N}^{-},\forall(w,z)\in B_{\epsilon}(0,0);$
- $ \langle (\xi^{-})'(0,0),(w,z)\rangle=[\|(u,v)\|^{2}-3A(u,v)]^{-1}[2\displaystyle\int_{\Omega}(\nabla u\nabla w+\lambda_{1}uw+\nabla v\nabla z+\lambda_{2}vz)-4\displaystyle\int_{\Omega}(\mu_{1}u^{3}w+\mu_{2}v^{3}z+\beta uv^{2}w+\beta u^{2}vz)-\displaystyle\int_{\Omega}(fw+gz)].$
We are in a position to give:\
\
**Lemma 4.6** *Assume that $f(x),g(x)\in
L^{\frac{4}{3}}(\Omega)$, both nonzero, and that $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$. Then there exists a sequence $\{(u_{n},v_{n})\}\subset
\mathcal{N}^{+}$ such that ($n \rightarrow \infty$)*:
1. $J(u_{n},v_{n})\rightarrow \theta^{+};$
2. $J'(u_{n},v_{n})\rightarrow 0.$
**Proof.** Notice $\mathcal{N}^{+}\cup \{(0,0)\}$ is closed in $H$, we use the Ekeland’s variational principle([@14]) on $\mathcal{N}^{+}\cup \{(0,0)\}$ to obtain a minimizing sequence $\{(u_{n},v_{n})\}\subset \mathcal{N}^{+}\cup \{(0,0)\}$ such that
(a)
: $J(u_{n},v_{n})< \inf\limits_{(u,v)\in \mathcal{N}^{+}\bigcup \{(0,0)\}}J(u,v)+\dfrac{1}{n}$;
(b)
: $J(w,z)\geq J(u_{n},v_{n}) -\dfrac{1}{n}\|(w-u_{n},z-v_{n})\|,\;\;\forall (w,z)\in \mathcal{N}^{+}\cup \{(0,0)\}$.
By Lemma 4.2, we know $\theta^{+} < 0$; since $ J(0,0)=0$, we get that $$\inf\limits_{(u,v)\in \mathcal{N}^{+}\bigcup
\{(0,0)\}}J(u,v)=\theta^{+}.$$
Thus $J(u_{n},v_{n})\rightarrow
\theta^{+}$, we may assume $\{(u_{n},v_{n})\}\subset
\mathcal{N}^{+}$, then the first assertion holds.
For the second assertion, Firstly we have that $\inf\limits_n\|(u_{n},v_{n})\|\geq m>0$, where $m$ is a constant. Indeed, if not, then $J(u_{n},v_{n})$ would converge to zero. Moreover, by Lemma 4.1 we know that $J$ is coercive on $\mathcal{N}^{+}$, then $\{\|(u_{n},v_{n})\|\}$ is bounded. i.e., $\exists M>0$ such that $$\label{}
0<m\leq \|(u_{n},v_{n})\| \leq M.$$
Now by contradiction we assume $\|J'(u_{n},v_{n})\|\geq C>0$ as $n$ is large, otherwise we may extract a subsequence to get the conclusion.
Now let us take $(u,v)=(u_{n},v_{n})$ in Lemma 4.4, and define a differentiable function $\xi_{n}^{+}:(-\epsilon,\epsilon)\rightarrow \mathbb{R}_{+}$ as:
$\xi_{n}^{+}(\delta ):= \xi^{+}\left(\dfrac{\delta J'(u_{n},v_{n})}{\|J'(u_{n},v_{n})\|}\right),$
then by Lemma 4.4, we know that $\xi_{n}^{+}(0)=
\xi^{+}(0,0)=1$, and for $ \delta \in (-\epsilon,\epsilon)$ we have
$(w,z)_{\delta}:=\xi_{n}^{+}(\delta )\cdot[(u_{n},v_{n})-\dfrac{\delta J'(u_{n},v_{n})}{\|J'(u_{n},v_{n})\|}]\in\mathcal{N}^{+}.$
Since $(u_{n},v_{n})$ satisfies (b), one has $$\label{}
J(u_{n},v_{n})-J((w,z)_{\delta})\leq \dfrac{1}{n}\|(w,z)_{\delta}-(u_{n},v_{n})\|.$$
Expanding the left hand side of (4.5) we get $$\begin{aligned}
\label{}
J(u_{n},v_{n})-J((w,z)_{\delta}) =& (1-\xi_{n}^{+}(\delta ))\langle J'((w,z)_{\delta}),(u_{n},v_{n})\rangle\notag\\
&+\delta\xi_{n}^{+}(\delta )\langle J'((w,z)_{\delta}),
\frac {J'(u_{n},v_{n})}{\|J'(u_{n},v_{n})\|}\rangle\notag\\
&+o(\|(w,z)_{\delta}-(u_{n},v_{n})\|).\end{aligned}$$
Combining (4.6) with (4.5) we obtain $$~~~~(1-\xi_{n}^{+}(\delta ))\langle J'((w,z)_{\delta}),(u_{n},v_{n})\rangle+\delta\xi_{n}^{+}(\delta )\langle J'((w,z)_{\delta}),
\dfrac {J'(u_{n},v_{n})}{\|J'(u_{n},v_{n})\|}\rangle$$ $$\leq o(\|(w,z)_{\delta}-(u_{n},v_{n})\|)+\dfrac{1}{n}\|(w,z)_{\delta}-(u_{n},v_{n})\|.~~~~~~~~~~~~~~~~~~~~~~~~$$
Divide the above inequality by $\delta$ for $\delta\neq0$ and let $\delta\rightarrow 0$, then we get $$-(\xi_{n}^{+})'(0)\langle J'(u_{n},v_{n}),(u_{n},v_{n})\rangle+
\|J'(u_{n},v_{n})\|$$ $$\leq (o(1)+\dfrac{1}{n})(1+|(\xi_{n}^{+})'(0)|\cdot\|(u_{n},v_{n})\|).~~~~~~~~~~~$$
That is,
$\|J'(u_{n},v_{n})\| \leq (o(1)+\dfrac{1}{n})\cdot(1+|(\xi_{n}^{+})'(0)|\cdot\|(u_{n},v_{n})\|).$
By (4.4), we only need to show $|(\xi_{n}^{+})'(0)|$ is uniformly bounded with respect to $n$. Noticing that
$(\xi_{n}^{+})'(0)=\langle(\xi^{+})'(0,0),\dfrac {J'(u_{n},v_{n})}{\|J'(u_{n},v_{n})\|}\rangle,$
by (4.4) and the third assertion of Lemma 4.4, we can get that there exists $C>0$ such that
$|(\xi_{n}^{+})'(0)|\leq\dfrac{C}{|\|(u_{n},v_{n})\|^{2}-3A(u_{n},v_{n})|}.$
Thus we only need to prove that $|\|(u_{n},v_{n})\|^{2}-3A(u_{n},v_{n})|$ has a positive lower bound. Assume the contrary, then up to a subsequence, $$\label{}
\|(u_{n},v_{n})\|^{2}-3A(u_{n},v_{n})=o(1).$$
Since $\{(u_{n},v_{n})\}\subset \mathcal{N}^{+}$, $$\label{}
\|(u_{n},v_{n})\|^{2}-A(u_{n},v_{n})=B(u_{n},v_{n}).$$
From (4.7) and (4.8) we have, $$\label{}
B(u_{n},v_{n})=\frac{2}{3} \|(u_{n},v_{n})\|^{2}+o(1).$$
For fixed $f(x),g(x)$, since $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$, there must exist a small positive $\tau$ such that $$\label{}
\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<(1-\tau)\Lambda.$$
By the proof of Lemma 2.1, one knows from the derivation of (2.3) that more accurate inequality will occur once (4.10) holds. That is, one has for $\|(u,v)\|=1$,
$B(u,v)<\dfrac{2}{3}(1-\tau)\sqrt{\dfrac{1}{3A(u,v)}}.$
Thus by homogeneity, $$\label{}
B(u_{n},v_{n})<\frac{2}{3}(1-\tau)\sqrt{\frac{\|(u_{n},v_{n})\|^{2}}{3A(u_{n},v_{n})}}\|(u_{n},v_{n})\|^{2}.$$
Dividing (4.11) by $\|(u_{n},v_{n})\|^{2}$ and letting $n \rightarrow \infty$, we reach a contradiction from (4.4), (4.7) and (4.9), that
${\dfrac{2}{3}}\leq\dfrac{2}{3}(1-\tau).$
This completes the proof.\
On the other hand, we have\
\
**Lemma 4.7** *Assume $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, both nonzero, and $\max\{\|f\|_{\frac{4}{3}},\|g\|_{\frac{4}{3}}\}<\Lambda$. Then there exists a sequence $\{(u_{n},v_{n})\}\subset \mathcal{N}^{-}$, such that when $n \rightarrow \infty$, it holds*:
1. $J(u_{n},v_{n})\rightarrow \theta^{-};$
2. $J'(u_{n},v_{n})\rightarrow 0.$
**Proof.** By Lemma 4.3, $\mathcal{N}^{-}$ is closed in $H$. We use Ekeland’s variational principle on $\mathcal{N}^{-}$ to obtain a minimizing sequence $\{(u_{n},v_{n})\}\subset \mathcal{N}^{-}$, such that
1. $J(u_{n},v_{n})<\inf\limits_{(u,v)\in \mathcal{N}^{-}}J(u,v)+\dfrac{1}{n};$
2. $J(w,z)\geq J(u_{n},v_{n}) -\dfrac{1}{n}\|(w-u_{n},z-v_{n})\|$ holds for all $(w,z)\in \mathcal{N}^{-}$.
From Lemma 4.1 and the remark before Lemma 4.3 we obtain estimates similar to (4.4):
\[\] $0<\tilde{m}\leq \|(u_{n},v_{n})\| \leq \tilde{M}$,
where $\tilde{m}$ and $\tilde{M}$ are positive constants. By Lemma 4.5, the rest of the proof is similar to that of Lemma 4.6, we omit it.\
Since system (1.1) is subcritical, it is not difficult to obtain compactness conditions for the functional $J$. That is,\
\
**Lemma 4.8** *Assume $f(x),g(x)\in L^{\frac{4}{3}}(\Omega)$, then $J$ satisfies $(PS)$ condition, i.e., for any $c\in \mathbb{R}$, any sequence $\{u_{n}\}\subset H$ for which $J(u_{n})\rightarrow c,J'(u_{n})\rightarrow 0$ as $n\rightarrow+\infty$ possesses a convergent subsequence*.\
We are now able to give\
**Proof of Theorem 1.1.** Firstly let us consider the minimization problem (3.2). By Lemma 4.6, there exists $\{(u_{n},v_{n})\}\subset \mathcal{N}^{+}$ such that as $n\rightarrow\infty$
$J(u_{n},v_{n})\rightarrow \theta^{+}$, $J'(u_{n},v_{n})\rightarrow 0$.
Since $J$ satisfies $(PS)$condition by Lemma 4.8, we find a $(w_{1},z_{1})\in cl~(\mathcal{N}^{+})\subset\mathcal{N}^{+}\cup \{(0,0)\}$ such that $J(w_{1},z_{1})= \theta^{+}$, $J'(w_{1},z_{1})=0 $. By Lemma 4.2, $J(w_{1},z_{1})<0$. Thus $(w_{1},z_{1})\neq(0,0)$, which implies $(w_{1},z_{1})\in \mathcal{N}^{+}$. We see $(w_{1},z_{1})$ is a nontrivial weak solution of (1.1) by Lemma 3.2.
Furthermore, if $f$ and $g$ are both positive, we show the minimizer can be chosen to be a multiple of $(|w_{1}|,| z_{1}|)$. Indeed, $\|(w_{1},z_{1})\|=\|(|w_{1}|,| z_{1}|)\|$. Let $$(|w_{0}|, |z_{0}|):=\dfrac{(|w_{1}|,| z_{1}|)}{\|(|w_{1}|,| z_{1}|)\|},\;\;\;(w_{0}, z_{0}):=\dfrac{(w_{1}, z_{1})}{\|(w_{1}, z_{1})\|}.$$
Since $(w_{1},z_{1})\in \mathcal{N}^{+}$, from the proof of Lemma 2.1 we know $$B(\dfrac{w_{1}}{\|(w_{1},z_{1})\|},\dfrac{z_{1}}{\|(w_{1},z_{1})\|})>0,$$ thus $B(|w_{1}|,| z_{1}|)\geq B(w_{1},z_{1})>0$, which yields $B(|w_{0}|, |z_{0}|)>0$. By the proof of Lemma 2.1, there exists $t_{1}>0$ such that $t_{1}\cdot(|w_{1}|,| z_{1}|)\in \mathcal{N}^{+}$. Since $t_{1}\|(w_{1},z_{1})\|\cdot(|w_{0}|, |z_{0}|)=t_{1}\|(|w_{1}|,|z_{1}|)\|\cdot(|w_{0}|, |z_{0}|)=t_{1}\cdot(|w_{1}|,| z_{1}|)\in \mathcal{N}^{+}$, we know $t_{1}\|(w_{1},z_{1})\|$ is the first stationary point of the fibering map in the direction $(|w_{0}|, |z_{0}|)$.
Moreover, $(w_{1},z_{1})\in \mathcal{N}^{+}$ is equivalent to $\|(w_{1},z_{1})\|(w_{0}, z_{0})\in \mathcal{N}^{+}$, so $\|(w_{1},z_{1})\|$ is the first stationary point of the fibering map in the direction $(w_{0}, z_{0})$. Since $B(|w_{0}|,| z_{0}|)\geq B(w_{0},z_{0})>0$ and $A(|w_{0}|,| z_{0}|)=A(w_{0},z_{0})$, we can compare the above two roots of the associated fibering map to infer $t_{1}\|(w_{1},z_{1})\|\geq\|(w_{1},z_{1})\|$. That is $$\label{}
t_{1}\geq 1.$$ Taking account of the graph of the fibering map in direction $(|w_{1}|,| z_{1}|)$, one has from (4.12) and the fact $t_{1}\cdot(|w_{1}|,| z_{1}|)\in \mathcal{N}^{+}$ that,
$J(t_{1}|w_{1}|,t_{1}| z_{1}|)\leq J(|w_{1}|,| z_{1}|)$.
Thus
$\theta ^{+}\leq J(t_{1}|w_{1}|,t_{1}| z_{1}|)\leq J(|w_{1}|,| z_{1}|)\leq J(w_{1}, z_{1})=\theta ^{+}$,
from which we know $(t_{1}|w_{1}|,t_{1}| z_{1}|)$ solves problem (3.2). By Lemma 3.2 we see $(t_{1}|w_{1}|,t_{1}| z_{1}|)$ is a weak solution of system (1.1).
Now we consider the minimization problem (3.3). By Lemma 4.7, there exists $\{(u_{n},v_{n})\}\subset \mathcal{N}^{-}$, such that as $n\rightarrow\infty$
$J(u_{n},v_{n})\rightarrow \theta^{-}$, $J'(u_{n},v_{n})\rightarrow 0$.
Since $J$ satisfies $(PS)$ condition by Lemma 4.8, we find a $\{(w_{2},z_{2})\}\in cl~(\mathcal{N}^{-})=\mathcal{N}^{-}$ such that $J(w_{2},z_{2})= \theta^{-}$, $J'(w_{2},z_{2})=0 $, and $(w_{2},z_{2})$ is a nontrivial weak solution of (1.1) by Lemma 3.2.
Furthermore, if $f$ and $g$ are both positive, we show the minimizer for (3.3) can be chosen to be a multiple of $(|w_{2}|,| z_{2}|)$. By the proof of Lemma 2.1, there exists $t_{2}>0$ such that $t_{2}\cdot(|w_{2}|,| z_{2}|)\in \mathcal{N}^{-}$. Moreover, one calculates that the two fibering maps in direction $(w_{2}, z_{2})$ and $(|w_{2}|,| z_{2}|)$ has the same turning point denoted $t_{0}=\sqrt{\frac{1}{3A(w_2,z_2)}}=\sqrt{\frac{1}{3A(|w_2|,|z_2|)}}$. Thus $t_{2}>t_{0}$, and by investigating the graph of the fibering map in direction $(w_{2},z_{2})$ one gets
$J(t_{2}w_{2},t_{2} z_{2})\leq J(w_{2}, z_{2})$.
Now we have
$\theta ^{-}\leq J(t_{2}|w_{2}|,t_{2}| z_{2}|)\leq J(t_{2}w_{2},t_{2} z_{2})\leq J(w_{2}, z_{2})=\theta ^{-}$,
so $(t_{2}|w_{2}|,t_{2}| z_{2}|)$ solves (3.3). By Lemma 3.2 we see $(t_{2}|w_{2}|,t_{2}| z_{2}|)$ is a weak solution of system (1.1).
We finish the proof by showing $\theta_{+}<\theta_{-}$. In fact, if $(w_{2},z_{2})$ is the minimizer of (3.3) satisfying $B(w_{2},z_{2})\leq0$, then the associated fibering map has only one stationary point, which implies $\theta_{-}>0$ by the proof of Lemma 2.1. So $\theta_{+}<\theta_{-}$ by Lemma 4.2. On the other hand, if $(w_{2},z_{2})$ satisfies $B(w_{2},z_{2})>0$, then the associated fibering map $\phi(t)$ has two stationary points: $t_{1},t_{2}(=1)$. Thus from the graph of this fibering map, we get immediately that $\theta_{+}\leq \phi(t_{1}) <\phi(t_{2})=\theta_{-}$. 0.4cm [**Remark**]{} For the definition $A(u,v)$ (2.1), we have by Hölder inequality $$\int_{\Omega}u^2v^2\leq(\int_{\Omega}u^4)^{\frac{1}{2}}(\int_{\Omega}v^4)^{\frac{1}{2}}=\|u\|_4^2\cdot\|v\|_4^2,$$ then $$A(u,v)\geq\mu_1\|u\|_4^4+\mu_2\|v\|_4^4-2|\beta|\|u\|_4^2\|v\|_4^2\geq 2(\sqrt{\mu_1\mu_2}-|\beta|)\|u\|_4^2\cdot\|v\|_4^2,$$ and the equality is satisfied for the second inequality if and only if $\mu_1\|u\|_4^4=\mu_2\|v\|_4^2$. Thus as $\|(u,v)\|=1, \beta>-\sqrt{\mu_1\mu_2}$, we have $$A(u,v)>0.$$ Therefore, as $\beta>-\sqrt{\mu_1\mu_2}$, all the proofs are valid, Theorem 1.1 above is still true.
[**Acknowledgements:**]{} The authors thank the referees for their careful reading and helpful suggestion.
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[^1]: Corresponding author, supported by NSFC 11325107, 11271353, 11331010.
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---
abstract: 'Subsequent to the ideas presented in our previous papers \[J.Phys.: Condens. Matter [**14**]{} (2002) 13777 and Eur. Phys. J. B [**42**]{} (2004) 529\], we discuss here in detail a new analytical approach to calculating the phase-diagram for the Anderson localization in arbitrary spatial dimensions. The transition from delocalized to localized states is treated as a generalized diffusion which manifests itself in the divergence of averages of wavefunctions (correlators). This divergence is controlled by the Lyapunov exponent $\gamma$, which is the inverse of the localization length, $\xi=1/\gamma$. The appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to correlators. The generalized diffusion can be described in terms of signal theory, which operates with the concepts of input and output signals and the filter function. Delocalized states correspond to bounded output signals, and localized states to unbounded output signals, respectively. Transition from bounded to unbounded signals is defined uniquely be the filter function $H(z)$. Simplifications in the mathematical derivations of the previous papers (averaging over initial conditions) are shown to be mathematically rigorous shortcuts.'
address:
- ' Institute of Solid State Physics, University of Latvia, 8 Kengaraga Street, LV – 1063 RIGA, Latvia'
- ' Institut für Physikalische und Theoretische Chemie, Technische Universität Braunschweig, Hans-Sommer-Stra[ß]{}e 10, 38106 Braunschweig, Germany'
author:
- 'V N Kuzovkov, W von Niessen'
date: Received
title: A new approach to the analytic solution of the Anderson localization problem for arbitrary dimensions
---
Introduction
============
Anderson localization [@Anderson58] has remained a hot topic in the physics of disordered systems for a long time (see review articles [@Kramer; @Janssen; @Abrahams2]). Typically the theoretical analysis of disordered systems is based either on approximations or on numerical methods. However, it is clear that this is cannot be sufficient for systems revealing a *phase transition* [@Baxter], e.g. metal-insulator transition as is the case with Anderson localization. In such a case an exact solution is greatly needed.
In our papers [@Kuzovkov02; @Kuzovkov04] we presented somewhat schematically and by leaving out many mathematical details an exact analytic solution of the Anderson localization problem. Subsequently there were critical comments [@Comment] that our proof is too short and condensed, and more details are needed, the more so, since the mathematical formalism used by us is quite new for this scientific community of Anderson localization. This is why we present in the present paper the detailed derivation of the analytical solution with illustrations. Note that under the exact solution we mean the calculation of the phase diagram for the metal-insulator system [@Kuzovkov02; @Kuzovkov04]. We do not calculate transport and other important properties. However, the knowledge of the phase diagram permits to understand, how these properties can be calculated. It is typical that any exact analytical solution is quite lengthy and uses frequently a non-standard mathematical formalism. If this presentation is put to the beginning of the paper the physical content of the new theory may be easily lost and drowned. A good illustration is the Onsager solution of the Ising problem [@Baxter]. This is why our theory is presented in a *series* of papers: the *mathematical idea* was presented in [@Kuzovkov02], with two illustrations: the one dimensional (1-D) case, where the solution is well known, and the two dimensional (2-D) case which is non-trivial. Several details of the proof were omitted or replaced by simplified arguments, but in all cases these simplifications were pointed out. The paper [@Kuzovkov04] dealt mainly with the *results* for the arbitrary spatial dimension (N-D).
The structure of the present paper is as follows. In Section 2 we explain how the localized states can be treated in terms of a *generalized diffusion*. This approach allows to understand why for defining the phase diagram it is sufficient to solve exactly only equations for the *joint correlators*. We present and solve here the equations for arbitrary dimensions. Section 3 deals with the *stability* of this solution. It is shown that the generalized diffusion arises due to the divergence of the *fundamental mode*. The determination of the stability range of this mode permits us to calculate the *phase diagram*. We explain our language of *input and output signals, filter function* which is new for the Anderson localization community.
Equations for correlators
=========================
General aspects
---------------
Strictly speaking, the disordered Anderson tight-binding model [@Anderson58] with Schrödinger equation $$\label{tight-binding}
\sum_{\mathcal{M}^{\prime}}\psi_{\mathcal{M}^{\prime}}=(E-\varepsilon_{\mathcal{M}})\psi_{\mathcal{M}}
,$$ where the summation over $\mathcal{M}^{\prime}$ runs over the nearest neighbours of site $\mathcal{M}=\{m_1,m_2,\dots,m_D\}$, cannot be solved exactly for arbitrary dimension D. The reason is that the random potential $\varepsilon_{\mathcal{M}}$ enters the equation as a product with the random amplitude $\psi_{\mathcal{M}}$ which corresponds to the multiplicative noise case [@Kuzovkov02]. Therefore, the *exact solution* of eq. (\[tight-binding\]) discussed in [@Kuzovkov02; @Kuzovkov04] is possible only under special circumstances to be discussed below.
The fundamental quantity of a disordered system - the localization length $\xi$ - was determined in [@Kuzovkov02; @Kuzovkov04] via the Lyapunov exponent $\gamma$. In these calculations the following contradictory conditions have to be satisfied. The phase diagram of the system with metal-insulator transition should be obtained in the *thermodynamical limit* [@Baxter] (the infinite system). However, the determination of the Lyapunov exponent needs the introduction of the coordinate system (starting point) which imposes certain limitations on the system size. Moreover, a direction for the growth of the divergent quantity should be chosen, despite the fact that all space directions are equivalent in eq. (\[tight-binding\]).
All these conditions are fulfilled for the *semi-infinite* system, or system with a boundary, where the index $n \equiv
m_D\geq 0$, but all $m_j\in(-\infty,\infty)$, $j=1,2,\dots,p$, with $p=D-1$. The boundary which is the layer $n=0$ defines the preferential direction (the axis $n$) along which the Lyapunov exponent $\gamma$ will be calculated.
It is convenient to interpret the index $n=0,1,\dots,\infty$ as the *discrete-time*, whereas all other indices combine in the vector $\mathbf{m}=\{m_1,m_2,\dots,m_p\}$. The Schrödinger equation (\[tight-binding\]) can be rewritten as a *recursion equation* (in terms of the discrete-time, and assuming summation over repeated indices) $$\label{recursion DN}
\psi _{n+1,\mathbf{m}}=-\varepsilon _{n,\mathbf{m}} \psi
_{n,\mathbf{m}}-\psi_{n-1,\mathbf{m}} +
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}\psi
_{n,\mathbf{m^{\prime}}} ,$$ where the operator $$\begin{aligned}
\label{L}
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}=E\delta_{\mathbf{m},\mathbf{m^{\prime}}}
-\sum_{\mathbf{m^{\prime \prime}}}\delta_{\mathbf{m^{\prime
\prime}},\mathbf{m^{\prime}}} ,\end{aligned}$$ is introduced for compactness (summation over $\mathbf{m^{\prime\prime}}$ includes the nearest neighbours of the site $\mathbf{m}$). The discrete-time equation (\[recursion DN\]) is a difference equation of the second order, which needs two initial conditions. The first natural condition is $\psi_{0,\mathbf{m}}=0$. The second initial condition can be presented in the general form $\psi_{1,\mathbf{m}}=\alpha_{\mathbf{m}}$ without any additional conditions for the arbitrary constants $\alpha_{\mathbf{m}}$, except that they are supposed to be finite. The rules for the treatment of the field $\alpha_{\mathbf{m}}$ will be explained below, section \[alpha\].
The recursion equation (\[recursion DN\]) reveals a general feature of the *causality* - in its formal solution $\psi
_{n+1,\mathbf{m}}$ depends only on the random variables $\varepsilon _{n^{\prime},\mathbf{m}^{\prime}}$ with $n^{\prime
}\leq n$. This gives us a hint for the *exact solution* of the equation provided that random variables $\varepsilon_{\mathcal{M}}$ on different sites *do not correlate*: $\left\langle \varepsilon_{\mathcal{M}}
\varepsilon_{\mathcal{M}^{\prime}}\right\rangle \propto
\delta_{\mathcal{M},\mathcal{M}^{\prime}}$. In this case all amplitudes $\psi$ on the r.h.s. of eq. (\[recursion DN\]) are statistically independent of $\varepsilon _{n,\mathbf{m}}$. In other words, while performing the mathematical operations in eq. (\[recursion DN\]), e.g. *taking the square* of both equation sides, with a further averaging over an ensemble of different realizations of the random potentials (symbol $\left\langle \dots \right\rangle $), the average of the product of amplitudes $\psi$ and potentials $\varepsilon _{n,\mathbf{m}}$ can be replaced by a product of the corresponding average quantities. Therefore, the exact solution can be obtained only when the one-sites potentials $ \varepsilon_{n,m} $ are *independently and identically* distributed. We assume hereafter existence of the two first moments, $\left\langle
\varepsilon _{n,\mathbf{m}} \right\rangle =0$ and $\left\langle
\varepsilon _{n,\mathbf{m}}^2\right\rangle =\sigma ^2$, where the parameter $\sigma$ characterizes the *disorder level*.
Generalized diffusion
---------------------
The ensemble averaging using recursion eq. (\[recursion DN\]) allows us to calculate averages of different functions containing the amplitudes $\psi$. The problem arises, *which* averages we can and should calculate? E.g. several even moments of the amplitude were calculated in the 1-D case [@Molinari], whereas only the second momentum was calculated for multidimensional systems in [@Kuzovkov02; @Kuzovkov04]. It is generally believed that the *complete information* is available from the complete set of *all* moments [@Pendry82]. However, it is not clear, how this information should be analyzed. In particular, how can the *one* parameter of our interest - the localization length $\xi$ - be extracted from the *infinite* set of amplitude momenta?
It was suggested in [@Comment] to calculate not only the second, $\left\langle |\psi|^2 \right\rangle $, and other momenta, but also *physical values*, such as $\left\langle \ln|\psi|
\right\rangle $. The average quantities are divided in [@Comment] into two categories: containing physical information (e.g. $\left\langle \ln|\psi| \right\rangle $) and containing no physical information (e.g. $\left\langle |\psi|^2
\right\rangle $).
In other words, one has to understand how the choice of a semi-infinite system with a selected direction ($n$ axis) and the boundary (layers in the transversal directions with $n=0,1$) allow us to detect the localized states under question.
Let us consider for simplicity a 1-D system, where the amplitude $\psi=\psi_n$ and the second initial condition $\psi_1=\alpha$. Again, for *simplicity* let us consider $\alpha$ to be a real quantity, i.e. $|\psi|^2=\psi^2$. The recursive equation reads: $$\begin{aligned}
\label{recursion1}
\psi_{n+1}=(E-\varepsilon_n)\psi_n-\psi_{n-1},\end{aligned}$$ where $\left\langle \varepsilon_n \right\rangle = 0$, $\left\langle \varepsilon^2_n \right\rangle = \sigma^2$.
Let us consider now the random walk problem which is described by the following equations $$\begin{aligned}
\label{recursion2}
\psi_{n+1}=\psi_n +\varepsilon_n .\end{aligned}$$
Considering $n$ as the discrete-time index, both eqs. (\[recursion1\]) and (\[recursion2\]) describe the dynamics of the system with a stochastic time-dependent perturbation $\varepsilon_n$. The only difference is that the dynamics of the system (\[recursion2\]) is trivial: $\psi_n \equiv \psi_0$ when there is no perturbation, $\varepsilon_n=0$. In contrast, the dynamical system (\[recursion1\]) even for $\varepsilon_n=0$ reveals proper dynamics (it is a second order equation!). In the band region, $|E|< 2$, this corresponds to the bounded motion. Therefore, the proper dynamics of both systems corresponds to the *bounded* trajectories, $\psi^2_n < \infty$.
Assuming now $\sigma \neq 0$, eq. (\[recursion2\]) describes the *diffusion*. The diffusion motion is characterized by *divergences*, e.g. for the mean time when the system returns to the initial state. In particular, the momenta of the amplitude $\psi_n$ are also divergent with the discrete-time $n$, e.g. $$\left\langle \psi^2_n \right\rangle = \psi^2_0+\sigma^2 n$$ reveals a *power law* divergence, Fig. \[fig: 1\]. To detect the diffusion, it is *sufficient* to demonstrate the divergence of the *second moment* of the amplitude and to establish its *law of time-dependence*. Generally speaking, other moments contain additional information, which however is not important. The divergence of the second moment defines the conditions of the diffusion appearance: *if* the second moment is divergent, so are *all other even moments* (this is why the choice of a particular moment for further analysis is *not unique*). The *law* of the time-dependence divergence allows us to distinguish between normal and abnormal diffusion. Important is the fact that the second moment $\left\langle \psi^2_n \right\rangle $ in eq. (\[recursion2\]) can be found exactly analytically and thus we prefer its use ($\psi^2$-definition). Speaking formally, the diffusion could also be classified using other averaged quantities, e.g. $\left\langle
|\psi_n| \right\rangle $, but such a choice is not convenient for mathematical reasons; it hinders an analytical solution.
As is well-known, in a 1-D system described by eq. (\[recursion1\]) any disorder $\sigma \neq 0$ leads to localization. This manifests itself in the simultaneous divergence, as $n \rightarrow \infty$, of different average quantities of $|\psi_n|$, e.g. the *linear* divergence of $\left\langle \ln |\psi_n| \right\rangle $ (log-definition of localization), whereas the *exponential* divergence occurs for the powers of $|\psi_n|$ [@Molinari]. The appearance of the localization in the approach based on eq. (\[recursion1\]) is equivalent to the appearance of diffusion. In eq. (\[recursion2\]) the random perturbation $\varepsilon_n$ is *additive*, which determines the *linear* (power-law) character of the second moment divergence (normal diffusion). In contrast, eq. (\[recursion1\]) contains the *product* of $\varepsilon_n$ and $\psi_n$ (*multiplication*) which determines the *exponential* character of the divergence. I.e., we can speak of the *generalized diffusion*, analyzing the $\left\langle \psi^2_n \right\rangle = f(n)$ divergence as a function of $n$. The exponential localization here corresponds to the exponential growth of $f(n) \propto \exp(2\gamma n)$, Fig. \[fig: 2\], where $\xi=1/\gamma$ can be interpreted as the *localization length* ($\psi^2$ - localization definition [@Kuzovkov02; @Kuzovkov04; @Molinari]). Note that the $\psi^2$-definition is convenient not only because the equations for the second moment can be exactly solved (see below). It is important that this definition works also for the non-exponential localization, which corresponds to the non-exponential behaviour of $f(n)$. In contrast, the traditional log-definition is valid only for the exponential localization, and does not allow an exact analytical solution.
It was analytically shown [@Molinari] for 1-D systems that if the higher moments of the random potential $\varepsilon_n$ can be neglected (for $\sigma \rightarrow 0$) and we can restrict ourselves to the only parameter $\sigma$, the exponents in *all* even moments (as well as in log-definition) are proportional to each other, they differ only by numerical factors.
In terms of the generalized diffusion, the results of Molinari [@Molinari] can be readily interpreted as follows. As the disorder increases, *all* average quantities of $|\psi_n|$ become simultaneously divergent (including powers and $\ln
|\psi_n|$). This is indication for the appearance of *generalized diffusion* with the critical disorder $\sigma_0=0$. Note that in the 1-D case the proper dynamics is *unstable* and the generalized diffusion arises already at infinitesimally small disorder. The proportionality of the different exponents (Lyapunov exponents) in the exponential growth indicates that different definitions of the generalized diffusion are in fact *identical* and one can use any of them, e.g the $\psi^2$-definition.
Comparison of eqs. (\[recursion DN\]) and (\[recursion1\]) shows that eq. (\[recursion DN\]) describes more complex dynamics, where the generalized diffusion can be detected through the second moment, $\left\langle \psi^2 _{n,\mathbf{m}}
\right\rangle $. As the space dimension D increases, the dynamics turns out to be *more stable* [@Kuzovkov04], i.e. the generalized diffusion arises at a higher disorder level, $\sigma >
\sigma_0 \neq 0$. Along with the exponential and non-exponential localizations (unstable motion), *stable* delocalized states (stable dynamics) can also arise. The definition of the phase-diagram of the system means the determination of the boundaries between the *stability and instability* of the system.
That is, the two points in [@Kuzovkov02]: *“(i) We are at present not interested in the shape of the distribution which is influenced by the higher moments of the on-site potentials but in the problem of localization (phase-diagram); (ii) In the analysis of the moments of the amplitudes the localization of states finds its expression in the simultaneous divergence of the even moments for $n \rightarrow \infty$.”* mean nothing else but the idea of generalized diffusion.
Note that the emergence of the generalized diffusion phenomenon and the existence of the phase diagrams in dynamical systems with the proper dynamics are analogous to *critical phenomena* in equilibrium and non-equilibrium systems. In this sense, it is clear that use of the $\psi^2$-definition for the determination of the phase-diagram is quite sufficient. Exactly solvable problems in physics of phase transitions [@Baxter], e.g. the Ising model, have demonstrated that in spite of the fact that the physical quantities, i.e. long range order parameter, magnetic susceptibility, specific heat, etc. are defined as *different averages*, they still behave *similarly*, when only the phase-diagram is considered. This is not surprising since one deals here with averages obtained for the *same statistical ensemble*. In particular, the critical temperature obtained for the order parameter coincides with that obtained for the susceptibility. This occurs for the *exact* solutions, although deviations can arise for the approximate solutions.
Similarly, the phase diagram for the system with a metal-insulator transition can be defined uniquely if one determines any non-trivial average quantity. This is why the choice of $\left\langle \ln|\psi| \right\rangle $ is not convenient. A product of a sum of variables can be represented as a polynomial, where each term can be calculated, whereas the logarithm of a sum cannot be represented this way and thus the causality principle cannot be used. On the other hand, the averages of the powers of the amplitude can be calculated exactly.
When we speak of the phase-diagram, we mean determination of the boundaries of metallic and insulating phases. In this respect, the results [@Kuzovkov02; @Kuzovkov04] demonstrate that the second order moments form a *closed and linear* set of equations and, where the only characteristic of the random potentials is the second momentum, $\sigma^2$, have a simple interpretation. The phase boundaries are defined by the only simple parameter $\sigma$ of the random potential. That is, calculation of second moments (*correlators*) is the *direct way* to obtaining the phase diagram.
Correlators
-----------
In general, the field $\alpha_{\mathbf{m}}$ is complex. We can introduce the simplest non-trivial correlators based on the $\psi^2$-definition: $$\begin{aligned}
x(n)_{\mathbf{m},\mathbf{l}}=\left\langle \psi^{*}
_{n,\mathbf{m}}\psi
_{n,\mathbf{l}} \right\rangle , \\
y(n)_{\mathbf{m},\mathbf{l}}=\frac{1}{2}[ \left\langle \psi^{*}
_{n,\mathbf{m}}\psi _{n-1,\mathbf{l}} \right\rangle +\left\langle
\psi^{*} _{n-1,\mathbf{m}}\psi _{n,\mathbf{l}} \right\rangle ] .\end{aligned}$$ Let us for *mathematical simplicity* consider real $\alpha_{\mathbf{m}}$. This is also justified by the fact that the final eqs. (\[chi3\]),(\[H\]) remain the same. For real $\alpha_{\mathbf{m}}$ definitions can be simplified: $$\begin{aligned}
x(n)_{\mathbf{m},\mathbf{l}}=\left\langle \psi _{n,\mathbf{m}}\psi
_{n,\mathbf{l}} \right\rangle , \\
y(n)_{\mathbf{m},\mathbf{l}}=\left\langle \psi _{n,\mathbf{m}}\psi
_{n-1,\mathbf{l}} \right\rangle .\end{aligned}$$ Taking square of the both sides of the eq. (\[recursion DN\]) and using the *causality principle*, one gets $$\begin{aligned}
\label{x(n)}
x(n+1)_{\mathbf{m},\mathbf{l}}=\delta_{\mathbf{m},\mathbf{l}}\sigma^2
x(n)_{\mathbf{m},\mathbf{m}}+x(n-1)_{\mathbf{m},\mathbf{l}}+\\
\nonumber
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}x(n)_{\mathbf{m^{\prime}},\mathbf{l^{\prime}}}
\mathcal{L}_{\mathbf{l^{\prime}},\mathbf{l}} -
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}y(n)_{\mathbf{m^{\prime}},\mathbf{l}}-
\mathcal{L}_{\mathbf{l},\mathbf{l^{\prime}}}y(n)_{\mathbf{l^{\prime}},\mathbf{m}}
.\end{aligned}$$ Analogously one obtains the additional relation $$\begin{aligned}
y(n+1)_{\mathbf{m},\mathbf{l}}=-y(n)_{\mathbf{l},\mathbf{m}}+
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}x(n)_{\mathbf{m^{\prime}},\mathbf{l}}
.\end{aligned}$$ Note that eq.(\[x(n)\]) contains explicitly the *diagonal correlators* with $\mathbf{l}=\mathbf{m}$: $\chi(n)_{\mathbf{m}}=x(n)_{\mathbf{m},\mathbf{m}}$. It is essential that diagonal correlators are *positive* numbers, $\chi(n)_{\mathbf{m}}\geq 0$.
Z-transform
-----------
It is convenient to use along the $n$ axis the Z-transform which is common in discrete-time systems [@Weiss]: $$\begin{aligned}
X(z)_{\mathbf{m},\mathbf{l}}=\sum_{n=0}^{\infty}\frac{x(n)_{\mathbf{m},\mathbf{l}}}{z^n}
, \\
Y(z)_{\mathbf{m},\mathbf{l}}=\sum_{n=0}^{\infty}\frac{y(n)_{\mathbf{m},\mathbf{l}}}{z^n}
.\end{aligned}$$ After transformation one gets $$\begin{aligned}
(z-z^{-1})X(z)_{\mathbf{m},\mathbf{l}}-\sigma^2\delta_{\mathbf{m},\mathbf{l}}\chi(z)_{\mathbf{m}}
-x(1)_{\mathbf{m},\mathbf{l}}=\\ \nonumber
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}X(z)_{\mathbf{m^{\prime}},\mathbf{l^{\prime}}}
\mathcal{L}_{\mathbf{l^{\prime}},\mathbf{l}} -
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}Y(z)_{\mathbf{m^{\prime}},\mathbf{l}}-
\mathcal{L}_{\mathbf{l},\mathbf{l^{\prime}}}Y(z)_{\mathbf{l^{\prime}},\mathbf{m}} ,\\
zY(z)_{\mathbf{m},\mathbf{l}}=-Y(z)_{\mathbf{l},\mathbf{m}}+
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}X(z)_{\mathbf{m^{\prime}},\mathbf{l}}
.\end{aligned}$$ Hereafter the symmetry properties of the operator $\mathcal{L}_{\mathbf{m},\mathbf{l}}$ are used.
The solution of the second equation is simple: $$\begin{aligned}
Y(z)_{\mathbf{m},\mathbf{l}}=\frac{z}{z^2-1}
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}X(z)_{\mathbf{m^{\prime}},\mathbf{l}}
-\frac{1}{z^2-1}X(z)_{\mathbf{m},\mathbf{l^{\prime}}}\mathcal{L}_{\mathbf{l^{\prime}},\mathbf{l}}
,\end{aligned}$$ which leads to the equation for $X$-correlators $$\begin{aligned}
\label{X(z)}
(z-z^{-1})X(z)_{\mathbf{m},\mathbf{l}}-\sigma^2\delta_{\mathbf{m},\mathbf{l}}\chi(z)_{\mathbf{m}}
-x(1)_{\mathbf{m},\mathbf{l}}=
\frac{z^2+1}{z^2-1}\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}X(z)_{\mathbf{m^{\prime}},\mathbf{l^{\prime}}}
\mathcal{L}_{\mathbf{l^{\prime}},\mathbf{l}} \\
\nonumber-\frac{z}{z^2-1}[
\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}\mathcal{L}_{\mathbf{m^{\prime}},\mathbf{l^{\prime}}}
X(z)_{\mathbf{l^{\prime}},\mathbf{l}}+
X(z)_{\mathbf{m},\mathbf{m^{\prime}}}\mathcal{L}_{\mathbf{m^{\prime}},\mathbf{l^{\prime}}}
\mathcal{L}_{\mathbf{l^{\prime}},\mathbf{l}}].\end{aligned}$$ Taking into account the second initial condition, $x(1)_{\mathbf{m},\mathbf{l}}=\alpha_{\mathbf{m}}\alpha_{\mathbf{l}}$.
Fourier transform
-----------------
The obtained equation can be easily solved formally via Fourier expansion. Indeed, one gets for diagonal correlators $$\begin{aligned}
\chi(z)_{\mathbf{m}}=\int
\frac{d^p\mathbf{k}}{(2\pi)^p}\chi(z,\mathbf{k})e^{-i\mathbf{k}\mathbf{m}}
,\\
\chi(z,\mathbf{k})=\sum_{\mathbf{m}}
\chi(z)_{\mathbf{m}}e^{i\mathbf{k}\mathbf{m}} .\end{aligned}$$ A general equation for $X$-correlators has to be solved using double Fourier expansion: $$\begin{aligned}
X(z,\mathbf{k},\mathbf{k^{\prime}})=\sum_{\mathbf{m},\mathbf{l}}
X(z)_{\mathbf{m},\mathbf{l}}e^{i\mathbf{k}\mathbf{m}+i\mathbf{k^{\prime}}\mathbf{l}}
.\end{aligned}$$ This leads to $$\begin{aligned}
\label{UX}
U(z,\mathbf{k},\mathbf{k^{\prime}})X(z,\mathbf{k},\mathbf{k^{\prime}})=
\alpha(\mathbf{k})\alpha(\mathbf{k^{\prime}})+\sigma^2
\chi(z,\mathbf{k}+\mathbf{k^{\prime}}) ,\end{aligned}$$ where $$\begin{aligned}
U(z,\mathbf{k},\mathbf{k^{\prime}})=(z-z^{-1})-
\frac{z^2+1}{z^2-1}\mathcal{L}(\mathbf{k})\mathcal{L}(\mathbf{k^{\prime}})+
\frac{z}{z^2-1}[\mathcal{L}^2(\mathbf{k})+
\mathcal{L}^2(\mathbf{k^{\prime}})] ,\end{aligned}$$ and $$\alpha(\mathbf{k})=\sum_{\mathbf{m}}
\alpha_{\mathbf{m}}e^{i\mathbf{k}\mathbf{m}} .$$ The function $$\begin{aligned}
\mathcal{L}(\mathbf{k})=E - 2\sum_{j=1}^p\cos (k_j) \label{Lk}\end{aligned}$$ arises as the Fourier transform of the operator $\mathcal{L}_{\mathbf{m},\mathbf{l}}$.
Diagonal correlators
--------------------
Assuming $\mathbf{k^{\prime
\prime}}=\mathbf{k}+\mathbf{k^{\prime}}$, eq.(\[UX\]) can be rewritten as $$\begin{aligned}
\label{UX2}
U(z,\mathbf{k},\mathbf{k^{\prime\prime}}-\mathbf{k})X(z,\mathbf{k},\mathbf{k^{\prime\prime}}-\mathbf{k})=
\alpha(\mathbf{k})\alpha(\mathbf{k^{\prime\prime}}-\mathbf{k})+\sigma^2
\chi(z,\mathbf{k^{\prime\prime}}) .\end{aligned}$$ Using the relation $$\begin{aligned}
\chi(z,\mathbf{k^{\prime \prime}})= \int
\frac{d^p\mathbf{k}}{(2\pi)^p}X(z,\mathbf{k},\mathbf{k^{\prime\prime}}-\mathbf{k})
,\end{aligned}$$ eq.(\[UX2\]) can be solved elementary for diagonal correlators: $$\begin{aligned}
\label{chi_k}
\chi(z,\mathbf{k})=\mathcal{H}(z,\mathbf{k})\chi^{(0)}(z,\mathbf{k}) ,\\
\chi^{(0)}(z,\mathbf{k})=\int
\frac{d^p\mathbf{k^{\prime}}}{(2\pi)^p}\frac{\alpha(\mathbf{k^{\prime}})\alpha({\mathbf{k}-\mathbf{k^{\prime}}})}
{U(z,\mathbf{k^{\prime}},\mathbf{k}-\mathbf{k^{\prime}})}
,\\
\frac{1}{\mathcal{H}(z,\mathbf{k})}=1-\sigma^2\int
\frac{d^p\mathbf{k^{\prime}}}{(2\pi)^p}\frac{1}
{U(z,\mathbf{k^{\prime}},\mathbf{k}-\mathbf{k^{\prime}})}
\label{Hk}.\end{aligned}$$
Therefore we have obtained for an arbitrary second initial condition (field $\alpha_{\mathbf{m}}$) an exact solution for fundamental averages - *diagonal correlators*. After use of the Fourier transform the solution is expanded in modes labelled by the $\mathbf{k}$-vector and describing the oscillations in the solutions in the transversal direction. It is easy to see that the $\psi^2$-definition leads to equations containing only the $\sigma$ parameter of the random potentials. If $\sigma=0$, all functions $\mathcal{H}(z,\mathbf{k})\equiv 1$ and thus $\chi(z,\mathbf{k})\equiv \chi^{(0)}(z,\mathbf{k})$. In other words, functions $\chi^{(0)}(z,\mathbf{k})$ correspond to solutions in a *completely ordered system*. Introduction of disorder, $\sigma > 0$, transforms the initial solution $\chi^{(0)}(z,\mathbf{k})$ into $\chi(z,\mathbf{k})$, where the operator $\mathcal{H}(z,\mathbf{k})$ describes this transformation.
In our derivation we assumed the system size in a transversal direction to be infinite, $m_j\in(-\infty,\infty)$, and thus the obtained equations correspond to the *thermodynamic limit* ($L=\infty$). In the case of finite size of the system, $m_j=0,1,\dots,L-1$ (cyclic conditions) the integrals should be replaced by series. Let us consider for illustration the 2-D case: $p=D-1\equiv 1$. Here the index $\mathbf{m}=m$ has $L$ values and the number of modes $\mathbf{k}=k$ also equals $L$. That is, analysis of the diagonal correlators leads to $L$-order matrices (vectors). Surprisingly, this obvious fact was interpreted in ref. [@Comment] that degrees of freedom of the initial problem were lost in our paper [@Kuzovkov02], thus questioning our proof of the Anderson theorem. This conclusion was based on the traditional approach of the transfer-matrix [@Comment; @Pendry82] which operates with much larger, $L^2 \times L^2$, matrices for 2-D. The transition from $L^2 \times L^2$ matrices to $L$-order matrices was interpreted as an error arising due to *averaging over initial conditions* [@Kuzovkov02] and thus artificially introducing a translation in the transversal directions. However, we do not use in the present study such an averaging (see also Section \[Averaging\]). The $L$-dimensional matrices are the *natural* mathematical tool for the description of diagonal correlators. Moreover, an excessive size of $L^2 \times L^2$ matrices indicates that the transfer-matrix formalism is not adequate. This is seen, in particular, from the existence of the so-called *trivial eigenvalues* of the transfer-matrix [@Comment; @Pendry82], which are $\sigma$-independent. Moreover, the transfer-matrix does not permit to make the transition to the thermodynamic limit and thus to calculate the phase-diagram (for more details see [@Reply]). These disadvantages are absent in the 1-D case, when formally $L=0$ and the matrix dimensions coincide. This is why we used in [@Kuzovkov02] the transfer-matrix method only in the 1-D case and exclusively for the illustration.
Anderson localization and stability
===================================
Signals and filters
-------------------
The set of equations for the correlators is linear, the same is true for eqs.(\[chi\_k\]) for each $\mathbf{k}$-mode. The modes are *normal* (no mixing of modes with different $\mathbf{k}$ values). The above mentioned divergence (exponential or non-exponential growth) of the diagonal correlators, arising due to localized states, transforms mathematically to the *instability* of set of linear equations. For linear discrete-time systems under study the most adequate formalism is *signal theory* [@Weiss]. Following [@Kuzovkov02; @Kuzovkov04; @Weiss], let us define $\chi^{(0)}(z,\mathbf{k})$ as *input signals*. In our particular case this is a mathematical characteristic of the $\mathbf{k}$-mode for an initial ordered systems. (Its sense is explained below). The inverse Z-transform of a given mode can be performed as follows: $$\chi^{(0)}(z,\mathbf{k})\Rightarrow \chi^{(0)}(n,\mathbf{k}) .$$ The input signal is a one dimensional numerical series. The $\chi(z,\mathbf{k})$ is the *output signal*; respectively after the inverse Z-transform the latter corresponds to the numerical series $\chi(n,\mathbf{k})$, associated with the *disordered system*, $\sigma > 0$. The relation between these two signals is given by eq.(\[chi\_k\]): the output signals are linear transformations of input signals performed through $\mathcal{H}(z,\mathbf{k})$ functions called the *system filter* [@Weiss].
Physical systems with time as independent variable are *causal* systems [@Weiss]. In our problem with discrete-time index $n$ the causality is of primary importance for the interpretation of the result. Note that in the general case the inverse Z-transform is defined through the complex integral $$\label{inverse}
h(n,\mathbf{k})=\frac{1}{2\pi i}\oint \mathcal{H}(z,\mathbf{k})
z^n \frac{dz}{z} .$$ The integration here is performed over the complex plane called the *region of convergence* (ROC) [@Kuzovkov02; @Weiss], well-defined for the causal filters. For the causal systems $h(n,\mathbf{k})=0$ for $n<0$ holds, thus after the inverse Z-transform eqs.(\[chi\_k\]) transform into the *convolution property* [@Weiss]: $$\label{inverse2}
\chi(n,\mathbf{k})=\sum_{l=0}^{n}h(n-l,\mathbf{k})\chi^{(0)}(l,\mathbf{k})
.$$
One can see here the linear transformation of input signals into output signals. The fundamental property of the signal theory is that the divergence of the output signals is related to the asymptotic behaviour of the filter coefficients $h(n,\mathbf{k})$ as $n \rightarrow \infty$, which is mathematically equivalent to the *poles* of the $\mathcal{H}(z,\mathbf{k})$ function of the complex argument $z$ [@Weiss].
Fundamental mode
----------------
Since the appearance of localized states leads to the divergence of the diagonal correlator, it is necessary to clarify, which *fundamental* mode $\mathbf{k}=\mathbf{k_0}$ looses its stability and thus is responsible for the divergence. Such a problem is quite general in many branches of physics. However, in our case it is quite obvious that the fundamental mode is *static*, $\mathbf{k_0=0}$.
The mode with $\mathbf{k\neq 0}$ describes transversal oscillations of the diagonal correlators, but the *sign* of the divergent oscillating solution is *not defined* (see also section \[Averaging\]). On the other hand, signs of the diagonal correlators are well defined, they are *non-negative*, $\chi(n)_{\mathbf{m}}\geq 0$. That is, in our particular case the solution is trivial, the fundamental mode is *static*. It is thus necessary to study the boundaries of the stability of the fundamental mode $\mathbf{k_0=0}$ only (the sign of the divergent non-oscillating solution is defined), which means *uniquely* the determination of the *phase diagram* of the system.
Assuming $\chi^{(0)}(z,\mathbf{0})\equiv S^{(0)}(z)$, $\chi(z,\mathbf{0})\equiv S(z)$, $\mathcal{H}(z,\mathbf{0})\equiv
H(z)$), we arrive at $$\begin{aligned}
\label{chi2}
S(z)=H(z)S^{(0)}(z) .\end{aligned}$$ After the inverse Z-transform one gets $$\label{inverse4}
s_n=\sum_{l=0}^{n}h_{n-l}s^{(0)}_{l} .$$ The fundamental input signal $S^{(0)}(z)$ here and the fundamental filter are defined by the integrals: $$\begin{aligned}
S^{(0)}(z)= \frac{(z+1)}{(z-1)}\int\frac{d^p\mathbf{k}}{(2\pi)^p}
\frac{|\alpha(\mathbf{k})|^2}{[(z+1)^2/z-\mathcal{L}^2(\mathbf{k})]}
,\label{chi3} \\
\frac{1}{H(z)} = 1-\sigma^2
\frac{(z+1)}{(z-1)}\int\frac{d^p\mathbf{k}}{(2\pi)^p}
\frac{1}{[(z+1)^2/z-\mathcal{L}^2(\mathbf{k})]} .\label{H}\end{aligned}$$ Note also the relation $$\label{inverse3}
h_n=\frac{1}{2\pi i}\oint H(z) z^n \frac{dz}{z} .$$ That is, we have got here a new derivation of relations earlier derived in Refs. [@Kuzovkov02; @Kuzovkov04]. Note that for the 1-D case the integral disappears, since $p=0$, and one gets $$\begin{aligned}
S^{(0)}(z)= \frac{(z+1)}{(z-1)} \frac{|\alpha|^2}{[(z+1)^2/z-E^2]}
,\label{chi3a} \\
\frac{1}{H(z)} = 1-\sigma^2 \frac{(z+1)}{(z-1)}
\frac{1}{[(z+1)^2/z-E^2)]} ,\label{Ha}\end{aligned}$$
Fundamental input signal {#alpha}
------------------------
Let us discuss the *physical sense* of the input signal signal $S^{(0)}(z)$ (or $s^{(0)}_{n}$). Note that this corresponds to an *ideal* system with $\sigma=0$. When there is no disorder, $\varepsilon_{\mathcal{M}}\equiv 0$, the particular solutions of the tight-binding eq. (\[tight-binding\]) are *bounded* functions, plane waves $\exp(i\mathcal{K}\mathcal{M})$ with $\mathcal{K}=\{k_1,k_2,\dots,k_D\}$, provided for a given energy $E$ $$\label{EK}
\mathcal{E}(\mathcal{K}) = E$$ holds, where $$\mathcal{E}(\mathcal{K}) = \sum_{j=1}^D 2 \cos(k_j) ,$$ otherwise particular solutions are *not bounded* functions, which lie beyond the band and have no physical interpretation. If we divide the wave vector into transversal or normal directions, $\mathcal{K}\equiv \{\mathbf{k},k_D\}$, and keeping in mind that $$\label{EK2}
\mathcal{L}(\mathbf{k}) = 2\cos(k_D) ,$$ for a given transversal mode $\mathbf{k}$ the bounded physical solution exists, provided $|\mathcal{L}(\mathbf{k})| \leq 2$. These relations are sufficient for understanding the physical sense of the input signal.
Eq. (\[chi3\]) contains in the integrand the function $$\begin{aligned}
\label{fun}
\frac{(z+1)}{(z-1)}\frac{1}{(z+1)^2/z-\mathcal{L}^2(\mathbf{k})}.\end{aligned}$$ Under the condition $|\mathcal{L}(\mathbf{k})| \leq 2$ its inverse Z-transform gives $$\begin{aligned}
\frac{\sin^2(k_D n)}{\sin^2(k_D)} ,\end{aligned}$$ where $k_D=k_D(\mathbf{k})$ is the solution of eq.(\[EK2\]). Assuming that the arbitrary field $\alpha_{\mathbf{m}}$ is selected in such a way that the Fourier transform coefficients $\alpha(\mathbf{k})$ are not zero *only* if $|\mathcal{L}(\mathbf{k})| \leq 2$, eq.(\[chi3\]) after the Z-transform reads $$\begin{aligned}
s^{(0)}_{n}= \int\frac{d^p\mathbf{k}}{(2\pi)^p}
\frac{|\alpha(\mathbf{k})|^2 \sin^2(k_D n)}{\sin^2(k_D)} .
\label{chi4}\end{aligned}$$ The $s^{(0)}_{n}$ is *bounded* for any $n$.
On the other hand, assuming that $|\mathcal{L}(\mathbf{k})| \leq
2$ is violated for certain $\mathbf{k}$, so the $\alpha(\mathbf{k})\neq 0$ for $|\mathcal{L}(\mathbf{k})| \geq 2$, the inverse Z-transform of eq.(\[fun\]) leads to the function $$\begin{aligned}
\frac{\sinh^2(\kappa_D n)}{\sinh^2(\kappa_D)} ,\end{aligned}$$ which increases without bounds as $n \rightarrow \infty$. Here $\kappa_D=\kappa_D(\mathbf{k})$ is the solution of equation $$\label{Ekappa}
|\mathcal{L}(\mathbf{k})| = 2\cosh(\kappa_D) .$$ The input signal $s^{(0)}_{n}$ is also increasing to infinity as a function of $n$. Note that both input and output signals are *real* values.
In other words, on can conclude that any *physical solution* of an ideal system ($\sigma=0$) with *bounded* wave functions has a one-to-one correspondence to the mathematical object - a *bounded* 1-D input signal $s^{(0)}_{n}$. Such solutions always exist, if the energy $E$ lies within the band interval. On the other hand, formal solutions without physical interpretation (*unbounded* solutions beyond the band) correspond to the 1-D *unbounded* input signals. Note that particular numerical values of input or output signals are not important from the point of view of the signal theory [@Weiss] whose main concern is a *qualitative discrimination* between bounded and unbounded signals.
Fundamental filter
------------------
The quantity characterizing the phase diagram of a disordered system is the *fundamental filter*, eq.(\[H\]). Its idea is quite clear [@Kuzovkov02; @Kuzovkov04]. Let us consider the physical states inside the band, $|E|<2D$. All these states are described by wave functions bounded in amplitude, which corresponds to a bounded 1-D input signal $s^{(0)}_{n}$. In a disordered system this signal transforms into $s_{n}$, according to eq. (\[inverse4\]). This output signal $s_{n}$ can be either bounded, or infinitely growing (generalized diffusion).
The natural interpretation of *bounded output signals* is that they - as before - correspond to the *delocalized states* bounded in amplitude. The *unbounded output signals* correspond to the *localized states*, respectively, which lead in the semi-infinite system to the divergence of the diagonal correlators. An important result of signal theory [@Weiss] is that the divergence does not result from properties of a particular bounded input signal; the cause lies in the filter, $H(z)$ or $h_n$. We define the phase diagram of the Anderson model based on a general concept of the signal theory known as a *BIBO stability* [@Weiss]. Namely, a system is BIBO stable if every **B**ounded **I**nput leads to a **B**ounded **O**utput. A stable system (delocalized states) is characterized by a *stable* filter $H(z)$; its main property is the *absence of poles* in the complex $z$-plane outside the circle $|z|>1$. An *unstable* system (localized states) is characterized by the *unstable* filter $H(z)$ *with poles* outside the circle $|z|>1$. As was shown in [@Kuzovkov02; @Kuzovkov04], in the Anderson model the pole $z_0$ of the unstable filter lies on the real axis. Consequently, $z_0=\exp(2\gamma)$, where $\gamma$ is the Lyapunov exponent ($\psi^2$-definition) [@Kuzovkov02; @Kuzovkov04]. Using the inverse Z-transform, one can easily find that asymptotically $h_n \sim \exp(2\gamma n)$, i.e. it is divergent. The localization length is defined as $\xi=\gamma^{-1}$.
The mathematical formalism of the signal theory allows to extend and complement this result for the energy range $|E|>2D$, where in the absence of disorder there existed only mathematical (divergent) solutions having no physical interpretation. It is known that disorder extends the band, i.e. new physical states arise also at $|E|>2D$. The question is: can delocalized states exist amongst these new states? The answer is simple. In the region $|E|>2D$ only *unbounded input signals* $s^{(0)}_{n}$ exist. Such a signal cannot be transformed into a *bounded output signal* $s_{n}$. The output signal $s_{n}$ is always unbounded with a dual interpretation: on the one hand it corresponds to mathematical solutions (no physical interpretation); on the other hand, the divergence of the output signal can be associated with an emergence of the localized states. I.e., an emergence of delocalized states in the region $|E|>2D$ (outside the band) is impossible [@Kuzovkov04].
Use of the filter function $H(z)$ is a general and abstract method for describing the metal-insulator transition, valid for any space dimension. Earlier [@Kuzovkov02] we used a less fundamental approach for the 1-D problem, based on the calculation of the Lyapunov exponent for the $\psi^2$-definition $$\label{gamma2}
\gamma=\lim_{n \rightarrow \infty} \frac{1}{2n} \ln s_n ,$$ where for the 1-D case $s_n \equiv \left\langle \psi_n^2
\right\rangle$. Taking into account that the divergence of the output signal $s_n$ is caused by the divergence of the filter $h_n$, one can use also the following equation $$\label{gamma3}
\gamma=\lim_{n \rightarrow \infty} \frac{1}{2n} \ln h_n .$$ In the 1-D case both definitions of $\gamma$ are equivalent, since for a fixed energy and disorder, for a second initial condition $\psi_1=\alpha$, one gets the *single* output signal $s_n$.
Let us use for illustration eqs.(\[chi3a\]), (\[Ha\]). Restricting ourselves by the band center $E=0$ (which simplifies the equations) and making the inverse Z-transform, one gets $$\begin{aligned}
s_n=|\alpha|^2 \frac{z_0}{(z^2_{0}+1)}[z^n_0-(-1/z_0)^n] ,\\
h_n=\delta_{n,0}+ \frac{(z^2_0-1)}{(z^2_{0}+1)}[z^n_0-(-1/z_0)^n]
, \label{h1D}\\
z_0=\frac{\sigma^2+\sqrt{4+\sigma^4}}{2} ,\end{aligned}$$ provided $s^{(0)}_n=|\alpha|^2 [1-(-1)^n]/2$ (bounded input signal). Indeed, simultaneous divergence of the output signal $s_n$ and the filter $h_n$ arises due to the fact that the asymptotic behaviour of both quantities is determined by the same parameter, $z_0=\exp(2\gamma)>1$.
However, this equivalence is no longer valid for a space dimension higher than one. Since the second initial condition is defined by the field $\alpha_{\mathbf{m}}$, this corresponds to a continuum of input signals $s^{(0)}_n$ and, respectively, a continuum of output signals $s_n$. That is, the definition (\[gamma2\]) is not valid, since it is not clear that different signals should correspond to the same Lyapunov exponent $\gamma$. Formally, a whole continuum of solutions should be analyzed. However, eq.(\[inverse4\]) demonstrates that the fundamental Lyapunov exponent $\gamma$ does *not depend* at all on the field $\alpha_{\mathbf{m}}$, it is sufficient to define the Lyapunov exponent using only eq.(\[gamma3\]).
Phase diagram and multiplicity of solutions
-------------------------------------------
A more detailed analysis [@Kuzovkov02; @Kuzovkov04; @Reply] reveals another problem of the definition (\[gamma2\]): this is valid *only* in the 1-D case where for any disorder $\sigma$ only the phase of the localized solutions exists.
As it was mentioned above, for the calculation of the fundamental filter $h_n$ using the inverse Z-transform, eq.(\[inverse3\]), the contour integration over the so-called *region of convergence* (ROC) [@Weiss] is necessary, provided the filter under consideration is causal. It is shown [@Kuzovkov02; @Kuzovkov04] that dependent on the energy $E$ and disorder $\sigma$, two general cases are possible for the Anderson model with $D \geq 2$. In the first case, the ROC lies *outside* the circle $|z|>1$. The filter $h_n$ is thus *unstable* and describes the localized states (insulating phase). In the second case, the ROC consists of *two domains* in the complex plane: one domain is *inside* the circle $|z|\leq 1$, another one - *outside* this circle, $|z|>1$. Respectively, there are *two ways* to calculate the integral using eq. (\[inverse3\]). The double solution arises, one describing delocalized states (filter $h^{(-)}_n$, metallic phase), the other localized states (filter $h^{(+)}_n$, insulating phase).
Consequently, in this range of parameters $E$ and $\sigma$ the *two phases* can co-exist, this is why the metal-insulator transition in the Anderson model has to be analyzed in terms of *first-order* phase transition theory [@Kuzovkov02; @Kuzovkov04; @Reply]. For arbitrary random potentials the wave function can be *either* localized *or* delocalized (no co-existence!). However, in terms of *statistics* of an ensemble of random potentials, *both* localized *and* delocalized solutions can arise with a *comparable probability*. Namely this comparability of probabilities is the main characteristic of the first-order phase transition, in contrast to the second-order transitions where either a pure metallic phase (no localized states) or a pure insulating phase (no delocalized states) should exist: co-existence is impossible. Strictly speaking, the type of phase transition in the *microscopical* Anderson model cannot be determined without exact solution of the problem. Despite the fact that first-order phase transitions are very common in nature, its realization in the Anderson model (under strong influence of the *phenomenological* scaling theory of localization [@Abrahams]) never was seriously discussed.
Phase co-existence creates a serious problem of the choice of an adequate mathematical formalism. Strictly speaking, when calculating the average quantities on the ensemble of different realizations of random potentials, the contributions from pure metallic and insulating phases are considered as equivalent. However, such *heterophase averages* have no physical sense [@Kuzovkov02; @Kuzovkov04; @Reply]. In a two-phase system with first-order phase transition one is interested in properties of *pure phases*. From this point of view, the earlier introduced output signals $s_n$ are also heterophase averages. This is why the derivation of equations for signals is not a goal, but the tool for obtaining a more fundamental property - the filter $H(z)$ ($h_n$). As was noted [@Kuzovkov02; @Kuzovkov04; @Reply], it is such a filter which reveals the *multiplicity of solutions* and which permits us to determine the phase diagram for the Anderson model.
Averaging over initial conditions {#Averaging}
---------------------------------
Let us discuss now the problem of *averaging over intitial conditions* [@Kuzovkov02; @Kuzovkov04]. This procedure was introduced in Ref. [@Kuzovkov02], its meaning is quite simple. First of all, we choose a particular second initial condition, i.e. a field $\alpha_{\mathbf{m}}$. Secondly, let us consider the field $\alpha^{\prime}_{\mathbf{m}}\equiv \alpha_{\mathbf{m+m_0}}$ obtained from the first field as the result of a trivial *translation* in transversal direction by the vector $\mathbf{m_0}$. It is obvious that the relevant diagonal correlators $\chi^{\prime}(n)_{\mathbf{m}} \equiv
\chi(n)_{\mathbf{m+m_0}}$ can also be obtained by the *argument shift* in a transversal direction. Both solutions are *physically equivalent*. The Fourier transform of the field $\alpha_{\mathbf{m}}$ with the vector shift $\mathbf{m_0}$ satisfies a simple relation: $\alpha^{\prime}(\mathbf{k})\equiv
\exp(-i\mathbf{k}\mathbf{m_0})\alpha(\mathbf{k})$.
Eqs. (\[chi\_k\]) to (\[Hk\]) clearly demonstrate that the vector shift affects only signals for particular modes $\mathbf{k}$, and all physically equivalent solutions differ from each other only by *phases*: $$\begin{aligned}
\chi^{\prime(0)}(z,\mathbf{k}) \equiv
\exp(-i\mathbf{k}\mathbf{m_0})\chi^{(0)}(z,\mathbf{k}) , \\
\chi^{\prime}(z,\mathbf{k}) \equiv
\exp(-i\mathbf{k}\mathbf{m_0})\chi(z,\mathbf{k}) .\end{aligned}$$ Note that only the fundamental mode $\mathbf{k=0}$ remains invariant. Averaging now the signal over all translations in transversal directions (averaging over initial conditions [@Kuzovkov02]) gives zero for all non-fundamental modes, $\mathbf{k\neq 0}$: $\overline{\chi^{(0)}(z,\mathbf{k}) }=0$, $\overline{\chi(z,\mathbf{k}) }=0$. Nonzero is only the fundamental mode ($\mathbf{k\equiv 0}$): $\overline{\chi^{(0)}(z,\mathbf{0}) }=S^{(0)}(z)$, $\overline{\chi(z,\mathbf{0}) }=S(z)$. The average diagonal correlator loses its dependence on the argument $\mathbf{m}$: $\overline{\chi(n)_{\mathbf{m}}}=s_n$. That is, averaging over initial conditions is an efficient tool for getting rid of all non-fundamental modes which are non-essential for the analysis.
Note that eq. (\[chi3\]) contains $\Gamma(\mathbf{k})=|\alpha(\mathbf{k})|^2$ which is a function of the field $\alpha_{\mathbf{m}}$, its Fourier transform is $\Gamma_{\mathbf{m}}$. The averaging over initial conditions corresponds mathematically in the initial eq. (\[X(z)\]) to the replacement of the initial condition $x(1)_{\mathbf{m},\mathbf{l}}=\alpha_{\mathbf{m}}\alpha_{\mathbf{l}}$ by the average, $\overline{x(1)_{\mathbf{m},\mathbf{l}}}=\Gamma_{\mathbf{m-l}}$. After such a *replacement* of initial conditions the system becomes much simpler: it is *translation-invariant* in transversal directions which makes use of the double Fourier transform unnecessary. In other words, averaging over initial conditions is nothing but a simple mathematical trick which permits to get quickly the fundamental property of system - the filter $H(z)$. Such tricks are based on the fact that the filters are universal system characteristics, describing the solution transformation from order to disorder. Such universal characteristics are *independent* of the initial conditions and other details. The initial conditions determine such non-universal properties as signals. Consequently, in order to obtain the fundamental filter $H(z)$ and then the phase diagram, one can perform *linear operations* with signals, in particular, averaging over initial conditions.
Another example of such operations is the following. Let us consider the field $\alpha_{\mathbf{m}}$ not as fixed (second initial condition), but as a random variable, with simultaneous averaging *over the field* $\alpha_{\mathbf{m}}$ and *over the ensemble of the random potential realizations* $\varepsilon_{\mathcal{M}}$. We assume the absence of correlations between the potentials $\varepsilon_{\mathcal{M}}$ and the field $\alpha_{\mathbf{m}}$; unlike the correlation between the field components characterized by the arbitrary correlation function $\left\langle \alpha_{\mathbf{m}} \alpha_{\mathbf{l}}\right\rangle
=\Gamma_{\mathbf{m-l}}$. The initial condition in eq.(\[X(z)\]) becomes $x(1)_{\mathbf{m},\mathbf{l}}=\Gamma_{\mathbf{m-l}}$. Taking into account that the operator $\mathcal{L}_{\mathbf{m},\mathbf{m^{\prime}}}$ by its definition, eq.(\[L\]), depends only on the on the argument difference, $\mathbf{m}-\mathbf{m^{\prime}}$, one gets the translation invariant system, where $X(z)_{\mathbf{m},\mathbf{l}} \equiv
\hat{X}(z)_{\mathbf{m-l}}$, provided for the diagonal correlators $X(z)_{\mathbf{m},\mathbf{m}}=\chi(z)_{\mathbf{m}} \equiv
\hat{X}(z)_{\mathbf{0}} = S(z)$ holds (the diagonal correlators are $\mathbf{m}$-independent).
In this case in order to solve eq. (\[X(z)\]), it is sufficient to use a single Fourier transform $$\begin{aligned}
\hat{X}(z)_{\mathbf{m}}=\int
\frac{d^p\mathbf{k}}{(2\pi)^p}\hat{X}(z,\mathbf{k})e^{-i\mathbf{k}\mathbf{m}}
.\end{aligned}$$ Instead of eq.(\[UX\]) one gets $$\begin{aligned}
\label{UX3}
U(z,\mathbf{k},\mathbf{-k})\hat{X}(z,\mathbf{k})=
\Gamma(\mathbf{k})+\sigma^2 S(z) ,\end{aligned}$$ which gives $$\begin{aligned}
S(z)=\int \frac{d^p\mathbf{k}}{(2\pi)^p}\hat{X}(z,\mathbf{k}) .\end{aligned}$$ As a result, one returns to the fundamental eqs.(\[chi2\]) to (\[H\]), with replacement of the $|\alpha(\mathbf{k})|^2$ for $\Gamma(\mathbf{k})$.
Conclusion
==========
We would like to stress that in this paper we presented a mathematically rigorous method for the calculation of the phase diagram for the Anderson localization in arbitrary dimensions, which was briefly discussed earlier [@Kuzovkov02; @Kuzovkov04]. The phase diagram for the metal-insulator transition is obtained using the Lyapunov exponent $\gamma$. Localized states correspond to values of $\gamma > 0$, i.e. a divergence of the averages over wavefunctions. This divergence is mathematically similar to the divergence of averages for the diffusion motion. That is, transition to the localized states can be treated as a generalized diffusion. From this viewpoint, in order to determine the range of the existence of localized states (i.e. the phase diagram) and the type of localization (exponential or non-exponential), it is sufficient to solve equations for the joint correlators. We have shown that these equations are *exactly* solvable analytically.
In its turn, the appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to correlators. The generalized diffusion can be described in terms of signal theory, which operates with the concepts of input and output signals and the filter function. Delocalized states correspond to bounded output signals, and localized states to unbounded output signals, respectively. Transition from bounded to unbounded signals is defined uniquely by the filter function $H(z)$, or more precisely, by the position of its poles in the complex plane. This function can be calculated for arbitrary space dimension D.
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---
abstract: 'We study the possibility of implementing a quantum switch and a quantum memory for matter wave lattice solitons by making them interact with “effective” potentials (barrier/well) corresponding to defects of the optical lattice. In the case of interaction with an “effective” potential barrier, the bright lattice soliton experiences an abrupt transition from complete transmission to complete reflection (quantum switch) for a critical height of the barrier. The trapping of the soliton in an “effective” potential well and its release on demand, without loses, shows the feasibility of using the system as a quantum memory. The inclusion of defects as a way of controlling the interactions between two solitons is also reported.'
author:
- 'V. Ahufinger'
- 'A. Mebrahtu'
- 'R. Corbalán'
- 'A. Sanpera'
title: 'Quantum switches and quantum memories for matter-wave lattice solitons'
---
Introduction {#sec:1}
============
Bose Einstein condensates (BEC) in optical lattices have attracted during the last years a lot of attention both in the mean field regime [@meanfield] as well as in the strongly correlated regime [@strongly]. One of the main reasons for this activity burst is the high level of control achieved in the experiments of ultracold gases in optical lattices which permits to explore a broad range of fundamental phenomena.
In the mean field regime, a huge interest has been devoted to nonlinear dynamics of matter-waves in periodic media and specifically to matter-wave solitons. Matter-wave solitons are self stabilized coherent atomic structures that appear in nonlinear systems due to the balance between the nonlinearity and the dispersive effects. The nature of the solitons supported by Bose Einstein condensates (BEC) is determined by the character of the interactions: attractive (repulsive) nonlinearity supports bright [@bright-soliton] (dark [@dark-soliton]) solitons. In the presence of an optical lattice, this scenario changes completely due to the appearance of a band structure in the spectrum and the possibility of having either bright or dark lattice solitons with either repulsive or attractive interactions arises. Very recently, the first experimental demonstration of bright lattice solitons in repulsive condensates was reported [@Oberthaler].
Since the first proposals of BEC lattice solitons [@lattfirst], there has been an explosion of contributions regarding generation, mobility and interactions of this novel type of matter-wave solitons both in one-dimensional systems [@oned; @nosaltres1; @nosaltres2] and in higher dimensions [@mesd]. The interest is mainly centered in bright matter-wave lattice solitons due to their potential applications in energy and information transport in nonlinear systems. The fact that matter wave solitons are massive permits to generate them at rest and to move them after an appropriate transfer of momentum. Proposals for controlling the dynamics of bright gap solitons are mainly devoted to the manipulation of the optical lattice [@control1] and to the modification of the nonlinearity [@control2]. Nevertheless, a complete control on the dynamics of bright matter-wave gap solitons also requires a profound knowledge of their interactions with defects.
The interaction of solitons with local inhomogeneities is a subject that appears in the literature in different contexts and has been studied in the framework of different nonlinear equations (see for instance [@review]). In particular, the nonlinear Schrödinger equation with point-like defects either in the continuum regime [@continu] or in discrete systems [@discrete] has deserved special attention. Extended defects have also been addressed in this framework [@extended]. In nonlinear optics, the coupled mode equations have been used to study collisions of moving Bragg solitons with finite size ([@coupled1],[@coupled2]) and point like defects [@coupled3]. Very recently, interactions with defects in the context of continuous matter-wave solitons have also been addressed [@defdark; @defbright].
In this paper we focus on bright lattice matter-wave solitons and propose different possibilities of control of their dynamics by making them interact with defects of arbitrary amplitude and width. Specifically we will show how to change the direction of movement (a complete bounce back) of the soliton and how it can be stored and retrieve on demand. In Sec. \[sec:2\] the physical system considered and the model used is introduced. In Section \[sec:3\] we present the results concerning the interaction of solitons with an “effective” potential barrier, where the possibility of implementing a quantum switch arises. Next, in Sec. \[sec:4\] the results regarding the interaction with an “effective” potential well will be shown and the use of the system as a quantum memory will be discussed. The possibility of controlling the interactions between two lattice solitons by placing a defect at the interaction point is discussed in Sec. \[sec:5\]. We will conclude in Sec. \[sec:6\].
Physical system {#sec:2}
===============
We consider a zero temperature $^{87}$Rb condensate confined in a one-dimensional geometry and in the presence of an optical lattice. The description of the system is performed within the one dimensional Gross Pitaevskii equation (GPE): $$i\hbar \frac{d\psi \left( x,t\right)}{dt} = \left(-\frac{\hbar^2}{2m}\triangle+
V \left( x\right)+g\vert \psi\left( x,t\right)\vert^2\right)\psi \left( x,t\right),
\label{GPE}$$ where $g=2 \hbar a_s \omega_{t}$, being $a_s$ the $s$-wave scattering length and $\omega_{t}$ the radial angular trapping frequency, is the averaged one-dimensional coupling constant. The external potential is given by: $$V \left( x \right)=\frac{m}{2} {\omega_x}^2 x^2+ V_0\sin^2(\frac{\pi x}{d}),$$ which describes both the axial trapping potential, with angular frequency $\omega_x$, and the optical lattice, with spatial period $d=\lambda/{2\sin(\theta/2)}$, being $\lambda$ the wavelength of the lasers forming the optical lattice and $\theta$ the angle between them. The depth of the optical lattice, $V_0$, is given in units of the lattice recoil energy $E_r=\hbar^2 k^2/2m$ where $k=\pi/d$ is the lattice recoil momentum.
The generation of the bright lattice soliton is performed as it is reported in [@nosaltres1] using parameters close to the experimental realizations [@Oberthaler]. The procedure is briefly summarized in what follows. The starting point is a $^{87}$Rb condensate ($a_s=5.8$nm, $m=1.45\times 10^{-25}$Kg) of $N=500$ atoms in the presence of a magnetic trap with frequencies $\omega_{t}=715\times2\pi$ Hz and $\omega_x=14\times2\pi$ Hz and an optical lattice, with potential depth $V_0=1E_r$ and period $d=397.5$nm. The axial magnetic trap is suddenly turned off and the appropriate phase imprinting, corresponding to phase jumps of $\pi$ in adjacent sites, is performed [@nosaltres1]. After the phase imprinting, the system evolves towards a negative mass, self-maintained staggered soliton at rest centered at $x=0$, which contains approximately $35\%$ of the initial atoms ($N=187$) and extends circa $11$ sites. The exceeding atoms are lost by radiation.
The total energy of the generated bright lattice soliton can be calculated using the energy functional of the GPE (\[GPE\]) that contains the total kinetic ($E^T_k$), interaction ($E_i$) and potential ($E_p$) energies:
$$E=\int\left[\frac{\hbar^2}{2m}|\nabla \psi(x)|^2 +\frac{g}{2}|\psi(x)|^4 +V(x)|\psi(\
x)|^2\right]dx .
\label{functional}$$
As clearly observed in the numerical solutions of the GPE, the density profile of the bright lattice soliton inside each well is shifted with respect to the minimum of the optical lattice. This shift has to be taken into account to properly calculate the energy. Assuming a constant shift $\delta$, the Ansatz $\psi(x)=A \exp(-(x-x_0)^2/2 \eta^2)\cos(2\pi x/(\lambda'))$, with $\lambda'=2d+\delta$, $A$ the amplitude and $\eta$ the width of the Gaussian envelope [@nosaltres2], leads to the following energy functional: $$\begin{aligned}
&&E=B \Bigg( \frac{\hbar^2}{m}\left[\frac{ 1+e^{-{k'}^2 \eta^2}\cos(2{k'}x_0)}{2\eta\
^2}+{k'}^2 \right] \nonumber \\
&&+\frac{g|A|^2}{4\sqrt{2}}\left[3+e^{-2 k'^2 \eta^2}\cos(4k'x_0)+4e^{-\frac{k'^2 \eta^2}{2}}\cos(2k'x_0)\right]\nonumber \\
&&+V_0\Big[1+e^{-k'^2 \eta^2}\cos(2k'x_0)-e^{-k^2 \eta^2}\cos(2k x_0) \nonumber \\
&& -\frac{1}{2}e^{-k_-^2 \eta^2}\cos(2k_- x_0) -\frac{1}{2}e^{-k_+^2 \eta^2}\cos(2k_+ x_0)\Big]
\Bigg).
\label{energies}\end{aligned}$$ where $B=|A|^2 \sqrt{\pi}\eta/4$, $k'=2\pi/\lambda'$ and $k_{\pm}=k' \pm k$. Fixing $N=187$, $\delta=0.07\mu$m and $x_0=0$ we obtain a minimum of Eq. (\[energies\]) corresponding to $1.31E_r$. Exact numerical integration of Eq. (\[functional\]) gives a total energy of $1.35E_r$. This shows the high level of accuracy that the used variational method provides. Also, an strong agreement is found when we evaluate each term of Eq.(\[functional\]) by using the variational method [@nosaltres2] (exact integration): $E^T_k=0.85E_r(0.85E_r)$, $E_i=0.12E_r(0.13E_r)$ and $E_p=0.34E_r(0.37E_r)$.
By calculating the linear band spectrum of the system we obtain an energy at the band edge of $1.25E_r$. This value is in good agreement with the total energy obtained previously (Eq.(\[functional\])) without the non linear term. The linear band spectrum predicts an effective mass at the edge of the first Brillouin zone corresponding to $m_{eff}=-0.15m$.
Once the lattice soliton is created, it is set into motion by applying an instantaneous transfer of momentum at $t=0$. It has to be large enough to overcome the Peierls-Nabarro (PN) barrier [@nosaltres1; @kivshar93] but sufficiently small to assure that the soliton remains in the region of negative effective mass, i.e., $0.009k\hbar<p<0.2k\hbar$. The soliton starts to move opposite to the direction of the kick, manifesting thus its negative effective mass. The concept of the effective mass is used through the paper to give an intuitive explanation of the observed dynamics. Nevertheless, all the results presented in what follows have been obtained, without any approximation, by direct integration of Eq. (\[GPE\]).
At a certain distance $x_{m}$ of the initial position of the soliton ($x=0$), the lattice potential $V(x)$ is modified in the following way:
$$\left\{ \begin{array}{ll}
V_0\sin^2(\frac{\pi x}{d})+ V_{m}(1-\frac{(x-x_{m})^2}{2\sigma^2})& \mbox{\small{if $x_m-l/2\leq x\leq x_m+l/2$}};\\
V_0\large\sin^2(\frac{\pi x}{d}) & \mbox{\small{otherwise}}.\end{array} \right.
\label{potmod}$$
where $V_{m}$ can be either positive or negative and $\sigma=6d$ for all the cases. For $ V_{m}<0$, the local decrease of the lattice potential corresponds to an “effective” potential barrier for the soliton due to its negative effective mass while if $V_{m}> 0$, i.e. a local increase of the potential acts as an “effective” well for the soliton. For the case of an effective barrier, $x_{m}$ is fixed to match exactly with a minimum of the optical lattice while in the effective well case, $x_{m}$ corresponds to a maximum of the optical lattice. We have checked that the results reported in the following do not depend strongly on the specific shape of the potential by reproducing them with Gaussian and square potentials for the defect.
To analyse the interaction of the bright lattice soliton with the defect, it is crucial to know the total energy of the soliton while it moves. The applied variational Ansatz with the shift in the periodicity is meaningful only in the static case and cannot be used to study soliton dynamics, since the soliton is always chirped with respect its center [@nosaltres2]. Therefore, to study dynamical behaviour one has to rely on numerical simulations. We have numerically calculated the contributions to the total energy of the soliton as a function of time. Immediately after the kick, the soliton expels atoms and its energy abruptly decreases becoming much smaller than the energy that it would need to remain at rest at the edge of the first Brillouin zone. In the framework of the linear band theory this would correspond to displace the particle from the edge of the band of the first Brillouin zone by changing its quasimomentum. To illustrate the dynamics of the system, we consider the case in which we give a kick of $p=0.1k\hbar$ to the generated soliton at rest. At $t=0$, just after the kick, the total energy of the soliton is its energy at rest plus the contribution of the transfer of momentum, i.e. $E=1.35E_r+(0.1)^2E_r=1.36E_r$. At $t=1$ms, the soliton energy has decreased already to $0.96E_r$, the rest of energy has been taken by the expelled atoms. A steady state is reached for a soliton energy of $E=0.92E_r$. While moving, some energy is devoted to cross the PN barrier (the soliton configuration changes its shape from a configuration centered in one well of the optical lattice to a configuration centered in one maximum and viceversa). This change of the shape of the soliton is reflected in the out of phase oscillations of the kinetic energy with respect to the potential plus non linear energy in such a way that the mean value of the energy remains constant.
![(a) Transmission coefficient, $T$, as a function of the amplitude of the “effective” potential barrier, $|V_{m}|$; (b) Transmission coefficient, $T$ and (c) reflection coefficient $R$ as a function of time for different amplitude defects: $|V_{m}|=0$ (solid line), $|V_{m}|=0.01 E_r$ (dashed line), $|V_{m}|=0.0115 E_r$ (dot-dashed line), $|V_{m}|=0.0117E_r$ (circles) and $|V_{m}|=0.018E_r$ (dotted line). In all the plots the soliton kinetic energy is $E_k=0.01E_r$ and the width of the defect $l=2d$[]{data-label="barrier1"}](fig1.eps){width="1.0\linewidth"}
“Effective” potential barrier {#sec:3}
=============================
We discuss first the interaction of a bright lattice soliton with an “effective” potential barrier. Scattering depends on the width of the defect ($l$) and the relevant energy scale, settled by the ratio $|V_{m}|/E_k$, where $E_k=<P>^2/2m $ is the fraction of the total kinetic energy $E^T_k$ devoted to move the soliton, and $< >$ denotes time average (before reaching the defect). The momentum $P$ is defined as: $$P(t)=\int -i \hbar \psi^*(x,t) \nabla \psi(x,t) dx.
\label{moment}$$ The rest of the kinetic energy is needed to keep the structure and cannot be used to overcome the “effective” potential barrier. We have checked that apart from the necessary change in shape to overcome the PN barrier, the soliton keeps its overall shape when it reaches the defect. This corroborates that there is no transfer between non linear energy and kinetic energy apart of the one corresponding to the already discussed PN barrier.
We distinguish two regimes of parameters: (i) when the amplitude of the “effective” potential barrier is on the order of the kinetic energy of the soliton ($|V_{m}|\sim E_k$) and (ii) when amplitude of the potential barrier is much larger than the kinetic energy of the soliton. In the former case, the potential barrier acts as a quantum switch, [*i.e.*]{}, either the entire soliton is transmitted or it is completely reflected depending on the amplitude of the barrier (Fig. \[barrier1\](a)). The transmission ($T$) and reflection ($R$) coefficients are calculated by integrating over space (and time) the density of the wavefunction in the region after and before the defect, respectively. Note that since only approximately 35% of the initial atoms form a soliton and since there are also loses of atoms during the kick, the merit figure for perfect transmission is well below $1$ and corresponds approximately to $0.27$ (N=135).
![(a) Reflection and transmission behavior of a soliton interacting with an effective potential barrier as a function of the potential width $l$ (in $d$ units) and potential amplitude $|V_{m}|$ (in $E_r$ units): the region of total transmission in gray, and the region of total reflection in white. Inside the reflection region, bands in which trapping and reflection occurs appear. Region of parameters for which tunneling and overbarrier reflection occur are also shown. The dashed line shows the value of the kinetic energy of the soliton. (b) Density profiles of the trapped structure that appears for $|V_{m}|=0.55E_r$ (lower plot), $|V_{m}|=0.8E_r$ (middle plot) and $|V_{m}|=1.1E_r$ (upper plot). In the three cases $l=8d$.[]{data-label="diagram"}](fig2.eps){width="1.0\linewidth"}
For a fixed width of the defect a drastic change of behavior occurs for a given height of the barrier $|V_{m}^c|$. The wider the defect is the lower the critical value of the height of the barrier $|V_{m}^c|$. The critical values, indicating the transition between complete transmission and complete reflection for different potential widths are shown in Fig.\[diagram\](a) by solid black squares. Below these values, depicted by a gray region in Fig.\[diagram\](a), complete transmission of the soliton occurs. As one approaches the critical value from below, the soliton experiences a time delay with respect to free propagation (i.e., in the absence of the defect). This delay increases as one gets closer to the critical point and eventually the time needed by the soliton to cross the barrier diverges (see Fig. \[delay\]). Above the transition line, reflection of the entire soliton occurs after a storage time inside the region of the barrier that also increases as one approaches the critical value. To illustrate this behavior, Fig. \[barrier1\](b,c) shows transmission and reflection coefficients as a function of time for a barrier of fixed width $l=2d$ and different values of the amplitude. In the situation shown in Fig. \[barrier1\], the critical value is nearly equal to the kinetic energy of the soliton but if the width of the barrier is reduced, this critical value can exceed the kinetic energy of the soliton. In this case, the soliton tunnels through the barrier, i.e., transmission is obtained for values of the amplitude of the barrier higher than the kinetic energy of the soliton (see Fig. \[diagram\](a)). On the other hand, for wider defects, a region of overbarrier reflection appears (Fig. \[diagram\](a)). There, the lattice soliton is completely reflected although it has a kinetic energy larger than the height of the potential barrier. This region extends for a wide range of widths of the defect. We have checked that overbarrier reflection occurs even in the limit when the width of the defect is much larger than the size of the soliton.
Fig. \[delay\] shows the delay time in transmission with respect to the absence of defect, $t_d$, for $l=d$ as a function of the height of the barrier, $|V_{m}|$, including the cases where tunneling occurs ($|V_{m}|> 0.01E_r$).
Up to now we have described the transition from complete transmission to complete reflection by fixing the width of the defect and varying the amplitude. It is worth noticing that a similar switching behavior can be obtained by fixing the amplitude of the effective potential and changing its width. This would correspond to horizontal lines in Fig. \[diagram\](a) crossing the transition line (solid black squares) in the region where $|V_{m}|$ is on the order of the kinetic energy of the soliton. This observed abrupt transition from complete reflection to complete transmission opens the possibility to use the system as a quantum switch. A similar switching behaviour has been predicted for optical Bragg solitons described with the coupled mode equations [@coupled2].
![Transmission delay time as a function of $|V_{m}|$ for a soliton with kinetic energy $E_k=0.01E_r$ crossing a defect of width $l=d$[]{data-label="delay"}](fig3.eps){width="1.0\linewidth"}
Let us now focus on the regime where the amplitude of the barrier is much larger than the kinetic energy of the soliton $|V_{m}|\gg E_k$, where the expected behavior is complete reflection of the soliton. Complete reflection occurs but there are specific values of the ratio $l/|V_{m}|$ for which the soliton splits into two parts: a fraction of the initial soliton becomes trapped inside the region of the barrier while the other part is reflected back keeping a solitonic structure. Fig. \[diagram\](a), for $|V_{m}| \gg E_k$, shows the regions where the soliton splits into two parts (trapping and reflection) embedded in the complete reflection regions. The fraction of atoms trapped inside the defect region has its origin on the atoms lost by radiation due to the repulsive force experienced by the soliton when it reaches the potential barrier. These radiated atoms enter the region of the barrier and for some specific ratios of the width and the height of the defect the fraction of trapped atoms increases. These trapping regions appear as bands as shown in Fig. \[diagram\](a). In each band, the trapped fraction exhibits different spatial distributions: for the first (lowest) one, the structure is a single hump; in the second one a double hump structure appears, and so on (see Fig. \[diagram\](b)). A noticeable feature of this trapped fraction is that the density maxima of the structure are located at the positions of the maxima of the optical potential. Increasing the amplitude of the barrier, the structure becomes more independent of the lattice periodicity. The extension of the trapped structure is the same independently of the features of the barrier but the number of trapped atoms differs for different widths of the barrier. The narrower the defect is the larger the number of trapped atoms. This number changes also with $|V_m|$ inside each band, being maximum at the center of the band. For all cases the number of atoms forming the reflected soliton is always larger than the trapped fraction. We have checked that these “resonance” bands like do not correspond to bound states of the linear case. We have also observed that this behaviour occurs for all the accessible initial transfers of momentum that allow motion of the soliton.
![(a) Temporal evolution of the trapped fraction of the soliton interacting with an “effective” well of depth $|V_{m}|=0.018E_r$ and width $l=8d$ after an instantaneous transfer of momentum of $p=0.05k\hbar$ (solid line), $p=0.1k\hbar$ (dashed line), $p=0.17k\hbar$ (dotted line) and $p=0.2k\hbar$ (circles). (b) Contour plot of the evolution in space and time of the lattice soliton with conditions corresponding to the dashed line case in (a).[]{data-label="kinetic"}](fig4.eps){width="1.0\linewidth"}
“Effective” potential well {#sec:4}
==========================
Let us now turn to the interaction of a lattice soliton with an “effective” potential well with a depth of the order of its kinetic energy. For a fixed depth of the well, the soliton exhibits different behaviors depending on its kinetic energy. For low kinetic energies, the soliton gets bound with the defect and exhibits oscillations while for kinetic energies overcoming a certain threshold, the soliton crosses the defect region. In the latter case, the only detectable effect of the potential well is the speed up of the soliton with respect to free propagation. It is important to note that as the width of the defect increases, the range of velocities for which transmission occurs decreases. To illustrate the described behavior, Fig. \[kinetic\](a) shows the time evolution of the trapped fraction density, $D$, for different initial transfer of momentum: $p=0.05k\hbar$ (solid line), $p=0.1k\hbar$ (dashed line), $p=0.17k\hbar$ (dotted line) and $p=0.2k\hbar$ (circles) keeping the depth ($|V_{m}|=0.018E_r$) and the width ($l=8d$) of the well fixed. The positions in time of the minima of the trapped fraction correspond to the turning points of the oscillating movement of the soliton around the “effective” well. The maxima indicates the times for which the soliton is completely inside the well. As expected, the amplitude of the oscillations increases with an increasing momentum transfer. If the amplitude of the oscillation is larger than the width of the defect, the turning points are located outside the potential well. This is reflected by a lower value of $D$. Fig. \[kinetic\](b) shows a contour plot of the evolution in space and time of a lattice soliton with $E_k=0.01E_r$ (dashed line case in Fig. \[kinetic\](a)). The width of the “effective” well is shown at the right hand part of the plot to illustrate that indeed the turning points are outside the defect.
![Temporal evolution of the trapped fraction density of a lattice soliton with $E_k=0.01E_r$ interacting with an “effective” well with (a)$|V_{m}|=0.018E_r$ and $l=4d$ (solid line), $l=8d$ (dashed line) and $l=12d$ (dotted line); (b) $l=4d$ and $|V_{m}|=0.018E_r$ (solid line), $|V_{m}|=0.03E_r$ (dashed line) and $|V_{m}|=0.08E_r$ (dotted line)[]{data-label="oscil"}](fig5.eps){width="1.0\linewidth"}
By fixing $E_k=0.01E_r$, we explore now the dependence of the oscillations on $l$ and $|V_m|$. Fig \[oscil\] (a) displays the temporal evolution of the trapped fraction, $D$, for $|V_{m}|=0.018E_r$ and different values of the width of the defect: $l=4d$ (solid line), $l=8d$ (dashed line) and $l=12d$ (dotted line). The frequency of the oscillations gives an indication of the width of the defect, decreasing as the width increases, while the amplitude remains approximately the same for all widths. In Fig. \[oscil\](b) we fix $l=4d$ and display the trapped fraction, $D$, as a function of time for $|V_{m}|=0.018E_r$ (solid line), $|V_{m}|=0.03E_r$ (dashed line) and $|V_{m}|=0.08E_r$ (dotted line). By inspection of Fig. \[oscil\](b), one can confirm that the frequency of the oscillations increases with the depth of the potential while the amplitude of the oscillations decreases. This is due to the fact that the soliton experiences a much larger attractive force as the depth of the defect increases limiting the displacements around the central position of the well.
The trapping of the entire lattice soliton around the position of the defect opens possibilities to use the system as a quantum memory because it provides the capacity of storage. Nevertheless, in order to perform a memory, one should also be able to release the trapped structure after a desirable time and with the minimum loses. We have checked that a soliton trapped in an “effective” potential well can be released with a certain velocity keeping the totality of its initial atoms if the defect amplitude is instantaneously set to zero. In fact the velocity of the lattice soliton after releasing it will depend on the amplitude of the oscillations it was performing while it was trapped. Specifically, the velocity of the structure, after releasing it, grows with the amplitude of the oscillations. Moreover, choosing appropriately the time at which the release takes place, one can vary the direction of the movement.
![Contour plot of the evolution in space and time of a collision between two identical solitons initially placed at symmetrical positions with respect $x=0$ and moving in opposite directions when an “effective” potential barrier with $l=2d$ and (a) $|V_{m}|=0.012E_r$, (b) $|V_{m}|=0.2E_r$ and (c) $|V_{m}|=0.5E_r$ is placed at $x=0$.[]{data-label="colisions"}](fig6.eps){width="1.0\linewidth"}
Control of the collisions {#sec:5}
=========================
Now we investigate if the inclusion of a defect in the lattice helps to control the interactions between two lattice solitons. It has been shown that collision of two identical lattice solitons (moving with the same velocity and with the same average phase) merge into a soliton with the same number of atoms as the initial ones [@nosaltres2]. The exceeding atoms are lost by radiation. If an “effective” potential barrier much narrower than the dimensions of the solitons is placed at the crossing point, we find the following behaviours: (i) for $|V_{m}|\leq E_k$ the merging behavior is maintained (Fig. \[colisions\] (a)); (ii) for $|V_{m}|\gg E_k$, each soliton reflects back (Fig. \[colisions\] (b)). Moreover, for some values of $|V_m|$, in addition to the reflection, a fraction of atoms trapped in the defect can appear (Fig. \[colisions\] (c)). The trapped fraction shows up the same features as in the single soliton case (section III). Modifying the features of the defect, different outcomes can be engineered. For instance, when the width of the barrier is of the order of the dimensions of the initial solitons, effects like the trapping of both solitons at the edges of the barrier can occur.
Conclusions {#sec:6}
===========
Summarizing, we have found that bright matter wave lattice solitons behave as “quantum” particles when colliding with an “effective” barrier/well, corresponding to a defect in the optical lattice. Among the rich dynamics exhibited by the system, we would like to remark two effects. The first one corresponds to the interaction of a soliton with an “effective” potential barrier which permits the implementation of a quantum switch. In this case, a sharp transition from complete reflection to complete transmission is present at a specific value of the height of the barrier. Although this resembles the classical particle behaviour, the quantum nature of the solitons is explicitly manifested in the appearance of overbarrier reflection and tunneling. The second effect we would like to stress appears when the defect acts as an “effective” potential well. We have shown that trapping of the entire soliton around the position of the defect and its release on demand with a given velocity and direction of motion is possible. This fact indicates the suitability of the system as a quantum memory. Finally, it has been also reported that the presence of a defect in the lattice can help to control the interactions of two lattice solitons.
We thank M. Lewenstein and A. Bramon for fruitful discussions. We acknowledge support from the Spanish Ministerio de Ciencia y Tecnología grants FIS2005-01369, FIS2005-01497 and Consolider Ingenio 2010 CSD2006-00019. A. M. acknowledges financial support from the Deutscher Akademischer Austausch Dienst (DAAD).
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---
abstract: 'In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we point out implications for measurement-based quantum computing and studies of quantum many-body models and suggest that a plethora of problems may indeed be undecidable.'
author:
- 'J. Eisert'
- 'M. P. M[ü]{}ller'
- 'C. Gogolin'
title: Quantum measurement occurrence is undecidable
---
\[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Observation]{} \[theorem\][Example]{} \[theorem\][Assumption]{}
At the heart of the field of quantum information theory is the insight that the computational complexity of similar tasks in quantum and classical settings may be crucially different. Here we present an extreme example of this phenomenon: an operationally defined problem that is undecidable in the quantum setting but decidable in an even slightly more general classical analog. While the early focus in the field was on the assessment of tasks of quantum information processing, it has become increasingly clear that studies in computational complexity are also very fruitful when approaching problems outside the realm of actual information processing, for example in the field of Hamiltonian complexity [@Kempe; @Terhal; @Dorit; @Osborne; @Area], or dynamical problems in channel theory [@ChannelNPHard]. In the meantime, a plethora of computationally hard tasks has been identified, both as far as *NP-hard problems* are concerned as well as their “quantum analogues,” the *QMA-hard* ones. These results show that it is presumably difficult to find an answer to those problems, but with sufficient computational effort, it can still be done.
Surprisingly, as will become clear, very natural decision problems in quantum theory may not only be computationally hard, but in fact even provably *undecidable* [@FrustratedRemark; @CubittPGWolf], i.e., there cannot be an algorithm, or for that matter a standard Turing machine, that always provides the correct answer in finite time. As such, this class of problems is in the same category as the halting problem that was famously shown to be undecidable in Alan Turing’s work from 1936 [@Turing]. The problem is to determine, given some program and an input, whether this program will eventually come to an end with that input – so will “halt” – or whether the program will continue running forever. The key insight of Alan Turing was to recognize that there cannot be a single algorithm that is able to correctly answer every instance of that problem. Of course, one can execute any algorithm for any finite time, but in case the program has then still not halted, one cannot judge in general whether or not it will ever do so. This seminal insight has had profound implications in the theory of computing and in fact even to mathematics: It implies G[ö]{}del’s first incompleteness theorem [@Goedel], which states that a consistent, complete, and sound axiomatization of all statements about natural numbers cannot be achieved.
In this work, we demonstrate that the very natural physical problem of determining whether certain outcome sequences cannot occur in repeated quantum measurements is undecidable, even though the same problem for classical measurements is readily decidable. We do so by employing a reduction: We show that if the problem that we introduce could always be solved, then one could find an algorithm that solved every instance of the halting problem – which cannot be true. At the same time we prove that the arguably most general classical analogue of the problem is always decidable, which shows that the undecidability is remarkably a genuine quantum mechanical feature.
We also suggest that it is reasonable to expect a number of further such results, in particular in the context of quantum information and quantum many-body theory.
![ (Color online) The setting of sequential application of Stern-Gerlach-type devices considered here, gives rise to a tree of possible outcomes. The problem is to decide whether there exists an *empty port* through which the particle will never fly.[]{data-label="fig_und"}](devicetree){width=".8\columnwidth"}
#### Setting.
The decision problem that we will prove undecidable is motivated by the following natural quantum measurement setting: Consider a measurement device that selectively measures a $d$-dimensional quantum system and has $K$ possible outcomes. Such a device is a generalization of a *Stern-Gerlach type* device that performs a nonprojective measurement. The $K$ outcomes of the device are associated with Kraus operators $A_1,\dots, A_K$. A measurement leading to outcome $j\in\{1,\dots, K\}$ occurs with probability $\operatorname{tr}(A_j \rho A_j^\dagger)$ and changes the state of the system according to $$\rho\mapsto \frac{A_j \rho A_j^\dagger}{\operatorname{tr}(A_j \rho A_j^\dagger)}.$$ In a sequence of $n$ measurements, the output state of such a device is repeatedly fed into an identical measurement device, leading to a tree of measurements (see Fig. \[fig\_und\]). Each path through this tree is associated with a sequence of outcomes $j_1,\ldots,j_n$. In order to have a meaningful decision problem, where each input can be described by finitely many bits, we restrict the problem to measurements whose Kraus operators are matrices of rational numbers:
Given a description of a quantum measurement device in terms of $K$ Kraus operators $\{A_1,\dots, A_K\}\subset {{\mathbbm{Q}}}^{d\times d}$, the task is to decide whether, in the setting described above, there exists any finite sequence of outcomes $j_1,\ldots,j_n$ that can never be observed, even if the input state has full rank.
Note that the notion of undecidability itself is independent of the physical theory: if a problem is undecidable for classical Turing machines, then it is also undecidable for quantum Turing machines, and vice versa since they can mutually simulate each other. Thus, our result says that the QMOP cannot be decided, no matter what physical resources we use to try to come to a decision.
Furthermore, note that in the QMOP one is supplied a perfect classical description of the quantum measurement device, and there is no “quantum uncertainty” in the description itself. Yet, we will see below that desctructive interference in the working of the quantum device, as encoded in the Kraus operators, renders the quantum measurement occurrence problem undecidable, in contrast to its classical counterpart.
#### Undecidability of the quantum problem.
Figuratively speaking in the metaphor of the Stern-Gerlach device with its tree of outcomes, the problem is to decide whether there exists an *empty port* somewhere in the tree through which the particle will never fly. Surprisingly this turns out to be undecidable:
\[thm:qmopisundecidable\] The QMOP for $K=9$ and $d=15$ is undecidable.
This statement is a consequence of the undecidability of the so-called *matrix mortality problem (MMP)*: *Given some finite set of integer matrices $\{M_1,\ldots, M_k\}$, is there any finite matrix product $M_{j_1}\ldots M_{j_n}$ that equals the zero matrix?* In other words, does the semigroup generated by $\{M_1,\ldots,M_k\}$ contain the zero matrix? One can show that the MMP is undecidable by reducing it to the so-called post correspondence problem (PCP) [@PCP; @PCPComment] (see the Appendix). More specifically:
The MMP is undecidable for $3\times 3$ integer matrix semigroups generated by $8$ matrices.
That is to say, there cannot be an algorithm that takes the input $ \{M_1,\dots, M_8\}\subset {{\mathbbm{Z}}}^{3\times 3}$ and computes in finite time whether or not there exists a sequence $j_1,\dots, j_n$ such that $
M_{j_1}\dots M_{j_n}=0.
$ In fact, in a variant of the argument, the above theorem is still valid for semigroups generated by $7$ integer $3\times 3$ matrices [@New]. Whether the MMP is also undecidable in the case of $2\times 2$ matrices is still an open problem [@Miller]. Turning back to the quantum problem, in terms of the Kraus operators, the probability for obtaining a particular sequence $\mathbf{j}=j_1,\dots ,j_n$ of outcomes $j_i\in \{1,\dots, K\}$ is $$\label{eqpw}
p_{\mathbf{j}}= \operatorname{tr}(A_{j_n}\dots A_{j_1}\rho A_{j_1}^\dagger \dots A_{j_n}^\dagger).$$ Now $\operatorname{tr}(A_{j_1}^\dagger \dots A_{j_n}^\dagger A_{j_n}\dots A_{j_1}\rho )=0$ for a full rank quantum state $\rho$ if and only if $A_{j_1}^\dagger \dots A_{j_n}^\dagger A_{j_n}\dots A_{j_1}=0$. Since this is a positive operator, the latter equality is true if and only if all of its singular values are zero, i.e., if and only if $A_{j_n}\dots A_{j_1}=0$.
Now we relate an instance of a MMP to a set of suitable Kraus operators $\{A_1,\dots,A_9\} \subset {{\mathbbm{Q}}}^{15\times 15}$. Our approach is to take an instance of the MMP, to encode it in Kraus operators having rational entries, and to complete them such that they form a trace-preserving completely positive map. The key point of the argument is that although we extend the dimension of the Kraus operators, a zero matrix is still found in the product of Kraus operators exactly if and only if the corresponding MMP contains a zero matrix in the semigroup. A slight detour in the argument is necessary as we wish to arrive at Kraus operators with rational entries.
For a given instance $\{M_1,\dots, M_8\} \subset {{\mathbbm{Z}}}^{3\times 3}$ of the MMP, define $$T\coloneqq \sum_{j=1}^8
M^\dagger_j M_j.$$ Using the three integer matrices $P_1\coloneqq\operatorname{diag}(-1,1,1)$, $P_2\coloneqq\operatorname{diag}(1,-1,1)$, $P_3\coloneqq\operatorname{diag}(1,1,-1)$, and for $j\in\{1,\dots, 8\}$ set $$\begin{aligned}
M_{8+j} &\coloneqq&M_j P_1,\label{1}\\
M_{16+j} &\coloneqq&M_j P_2,\label{2}\\
M_{24+j} &\coloneqq&M_j P_3\label{3}.\end{aligned}$$ This gives $$\sum_{j=1}^{32}
M_j^\dagger M_j = 4 \operatorname{diag}(T_{1,1},T_{2,2},T_{3,3}).$$ Define $c\in {{\mathbbm{N}}}$ as $
c \coloneqq \left\lceil 2 \left( \max\{T_{1,1},T_{2,2},T_{3,3}\} \right)^{1/2} \right\rceil .
$ By virtue of *Lagrange’s four-square theorem* [@foursquaretheorem], every natural number can be written as the sum of four integer squares. Hence, there exist four diagonal matrices $M_{33},\dots, M_{36}$ such that $
\sum_{j=1}^{36}
M_j^\dagger M_j = c^2\, {\mathbbm{1}}_3 .
$ We now set for $j=1,\dots, 8$, $$A_j\coloneqq \frac 4 {5c}\left[
\begin{array}{c|c}
M_j &\\
M_{8+j} &\\
M_{16+j} & 0_{15\times12}\\
M_{24+j} &\\
M_{32+j} &\\
\end{array}
\right]$$ with $M_{37}, \ldots, M_{40}\coloneqq 0_3$ and $$A_{9}\coloneqq \left(\frac 3 5{\mathbbm{1}}_3\right)\oplus {\mathbbm{1}}_{12}.$$ The matrices $\{A_1,\dots,A_9\} \subset {{\mathbbm{Q}}}^{15\times 15}$ satisfy $\sum_{j=1}^{9} A_j^\dagger A_j={\mathbbm{1}}_{15}$, as a simple calculation shows, and thus describe a quantum measurement device.
We are now in the position to reduce the quantum measurement occurrence problem to the problem of deciding whether the given semigroup contains the zero matrix. If this is the case, i.e., if there exists a sequence $\mathbf{j}$ for which $M_{j_n} \dots M_{j_1}=0$, $j_1,\ldots,j_n\in\{1,\dots,8\}$, then $A_{j_n}\ldots A_{j_1}$ has the zero matrix as its upper-left $3\times 3$ block. Moreover, the whole upper triangular matrix (including the diagonal) is zero as well, which means that the matrix is nilpotent: there is some $m\leq15$ such that $
(A_{j_n}\ldots A_{j_1})^m = 0.
$ Conversely, let us assume that there exists an outcome sequence that is never observed, so there exists a sequence $\mathbf{j}$ such that $A_{j_n} \dots A_{j_1}=0$. Let $\mathbf{v}$ be the sequence that is obtained from $\mathbf{j}$ by omitting all $j_i$ for which $j_i=9$. Then, by construction, $M_{v_{|v|}}\dots M_{v_1}=0$. Therefore, the semigroup generated by $\{M_1,\dots, M_8\}$ contains the zero matrix. The QMOP as described so far asks whether certain outcome sequences have probability *exactly* equal to zero. From a physical point of view, it is interesting to note that this result is to some extent robust, in the sense that it remains valid if small nonzero probabilities are allowed. To see this, write every Kraus operator in the form $A_j=Z_j/N_j$, where $N_j\in{{\mathbbm{N}}}$ and $Z_j\in{{\mathbbm{Z}}}^{d\times d}$. Then the probability of the sequence $\mathbf{j}$ from Eq. becomes $$\begin{aligned}
p_{\mathbf{j}}&=&(N_{j_1}\ldots N_{j_n})^{-2}\operatorname{tr}(\rho Z)\geq \frac 1 d N^{-n} \operatorname{tr}(Z)\nonumber\\
&\geq &(d N)^{-n}\operatorname{tr}(Z),\end{aligned}$$ where $Z\in {{\mathbbm{Z}}}^{d\times d}$ fulfils $Z\geq 0$ and $N:=\max_j N_j^2\in{{\mathbbm{N}}}$. Thus, $p_{\mathbf{j}}$ is either exactly zero or not less than $\delta^n$, where $\delta:=1/(dN)$ is a function of the Kraus operators. Therefore, the QMOP is equivalent to the following problem: *Given $K$ Kraus operators $\{A_1,\ldots,A_K\}\subset{{\mathbbm{Q}}}^{d\times d}$, is there a finite sequence of outcomes $j_1,\ldots,j_n$ which has probability less than $\delta^n$ (with $\delta>0$ defined above) if the input is the maximally mixed state?*
#### Decidability of the classical problem.
We now turn to a corresponding classical problem, the *classical measurement occurrence problem* (CMOP). A classical channel is described by a stochastic matrix $Q$ acting on $d$-dimensional probability vectors $\vec{q}$. A description of a classical selective measurement device with $K$ outcomes, is given by a decomposition $Q=\sum_{j=1}^K Q_j$ into matrices $Q_1,\dots , Q_K$ with non-negative entries (such matrices are sometimes called *substochastic*), that specify the action of the device on the classical system. That is, on outcome $j$ the probability vector is transformed according to $$\vec{q} \mapsto \frac{Q_j \vec{q}}{\sum_{i=1}^d (Q_j \vec{q})_i} .$$ This is arguably the most general classical analog of the QMOP. The probability for obtaining a particular sequence $j_1,\dots, j_n$ of outcomes $j_i\in \{1,\dots, K\}$ on an input probability vector $\vec{q}$ is $\sum_{i=1}^d (Q_{j_n}\ldots Q_{j_1} \vec{q})_i$. This is zero for an input vector $\vec{q}$ with all $(\vec{q})_i>0$ if and only if $Q_{j_n}\ldots Q_{j_1}=0$. The CMOP is thus obviously equivalent to the MMP with entrywise non-negative matrices. For this case the MMP is decidable, which was shown in Ref. [@Blondel] for $K=2$, and the general case follows by an essentially equivalent argument.
It shall be noted that our definition of classical devices is even more general than that of the quantum devices considered before; it represents the most general form of any conceivable classical measurement device. Namely, we allow for mixing in each outcome, which would in the quantum case correspond to a device that applies a whole quantum channel, not just a single Kraus operator, per outcome.
We now turn to proving decidability of the MMP with elementwise non-negative Kraus operators from which decidability of the classical case and for a subclass of quantum measurement devices follows.
\[thm:decidabiliy\] The MMP is decidable for any $d\times d$ matrix semigroup generated by $K$ matrices with non-negative rational entries.
Although the MMP is decidable for matrices with non-negative entries, it is still a hard problem: even in the case of $K=2$ matrices, this problem is NP-complete [@Blondel].
For any $K$ and $d$, both the *quantum measurement occurrence problem (QMOP)* with Kraus operators $\{A_1,\dots, A_K\} \subset {{{\mathbbm{Q}}}_0^+}^{d\times d}$ with non-negative entries and the *classical measurement occurrence problem (CMOP)* are decidable.
In order to prove Theorem \[thm:decidabiliy\] we introduce some notation first. For an elementwise non-negative matrix $M$ we define the matrix $M'$ elementwise by $$\begin{aligned}
M'_{a,b}\coloneqq \begin{cases}
0 & \text{if } M_{a,b}=0,\\
1 & \text{if } M_{a,b}>0 .
\end{cases}\end{aligned}$$ For two such binary matrices $M',N'$ we define their associative binary matrix product by $M' * N'\coloneqq (M' N')'$. Note that $M_{j_1} \ldots M_{j_n} = 0$ if and only if $(M_{j_1} \ldots M_{j_n})' = 0$, which in turn holds if and only if $
M'_{j_1} * \ldots * M'_{j_n} = 0.
$ As all matrices in the semigroup $\cal S$ generated by $S=\{M'_1, \ldots , M'_K\}$ under the matrix multiplication $\ast$ are binary matrices, hence $|{\cal S}| \leq 2^{(d^2)}$. We finish the proof by arguing that every element $M'$ of $\cal S$ can be written in terms of at most $|\cal S|$ elements from $S$. Fix some $M'$ and let $j_1,\ldots,j_m$ be the shortest sequence of indices such that $M'= M'_{j_m}\ast\ldots\ast M'_{j_1}$. Then for all $k < l \leq m$ we have $M'_{j_l}\ast \ldots \ast M'_{j_1} \neq M'_{j_k}\ast \ldots \ast M'_{j_1}$, because otherwise we would obtain a shorter representation of $M$ by replacing the former product with the latter. Therefore, for each $l\leq m$ the product $M'_{j_l}\ast \ldots \ast M'_{j_1}$ yields a different elements of $\cal S$ and hence $m\leq |{\cal S}| \leq 2^{(d^2)}$.
#### Outlook and implications for quantum many-body problems.
We have seen in this work that very natural decision problems in quantum measurement theory can be undecidable, even if their classical counterparts are decidable. In the specific problem that we considered (quantum measurement occurrence problem), the existence of negative transition matrix elements renders the quantum problem more complex than its classical counterpart – that is, the effect of *destructive interference*. We conclude by a number of further comments:
Firstly, note that mild variants of the above problem can easily lead to problems that have efficient solutions. For example, if one considers trace-preserving quantum channels, one can give upper bounds to the number of times a channel must be applied, so that it maps any density operator to one with full rank, by virtue of the quantum Wielandt theorem [@Sanz]. Thus, the problem whether there is some $n$ such that the $n$-fold application of a nonselective channel yields nonzero probabilities, for all subsequent measurements and for all inputs, is efficiently decidable.
Second, the above statement has immediate implications to undecidability in quantum many-body physics [@MBP] and quantum computing. Interpreting the above matrices $\{A_1,\dots, A_K \}$ as those defining matrix-product states [@Area; @Gross; @MPS], several other natural undecidable problems open up.
As an example, consider a family of one-dimensional quantum wires for *measurement-based quantum computing* in the sense of Refs. [@Gross]. These wires are described by families of matrix-product states of length $n$, being defined by products of matrices $\{A_1,\dots, A_K\}$ (the same set of matrices is taken for each site), associated with measurement outcomes $1,\dots, K$ in the computational basis. The left and right boundary conditions are fixed as $|L\rangle=|R\rangle= [1 \, 0 \,\dots 0]^T$. The task is to determine whether there exists a sequence of measurement outcomes $j_1,\dots, j_n$ that will never occur [@MPSFoot]. The subsequent result is a consequence of the above reasoning, together with the fact that the problem whether the semigroup generated by integer matrices contains a matrix with a zero element in the left upper corner is undecidable [@New].
Given a description of a family of matrix-product states defined by the matrices $\{A_1,\dots, A_K\}\subset {{\mathbbm{Q}}}^{d\times d}$, the task is to decide whether there exists an $n$ and a sequence of outcomes $j_1,\ldots,j_n$ for a wire of length $n$ of local measurements in the computational basis that will never be observed. This problem is undecidable.
Similar reasoning as in the proof of the undecidability of the quantum measurement occurrence problem suggests that other questions concerning the characterization of measurement outcomes are undecidable as well. These observations indicate that undecidability may be a natural and frequent phenomenon in many-body quantum physics and computation.
Similarly interestingly, a number of problems in quantum information theory seem to be natural candidates for being potentially undecidable. This applies notably to the problem of deciding whether a quantum state is distillable, giving a new perspective to the notorious question of deciding whether bound entangled states with a negative partial transposition exist.
*Note added.* Compare also the recent related independent work Ref. [@WolfCubittPerezGarcia].
#### Acknowledgments.
We would like to thank M. Kliesch for discussions and the EU (QESSENCE, MINOS, COMPAS), the German National Academic Foundation, the BMBF (QuOReP), the Government of Canada (Industry Canada), the Province of Ontario (Ministry of Research and Innovation), and a EURYI for support.
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Appendix {#appendix .unnumbered}
========
For the readers convenience we very briefly sketch the elements of the argument relating the MMP to the PCP. We consider the PCP over the two alphabets $\Sigma$ and $\Delta$, where $\Sigma$ is arbitrary and $\Delta=\{2,3\}$. Even though $\Delta$ is fixed, this version of the PCP is still undecidable [@Halava]. In order to relate this problem to a matrix problem, set $\Gamma\coloneqq\{1,2,3\}$ and consider the map $f:\Gamma^\ast \rightarrow {{\mathbbm{N}}}$ defined as $$f(w) = \sum_{j=1}^{|w|} w_j 3^{|w|-j}$$ for all nonempty words $w$ over $\Gamma$, where $|w|$ denotes the length of $w$. $f(w)$ is the $3$-adic representation of $w$. Now continue to define the function $F: \Gamma^\ast \times \Gamma^\ast\rightarrow {{\mathbbm{N}}}^{3\times 3}$ as $$F(u,v) = \left[
\begin{array}{ccc}
1 & 0 & 1\\
1 & 1 & 0\\
0 & 0 & 1\\
\end{array}
\right]
\left[
\begin{array}{ccc}
3^{|u|} & 0 & 0\\
0 & 3^{|v|} & 0\\
f(u) & f(v) & 1\\
\end{array}
\right]
\left[
\begin{array}{ccc}
1 & 0 & 1\\
1 & 1 & 0\\
0 & 0 & 1\\
\end{array}
\right]^{-1}.$$ Let now $(h,g)$ be an instance of PCP, $h,g:\Sigma^\ast\rightarrow \Delta^\ast$. For each such instance, define the $3\times 3$-matrices $$X_a= F(h(a),g(a)),\,
Y_a= F(h(a),1 g(a))$$ for $a\in \Sigma$. Let $S$ be the matrix semigroup generated by $\{X_w,Y_w: w\in \Sigma\}$. One then continues to consider matrix products $$M= M_{w_1}\dots M_{w_n}\in S
\label{eqProduct}$$ for a given word $w=w_1\dots w_n$, where $M_{w_j}= X_{w_j}$ or $M_{w_j}= Y_{w_j}$. The key step of the proof of Ref. [@Halava], deriving from the encoding of Ref. [@Paterson], is to show that $M_{1,1}=0$ (denoting the upper left element of the matrix) holds true if and only if $w$ is a solution of the instance $(h,g)$. This shows that the problem to decide whether the semigroup contains an element the upper left element of which is zero is undecidable. By adding the idempotent matrix $$B= \left[
\begin{array}{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}
\right]$$ as an additional generator to the set of matrices $\{X_w,Y_w: w\in \Sigma\}$, it is then a simple step to reduce the MMP to the PCP. In Ref. [@Halava], it is shown that we may choose $|\Sigma|=7$ and specific forms of the product , which gives a count of exactly $8$ matrix generators.
|
---
abstract: 'There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph $H$ is its extremal density, as an induced subgraph of $G$, where $|G| \rightarrow \infty$. Already for $4$-vertex graphs many questions are still open. Thus, the inducibility of the $4$-path was addressed in a construction of Exoo (1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (1997) constructed graphs with a small density of both $4$-cliques and $4$-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.'
author:
- 'Chaim Even-Zohar[^1]'
- 'Nati Linial[^2]'
title: 'A Note on the Inducibility of $4$-vertex Graphs'
---
Introduction {#introsect}
============
Let $H$ and $G$ be simple graphs. Denote by $P(H,G)$ the proportion of $H$ in $G$, i.e., the number of induced copies of $H$ in $G$, divided by [$\binom{|G|}{|H|}$]{}. More generally, consider the *local $t$-profile* of $G$, the vector $\mathbf{P}_t(G) = \{P(H,G)\}_H$, where $H$ runs over all isomorphism types of $t$-vertex graphs.
It is a major challenge to understand the limit points of $\mathbf{P}_t(G)$ as $|G| \rightarrow \infty$, since in full generality this question includes large portions of extremal graph theory. An important example is the study of graph inducibility, that was started by Pippenger and Golumbic [@pippenger1975]. To the best of our knowledge, the general concept of a local profile was essentially first considered in [@erdos1979]. More recently, the study of graph limits [@lovasz2012] has brought to the fore the significance of local $t$-profiles of large graphs. The relevance of these concepts in the computational realm is illustrated by the study of graph property testing [@goldreich1998]. A key advance that enabled much of the recent progress in this area is Razborov’s theory of flag algebras [@razborov2007; @razborov2013]. Indeed, there is substantial recent activity in this domain [@huang2014; @huang2012; @linial2014graphs], and additional combinatorial structures with natural notions of local profile and inducibility are being investigated as well, e.g. tournaments [@linial2014tournaments] trees [@bubeck2013], and permutations [@wilf2002].
The *(maximal) inducibility* of a graph $H$ is $$I(H) = \lim\limits_{n \rightarrow \infty} \max\limits_{|G|=n} P(H,G).$$ An averaging argument shows that the sequence decreasingly converges to a limit, and the implicit error term is bounded by $|H|^2/|G|$ ([@pippenger1975], Theorem $3$). We briefly review a few facts and figures about inducibility.
The inducibility of some graphs is known precisely. Clearly cliques and anticliques have inducibility $1$. In general, by passing from $G$ to its complement $G^C$ one can easily obtain $I(H) = I(H^C)$. The inducibility of complete partite graphs has been considered in the literature [@pippenger1975; @exoo1986; @bollobas1986; @brown1994; @bollobas1995]. For example, the inducibility of the complete bipartite graph $K_{t,t}$, is $I(K_{t,t}) = \binom{2t}{t}/4^t \approx 1/\sqrt{\pi t}$, as attained by larger bipartite graphs.
Every $t$-vertex graph $H$ has inducibility $\ge t!/(t^t-t) \approx \sqrt{2 \pi t} \exp(-t)$, as shown by a nested blow-up of $H$ ([@pippenger1975], see Section \[constsect\]). Sometimes, one can exploit symmetries of $H$, and adjust this construction to yield better lower bounds [@siran1984]. However, generically the bound $\exp(-t)$ is nearly tight. Almost every $H$ contains a set $S$ of about $3\log t$ vertices that “separates” between the rest of the vertices. Namely for every two vertices $x, y \in V(H) \setminus S$ there holds $\Gamma(x)\cap S\neq\Gamma(y)\cap S$, where $\Gamma(v)$ is $v$’s set of neighbors. This implies that $I(H) \leq \exp( - t + O(\log^2 t))$.
Can we determine the exact inducibility of small graphs? For $3$ vertices, we only need to consider $K_{1,2}$ or its complement. Large bipartite graphs yield $I(K_{1,2}) \geq 3/4$, which is optimal by the following classic result.
In a graph with $2n$ vertices, at least $2\binom{n}{3}$ triples form a triangle or an anti-triangle.
There are eleven isomorphism types of $4$-vertex graphs, the inducibilities of ten of which are known, and summarized in Table \[exootable\], an updated version of Exoo’s [@exoo1986]. Some of these numbers follow from general results concerning complete bipartite graphs [@pippenger1975; @bollobas1986; @brown1994; @bollobas1995]. For others, various extremal constructions were found, and their optimality was proved using flag algebra [@hirst2011].
[l l l l c c l]{} $H$ & & $H^C$ & & $I(H)$ & & Extremal Construction\
$K_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& $A_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& 1 & & A complete graph\
$S_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
$\;\;\;\;$& $T_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
$\;\;\;\;$& 1/2 & & A complete bipartite graph\
$C_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& $M_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& 3/8 & & A complete bipartite graph\
$V_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& $Q_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& 3/8 & & Two disjoint complete bipartite graphs\
$D_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& $E_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& 72/125 & & A complete $5$-partite graph\
$P_4$ &
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt\]
at(-0.2,0.2)(0); at(-0.2,-0.2)(1); at (0.2,0.2)(2); at (0.2,-0.2)(3); ;
& & & ? & & Unknown\
The case of $I(P_4)$ is intriguing. Exoo’s construction gives a lower bound of $60/307 \approx 0.1954$ (See Section \[constsect\]). We comment that [@exoo1986] and [@hirst2011] state the inaccurate bound $960/4877 \approx 0.1968$, presumably by mistake. Upper bounds on $I(P_4)$ were first given by Exoo [@exoo1986], then by Hirst [@hirst2011], and finally Vaughan [@flagmatic] set the current record at $0.204513$, using flag algebra calculus. Here we present a new construction, which implies:
\[IP4\] $ I(P_4) \geq 1173/5824 \approx 0.2014$
We proceed with some additional definitions. Inducibility naturally extends to formal linear combinations of graphs, also known as *quantum graphs*. Namely, we interpret $P(H+ \nobreak H',G)$ as $P(H,G) + P(H',G)$, and define the inducibility $I(H+H')$ accordingly. The *minimal inducibility* can be defined as the former, $\max$ replaced by $\min$: $$i(H) = \lim\limits_{n \rightarrow \infty} \min\limits_{|G|=n} P(H,G) .$$
By Ramsey’s theorem a monochromatic $K_t$ exists in every red/blue edge-coloring of a large enough $K_n$. Goodman [@goodman1959] and Erdős [@erdos1962] asked how many monochromatic $K_t$-s such a coloring must contain. The asymptotic nature of this quantity is captured in the minimal inducibility $i(K_t+A_t)$, where $A_t = K_t^C$ is an anticlique on $t$ vertices. By the above theorem of Goodman, $i(K_3+A_3) = 1/4$. Interestingly, this is attained not only by bipartite graphs but also by random ones. Using Ramsey numbers on the one hand and random graphs on the other, Erdős [@erdos1962] showed $$\frac{1}{\binom{\binom{2t-2}{t-1}}{t}}
\;\leq\; i(K_t+A_t) \;\leq\; \frac{2}{2^{\binom{t}{2}}} \;,$$ and added that the upper bound seems likely to be tight.
This conjecture was refuted for all $t \geq 4$ by Thomason ([@thomason1989], see also [@jagger1996; @thomason1997]). For $t= \nobreak 4$, Thomason showed that $i(K_4+A_4)$ is *strictly* smaller than $3769/124416 \approx 1/33.0103$. His construction will be described below. Another family of counterexamples for $4 \leq t \leq 8$ was started by Franek and Rödl [@franek1993; @franek2002; @deza2012; @shen2012] (see Section \[discsect\]). Lower bounds on $i(K_4+A_4)$ were obtained by several authors [@giraud1979; @evans1981; @wolf2010; @sperfeld2011; @niess2012; @flagmatic], with Vaughan’s $0.0294343 \approx 1/33.9739$ being the current record. Here we slightly improve Thomason’s bound:
\[iK4A4\] $ i(K_4+A_4) \leq 1411/46592 \approx 1/33.0205$
*Overview:* Propositions \[IP4\] and \[iK4A4\] are proved by construction of graph sequences. In the next section we survey several constructive tools, which provide the framework for the subsequent presentation of the graphs. Then we prove in Section \[compsect\], that these sequences indeed approach the densities prescribed in the propositions. In Section \[discsect\] we discuss the inducibility of longer paths, larger monochromatic cliques and all $5$-vertex graphs.
Construction {#constsect}
============
We start with a technical reduction. Let $\mathbf{P}_t(G)$ be the $t$-profile of a graph $G$. The probability that $t$ random vertices induce a copy of $H$ is $P(H,G)$. It is useful to define a similar number, $R(H,G)$, the same probability when the $t$ vertices are sampled with replacements. The vector $\mathbf{R}_t(G) = \{R(H,G)\}_H$ is called the *repetitive $t$-profile* of $G$. Note that $R(H,G)$ may depend on whether there is an edge from some vertex of $G$ to itself. Therefore, mention of $R(H,G)$ will hereafter imply that $G$ allows loops. Note also that $R(H,G)$ can replace $P(H,G)$ in the definition of inducibility, because in large graphs repeated sampling of vertices gets rare.
Several extremal constructions for inducibility fall under the following simple definition. Let $G$ be a graph, with or without loops. A *blow-up* of $G$ of order $m$, is a graph on the vertex set $V(G) \times \{1,...,m\}$, with edges given by $$(g,h) \sim (g',h') \;\;\;\;\; \Leftrightarrow \;\;\;\;\; g \sim g' \;.$$ For example, a complete graph is a blow-up of a loop and a complete bipartite graph is a blow-up of an edge. The reader may verify that if $G'$ is a blow-up of $G$ then $\mathbf{R}_t(G') = \mathbf{R}_t(G)$ for all $t$. Thus a sequence of blow-ups of $G$ yields,
\[qlemma\] For every two graphs $G, H$ $$i(H) \leq R(H,G) \leq I(H) \;.$$
Each of the exactly known inducibilities in Table \[exootable\] is attained by a sequence of blow-ups of some graph, but as the next lemma shows this is not always the case. A graph is called *twin-free* if no two vertices have the same set of neighbors among the other vertices. Of course, most graphs are twin-free. An example of relevance here is the $4$-path $P_4$.
\[twins\] For every twin-free graph $H$ and for every graph $G$, $$R(H,G) < I(H) \;.$$
Let $G'$ be a blow-up of $G$ of order $t = |H|$. If $U = \{(u,1),...,(u,t)\}$ is the blow-up of a vertex $u$ of $G$, then $U$ induces either a clique or an anticlique in $G'$. No copy of $H$ in $G'$ can have more than one vertex from $U$, since $H$ is twin-free and every two vertices in $U$ are twins in $G'$. We next define $G''$: Start from $G'$ and modify the edges among $U$’s vertices to create there a new copy of $H$. Note that this cannot jeopardize the existing induced copies of $H$. Therefore $R(H,G) = R(H,G') < R(H,G'') \leq I(H)$ as required.
See [@hatami2011] for conditions under which inducibility is attained by blow-ups.
Lemma \[twins\]’s proof naturally leads to the following construction, by Pippenger and Golumbic [@pippenger1975]. The *composition* of two graphs $G$ and $H$, which we denote $G \odot H$, is essentially a blow-up of $G$, with a copy of $H$ corresponding to every vertex of $G$. Formally, $G \odot H$ is a graph on $V(G) \times V(H)$, with edges $$(g,h) \sim (g',h') \;\;\;\;\; \Leftrightarrow \;\;\;\;\;
(g \sim g') \text{ or } (g = g' \text{ and } h \sim h') \;.$$ Note that this operation is non-commutative but associative. We use the shorthand $G^{\odot n}$ for the iterated composition of $n$ copies of $G$. The sequence $\left\{ G^{\odot n} \right\}_{n=1}^{\infty}$ is called the *nested blow-up* of $G$.
The nested blow-up of a graph $H$ yields the above-mentioned lower bound $I(H) \geq t!/(t^t-\nobreak t)$, where $t=|H|$. For $H = C_t$ $(t \geq 5)$, Pippenger and Golumbic showed that this bound is tight up to a constant factor, and conjectured it to be sharp. For example, by nested blow-up $I(C_5) \geq\nobreak 5!/(5^5- \nobreak 5) = 1/26 \approx 0.03846154$, rather than $5!/5^5 = 0.0384$ obtained by regular blow-ups. Flag algebra calculus yields a different but close upper bound of $0.03846157$ [@flagmatic].
Exoo [@exoo1986] noticed that sometimes the nested blow-up of some $H' \neq H$ beats the nested blow-up of $H$. Indeed, suppose that a $t$-vertex graph $H$ is obtained from a vertex-transitive graph $H'$ by removal of a single vertex. In this case, the nested blow-up of $H'$ gives $$I(H) \;\geq\; \frac{t!}{(t+1)^{t-1}-1}
\;\approx\; \sqrt{2\pi} \cdot (t+1)^{3/2}e^{-t-1} ,$$ which is superior to the above bound by about $(t+1)/e$. In particular, taking $H'=C_5$ he obtained $I(P_4) \geq 6/31 \approx 0.1935$. By computer search he found an even better candidate, the Paley graph $Q_{17}$ in which two elements of the finite field $\F_{17}$ are neighbors iff their difference is a square. The nested blow-up of $Q_{17}$ yields $I(P_4) \geq 60/307 \approx 0.1954$.
We turn to describe the other main building block, which is due to Thomason [@thomason1997]. Let $G$ and $H$ be graphs, possibly with loops. Their *tensor product*, $G \otimes H$, has vertex set $V(G) \times V(H)$, where $$(g,h) \sim (g',h') \;\;\;\;\; \Leftrightarrow \;\;\;\;\;
(g \sim g' \text{ and } h \not \sim h') \text{ or }
(g \not \sim g' \text{ and } h \sim h') \;.$$ In particular, $G \otimes A_n$ is a blow-up of $G$. A less trivial example is $K_3 \otimes K_3$ which is isomorphic to the Paley graph of order $9$. Note that the tensor product is associative and commutative.
As Thomason observed, and we explain below, the $t$-profile of $G \otimes H$ can be easily computed from those of $G$ and $H$. This considerably simplified his original refutation of the Erdős conjecture. For the special case of $i(K_4 + A_4)$, he found plenty of further counterexamples. In particular, a computer investigation of products of small graphs yielded $$R(K_4 + A_4,\; M_4 \otimes K_4 \otimes K_3 \otimes K_3 ) \;=\;
\frac{11411}{373248} \;\approx\; \frac{1}{32.7095} \;.$$ Then, a broader search on larger Cayley graphs enhanced the bound to $$R(K_4 + A_4,\; M_4 \otimes K_4 \otimes G_{18} ) \;=\;
\frac{3769}{124416} \;\approx\; \frac{1}{33.0103} \;,$$ where we define $G_{18} = (K_3 \otimes K_3) \odot K_2$. A final remark in [@thomason1997], attributed to a referee, states that this construction is still not optimal. Replacing $K_2$ by a randomly perturbed blow-up of $K_2$ yields a slight improvement ($< 10^{-7}$).
The above chain of events makes it natural to ask, what would happen if $(K_3 \otimes K_3) \odot K_2$ were replaced by a nested blow-up of $K_3 \otimes K_3$. Indeed, as we explain in the next section $$R\left(K_4 + A_4,\; M_4 \otimes K_4 \otimes (K_3 \otimes K_3)^{\odot n} \right)
\;\xrightarrow{\;n\rightarrow\infty\;}\; \frac{1411}{46592} \;\approx\; \frac{1}{33.0205} \;.
\label{RK4A4}$$ Note that Proposition \[iK4A4\] follows from this by Lemma \[qlemma\]. In fact, the graph sequence that demonstrates Proposition \[IP4\] hides in the $4$-profiles along the way: $$I(P_4) \;\geq\; R\left(P_4,\; K_4 \otimes (K_3 \otimes K_3)^{\odot n} \right)
\;\xrightarrow{\;n\rightarrow\infty\;}\; \frac{1173}{5824} \;\approx\; 0.2014 \;.
\label{RP4}$$
Computation {#compsect}
===========
We next develop general tools, which may also provide some insight on the construction. We begin with a slightly different formulation of Thomason’s analysis from [@thomason1997].
Recall that $\mathbf{R}_t(G)$ and $\mathbf{P}_t(G)$ are the distributions of induced graphs on $t$ random vertices, sampled with and without replacements, respectively. It is convenient to define the *labeled $t$-profile*, which accounts also for the ordering of the sample. For a labeled $t$-vertex graph $H$ we define $r(H,G) = R(H,G)/|orbit(H)|$, where $orbit(H)$ corresponds to the action of the symmetric group $S_t$ through relabeling of the vertices. Equivalently $R(H,G) = r(H,G) \cdot [S_t:Aut(H)]$, where $Aut(H)$ is the group of automorphisms of $H$. To complete the picture, we similarly define $p(H,G) = P(H,G)/|orbit(H)|$ in the non-repetitive case.
The labeled graphs of order $t$ admit a natural group structure, with the operation of symmetric difference of the edges. One can apply the discrete Fourier transform over this group, and obtain $\mathbf{\hat{r}}_t(G)$, the *spectral $t$-profile*: $$\hat{r}(H,G) = \sum\limits_{H'}(-1)^{e(H \cap H')} \cdot r(H',G)$$ The summation is over all $2^{\binom{t}{2}}$ labeled graphs with $t$ vertices, and $e(H \cap H')$ is the number of edges which appear in both $H$ and $H'$. Observe that $\hat{r}(A_t,G) = 1$ for every $G$. Note also that $\hat{r}(H,G)$ is constant on classes of graph isomorphism, exactly like $r(H,G)$. Therefore we can refer to, e.g., $\hat{r}(C_4,G)$ without specifying labels on the vertices of $C_4$.
The basic observation about the tensor product is that sampling vertices in $G \otimes G'$ can be separated, in a sense, to sampling each factor independently. More precisely, if the $G$ components of the sample create $H$ and the $G'$ components create $H'$, then the sampled graph is the symmetric difference $H \triangle H'$. Consequently, $$\mathbf{r}_t(G \otimes G') \;=\; \mathbf{r}_t(G) \ast \mathbf{r}_t(G')$$ where $\ast$ stands for convolution over the group of labeled graphs. From this immediately follows,
\[pointwise\] $$\hat{r}(H,\;G \otimes G') \;=\; \hat{r}(H,G) \; \hat{r}(H,G')
\label{pointwisewise}$$
We are now in position to explain Thomason’s method. The spectral $4$-profiles of many tensor products of small graphs were computed fairly easily by Corollary \[pointwise\]. By the inverse transform, $r(K_t + A_t, G)$ is the average of $\mathbf{\hat{r}}_t(G)$ over all graphs of even number of edges. If $t=4$ then this amounts to the following linear functional: $$r(K_4 + A_4, G) = \frac{
1 + \hat{r}(K_4,G) + 3\hat{r}(M_4,G) + 3\hat{r}(C_4,G) + 12\hat{r}(Q_4,G) + 12\hat{r}(V_4,G)
}{32} \;.
\label{funcK4A4}$$ We are interested also in $$r(P_4, G) = \frac{
1 - \hat{r}(K_4,G) + \hat{r}(M_4,G) - \hat{r}(C_4,G) + 4\hat{r}(Q_4,G) - 4\hat{r}(V_4,G)
}{64} \;.
\label{funcP4}$$
The following lemma applies a similar line of reasoning, in order to find the $t$-profile of the composition of graphs.
\[bilinear\] Let $s,t \geq 2$, and let $G$ and $G'$ be two graphs where $|G| = s$. Then, there exists a bilinear operator $B_{s,t}$ such that $$\mathbf{r}_t(G \odot G') \;=\; B_{s,t}(\mathbf{p}_m(G),\mathbf{r}_t(G'))$$ where $m = \min(s,t)$.
The proof is a matter of straightforward computation of $B_{s,t}$, which we shortly spell out for completeness of exposition and for future reference. All the necessary terminology is developed, but some easy verifications are left to the reader.
Suppose that $H$ is a labeled graph on the vertex set $V = V(H)$. The induced labeled graph on a vertex subset $V' \subseteq V$ is denoted $H[V']$. In the other direction, if $H'$ is a labeled graph on some $V' \subseteq V$, then we denote $\Gamma(H',V) = \{H : V(H)=V,\;H[V'] = H'\}$. The following observation is obvious: $$p(H',G) \;=\; \sum\limits_{H \in \Gamma(H',V)} p(H,G) \;.
\label{ppp}$$ In other words, for every $j < m$ the $j$-profile $\mathbf{p}_j(G)$ is given by a linear projection of $\mathbf{p}_m(G)$ on $j$ fixed vertices. This relation will permit us some flexibility when we work with the first argument of $B_{s,t}$.
Let $\Lambda(V)$ be the collection of all partitions of $V$ into disjoint sets. Given a partition $\lambda \in \Lambda(V)$, we denote the number of the parts by $\ell = \ell(\lambda)$, and arbitrarily fix their indexes: $ V = \lambda_1 \cup \lambda_2 \cup ... \cup \lambda_{\ell} $. By abuse of notation, we write $ \Gamma(H,\lambda) = \{H' : V(H')=V(H), \;\forall i\; H'[\lambda_i] = H[\lambda_i] \} $. For example, if $\lambda = \{V\}$, the trivial one-part partition, then $\Gamma(H,\lambda) = \{H\}$, while the other trivial partition into singletons $\lambda = \{\{v\}:v\in V\}$ yields all graphs on $V$.
The *transversals* of $\lambda$ are the ordered sets of representatives, one from each part: $ \text{Tr}(\lambda) \;=\;
\left\{ (v_1,v_2,...,v_{\ell}) : \forall i\; v_i \in \lambda_i \right\} $. A partition $\lambda \in \Lambda(V)$ is said to be *admissible* with respect to $H$ if $H[V'] = H[V'']$ for every $V',V'' \in \text{Tr}(\lambda)$, where $V'$ and $V''$ are naturally identified according to the partition. For example, the two above-mentioned trivial partitions are $H$-admissible for every $H$. The reader may verify that if $H=P_4$ then no other partition is admissible. The set of all $H$-admissible partitions in denoted $\Lambda(H)$. By further abuse of notation, if $\lambda \in \Lambda(H)$ we define $H[\lambda] = H[V']$ for some $V' \in \text{Tr}(\lambda)$.
Let $G$ and $G'$ be as in the lemma, and let $H$ be a labeled graph on $t$ vertices. We carry out the counting of induced $H$-s in $G \odot G'$, by applying the law of total probability over the possible partitions of $V(H)$ according to the $G$ component. The reader may verify that this leads to an expression of the following form: $$r(H,\;G \odot G') \;=\;
\sum\limits_{\lambda \in \Lambda(H)} \frac{(s)_{\ell(\lambda)}}{s^t} \cdot
p(H[\lambda], G) \sum\limits_{H' \in \Gamma(H, \lambda)} r(H',G') \;.
\label{rrr}$$ Recall that the falling factorial $(s)_\ell = s(s-1)(s-2)...(s-\ell+1)$ is zero if $\ell > s$. Together with (\[ppp\]), this implies that $r(H,\;G \odot G')$ is bilinear in $\mathbf{p}_{\min(s,t)}(G)$ and $\mathbf{r}_t(G')$, as required.
Let $G'=K_1$, the graph with a single vertex and no edges, and suppose that $G$ doesn’t contain loops. Then Lemma \[bilinear\] establishes a useful relation between $\mathbf{r}_t(G)$ and $\mathbf{p}_t(G)$, the profiles with and without replacements. The inner sum in (\[rrr\]) reduces, in this case, to $0$ or $1$.
Consider the linear map $f_t(G)$, defined by $f_t(G)\mathbf{v} = B_{s,t}(\mathbf{p}_m(G),\mathbf{v})$. Note that $f_t(G)$ is represented by a stochastic matrix, and in non-degenerate cases it has a single stationary state vector $\mathbf{q}_t(G)$, such that $f_t(G) \mathbf{q}_t(G) = \mathbf{q}_t(G)$. This implies $$\mathbf{q}_t(G) \;=\;
\lim\limits_{n \rightarrow \infty} \mathbf{r}_t\left(G^{\odot n}\right)\;.
\label{lim}$$ Thus $\mathbf{q}_t(G)$ is the limiting $t$-profile of the nested blow-up of $G$, or shortly the *nested $t$-profile* of $G$. Finding $\mathbf{q}_t(G)$ reduces to computing the Perron-Frobenius eigenvector of $f_t(G)$. We also define an unlabeled variant, $Q(H,G) = q(H,G) \cdot |orbit(H)|$.
We now demonstrate the above notions on the graph $G = K_3 \otimes K_3$. Here Lemma \[bilinear\] reads $\mathbf{r}_4(G \odot\nobreak G') = B_{9,4}(\mathbf{p}_4(G),\mathbf{r}_4(G'))$, which lets us compute $f_4(G)$ using (\[rrr\]). Instead of $\mathbf{q}_4(G)$, we prefer to consider the equivalent unlabeled nested $4$-profile $\mathbf{Q}_4(G)$, which is the eigenvector of another matrix $F_4(G)$, easily derived from $f_4(G)$. This reduces the order of the matrix from $64$ to $11$, which enables us to explicitly record it here. The ordering of the basis is fixed to $ K_4, A_4, T_4, S_4, M_4, C_4, Q_4, V_4, D_4, E_4, P_4 $ (See Table \[exootable\]). $$F_4(K_3 \otimes K_3) \;=\; \frac{1}{729} \left( \begin{array}{ccccccccccc}
53 & 0 & 16 & 12 & 12 & 24 & 24 & 8 & 36 & 4 & 16 \\
0 & 53 & 12 & 16 & 24 & 12 & 8 & 24 & 4 & 36 & 16 \\
112 & 0 & 53 & 48 & 32 & 64 & 68 & 32 & 88 & 16 & 48 \\
0 & 112 & 48 & 53 & 64 & 32 & 32 & 68 & 16 & 88 & 48 \\
84 & 24 & 48 & 48 & 45 & 64 & 60 & 40 & 72 & 32 & 52 \\
24 & 84 & 48 & 48 & 64 & 45 & 40 & 60 & 32 & 72 & 52 \\
192 & 96 & 156 & 144 & 144 & 160 & 165 & 136 & 176 & 120 & 152 \\
96 & 192 & 144 & 156 & 160 & 144 & 136 & 165 & 120 & 176 & 152 \\
48 & 24 & 48 & 60 & 32 & 56 & 56 & 44 & 57 & 32 & 48 \\
24 & 48 & 60 & 48 & 56 & 32 & 44 & 56 & 32 & 57 & 48 \\
96 & 96 & 96 & 96 & 96 & 96 & 96 & 96 & 96 & 96 & 97
\end{array} \right)$$ which implies $$\mathbf{Q}_4(K_3 \otimes K_3) \;=\; \frac{1}{728} \left( \begin{array}{ccccccccccc}
17 & 17 & 50 & 50 & 51 & 51 & 150 & 150 & 48 & 48 & 96
\end{array} \right)^T\;.$$ We now proceed to the spectrum. Following Thomason, we write down the relevant Fourier coefficients of each factor of the tensor products in question.
[l\*[5]{}[c]{}]{} $H$ & $K_4$ & $M_4$ & $C_4$ & $Q_4$ & $V_4$\
$\hat{r}(H,K_4)$ & -1/2 & 1/4 & 1/4 & -1/8 & 1/4\
$\hat{r}(H,M_4)$ & 1/2 & 1/4 & 1/4 & 1/8 & 1/4\
$\hat{q}(H,K_3 \otimes K_3)$& 18/91 & 0 & 9/91 & 0 & 0\
The desired limits in (\[RK4A4\]) and (\[RP4\]) then follow by plugging these numbers into (\[funcK4A4\]) and (\[funcP4\]) by means of (\[pointwisewise\]) and (\[lim\]).
Discussion {#discsect}
==========
Propositions \[IP4\] and \[iK4A4\] run new candidates for $I(P_4)$ and $i(K_4+A_4)$ respectively. The constructed graphs have many symmetries, and in particular are vertex-transitive. It is unclear whether an extremal construction for these problems has to be symmetric at all. The authors have no particular reason to expect these graphs to be best possible, even within the capability of the methods in use. In this section we give a brief account of the state of these questions for graphs with more than $4$ vertices, which we believe to support our skepticism.
Razborov’s flag algebra calculus [@razborov2007; @razborov2013] is the most powerful currently available method for proving that some candidate construction is optimal. However, as discussed in Section $4$ of [@falgas2013], it is not so good in dealing with nested constructions like ours. This, too, can at least partly explain the current gaps between the bounds.
What is the inducibility of longer paths? Let $P_t$ be a path on $t \geq 5$ vertices. As shown by Exoo (see Section \[constsect\]), the nested blow up of $C_{t+1}$ yields $ I(P_t) \geq t!/((t+1)^{t-1}-1) $. An appropriate modification of the counting argument by Pippenger and Golumbic ([@pippenger1975], Theorem $9$) gives $I(P_t) \leq t!/2(t-1)^{t-1}$. In conclusion, the inducibility of the $t$-path is $\Theta\left(t^{3/2}\exp(-t)\right)$, and the ratio between the two bounds is asymptotically $e^2/2$.
As for monochromatic cliques, the asymptotic behavior of $i(K_t+A_t)$ remains an intriguing open problem, related to bounds on the Ramsey number $R(t,t)$. The best-known bounds for $t \geq 6$ are $$(2.18)^{-(1+o(1))t^2} \;\leq\; i(K_t+A_t) \;\leq\; 0.835 \cdot 2^{1-\binom{t}{2}}$$ by Conlon [@conlon2012] and Thomason [@jagger1996] respectively. Conlon conjectures that $2.18$ can be replaced with a lower constant, maybe even $\sqrt2$, and remarks that the Erdős conjecture may still be true within a constant factor. Rödl, however, conjectures that the upper bound can be improved at least exponentially in $t$ [@franek2002].
Let’s look closer at Thomason’s construction. In terms of the tensor product, it is given in [@thomason1997] by blow-ups of $G_t = K_4 \otimes M_4^{~\otimes~t-1}$. Note that $M_4$ and $K_4$ are Cayley Graphs of $\F_2^2$, whose sets of generators are characterized by the functions $m(x) = x_1x_2$ and $k(x) = x_1 + x_2 + x_1x_2$ respectively. Indeed, Thomason’s original representation of $G_t$ was as a Cayley graph of $\F_2^{2t}$, generated by the quadratic form $q(x) = (x_1 + x_2 + x_1x_2) + x_3x_4 + ... + x_{2t-1}x_{2t}$ over $\F_2$. Suppose that $t$ is not divisible by $4$. Then, by linear automorphisms of $\F_2^{2t}$, $q(x)$ is further equivalent to at least one of the following symmetric quadratic forms over $\F_2$: $$s_1(x) = \sum\limits_{1 \leq i \leq j \leq 2t} x_i x_j \;,\;\;\;\;\;\;\;\;\;\;\;
s_2(x) = \sum\limits_{1 \leq i < j \leq 2t} x_i x_j \;.$$ Note that $s_k(x)=1$ if and only if the Hamming weight of $x$ equals $k$ or $k+1$ mod $4$. This can simplify our choice of generators. For instance, $G_5$ can be generated by all vectors in $\F_2^{10}$ of Hamming weight $\in\nobreak\{1,2,5,6,9,10\}$.
Franek and Rödl [@franek1993] introduced further counter-examples to the Erdős conjecture, using other Cayley graphs of $\F_2^n$ which are generated by sets of Hamming distances. A computer search for such graphs provided the best-known upper bounds on $i(K_t+A_t)$ for $6 \leq t \leq 8$ [@franek2002; @deza2012; @shen2012]. For example, in $\F_2^{10}$ the weights set $\{0,2,5,6,9,10\}$ yields $i(K_6+A_6) \leq 0.74444 \cdot 2^{-14}$, improving $0.76414 \cdot 2^{-14}$ obtained by $G_5$. Note that the two graphs differ in about $1\%$ of the edges. In general, these new constructions can be interpreted as minor modifications of the ones derived from $s_1(x)$ and $s_2(x)$. It would be interesting to understand why this method generates better graphs, and to find how its performance grows with $t$. Meanwhile, it indicates that our graph construction toolbox is still far from sufficient.
We close this note with a quick glance at $5$-vertex inducibilities. Our current state of knowledge is collected in a table in Appendix \[app1\]. Lower and upper bounds on the inducibility of all $34$ graphs with $5$ vertices are listed there, together with a short description of the construction leading to the lower bound. The quoted upper bounds were generated by the excellent freeware *Flagmatic*, created by Vaughan [@flagmatic], which has reduced such calculations to typing one line on the computer. We used the method `GraphProblem(7,density=...).solve_sdp()`, but didn’t try to round the results to rational numbers. Consequently, the upper bounds should only be viewed as well-established conjectures.
Some trends emerge in Table \[exoo5table\]. In the first eight lines, the lower bound comes from blow-up of small graphs. The choice of the blown-up graph is rather natural in all these cases. Only in two of the cases are the upper and the lower bounds different, and then, too, the difference is quite small. In this view we suspect that the lower bound is the correct value.
The next five constructions are again either tight or at least plausibly so. They each consist of two disjoint blow-ups of the same small graph, with size-ratios optimized to $\alpha = 2+\sqrt{3}$. This number comes up as the ratio $p:q$ that maximizes $pq^4+p^4q$ subject to $p+q=1$. In all five cases the target graph is a connected graph plus an isolated vertex. Moreover, the graph that we duplicate and blow up is the best construction for the inducibility of the $4$-vertex component, as in Table \[exootable\]. The only exception is, again, $P_4$. It is plausible that such relations between constructions carry on in larger graphs.
For the cycle $C_5$, nested blow-up has been long conjectured, as discussed in Section \[constsect\].
The remaining four cases seem to exhibit more complex behavior. In two cases, our current constructions use random graphs. One of them is tight and in the other one there is still a considerable gap, which make us doubt its optimality. It would be interesting to explore the role of random constructions in the study of inducibility. The challenge of derandomization suggests itself as well. This may require the introduction of new machinery in the realm of constructions beyond blow-ups, nesting and products.
Our best constructions for the two last cases, the $5$-path and the self-complementary “bull” graph, combine nested blow-up with tensor products. This situation is similar to $P_4$ and $K_4+\nobreak A_4$, the heroes of this note. Thus we believe these cases to be relevant as well to the search for new interesting graph constructions.
The inducibility of 5-vertex graphs {#app1}
===================================
[c c l l l l]{} $H,\;\overline{H}$ & orbit & FA bound & & Lower bound & Construction\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 1 & 1 & = & 1 & $A_n$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 10 & 0.625 & = & 5/8 & $2 \times K_n$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 15 & 0.37037037 & = & 10/27 & $3 \times K_n$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 10 & 0.51367669 & $\geq$ & 0.5126953125 & $8 \times K_n$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 30 & 0.27777778 & = & 5/18 & $2 \times K_{n,n,n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 60 & 0.192 & = & 24/125 & $C_5 \otimes A_n$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 10 & 0.3456 & = & 216/625 & $K_{3n} \cup A_{2n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 30 & 0.27886864 & $\geq$ & 0.2784 & $K_{n,n} \cup K_{2n,2n} \cup K_{2n,2n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 5 & 0.41666667 & = & 5/12 & $K_n \cup K_{\alpha n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 20 & 0.20833333 & = & 5/24 & $K_{n,n} \cup K_{\alpha n,\alpha n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 15 & 0.15625 & = & 5/32 & $K_{n,n} \cup K_{\alpha n,\alpha n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 60 & 0.1562855 & $\geq$ & 0.15625 & $(2 \times K_{n,n})^C \cup (2 \times K_{\alpha n,\alpha n})^C$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 30 & 0.24001625 & $\geq$ & 0.24 & $K_{n,n,n,n,n} \cup K_{\alpha n,\alpha n,\alpha n,\alpha n,\alpha n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 12 & 0.038462591 & $\geq$ & 1/26 $\approx$ 0.0384615 & $C_5^{\odot n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 60 & 0.25117348 & = & 15625/62208 & $G(n,n,5/6)$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 60 & 0.14470304 & $\geq$ & 0.133413966 & $G(n,0.3)$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 60 & 0.095475179 & $\geq$ & 1968/20995 $\approx$ 0.09373 & $(K_3 \otimes K_3 \otimes K_2)^{\odot n}$\
=\[circle, draw, fill=black!50, inner sep=0pt, minimum width=3pt\]
at (-0.2,-0.25)(1); at (-0.2, 0.05)(2); at ( 0.0, 0.25)(3); at ( 0.2, 0.05)(4); at ( 0.2,-0.25)(5); ;
& 60 & 0.077634203 & $\geq$ & 813/11111 $\approx$ 0.07317 & $(K_3 \otimes K_3)^{\odot n}$\
As usual, $K_n$ is a clique, $A_n$ is an anticlique, $C_n$ is an cycle and $K_{n,n,...}$ is a complete multi-partite graph. The graph products $\odot$ and $\otimes$ are as defined in Section \[constsect\]. The disjoint union of $G$ and $H$ is denoted $G \cup H$, and $k \times G$ is the union of $k$ copies of $G$, i.e., $A_k \odot G$. Let $\alpha = 2+\sqrt{3}$, where “$\alpha n$” should be rounded to the closest integer. The graph $G^C$ is the complement to $G$. $G(n,p)$ is the Erdős-Rényi random graph, with $n$ vertices and edge-probability $p$, and $G(n,m,p)$ is, similarly, a random bipartite graph.
[^1]: Department of Mathematics, Hebrew University, Jerusalem 91904, Israel. e-mail: <chaim.evenzohar@mail.huji.ac.il> .
[^2]: Department of Computer Science, Hebrew University, Jerusalem 91904, Israel.e-mail: <nati@cs.huji.ac.il> . Supported by grants from the ERC and from the ISF.
|
---
abstract: 'A study on the effects of optical gain nonuniformly distributed in one-dimensional random systems is presented. It is demonstrated numerically that even without gain saturation and mode competition, the spatial nonuniformity of gain can cause dramatic and complicated changes to lasing modes. Lasing modes are decomposed in terms of the quasi modes of the passive system to monitor the changes. As the gain distribution changes gradually from uniform to nonuniform, the amount of mode mixing increases. Furthermore, we investigate new lasing modes created by nonuniform gain distributions. We find that new lasing modes may disappear together with existing lasing modes, thereby causing fluctuations in the local density of lasing states.'
author:
- Jonathan Andreasen
- Christian Vanneste
- Li Ge
- Hui Cao
title: Effects of Spatially Nonuniform Gain on Lasing Modes in Weakly Scattering Random Systems
---
Introduction
============
Lasing modes in random media behave quite differently depending on the scattering characteristics of the media [@caoLRM]. In the strongly scattering regime, lasing modes have a nearly one-to-one correspondence with the localized modes of the passive system [@VannestePRL01; @souk02]. Due to small mode volume, different localized modes may be selected for lasing through local pumping of the random medium [@caoprl00; @VannestePRL01]. The nature of lasing modes in weakly scattering open random systems (e.g., [@frolov; @ling01; @Mujumdar]) is still under discussion [@wiersman]. In systems which are diffusive on average, prelocalized modes may serve as lasing modes [@apalkov]. In general, however, the quasi modes of weakly scattering systems are very leaky. Hence, they exhibit a large amount of spatial and spectral overlap. For inhomogeneous dielectric systems with uniform gain distributions, even linear contributions from gain induced polarization bring about a coupling between quasi modes of the passive system [@deych05PRL]. Thus, lasing modes may be modified versions of the corresponding quasi modes. However, Vanneste *et al.* found that when considering uniformly distributed gain, the first lasing mode appears to correspond to a single quasi mode [@VannestePRL]. The study was done near the threshold pumping rate and nonlinear effects did not modify the modes significantly. Far above threshold, it was found that lasing modes consist of a collection of constant flux states [@tureciSci]. Mode mixing in this regime is largely determined by nonlinear effects from gain saturation.
Remaining near threshold, pumping a local spatial region, and including absorption outside the pumped region found lasing modes to differ significantly from the quasi modes of the passive system [@yamilovOL]. This change is attributed to a reduction of the effective system size. More surprisingly, recent experiments [@polson; @wu06] and numerical studies [@wu07] showed the spatial characteristics of lasing modes change significantly by local pumping even without absorption in the unpumped region. It is unclear how the lasing modes are changed in this case by local pumping. In this paper, we carry out a detailed study of random lasing modes in a weakly scattering system with a nonuniform spatial distribution of linear gain. Mode competition depends strongly on the gain material properties, e.g., homogeneous vs. inhomogeneous broadening of the gain spectrum. Ignoring gain saturation (usually responsible for mode competition) and absorption, we find that spatial nonuniformity of linear gain alone can cause mode mixing. We decompose lasing modes in terms of quasi modes and find them to be a superposition of quasi modes close in frequency. The more the gain distribution deviates from being uniform, more quasi modes contribute to a lasing mode.
Furthermore, still considering linear gain and no absorption outside the gain region, we find that some modes stop lasing no matter how high the gain is. We investigate how the lasing modes disappear and further investigate the properties of new lasing modes [@andreasenOL] that appear. The new lasing modes typically exist for specific distributions of gain and disappear as the distribution is further altered. They appear at various frequencies for several different gain distributions.
In Section \[sec:method\], we describe the numerical methods used to study the lasing modes of a one-dimensional (1D) random dielectric structure. The model of gain and a scheme for decomposing the lasing modes in terms of quasi modes is presented. A method to separate traveling wave and standing wave components from the total electric field is introduced. The results of our simulations are presented and discussed in Section \[sec:results\]. Our conclusions concerning the effects of nonuniform gain on lasing modes are given in Section \[sec:conclusion\].
Numerical Method\[sec:method\]
==============================
The 1D random system considered here is composed of $N$ layers. Dielectric layers with index of refraction $n>1$ alternate with air gaps ($n=1$) resulting in a spatially modulated index of refraction $n(x)$. The system is randomized by specifying different thicknesses for each of the layers as $d_{1,2} = \left<d_{1,2}\right>(1+\eta\zeta)$ where $\left<d_1\right>$ and $\left<d_2\right>$ are the average thicknesses of the layers, $0 < \eta < 1$ represents the degree of randomness, and $\zeta$ is a random number in (-1,1).
A numerical method based on the transfer matrix is used to calculate both the quasi modes and the lasing modes in a random structure. Electric fields on the left (right) side of the structure $p_0$ ($q_N$) and $q_0$ ($p_N$) travel toward and away from the structure, respectively. Propagation through the structure is calculated via the $2\times 2$ matrix $M$ $$\left(
\begin{array}{c}
p_N \\
q_N \\
\end{array}
\right)=
M
\left(
\begin{array}{c}
p_0 \\
q_0 \\
\end{array}
\right). \label{eq:transfermatrix}$$ The boundary conditions for outgoing fields only are $p_0=q_N=0$, requiring $M_{22}=0$.
For structures without gain, wavevectors must be complex in order to satisfy the boundary conditions. The field inside the structure is represented by $p(x)\exp[i n(x) \tilde{k} x] + q(x)\exp[-i n(x) \tilde{k} x]$, where $\tilde{k}=k+ik_i$ is the complex wavevector and $x$ is the spatial coordinate. For outgoing-only boundary conditions ($M_{22}=0$), $k_i < 0$. The resulting field distributions associated with the solutions for these boundary conditions are the quasi modes of the passive system.
Linear gain is simulated by appending an imaginary part to the dielectric function $\epsilon(x)=\epsilon_r(x)+i\epsilon_i(x)$, where $\epsilon_r(x)=n^2(x)$. This approximation is only valid at or below threshold [@souk99]. In Appendix \[ap:lineargain\], the complex index of refraction is calculated as $\tilde{n}(x) = \sqrt{\epsilon(x)} = n_r(x) + in_i$, where $n_i < 0$. We consider $n_i$ to be constant everywhere within the random system. This yields a gain length $l_g=1/k_i=1/n_ik$ ($k=2\pi/\lambda$ is the vacuum wavenumber of a lasing mode) which is the same in the dielectric layers and the air gaps. Note that this gain length is not to be confused with the length of the spatial gain region, which is described below. The real part of the index of refraction is modified by the imaginary part as $n_r(x)=\sqrt{n^2(x)+n_i^2}$. A real wavevector $k = 2\pi/\lambda$ describes the lasing frequency. The field inside the structure is now represented by $p(x)\exp[i \tilde{n}(x) k x] + q(x)\exp[-i \tilde{n}(x) k x]$. The frequencies and thresholds are located by determining which values of $k$ and $n_i$, respectively, satisfy $M_{22}=0$.
Nonuniform gain is introduced through an envelope function $f_E(x)$ multiplying $n_i$. The envelope considered here is the step function $f_E(x)=H(-x+l_G)$, where $x=0$ is the left edge of the random structure and $x=l_G$ is the location of the right edge of the gain region. $l_G$ may be chosen as any value between 0 and $L$.
The solutions of the system are given by the points at which the complex transfer matrix element $M_{22}=0$. Where $\operatorname{Re}[M_{22}]=0$ or $\operatorname{Im}[M_{22}]=0$, “zero lines” are formed in the plane of either complex $\tilde{k}$ (passive case) or ($k$, $n_i$) (active case). The crossing of a real and imaginary zero line results in $M_{22}=0$ at that location, thus revealing a solution. We visualize these zero lines by plotting $\log_{10}\left|\operatorname{Re}M_{22}\right|$ and $\log_{10}\left|\operatorname{Im}M_{22}\right|$ to enhance the regions near $M_{22}=0$ and using various image processing techniques to enhance the contrast.
The solutions are pinpointed more precisely by using the Secant method. Locations of minima of $|M_{22}|^2$ and a random value located closely to these minima locations are used as the first two inputs to the Secant method. Once a solution converges or $|M_{22}| < 10^{-12}$, a solution is considered found. This method has proved extremely adept at finding genuine solutions when suitable initial guesses are provided.
Verification of these solutions is provided by the phase of $M_{22}$, calculated as $\theta=\operatorname{atan2}(\operatorname{Im}M_{22},\operatorname{Re}M_{22})$. Locations of vanishing $M_{22}$ give rise to phase singularities since both the real and imaginary parts of $M_{22}$ vanish. The phase change around a path surrounding a singularity in units of $2\pi$ is referred to as topological charge [@halperin; @ZhangJOSAA]. In the cases studied here, the charge is $+ 1$ for phase increasing in the clockwise direction and $- 1$ for phase increasing in the counterclockwise direction.
Once a solution is found, the complex spatial field distribution may be calculated. Quasi modes $\psi(x)$ are calculated in the passive case and lasing modes $\Psi(x)$ are calculated in the active case. In order to determine whether or not mode mixing occurs (and if so, to what degree) in the case of nonuniform gain, the lasing modes are decomposed in terms of the quasi modes of the passive system. It was found [@LeungJPA; @LeungJPA2] that any spatial function defined inside an open system of length $L$ \[we consider the lasing modes $\Psi(x)$ here\] can be expressed as $$\Psi(x) = \sum_m a_m \psi_m(x),\label{eq:recon}$$ where $\psi_m(x)$ are a set of quasi modes characterized by the complex wavevectors $\tilde{k}_m$. The coefficients $a_m$ are calculated by $$\begin{aligned}
a_m =& i \int_0^L\left[\Psi(x)\hat{\psi}_m(x)+\hat{\Psi}(x)\psi_m(x)\right]dx
\nonumber\\
&+ i\left[\Psi(0)\psi_m(0)+ \Psi(L)\psi_m(L)\right],\label{eq:coeffs}\end{aligned}$$ where
$$\hat{\psi}_m(x)=-i\tilde{k}_m n^2(x)\psi_m(x)\label{eq:conjmoma}$$
$$\hat{\Psi}(x)=-ik\left[n_r(x)+in_if_E(x)\right]^2\Psi(x).\label{eq:conjmomb}$$
The normalization condition is $$\begin{aligned}
1 =&i \int_0^L 2\psi(x)\hat{\psi}(x) dx
+ i\left[\psi^2(0)+ \psi^2(L)\right]\nonumber\\
=& i \int_0^L 2\Psi(x)\hat{\Psi}(x) dx
+ i\left[\Psi^2(0)+ \Psi^2(L)\right].\label{eq:normalization}\end{aligned}$$
An advantage of this decomposition method is that a calculation over an infinite system has been reduced to a calculation over a finite system. Error checking is done by using the coefficients found in Eq. (\[eq:coeffs\]) to reconstruct the lasing mode intensity distribution with Eq. (\[eq:recon\]) yielding $R(x) \equiv \sum_m a_m \psi_m(x)$. We define a reconstruction error $E_R$ to monitor the accuracy of the decomposition: $$E_R = \frac{\int | \Psi(x) - R(x)|^2 dx}{\int |\Psi(x)|^2 dx}.$$
In general, the field distribution of a mode in a leaky system consists of a traveling wave component and a standing wave component. In Appendix \[ap:sttr\] we introduce a method to separate the two. This method applies to quasi modes as well as lasing modes; we shall consider lasing modes here. At every spatial location $x$, the right-going complex field $\Psi^{(R)}(x)$ and left-going complex field $\Psi^{(L)}(x)$ are compared. The field with the smaller amplitude is used for the standing wave and the remainder for the traveling wave. If $|\Psi^{(R)}(x)| < |\Psi^{(L)}(x)|$ then the standing wave component $\Psi^{(S)}(x)$ and traveling wave component $\Psi^{(T)}(x)$ are
$$\Psi^{(S)}(x) = \Psi^{(R)}(x) + [\Psi^{(R)}(x)]^*$$
$$\Psi^{(T)}(x) = \Psi^{(L)}(x) - [\Psi^{(R)}(x)]^*.$$
\[eq:standtrav\]
Further physical insight on lasing mode formation and disappearance, as well as new lasing mode appearance is provided by a mapping of an “effective potential” dictated by the random structure. Local regions of the random medium reflect light at certain frequencies but are transparent to others [@kuhl08]. The response of a structure to a field with frequency $\omega=ck$ can be calculated via a wavelet transformation of the real part of the dielectric function $\epsilon_r(x) = n^2(x)$ [@bliojosab]. The relationship between the local spatial frequency $q_{res}$ and the optical wavevector $k$ is approximately $q_{res}=2k$ in weakly scattering structures. The Morlet wavelet $\chi$ is expressed as $$\chi\left(\frac{x'-x}{s}\right) =
\frac{\pi^{-1/4}}{\sqrt{s}}e^{i\omega_0(x'-x)/s}
e^{-(x'-x)^2/2s^2},$$ with nondimensional frequency $\omega_0$ and Gaussian envelope width $s$ [@torrencecompo]. With $\omega_0$ fixed, stretching the wavelet through $s$ changes the effective frequency. Wavelets with varying widths are translated along the spatial axis to obtain the transformation $$W(x,q_{res}) = \int \epsilon_r(x')
\chi^*\left(\frac{x'-x}{s}\right)dx'.\label{eq:wavelett}$$ where $$q_{res} = \frac{\omega_0 + \sqrt{2+\omega_0^2}}{2s}.$$ The wavelet power spectrum $|W(x,q_{res})|^2$ is interpreted as an effective potential. Regions of high power indicate potential barriers and regions of low power indicate potential wells for light frequency $\omega = q_{res}c/2$.
Results\[sec:results\]
======================
A random system of $N=161$ layers is examined in the following as an example of a random 1D weakly scattering system. The indices of refraction of the dielectric layers are $n_1 = 1.05$ and the air gaps $n_2 = 1$. The average thicknesses are $\left<d_1\right> = 100$ nm and $\left<d_2\right> = 200$ nm giving a total average length of $\left<L\right> =$ 24100 nm. The grid origin is set at $x=0$ and the length of the random structure $L$ is normalized to $\left<L\right>$. The degree of randomness is set to $\eta = 0.9$ and the index of refraction outside the random media is $n_0 = 1$. The localization length $\xi$ is calculated from the dependence of transmission $T$ of an ensemble of random systems of different lengths as $\xi^{-1} = -d\left<\ln T\right>/dL$. The above parameters ensure that the localization length is nearly constant at 200 $\mu$m $\le \xi \le 240$ $\mu$m over the wavelength range 500 nm $\le \lambda \le$ 750 nm. With $\xi \gg L$, the system is in the ballistic regime.
![\[fig:wavelet\] (Color online) Effective potential (wavelet power spectrum) $|W(x)|^2$ of the dielectric function $\epsilon_r(x)$ as a function of position $x$ and wavelength $\lambda$. Regions of high power indicate potential barriers and regions of low power indicate potential wells where intensities are typically trapped. The black lines on the top represent the spatial distribution of dielectric constant $\epsilon_r(x)=n^2(x)$. ](fig1.jpg){width="8.5cm"}
Figure \[fig:wavelet\] shows the effective potential of the structure within the wavelength range of interest via a wavelet transformation. We use a nondimensional frequency of $\omega_0=6$ [@farge] and a spatial sampling step of $\Delta x = 2$ nm. The power spectrum $|W(x)|^2$ reveals the landscape of the effective potential dictated by the locations and thicknesses of the dielectric layers.
![\[fig:phasemap161\] (Color online) A mapping of the phase $\theta$ of $M_{22}$ for the passive 1D random system without gain. The topological charge of all quasi modes seen here is $-1$. Modes are enumerated from left to right. Quasi mode 27 is encircled in black. ](fig2.jpg){width="8.5cm"}
Figure \[fig:phasemap161\] is a phase map of $M_{22}$ in the passive case (without gain). The phase singularities mark the quasi modes’ $\tilde{k}$ values and are indicated by phase changes from $-\pi$ to $\pi$ along any lines passing through. The topological charge of all quasi modes is $-1$. Adjacent modes are formed by real and imaginary zero lines of $M_{22}$ that are not connected to one another. We calculated $M_{22}$ for increasingly large $|k_i|$ values until machine precision was reached and no additional modes appeared. As previously found [@wu07], mode frequency spacing is fairly constant in the ballistic regime. The nearly equal spacing of phase singularities in Fig. \[fig:phasemap161\] attests to this.
![\[fig:wf27\] Normalized intensity $|\psi(x)|^2$ of a leaky mode–quasi mode 27 ($\tilde{k} = 11.8 - i 1.03$ $\mu$m$^{-1}$) of the passive random 1D structure. The intensity is peaked at the left boundary of the structure, similar to doorway states [@oko03]. ](fig3.jpg){width="8.5cm"}
Most quasi modes have similar decay rates except for a few which have much larger decay rates. Modes are enumerated here starting with the lowest frequency mode in our wavelength range of interest. Mode 1 has a wavelength of 748 nm and mode 33 has a wavelength of 502 nm. Most quasi modes have $k_i$ values around $-0.1$ $\mu$m$^{-1}$. But a few have much larger decay rates, such as mode 27 at $\lambda=532$ nm which has $k_i=-1.03$ $\mu$m$^{-1}$ (encircled in black in Fig. \[fig:phasemap161\]). Figure \[fig:wf27\] shows the intensity of mode 27 to be concentrated on one side of the open structure. We observe that it bears similarity to “doorway states” common to open quantum systems [@oko03]. Doorway states are concentrated around the boundary of a system and strongly couple to the continuum of states outside the structure. Therefore, they have much larger decay rates.
For the case of uniform gain, only the lasing modes with large thresholds change significantly from the quasi modes of the passive system. Finding the corresponding quasi modes for lasing modes with large thresholds is challenging due to changes caused by the addition of a large amounts of gain. Thus, we neglect them in the following comparisons. However, there is a clear one-to-one correspondence with quasi modes for the remaining lasing modes. The average percent difference between quasi mode frequencies and lasing mode frequencies is 0.026%. The average percent difference between quasi mode decay rates $k_i$ and lasing thresholds (multiplied by $k$ for comparison) $n_ik$ is 4.3%. The normalized intensities of the quasi modes $I_Q(x)\equiv |\psi(x)|^2$ and lasing modes $I_L(x)\equiv |\Psi(x)|^2$ are also compared. The spatially averaged percent difference between each pair of modes is calculated as $(2/L)\int\left\{ |I_{Q}(x)-I_{L}(x)|/[I_{Q}(x)+I_{L}(x)]\right\}dx\times 100$. The averaged difference between intensities of the 3 lasing modes with the largest thresholds (of the large threshold modes not neglected) compared to the quasi modes is 68% while the remaining pairs average a difference of 4.0%.
Nonuniform Gain Effects on Lasing Mode Frequency, Threshold, and Intensity Distribution
---------------------------------------------------------------------------------------
Figure \[fig:follow\] maps the ($k$, $n_i$) values of lasing modes as nonuniform gain is introduced by reducing the gain region length from $l_G=L$. In this weakly scattering system the intensity distributions of modes are spatially overlapping. This results in a repulsion of mode frequencies [@kramer93]. As the size of the gain region changes, the envelopes of the intensity distributions change, but for most modes $n_i$ is small enough to leave the optical index landscape unchanged. Thus, the modes continue to spatially overlap as the size of the gain region changes and their frequencies remain roughly the same as in the uniform gain case. Similar behavior of lasing mode frequencies can be seen as the gain region length is varied in a simpler cavity with uniform index. Thus, the robustness of frequency is not due to inhomogeneity in the spatial dielectric function. However, the threshold values of the lasing modes change as $l_G$ decreases. Due to the limited spatial region of amplification, the thresholds increase. The increase of $n_i$ due to the change of threshold, though considerable, is not large enough to significantly impact the lasing frequencies as evidenced by the small change of frequencies as $l_G$ decreases.
![\[fig:follow\] (Color online) Frequencies and thresholds ($k$, $n_i$) of the lasing modes of the 1D random structure with gain. Lasing modes 1, 17, 18, and 33 are explicitly marked. The gain region length $l_G$ reduces from uniform gain ($l_G=L$) to nonuniform gain $l_G < L$. The color indicates the value of $l_G$ (units of nm) decremented along the layer interfaces. Due to the random thicknesses of the layers, the $l_G$ decrements are unequal. Hence, the reason for the unequal spacing of the color code. ](fig4.jpg){width="8.5cm"}
![\[fig:wfs1\] (Color online) Normalized intensity of lasing mode 17 with uniform gain $l_G=24100$ nm (red solid lines) and nonuniform gain $l_G=14284$ nm (black dashed lines). Gain is only located in the region $0 \le x \le l_G$. (a) Total intensity $|\Psi(x)|^2$, (b) traveling wave intensity $|\Psi^{(T)}(x)|^2$, and (c) standing wave intensity $|\Psi^{(S)}(x)|^2$. Nonuniform gain significantly changes the spatial intensity envelope as well as the standing wave and traveling wave components. ](fig5.jpg){width="8.5cm"}
The intensity distributions of the lasing modes also change considerably as $l_G$ is reduced. Normalized spatial intensity distributions are given by $|\Psi(x)|^2$ after $\Psi(x)$ has been normalized according to Eq. (\[eq:normalization\]). The intensities are sampled with a spatial step of $\Delta x = 2$ nm. With uniform gain ($l_G=24100$ nm), the intensity of lasing mode 17 ($\lambda = 598$ nm) in Fig. \[fig:wfs1\](a) increases toward the gain boundaries due to weak scattering and strong amplification. When the gain boundary is changed to $l_G=14284$ nm, the envelope of the spatial intensity distribution changes dramatically. The intensity increases more rapidly toward the boundaries of the gain region and stays nearly constant outside the gain region but still inside the structure. This change can be understood as $|n_i|$ inside the gain region causes the intensity to become larger, while outside the gain region $n_i=0$ and the wavevector is real.
To monitor the change in the trapped component of the intensity, $\Psi(x)$ is separated into a traveling wave and a standing wave component via Eq. (\[eq:standtrav\]) (see Appendix \[ap:sttr\]). Figures \[fig:wfs1\](b) and (c) show the traveling wave and standing wave components of lasing mode 17, respectively. For $l_G=L$, the intensity increase toward the structure boundaries is caused mostly by the growth of the traveling wave. The standing wave part is strongest near the center of the system. For $l_G=14284$ nm, the standing wave exhibits two peaks, one concentrated near the center of the gain region and the other outside the gain region. However, the standing wave intensity outside the gain region should not be directly compared to the standing wave intensity inside the gain region. The total intensity inside the gain region increases toward the gain boundary in this weakly scattering system. Thus, the amplitude of the field outside the gain region, where there is no amplification, is determined by the total field amplitude at the gain boundary. The randomness of the dielectric function outside the gain region traps part of the wave which results in a relatively large standing wave intensity compared to inside the gain region. However, outside the gain region, there is a net flux toward the right boundary of the system meaning the traveling wave intensity in this region is large as well.
![\[fig:wfs2\] (Color online) Standing/traveling wave ratio $A_{ST}(x)$ (red solid lines) of lasing mode 17 with uniform gain $l_G=L$ (a) and nonuniform gain $l_G= 14284$ nm (b). Gain is located in the regions left of the vertical black solid line. We term the location at which the ratio diverges ($A_{ST}(x)\rightarrow\infty$), the standing wave center (SWC). The potential profile $|W(x)|^2$ (black dashed lines) of the dielectric function at the wavelength of mode 17 is overlaid in both (a) and (b). ](fig6.jpg){width="8.5cm"}
The relative strength of the standing wave is calculated through the ratio of standing wave amplitude to traveling wave amplitude. The amplitudes are calculated in Appendix \[ap:sttr\]. Depending on whether the prevailing wave is right-going or left-going, the standing/traveling wave ratio is given by $$A_{ST}(x) =
\left\{
\begin{array}{l}
\left|\frac{2\Theta(x)}{\Pi(x)-\Theta(x)}\right|^2, \Pi(x) > \Theta(x)\\
\left|\frac{2\Pi(x)}{\Theta(x)-\Pi(x)}\right|^2, \Pi(x) < \Theta(x).
\end{array}
\right.$$ Results from considering uniform and nonuniform gain for lasing mode 17 are shown in Fig. \[fig:wfs2\]. Where the standing wave is largest inside the gain region, $|\Psi^{(T)}(x)|=|\Theta(x)-\Pi(x)|=0$ and the ratio $A_{ST}(x)$ is infinite. The location where the ratio is diverging is the position of pure standing wave. Fields are emitted in both directions from this position. The prevailing wave to the right of this standing wave center (SWC) is right-going. The prevailing wave to the left of this SWC is left-going. The SWC of the lasing mode is located near the center of the total system when considering uniform gain in Fig. \[fig:wfs2\](a). With the size of the gain region reduced in Fig. \[fig:wfs2\](b), we see that the SWC of the lasing mode (where $A_{ST}(x)\rightarrow \infty$) moves to stay within the gain region. Furthermore, note that this mode has a relatively small threshold (see Fig. \[fig:follow\]). We have found that in general, modes with low thresholds have a SWC near the center of the gain region while high threshold modes have a SWC near the edge of the gain region.
The cause for the small peak of $A_{ST}(x)$ outside the gain region can be found in the potential profile of Fig. \[fig:wavelet\]. A slice of the potential profile $|W(x)|^2$ at the wavelength of mode 17 ($\lambda = 598$ nm) is overlaid on the intensities in Fig. \[fig:wfs2\]. This suggests the standing wave is weakly trapped in a potential well around $x=20500$ nm \[marked by an arrow in Fig. \[fig:wfs2\](b)\].
Mode Mixing
-----------
Lasing modes can be expressed as a superposition of quasi modes of the passive system via Eq. (\[eq:recon\]) for any distribution of gain. Coefficients obtained from the decomposition of the lasing modes in terms of the quasi modes by Eq. (\[eq:coeffs\]) offer a clear and quantitative way to monitor changes of lasing modes by nonuniform gain. Using Simpson’s rule for the numerical integrations and a basis consisting of at least 15 quasi modes at both higher and lower frequencies than the lasing mode being decomposed, we consistently find $E_R \approx 10^{-4}$.
![\[fig:c1732\] (Color online) Decomposition of lasing mode 17 in terms of passive quasi modes. (a) Decomposition with uniform gain (red crosses) and nonuniform gain (black circles). Leaky quasi modes, i.e., modes with large $|k_i|$ such as modes 7, 14, and 23, contribute to lasing modes markedly different than the others. (b) Five largest coefficients from the decomposition of lasing mode 17. As $l_G$ is reduced, the amount of mode mixing increases dramatically. The reconstruction error $E_R$ for lasing mode 17 is close to $10^{-4}$ until $l_G = 11000$ nm then rises to $10^{-2}$ at $l_G=3200$ nm. Some coefficients are greater than one, which is possible in open systems. ](fig7a.jpg "fig:"){width="8.5cm"} ![\[fig:c1732\] (Color online) Decomposition of lasing mode 17 in terms of passive quasi modes. (a) Decomposition with uniform gain (red crosses) and nonuniform gain (black circles). Leaky quasi modes, i.e., modes with large $|k_i|$ such as modes 7, 14, and 23, contribute to lasing modes markedly different than the others. (b) Five largest coefficients from the decomposition of lasing mode 17. As $l_G$ is reduced, the amount of mode mixing increases dramatically. The reconstruction error $E_R$ for lasing mode 17 is close to $10^{-4}$ until $l_G = 11000$ nm then rises to $10^{-2}$ at $l_G=3200$ nm. Some coefficients are greater than one, which is possible in open systems. ](fig7b.jpg "fig:"){width="8.5cm"}
Figure \[fig:c1732\](a) shows the decomposition of lasing mode 17 with uniform and nonuniform gain. Beginning with the case of uniform gain ($l_G=L$), the largest contribution to lasing mode 17 is from corresponding quasi mode 17. There is a nonzero contribution from other quasi modes on the order $10^{-3}$. This reflects slight differences between the lasing mode profile in the presence of uniform gain and the quasi mode profile [@deych05PRL; @wu07]. With the gain region length reduced to $l_G=14284$ nm, the coefficients $|a_m|^2$ from quasi modes closer in frequency to the lasing modes increase significantly; i.e., quasi modes closer in frequency are mixed in. The exceptions are the very leaky quasi modes 7, 14, and 23. Unlike leaky quasi mode 27 shown in Fig. \[fig:wf27\], quasi modes 7, 14, and 23 have intensities which are peaked at the right boundary of the structure. It has been observed that when $l_G$ reduces and the intensity distribution of lasing mode 17 shifts to the left boundary of the structure, there is less overlap with these leaky quasi modes. Thus, the magnitude of the coefficients associated with the leaky modes reduces as shown in Fig. \[fig:c1732\](a).
Figure \[fig:c1732\](b) reveals the five largest coefficients $|a_m|^2$ for lasing mode 17 as $l_G$ is incrementally reduced along the dielectric interfaces. While the lasing mode remains dominantly composed of its corresponding quasi mode, neighboring quasi modes mix into the lasing mode significantly. It was shown [@deych05PRL] that linear contributions from gain induced polarization bring about a coupling between quasi modes of the passive system. This coupling arises solely due to the inhomogeneity of the dielectric function, not the openness of the system. While this interaction may play a role in mode mixing with uniform gain, the effect is small compared to the mode mixing caused by the nonuniformity of the gain. This is clearly demonstrated in Fig. \[fig:c1732\](b), where the coefficients from quasi modes close in frequency are orders of magnitude larger for small $l_G$ than for $l_G=L$.
Lasing Mode Disappearance and Appearance
----------------------------------------
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As the size of the gain region reduces we observe that some lasing modes disappear and new lasing modes appear. The existence of new lasing modes in the presence of gain saturation has been confirmed [@andreasenOL]. This phenomenon is not limited to random media, but its occurrence has been observed in a simple 1D cavity with a uniform index of refraction. New lasing modes, to the best of our knowledge, are always created with larger thresholds than the existing lasing modes adjacent in frequency. The disappearance of lasing modes is not caused by mode competition for gain because gain saturation is not included in our model of *linear gain*. Disappearance/appearance events occur more frequently for smaller values of $l_G$. New lasing modes appear at frequencies in between the lasing mode frequencies of the system with uniform gain. These new modes exist only within small ranges of $l_G$. We also find that the disappearance events exhibit behavioral symmetry (as explained below) around particular values of $l_G$. This disappearance and subsequent reappearance causes a fluctuation of the local density of lasing states as $l_G$ changes.
We examine the progression of one representative event in detail. The gaps in the decomposition coefficients for lasing mode 17 in Fig. \[fig:c1732\](b), in the range 10500 nm $\le l_G \le 14500$ nm, indicate lasing mode 17 does not exist for those distributions of gain. Figure \[fig:ssezls\] shows the real and imaginary zero lines of $M_{22}$ and their accompanying phase maps for $l_G=$ 14961 nm, 14553 nm, 14523 nm, 14472 nm, 14284 nm, and 14042 nm. As $l_G$ decreases, the zero lines of lasing modes 17 and 18 join as seen in the transition from Fig. \[fig:ssezls\](a) to (c). This creates a new mode solution (marked by a white circle) with a frequency between lasing modes 17 and 18 and a larger threshold. The existence of a new lasing mode is confirmed by the phase singularity in Fig. \[fig:ssezls\](d). The new mode is close to mode 17 in the ($k$, $n_i$) plane and its phase singularity has the opposite topological charge as seen in Fig. \[fig:ssezls\](d). As $l_G$ decreases further, the joined zero lines forming mode 17 and the new mode pull apart. This causes the two solutions to approach each other in the ($k$, $n_i$) plane, i.e., the frequency and threshold of mode 17 increase while the frequency and threshold of the new mode decrease. In Figs. \[fig:ssezls\](f) and (g), the solutions are so close that they are nearly identical, yet they still represent two separate solutions. Further decreasing $l_G$ makes the solutions identical. The zero lines then separate and the phase singularities of opposite charge annihilate each other in Figs. \[fig:ssezls\](h) and (i). This results in the disappearance of mode 17 and the new mode. The process then reverses itself as $l_G$ is decreased further \[Figs. \[fig:ssezls\](j)-(m)\] yielding the reappearance of mode 17 and the new mode and their subsequent separation in the ($k$, $n_i$) plane. This is the aforementioned behavioral symmetry around $l_G=14472$ nm.
![\[fig:wfs094\] (Color online) (a) Standing/traveling wave ratio $A_{ST}(x)$ of the new lasing mode (red) and lasing mode 17 (black) for $l_G=14553$ nm (solid lines) and $l_G=14523$ nm (dotted lines). The potential profile $|W(x)|^2$ (black dashed line) of the dielectric function for this wavelength is overlaid in both (a) and (b) and major potential barriers are marked through . The ratio $A_{ST}(x)$ of the new lasing mode and lasing mode 17 become more similar an converge on each other as $l_G$ reduces. Reducing $l_G$ further causes these two lasing modes to first disappear then reappear as the process reverses itself. (b) Standing/traveling wave ratio $A_{ST}(x)$ of the new lasing mode (red) and lasing mode 17 (black) after they have reappeared for $l_G=14284$ nm. The ratios are similar to the ratios for $l_G=14553$ nm in (a). The vertical black solid line marks the gain edge. The ratios of the modes diverge now as $l_G$ is reduced. ](fig9.jpg){width="8.5cm"}
Examining the standing/traveling wave ratio of lasing mode 17 and the new lasing mode together with the potential profile $|W(x)|^2$ offers some insight of mode annihilation and reappearance in real space. Figure \[fig:wfs094\] shows the ratio $A_{ST}(x)$ for the new mode and mode 17 along with $|W(x)|^2$. The potential profile is very similar for the new mode and mode 17 since their wavelengths are very close. There are four major potential barriers at the mode 17 wavelength ($\lambda = 598$ nm) for $x < 15000$ nm. This is the spatial region associated with the gain distributions in Fig. \[fig:ssezls\] where $l_G$ is always smaller than 15000 nm. Figure \[fig:wfs094\] shows them at: $x = $ 927 nm, 5200 nm, 8700 nm, and 14519 nm. Due to oscillations, the centers of barriers and are less well defined. The right edge of the gain region at $l_G=14553$ nm is located just to the right of barrier . For $l_G=14523$ nm, the right edge of the gain region nears the maximum of barrier . Figure \[fig:wfs094\](a) shows that for $l_G=14553$ nm, the SWC of the new mode is between barrier and barrier . The SWC of mode 17 is in the middle of the gain region at $x = 5300$ nm and its SWC is between barrier and barrier . Before disappearing, the modes approach each other in the ($k$, $n_i$) plane, eventually merge, and their intensity distributions become identical (as evidenced by the trend of their standing/traveling wave ratios). As $l_G$ is further reduced and the modes reappear, the behavior of the modes’ ratios $A_{ST}(x)$ (or equivalently, intensity distributions) reverses itself as expected from the behavioral symmetry shown in Fig. \[fig:ssezls\]. At $l_G=14284$ nm, the right edge of the gain region has passed barrier and Fig. \[fig:wfs094\](b) \[with a different horizontal scale than Fig. \[fig:wfs094\](a)\] shows the SWC of the new mode is in roughly the same location as it was for $l_G=14553$ nm. The SWC of mode 17 is also in roughly the same location as it was for $l_G=14553$ nm.
![\[fig:c17nm1nm2\] (Color online) Decomposition at $l_G=14284$ nm of lasing mode 17 (black circles), lasing mode 18 (blue open diamonds), and the new lasing mode (red crosses) in terms of the quasi modes of the passive system. Lasing modes 17 and 18 are mostly composed of their respective quasi modes while the new mode is dominated by a mixture of both quasi mode 17 and 18. The inset shows the decomposition coefficients of outlying quasi modes for lasing mode 17 (black line), lasing mode 18 (blue line), and the new lasing mode (red line). ](fig10.jpg){width="8.5cm"}
The appearance of new lasing modes is unanticipated. In the passive system, the number of standing wave peaks for quasi modes increases incrementally by 1, e.g., quasi mode 17 has 82 peaks and quasi mode 18 has 83 peaks. Lasing modes 17 and 18 behave the same way. How exactly does a new lasing mode fit into this scheme? Though closer in frequency and threshold to lasing mode 17, counting the total number of standing wave peaks of the new lasing mode yields the same number as for lasing mode 18. However, the new lasing mode is somewhat compressed in the gain region having one more peak than lasing mode 18. It is decompressed in the region without gain having one less peak than lasing mode 18.
Comparing the decompositions of the lasing modes in terms of quasi modes helps reveal the character of the new lasing mode. Figure \[fig:c17nm1nm2\] shows the decomposition of the new lasing mode together with the decomposition of lasing modes 17 and 18 at $l_G=14284$ nm. The new mode has a slightly larger coefficient amplitude associated with quasi mode 17 than quasi mode 18, but the two amplitudes are nearly equal. We found that as mode 17 and the new mode solutions approach each other by varying $l_G$, their coefficient distributions also approach each other until becoming equal as expected from Figs. \[fig:ssezls\] and \[fig:wfs094\].
Conclusion\[sec:conclusion\]
============================
We have demonstrated the characteristics of lasing modes to be strongly influenced by nonuniformity in the spatial gain distribution in 1D random structures. While the entire structure plays the dominant role in determining the frequency of the lasing modes, the gain distribution mostly determines the lasing thresholds and spatial distributions of intensity. The gain distribution also appears to be solely responsible for the creation of new lasing modes. We have verified the existence of new lasing modes in numerous random structures as well as dielectric slabs of uniform refractive index. A more thorough investigation of the latter will be described in a future work. All of these changes caused by nonuniform gain take place without the influence of nonlinear interaction between the field and gain medium. Our conclusion is that nonuniformity of the gain distribution alone is responsible for the complicated behavior observed here.
By decomposing the lasing modes in terms of a set of quasi modes of the passive system, we illustrated how the lasing modes change. The contribution of a quasi mode to a lasing mode was seen to depend mostly on its proximity in frequency $k$ and the spatial distribution of gain. The more the gain changed from uniformity, the greater the mixing in of neighboring quasi modes. Thus, great care must be taken even close to the lasing threshold when using the properties of quasi modes to predict characteristics of lasing modes in weakly scattering systems with nonuniform gain or local pumping.
The change of intensity distributions of lasing modes as the size of the gain region is varied appears to be general. With reduction of the size of the gain region, the peak of the standing/traveling wave ratio $A_{ST}(x)$, or the standing wave center (SWC) of the mode, moves to stay within the gain region. Modes with low thresholds have a SWC near the middle of the gain region while high threshold modes have a SWC near the edge of the gain region. Changing the gain distribution thus changes the intensity distributions of lasing modes. The exact modal distributions, however, appear correlated with the potential profile. In the cases studied here, the new lasing mode and lasing mode 17 lay in between two large potential barriers. Decreasing the size of the gain region brought the intensity distributions closer together until they disappeared. These changes took place by varying the edge of the gain region only hundreds of nanometers. Thus, even a slight change in the gain distribution may have drastic consequences for lasing modes.
The authors thank Patrick Sebbah, Alexey Yamilov, A. Douglas Stone, and Dimitry Savin for stimulating discussions. This work was supported partly by the National Science Foundation under Grant Nos. DMR-0814025 and DMR-0808937.
Linear Gain Model \[ap:lineargain\]
===================================
In this appendix, we describe the model used to simulate linear gain in a 1D system. The gain is linear in the sense that it does not depend on the electromagnetic field intensity. The lasing solutions $\Psi(x)$ must satisfy the time-independent wave equation $$\left[\frac{d^2}{dx^2} + \epsilon(x,\omega)k^2\right]\Psi(x) = 0,$$ with a complex frequency-dependent dielectric function $$\epsilon(x,\omega) = \epsilon_r(x) + \chi_g(x,\omega),$$ where $\epsilon_r(x)=n^2(x)$ is the dielectric function of the non-resonant background material. The frequency dependence of $\epsilon_r(x)$ is negligible. $\chi_g(x,\omega)$, corresponding to the susceptibility of the resonant material, is given by $$\chi_g(x,\omega)=\frac{A_eN_A(x)}{\omega_a^2-\omega^2
-i\omega\Delta\omega_a},
\label{eq:chia1}$$ where $A_e$ is a material-dependent constant, $N_A(x)$ is the spatially dependent density of atoms, $\omega_a$ is the atomic transition frequency, and $\Delta\omega_a$ is the spectral linewidth of the atomic resonance. Equation (\[eq:chia1\]) may be simplified by assuming the frequencies of interest $\omega$ are within a few linewidths of the atomic frequency $\omega_a$, i.e., $\omega^2-\omega_a^2=(\omega+\omega_a)(\omega-\omega_a)
\approx 2\omega_a(\omega-\omega_a)$. Equation (\[eq:chia1\]) then reduces to $$\chi_g(x,\omega)\approx \frac{iA_eN_A(x)}{\omega_a\Delta\omega_a
[1+2i(\omega-\omega_a)/\Delta\omega_a]}.
\label{eq:chia2}$$ The frequency-dependent index of refraction is $$\begin{aligned}
\tilde{n}(x,\omega) =&
\sqrt{\epsilon(x,\omega)} = \sqrt{\epsilon_r(x)+\chi_g(x,\omega)} \nonumber\\
=& n_r(x,\omega) + in_i(x,\omega),\end{aligned}$$ which may then be implemented in the transfer matrix method. At this point, let us note that only 2 steps are needed to convert this classical electron oscillator model to real atomic transitions [@siegbook]. First, the radiative decay rate $\gamma_{\parallel}$ may be substituted in to Eq. (\[eq:chia2\]) in place of a few constants. Second, and more importantly, real quantum transitions induce a response proportional to the population difference density $\Delta N_A$. Thus, $N_A(x)$ should be replaced by $\Delta N_A$, the difference in population between the lower and upper energy levels.
Linear gain independent of $\omega$ is obtained by working in the limit $\omega-\omega_a \ll \Delta\omega_a$, yielding $$\chi_g(x)\approx i\frac{A_e\Delta N_A(x)}{\omega_a\Delta\omega_a}
\label{eq:chia3},$$ a purely imaginary susceptibility. We can make the definition $\chi_g(x)\equiv i\epsilon_i(x)$, where $\epsilon_i(x)$ is the imaginary part of $\epsilon(x)$. Note that $\epsilon(x)$ may include absorption \[$\epsilon_i>0$\] or gain \[$\epsilon_i<0$\]. We shall only consider gain here. The complex frequency-*independent* dielectric function now yields a frequency-independent index of refraction $\tilde{n}(x)=n_r(x)+in_i(x)$ which may be expressed explicitly as $$\begin{aligned}
n_r(x) =& \frac{n(x)}{\sqrt{2}}
\left[\sqrt{1+\frac{\epsilon_i^2(x)}{n^4(x)}}+1\right]^{1/2}\nonumber\\
n_i(x) =& \frac{-n(x)}{\sqrt{2}}
\left[\sqrt{1+\frac{\epsilon_i^2(x)}{n^4(x)}}-1\right]^{1/2}\label{eq:lgni}.\end{aligned}$$ Furthermore, in the main text, we assume $n_i$ to be spatially independent. Thus, by solving for $n_r(x)$ in terms of $n(x)$ and $n_i$, the index of refraction used throughout this paper is given by $$\begin{aligned}
\tilde{n}(x) =& n_r(x) + in_i\nonumber\\
=& \sqrt{n^2(x)+n_i^2} + in_i.\end{aligned}$$
Standing wave and traveling wave components of the total field \[ap:sttr\]
==========================================================================
In this appendix, we describe the method that enables one to define a standing wave component and a traveling wave component of the field at each point $x$ of a 1D system.
For an open structure without gain, the field reads $$\psi(x) = p(x)\exp[in(x)\tilde{k}x]+q(x)\exp[-in(x)\tilde{k}x],$$ where $\tilde{k}$ is the complex wavevector and $n(x)$ is the index of refraction, the value of which alternates between $n(x)=n_1>1$ in dielectric layers and $n(x)=n_2=1$ in air gaps. For structures with gain, the field reads $$\Psi(x) = p(x)\exp[i\tilde{n}(x)kx]+q(x)\exp[-i\tilde{n}(x)kx],$$ where $\tilde{n}(x)=n(x)+in_i$ is the complex index of refraction. We rewrite both equations in the single form $$E(x) = p(x)\exp[i\tilde{K}(x)x]+q(x)\exp[-i\tilde{K}(x)x],\label{eq:EKx}$$ where $\tilde{K}(x)=K_r(x)+iK_i(x)$ and $E(x)$ may be either $\psi(x)$ or $\Psi(x)$.
For now, we will consider the field within a single layer in order to simplify the notation. The following results will be valid within any layer. Since within a layer, the coefficients $p(x)$, $q(x)$ and the wavevector $\tilde{K}(x)$ do not depend on $x$, we rewrite Eq. (\[eq:EKx\]) as $$E(x) = p\exp[i\tilde{K}x]+q\exp[-i\tilde{K}x].\label{eq:EK}$$ The complex amplitudes $p$ and $q$ of the right-going and left-going fields, respectively, can be written as $p=P\exp[i\varphi]$ and $q=Q\exp[i\phi]$ where $P$ and $Q$ are the real amplitudes which can be chosen positive. The field becomes $$\begin{aligned}
E(x) =& P\exp[-K_ix]\exp[i(K_rx+\varphi)]\nonumber\\
&+ Q\exp[K_ix]\exp[-i(K_rx-\phi)]\nonumber\\
=& \Pi(x)\exp[i(K_rx+\varphi)] \nonumber\\
&+\Theta(x)\exp[-i(K_rx-\phi)],\label{eq:tot}\end{aligned}$$ where $\Pi(x)\equiv P\exp[-K_ix]$ and $\Theta(x)\equiv Q\exp[K_ix]$. Introducing the global phase $\Phi\equiv [\varphi+\phi]/2$ and the difference $\Delta\equiv [\varphi-\phi]/2$, the field reads $$\begin{aligned}
E(x)=&\exp[i\Phi]\{\Pi(x)\exp[i(K_rx+\Delta)]\nonumber\\
&+ \Theta(x)\exp[-i(K_rx+\Delta)]\} .\end{aligned}$$ Within a single layer, we can set $\Phi=0$ so that the field becomes $$\begin{aligned}
E(x)=&\Pi(x)\exp[i(K_rx+\Delta)]+\Theta(x)\exp[-i(K_rx+\Delta)]\nonumber\\
=&E^{(R)}(x) + E^{(L)}(x), \label{eq:RL}\end{aligned}$$ where $E^{(R)}(x)$ and $E^{(L)}(x)$ are the right-going and left-going waves, respectively.
We can build a standing wave component with $E^{(R)}(x)$ as $$\begin{aligned}
E^{(S)}(x)=&E^{(R)}(x) + [E^{(R)}(x)]^*\nonumber\\
=& 2\Pi(x)\cos[K_rx+\Delta]\label{eq:RS}\end{aligned}$$ and define the traveling wave component as the remaining part of the total field $$\begin{aligned}
E^{(T)}(x)=&E(x)-E^{(S)}(x)\nonumber\\
=&E^{(L)}(x) - [E^{(R)}(x)]^*\nonumber\\
=& [\Theta(x)-\Pi(x)]\exp[-i(K_rx+\Delta)].\label{eq:RT}\end{aligned}$$ Hence, $2\Pi(x)$ and $[\Theta(x)-\Pi(x)]$ are the amplitudes of the standing wave and traveling wave components, respectively. It is also possible to build a standing wave component with $E^{(L)}(x)$ as $$\begin{aligned}
E^{(S)}(x)=&E^{(L)}(x) + [E^{(L)}(x)]^*\nonumber\\
=& 2\Theta(x)\cos[K_rx+\Delta]\label{eq:LS}\end{aligned}$$ so that the traveling wave component reads $$\begin{aligned}
E^{(T)}(x)=&E(x)-E^{(S)}(x)\nonumber\\
=&E^{(R)}(x) - [E^{(L)}(x)]^*\nonumber\\
=& [\Pi(x)-\Theta(x)]\exp[i(K_rx+\Delta)].\label{eq:LT}\end{aligned}$$
Comparing both ways of resolving the total field into its two components, we see that in Eq. (\[eq:RT\]) the traveling wave component is a left-going wave while in Eq. (\[eq:LT\]) it is a right-going wave. Hence, if in the expression of the field in Eq. (\[eq:RL\]), the prevailing wave is the right-going wave $\Pi(x)\exp[i(K_rx+\varphi)]$ (i.e., $\Pi(x) > \Theta(x)$), we choose the standing and traveling wave components of Eqs. (\[eq:LS\]) and (\[eq:LT\]). In the opposite case of $\Pi(x) < \Theta(x)$, we choose the standing and traveling wave components of Eqs. (\[eq:RS\]) and (\[eq:RT\]).
Let us note that the imaginary part of the total field $E(x)$ is given in both cases by $$\operatorname{Im}[E(x)] = [\Pi(x)-\Theta(x)]\sin[K_rx+\Delta].$$ As expected, the presence of a traveling wave component, i.e., $|\Pi(x)-\Theta(x)|\ne 0$, makes $E(x)$ become complex instead of being real for a pure standing wave.
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|
---
author:
- |
[**Giovanni AMELINO-CAMELIA**]{}$^{\dag}$ and [**Joseph KAPUSTA**]{}$^{\dag,\ddag}$\
[*$^{\dag}$Theory Division, CERN*]{}\
\
[*$^{\ddag}$School of Physics and Astronomy, University of Minnesota*]{}\
[*Minneapolis, MN 55455, USA*]{}
title: |
[CERN-TH/99-228\
NUC-MINN-99/12-T\
July 1999\
]{}
---
-0.1cm -1cm
Abstract
Above a critical matter density the propagating modes of the neutral kaon system are essentially eigenstates of strangeness, but below it they are almost complete eigenstates of CP. We estimate the real and imaginary parts of the energies of these modes and their mixing at all densities up to nuclear matter density $2\times 10^{14}$ g/cm$^3$. In a heavy ion collision the strong interactions create eigenstates of strangeness, and these propagate adiabatically until the density has fallen to the critical value, whereupon the system undergoes a sudden transition to (near) eigenstates of CP. We estimate the critical density to be 20 g/cm$^3$, and that this density will be reached about $2\times 10^5$ fm/c after the end of the collision.
Neutral kaon systems are extremely interesting for a variety of reasons [@texts1]. When they are created in a collision among hadrons or leptons they are essentially in eigenstates of strangeness because the strong and electromagnetic interactions conserve flavor. When they propagate freely in vacuum the weak interactions are operative and both C and P are violated. The eigenstates of the full Hamiltonian are then almost completely flavor mixed as short- and long-lived kaons: $$\begin{aligned}
|K_S\rangle &=& \left[ (1+\epsilon)|K^0\rangle - (1-
\epsilon)|\bar{K}^0\rangle\right]
/\sqrt{2(1+|\epsilon|^2} \nonumber \\
|K_L\rangle &=& \left[ (1+\epsilon)|K^0\rangle + (1-
\epsilon)|\bar{K}^0\rangle\right]
/\sqrt{2(1+|\epsilon|^2} \, .\end{aligned}$$ Here $|\epsilon| \approx 2\times 10^{-3}$ is the measure of CP violation. When a beam of long-lived kaons is sent through ordinary matter, short-lived kaons are generated due to the different interactions between the components of the former, namely $K^0$ and $\bar{K}^0$, and atomic nuclei. This is called kaon regeneration. The goal of this paper is to study the collective modes of propagation of the neutral kaons in [*dense*]{} matter, and to determine the fate of these modes after they are created in a collision between large nuclei at high energy.
We will use a strong interaction basis with $$\begin{aligned}
|K^0\rangle &=& | d\bar{s}\rangle = |1\rangle = \left( \begin{array}{c}
1 \\ 0 \end{array} \right) \nonumber \\
|\bar{K}^0\rangle &=& |\bar{d}s \rangle = |2\rangle = \left( \begin{array}{c}
0 \\ 1 \end{array} \right) \, .\end{aligned}$$ Because it is so small and plays no special role in our analysis we shall set $\epsilon = 0$. Then $|K_S\rangle$ and $|K_L\rangle$ are eigenstates of CP with eigenvalues $+1$ and $-1$, respectively.
First we do a relativistic analysis. Poles of the propagator determine the time evolution of small amplitude excitations. These are obtained from solutions to the equation $$\omega^2 -k^2 -\Pi_{vac}(\omega^2-k^2) - \Pi_{mat}(\omega,k) = 0 \, ,$$ where the vacuum and matter contributions to the self-energy are indicated. Generally it is an excellent approximation to evaluate these $2\times2$ matrices on the mass shell. This means that $\Pi_{vac}$ is a constant and $\Pi_{mat}$ depends on the momentum $k$ only. The usual analysis gives $$\Pi_{vac} = \left( \begin{array}{cc} A & B \\ B & A \end{array} \right)^2$$ where $A$ and $B$ are complex numbers. In terms of measurables they are [@PDG]: $$\begin{aligned}
A &=& m_K -i(\Gamma_S+\Gamma_L)/4 \nonumber \\
B &=& \Delta m/2 +i\Delta \Gamma \nonumber \\
m_K &=& (m_S+m_L)/2 = 497.67 \; {\rm MeV} \nonumber \\
\Delta m &=& 3.52\times10^{-12} \; {\rm MeV} \nonumber \\
\Delta \Gamma &=& \Gamma_S-\Gamma_L \nonumber \\
\Gamma_S &=& 7.38\times10^{-12} \; {\rm MeV} \nonumber \\
\Gamma_L &=& 1.27\times10^{-14} \; {\rm MeV} \nonumber \\
\Delta m &=& (0.478\pm0.003) \Delta \Gamma \, .\end{aligned}$$ The matter contribution is diagonal in flavor and is expressed as $$\Pi_{mat} = \left( \begin{array}{cc} F & 0 \\ 0 & \bar{F} \end{array}
\right) \, .$$ In matter with an excess of baryons over antibaryons the strange and antistrange components, $F$ and $\bar{F}$, are different. Of course, this is the origin of kaon regeneration in matter. One reason they are different is because the valence antiquark $\bar{d}$ in the $\bar{K}^0$ can annihilate on a valence quark $d$ in the proton or neutron, producing a hyperon. This is not possible with a $K^0$.
Diagonalization of the equation $$(k^2 + \Pi) \left( \begin{array}{cc} \alpha \\ \beta \end{array} \right) =
\omega^2 \left( \begin{array}{cc} \alpha \\ \beta \end{array} \right)$$ yields the energy eigenvalues $$\omega^2_{\pm} = k^2 + A^2 + B^2 + \frac{1}{2} (F+\bar{F})
\pm \sqrt{4A^2B^2 + \frac{1}{4} (F-\bar{F})^2}$$ and eigenstates $$\beta_{\pm} = \left( -\chi \pm \sqrt{1+\chi^2} \right) \alpha_{\pm}$$ where $\chi = (F-\bar{F})/4AB$. The states evolve in time as $\exp(-
i\omega_{\pm}t)$. In baryon-free matter, even at finite temperature, there is no difference between $F$ and $\bar{F}$ (from the point of view of the strong and electromagnetic interactions). Then, with the above conventions, $\beta_{\pm} = \pm \alpha_{\pm}$, and the upper sign corresponds to $K_L$ and the lower sign to $K_S$. These are the familiar modes in vacuum except that the square of the mass is shifted by $F$, which generally has both real and imaginary parts.
At nuclear matter density, 0.155 nucleons/fm$^3$, the situation is very different. One should expect that $|F-\bar{F}|$ is only somewhat smaller than typical nuclear energies, implying something on the order of (100 MeV)$^2$. Since $$2AB = m_K(\Delta m + i\Delta \Gamma/2) = (4.19\times10^{-5} \; {\rm MeV}
)^2 + i(4.28\times10^{-5} \; {\rm MeV})^2$$ this means that $|\chi| >> 1$. In this case, the off-diagonal matrix elements in the self-energy, due to the weak interactions, are totally ignorable, and the energy eigenstates are also eigenstates of strangeness.
Let us now estimate $F-\bar{F}$ as a function of density. At low to moderate densities the self-energy may be expressed in terms of the scattering amplitude for kaons scattering on the constituents of the medium, here taken to be an equal mixture of protons and neutrons [@selfe]. $$\begin{aligned}
F(\omega,k) &=& - 4\pi \int \frac{d^3q}{(2\pi)^3} \,
4 n_{FD}(q) \, \sqrt{\frac{s}{m_N^2+q^2}}
\, f(s) \nonumber \\
&=& -4\pi \rho_N \left(1+\frac{m_K}{m_N} \right) \langle f \rangle \, .\end{aligned}$$ In the first line, $n_{FD}(q)$ is the Fermi-Dirac distribution for a nucleon with momentum $q$, the extra factor of 4 appears because of summation over spin and isospin of the nucleons, $f$ is the forward scattering amplitude in the center-of-momentum frame, and $s$ is the usual Mandelstam variable. The angular brackets in the second line denote the particular momentum averaging, and $\rho_N$ is the spatial density of nucleons. If the nucleons are not distributed according to a Fermi-Dirac distribution then one ought to use the actual momentum distribution; this will affect the numerical value of $\langle f
\rangle$ to some extent. Formula (11) is simply a version of the low density virial expansion familiar in statistical physics. Its applicability has been discussed in [@selfe]. Essentially it requires that the average separation of nucleons be larger than both the real part of the forward scattering amplitude and the inverse of the average relative momentum between a nucleon and a kaon. One may think of this formula quantum mechanically as representing indices of refraction (real part) and attentuation (imaginary part). Indeed, it is only a variation of the usual formulae used in kaon regeneration sudies.
There is an identical expression for antikaons with $\bar{F}$ replacing $F$ and $\bar{f}$ replacing $f$. The difference $\bar{f}-f$ has been estimated, or may be inferred, from several sources. For example, Eberhard and Uchiyama [@EU] calculated the real and imaginary parts of the forward scattering amplitudes for neutral kaons incident on proton and nuclear targets. These calculations were based on measured total cross sections for $K^+$ and $K^-$ on protons and neutrons, on published values of $K$-nucleon elastic scattering, and charge symmetry. From their figures 2 and 3 we estimate that $$\langle \bar{f}-f \rangle \approx 0.3(1+i) \; {\rm fm} \, .$$ Here the averaging is for kaon momenta ranging from 0.3 to 1 GeV, relevant for the conditions in hot and dense nuclear matter created in a heavy ion collision. Shuryak and Thorsson [@ST] have calculated the scattering amplitudes for charged kaons incident on nucleons based on partial wave analyses of data. From their figures 1 and 2 we estimate that $${\rm Re}\, \langle \bar{f}-f \rangle \approx 1.5 \; {\rm fm} \, .$$ Here the averaging is for $\sqrt{s}$ ranging between 1.5 to 1.8 GeV corresponding to the range of kaon momenta quoted above. (Unfortunately they have only displayed the real parts explicitly.) We do not know the reason for this discrepancy. Fortunately the precise numbers are totally irrelevant for the mixing of the neutral kaons in dense matter, since even the smallest estimate of the strong interaction induced self-energy overwhelms that due to the weak interactions. For example, using the estimate from Eberhard and Uchiyama, we obtain $$F-\bar{F} \approx (150 \; {\rm MeV})^2 (1+i) \left( \frac{\rho_N}{0.1 / {\rm
fm}^3}\right) \, ,$$ which ought to be compared to eq. (10). The fact that the real part of $F-
\bar{F}$ is positive implies that the nonrelativistic one-body potential, $U =
F/2m_K$, is relatively more repulsive for kaons than for antikaons. This is quite natural for the reasons stated earlier. The fact that the imaginary part is positive is also to be expected because the imaginary part of the scattering amplitude and cross section are related by the optical theorem, $\sigma =
(4\pi/k_{cm}){\rm Im}\, f$, and antikaons have a much bigger inelastic cross section with nucleons than kaons.
An alternative approach to the kaon self-energy at moderate to high density is to use an effective Lagrangian incorporating the relevant degrees of freedom and symmetries [@KN]. One finds that [@pie] $$\begin{aligned}
F(\omega,k) &=& -\frac{\Sigma_{KN}}{f^2_{\pi}}\rho_S +
\frac{3}{4}\frac{\rho_N}{f_{\pi}^2} \omega \nonumber \\
\bar{F}(\omega,k) &=& -\frac{\Sigma_{KN}}{f^2_{\pi}}\rho_S -
\frac{3}{4}\frac{\rho_N}{f_{\pi}^2} \omega \, .\end{aligned}$$ Here $\Sigma_{KN}$ is the kaon-nucleon sigma term, estimated to be of order 350 MeV. One distinquishes the scalar nucleon density, $\rho_S
= \langle \bar{N}N \rangle$, from the conserved vector density, $\rho_N = \langle \bar{N} \gamma^0 N \rangle$, although they only begin to differ significantly above twice nuclear density. The difference in sign in the above equations results from the fact that the scalar density treats particles and antiparticles the same whereas the vector density distinguishes particles from antiparticles. Thus $${\rm Re}\, (F-\bar{F}) = \frac{3}{2}\frac{\rho_N}{f_{\pi}^2} \omega = (260 \,
{\rm MeV})^2 \left(\frac{\omega}{m_K}\right) \left(\frac{\rho_N}{0.1 / {\rm
fm}^3}\right)$$ which lies between the estimates from Eberhard-Uchiyama and Shuryak-Thorsson. So far, to our knowledge, no one has calculated the imaginary part in this approach.
What happens to the neutral kaons after production in a heavy ion collision? Based on the above analysis of $F-\bar{F}$ we see that the dimensionless ratio $\chi$ is proportional to the nucleon density and has a very small imaginary part relative to the positive real part. Using the results of Eberhard-Uchiyama we get $$\chi = 6.3\times10^{12} \, \left(\frac{\rho_N}{0.1 / {\rm fm}^3}\right) \, .$$ From eq. (9) one finds that $|\alpha_+| \gg |\beta_+|$ and $|\beta_-| \gg
|\alpha_-|$. Hence the eigenstates are $|+\rangle = |1\rangle$ and $|-\rangle =
|2\rangle$ with corrections of order $1/\chi$. As the matter expands the density decreases until $\chi$ becomes very small. Then the eigenstates are the vacuum ones, $|K_L\rangle$ and $|K_S\rangle$. [*If*]{} this evolution is adiabatic it would have extremely interesting consequences. A $K^0$ produced in the collision would evolve into a $K_L$ (rather than into an equal mixture of $K_L$ and $K_S$), and a $\bar{K}^0$ would evolve into a $K_S$. Based on valence quark counting [@chem] one expects that $$\left( \frac{K_L}{K_S} \right)_{\rm observed} =
\left( \frac{K^0}{\bar{K}^0} \right)_{\rm in \; matter} =
\left( \frac{K^+}{K^-} \right) \left( \frac{\pi^-}{\pi^+} \right) \, .$$ That is, the observed ratio of $K_L$ to $K_S$ should be equal to the ratio of $K^0$ to $\bar{K}^0$ produced in dense matter, which then is equal to the ratio of charged kaons times the ratio of charged pions. The latter product of ratios should not change during the very late dilute expansion phase of a heavy ion collision. It has been measured [@AGS] in central Au-Au collisions at the AGS at Brookhaven National Laboratory (E$_{\rm beam}$ = 11 GeV/nucleon) with the value 6. It has also been measured [@SPS] in central Pb-Pb collisions at the SPS at CERN (E$_{\rm beam}$ = 160 GeV/nucleon) with the value 2. Hence, contrary to all other high energy accelerator experiments, the ratio of long-lived to short-lived kaons would be far from one! But does the neutral kaon system evolve adiabatically?
To answer this question, consider what happens in the kaon’s own frame of reference. As time goes on, the surrounding density of matter decreases as $1/t^3$. This naturally follows from dimensional analysis, and it also emerges from a calculation of the free expansion of an ideal relativistic gas [@Dirk]. A good estimate of the local nucleon density is $\rho_N(t) =
\rho_0 (t_0/t)^3$. For numerical estimates we shall use $\rho_0 = 0.1$ nucleons/fm$^3$ and $t_0 = 10$ fm/c, which are characteristic of central collisions between nuclei of atomic number near 200 at the AGS and the SPS [@qm]. Furthermore, the average relative speed in an encounter between a kaon and a nucleon will decrease with time. The reason is that the fireball has an initial radius of $R \approx 10$ fm or so. After an elapsed time $\Delta t = t-t_0$ nucleons with a relative speed greater than $R/\Delta t$ are unlikely to ever encounter the kaon. If we are interested in what happens at late times in the expansion, when the strong and weak interactions affecting the neutral kaon system become comparable, the relevant averaged scattering amplitudes are not those discussed above. Rather, one can use only the s-wave scattering lengths. A compilation of data yields [@a0] $$\bar{f}_0 - f_0 = 0.1 + 0.6 i \; {\rm fm} \, ,$$ with an uncertainty of about 0.1 fm in both the real and imaginary parts. This all leads to the difference of one-body potentials being $$U(t) - \bar{U}(t)
= (7.5+45i)\left( \frac{10 \; {\rm fm/c}}{t} \right)^3 \; {\rm MeV} \, .$$
The problem can be reduced to a two-level Schrödinger equation with a time-dependent complex potential. Taking out the kaon mass leads to $$i\frac{\partial}{\partial t} \psi(t) = H(t) \psi(t) = \left( \begin{array}{cc}
-\frac{i}{4}(\Gamma_S + \Gamma_L) + U(t) &
\frac{1}{2}\Delta m + \frac{i}{4}\Delta \Gamma \\
\frac{1}{2}\Delta m + \frac{i}{4}\Delta \Gamma &
-\frac{i}{4}(\Gamma_S + \Gamma_L) + \bar{U}(t) \end{array}
\right) \psi(t) \, .$$ The instantaneous energy eigenvalues are $$E_{\pm}=\frac{1}{2}(U+\bar{U}) -\frac{i}{4}(\Gamma_S + \Gamma_L)
\pm \sqrt{\left(\frac{1}{2}\Delta m + \frac{i}{4}\Delta \Gamma \right)^2
+ \frac{1}{4} (U-\bar{U})^2} \, .$$ The instantaneous eigenstates are given by eq. (9). The exact solutions can be expanded in terms of these, with time-dependent coefficients $a_{\pm}(t)$, as $$\psi(t) = a_+(t) |+,t\rangle + a_-(t) |-,t\rangle$$ where $$|\pm,t \rangle = \left( \begin{array}{cc} \alpha_{\pm}(t) \\ \beta_{\pm}(t)
\end{array} \right) \exp\left\{ -i\int_0^t E_{\pm}(t') dt' \right\} \, .$$ There exists a critical density $\rho_c$ defined by the condition that $|\chi| = 1$. With the above input its numerical value is $\rho_c = 1.1 \times 10^{-14}$ nucleons/fm$^3$ or about 19 g/cm$^3$. For densities much greater than $\rho_c$ the strong interactions dominate and the eigenstates of the system are eigenstates of strangeness. For densities much less than $\rho_c$ the weak interactions dominate and the eigenstates of the system are eigenstates of CP. The critical region may be conservatively defined as $8 > |\chi| > 1/8$ or $8 > \rho_N/\rho_c > 1/8$.
The picture that emerges is as follows. For $t < 10^5$ fm/c the matter expands freely, with frequent (on the time scale of $1/\Delta m$ and $1/\Delta \Gamma$) interactions of the kaons with the nucleons maintaining the kaons in eigenstates of strangeness, namely, $K^0$ and $\bar{K}^0$. For $10^5 < t < 4\times 10^5$ fm/c the system is in a transition region where $|\chi| \approx 1$, meaning that interactions with the nucleons are comparable in strength with the internal weak interactions of the kaons. In this region there is nothing to prevent transitions between the states. For $t > 4\times 10^5$ fm/c the matter is so dilute that the internal weak interactions of the kaons dominate and they propagate essentially as in vacuum, namely, as $K_L$ and $K_S$. Thus the states of the neutral kaon system evolve adiabatically at both high and low density. The time to pass through the transition region, about $3\times 10^5$ fm/c, is so short compared to the natural oscillation time of the neutral kaons, $1/\Delta m
= 5.6\times 10^{13}$ fm/c, that this transition may be treated with the sudden approximation. It is at the time $t \approx 2\times 10^5$ fm/c and the density $\rho_c \approx 20$ g/cm$^3$ that the strangeness eigenstates $K^0$ and $\bar{K}^0$ decompose into the eigenstates $K_L$ and $K_S$ of CP. This may be shown mathematically from the equations of motion of $a_{\pm}$.[^1] As a consequence of this sudden transition the observed ratio of long-lived to short-lived kaons should be 1. Finally we should remark that the probability for a kaon to decay before the transition region is reached is negligible because the lifetime of even $K_S$ is much greater than the time to reach the critical density.
The phenomenon described here does not happen in elementary particle collisions such as $e^+e^-$, $p\bar{p}$, and $pp$ because the net baryon number is either zero or negligibly small. It should be noted that the critical density of 20 g/cm$^3$ is characteristic of heavy metals[^2], perhaps opening the window on new types of experiments with neutral kaons.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank U. Heinz and P. J. Ellis for useful conversations. J.K. thanks the Institute of Technology at the University of Minnesota for granting a single quarter leave in the spring of 1999 and the Theory Division at CERN for hospitality and support during that time. This work was supported by the US Department of Energy under grant DE-FG02-87ER40328 and by the European Union under a TMR grant.
[99]{}
Most textbooks on particle physics have a discussion of the neutral kaons. An especially nice review is given by K. Kleinknecht in the [*Advanced Series on Directions in High Energy Physics - Vol. 3: CP Violation*]{}, ed. C. Jarlskog (World Scientific, Singapore, 1989).
Particle Data Group, Eur. Phys. J. [**3**]{}, 1 (1998).
E. V. Shuryak, Nucl. Phys. [**A533**]{}, 761 (1991); V. L. Eletsky and B. L. Ioffe, Phys. Rev. Lett. [**78**]{}, 1010 (1997); S. Jeon and P. J. Ellis, Phys. Rev. D [**58**]{}, 045013 (1998); V. L. Eletsky and J. I. Kapusta, Phys. Rev. C [**59**]{}, 2757 (1999).
P. H. Eberhard and F. Uchiyama, Nucl. Instr. and Meth. in Phys. Res. A [**350**]{}, 144 (1994).
E. Shuryak and V. Thorsson, Nucl. Phys. [**A536**]{}, 739 (1992).
D. B. Kaplan and A. E. Nelson, Phys. Lett. B [**175**]{}, 57 (1986); A. E. Kaplan and D. B. Nelson, Phys. Lett. B [**192**]{}, 193 (1987); H. D. Politzer and M. B. Wise, Phys. Lett. B [**273**]{}, 156 (1991).
G. Q. Li, C. M. Ko, and Bao-An Li, Phys. Rev. Lett. [**74**]{}, 235 (1995); R. Knorren, M. Prakash, and P. J. Ellis, Phys. Rev. C [**52**]{}, 3470 (1995).
This relation may also be obtained under the assumption of chemical equilibrium if one neglects the small effect of quantum statistics: it is basically just the ratio of fugacities. It is known that chemical equilibrium is attained in these collisions at the accuracy of 10-20%. See, for example: H. Dobler, J. Sollfrank, and U. Heinz, preprint nucl-th/9904018, April 1999; P. Braun-Munzinger, I. Heppe, and J. Stachel, preprint nucl-th/9903010, March 1999. The naive expectation that the average number of $K^0 + \bar{K}^0$ is the same as the average number of $K^++K^-$ in chemical equilibrium is not true in general.
L. Ahle [*et al.*]{} (E-802 Collaboration), Phys. Rev. C [**58**]{}, 3523 (1998).
I. G. Bearden [*et al.*]{} (NA44 Collaboration), Nucl. Phys. [**A638**]{}, 419c (1998).
C. Greiner and D. Rischke, Phys. Rev. C [**54**]{}, 1360 (1996).
For a review of the field see the proceedings of the Quark Matter conferences, the most recent in print being [*Proceedings of Quark Matter ‘97*]{}, ed. T. Hatsuda, Y. Miake, S. Nagamiya, and K. Yagi, Nucl. Phys. [**A638**]{}, 1 (1998).
T. Barnes and E. S. Swanson, Phys. Rev. C [**49**]{}, 1166 (1994); O. Dumbrais [*et al.*]{}, Nucl. Phys. [**B216**]{}, 277 (1982). These values were used to study kaonic atoms and kaon condensation by: C.-H. Lee, G. E. Brown, D.- P. Min, and M. Rho, Nucl. Phys. [**A585**]{}, 401 (1995).
[^1]: However, the analysis is more complicated than that given in textbook discussions of time-dependent perturbation theory and the adiabatic and sudden approximations because the Hamiltonian is not Hermitian, and the instantaneous eigenstates are not orthogonal in the transition region.
[^2]: In this case $f-\bar{f}$ is the difference of scattering amplitudes on a nucleus and $\rho_N$ is replaced by the density of these nuclei
|
---
author:
- |
Christophe Grojean\
Department of Physics University of California\
Berkeley, CA 94720 USA, and\
Theoretical Physics Group, LBL\
Berkeley CA 94720, USA\
E-mail:
- |
Fernando Quevedo and Ivonne Zavala C.\
Centre for Mathematical Sciences, DAMTP, University of Cambridge,\
Cambridge CB3 0WA UK\
E-mail: ,
- |
Gianmassimo Tasinato\
SISSA, Via Beirut 2-4, 34013 Trieste, and\
INFN, Sezione di Trieste, Italy\
E-mail:
title: 'Branes on charged dilatonic backgrounds: self-tuning, Lorentz violations and cosmology'
---
Introduction
============
The brane-world scenario is providing new ideas to approach old questions such as the hierarchy problem, the cosmological constant problem and early universe cosmology [@early]–[@ccXdim]. The simple idea that our Universe is a brane trapped in a higher dimensional space allows for a great amount of possible realizations depending on the distribution of matter on the brane and the bulk as well as their relative dimensionality.
As usual, simplicity has been the main guideline when considering explicit realizations of the brane world. Most discussions in the current literature refer to 3-branes inside a five dimensional bulk with only gravity propagating in the bulk. Adding extra fields to solve some of the problems such as radius stabilization and to ameliorate the cosmological constant problem has also been considered [@ccXdim]–[@CEG]. But in principle there is a great degree of arbitrariness and in order to go beyond the simplest realizations we need to have a general guideline.
Clearly the best motivated brane world scenarios are those that can naturally be obtained from string theory [@fromstrings]. There are actually at present few explicit realizations of quasi-realistic brane world models derived from string theory [@models]. We can try to extract the general properties of those models to incorporate them on a particular framework to approach different phenomenological and cosmological properties of these scenarios.
Following this guideline we will consider in this note a system consisting of a $q$-brane[^1] singularity in a $d=n+q+2$ dimensional bulk with gravity, dilaton and antisymmetric tensor fields. We find explicit solutions for the field equations for which the [$n$-dimensional]{} slices of the spacetime have constant curvature $n(n-1)k$, $k=1,0,-1$ (for a related discussion see for instance [@mirjam2]). A motivation for the study of these geometries is their potential application to brane cosmology. One of the most interesting results emerging in this field has been the realization, through Birkhoff’s theorem, that an additional $n$-brane[^2] carrying ordinary matter and gravitationally coupled to the bulk, when moving in a static black hole background, actually feels a time dependent cosmology [@kraus; @KeKi; @BCG]. Therefore it is clear that the solutions presented in this work provide interesting brane cosmology backgrounds even in the regions where they are static.
=-1The causal structure of the spacetime depends on the topology of the dimensions parallel to the external $n$-brane. The $k=1$ case has been studied in the past and it corresponds to black $q$-branes [@HS], with $n+q+2=10$. The global geometry consists on an asymptotically flat spacetime with a horizon and two singularities.[^3] The $k=0,-1$ solutions we find are not black $q$-branes. They have an interesting global structure with a horizon and only one singularity at the origin. The region between the singularity and the horizon is static, unlike the standard black hole case. Beyond the horizon it becomes time dependent, therefore corresponding to a cosmological solution for which there is no singularity at any surface of constant time.
For the $q=0$ case, the motion of the external $n$-brane (a codimension one brane) can be studied easily using the usual Israel junction equations. We mostly concentrate in the regions of the bulk spacetime which are static and find the possible places where the brane can be located, cutting the space in the transverse dimension such that a $\mathbb{Z}_2$ symmetry around the brane can be imposed and a finite Planck scale in four dimensions is guaranteed.
=-1We find that for $k=1$ the $n$-brane can be located in the region of the bulk spacetime which is outside of the singularities and the horizon and therefore the extra dimension can be naturally restricted to be the region between the black hole horizon and the location of the brane, avoiding naked singularities completely. The matter in the brane is not exotic because the value of the parameter $\omega$ relating the pressure and energy density on the brane ${p}=\omega
\rho$ lies in the physically allowed region $0>\omega>-1$.
For $k=-1,0$ the situation is different. The static region lies between the singularity and the horizon, therefore there are two possibilities. The space can be taken between the brane and the horizon or between the brane and the singularity. In the first case it turns out that the energy density of the brane has to be negative and in the second it is positive. In both cases $\omega$ can take physically allowed values.
We will consider in more detail the simplest system of a 3-brane in a five-dimensional bulk with dilaton and a two-index antisymmetric tensor field $B_{\mu\nu} $. Since in five-dimensions the antisymmetric tensor is dual to a vector field $\partial_\mu A_\nu =
\epsilon_{\mu\nu\rho\sigma\tau}\partial^\rho B^{\sigma\tau}$, this situation is equivalent to consider a gauge field $A_\mu $ instead of $B_{\mu\nu} $. Interestingly enough the introduction of a dilaton field has been studied to ameliorate the cosmological constant problem [@ADKS; @KSS] and more recently a similar proposal was made regarding the introduction of a gauge field [@CEG]. It is then natural to consider both fields together.
This generalization improves the situation with only one of the fields in several respects regarding the self-tuning of the cosmological constant.
- For vanishing gauge fields we reobtain the solution of [@ADKS; @KSS] as a particular case of our $k=0$ solution. However our solutions also include possibilities not considered in [@ADKS; @KSS] since we consider asymmetrically warped geometries which are not 4D Poincaré invariant from the bulk point of view but are so at each location of the brane, by suitably redefining the speed of light. These cases share the property with [@ADKS; @KSS] that there are always naked singularities and require a fine-tuning of the dilaton couplings and the parameter $\omega$ defining the equation of state of the matter on the brane.
- For non vanishing gauge fields and $k=0$, $\omega$ is still a constant related to the dilaton couplings. Indeed, contrary to the model [@CEG] with a pure Reissner-Nördstrom black hole in the bulk, the presence of the scalar field requires a fine-tuning of the dilaton couplings as in the original scalar self-tuning models [@ADKS; @KSS]. The improvement on the self-tuning relies on the fact that, as mentioned above, the naked singularities can be avoided as long as the energy density on the brane is negative. Also $\omega$ can lie in the physically allowed region.
Since our solutions correspond to asymmetrically warped metrics they may induce explicit violations of Lorentz invariance on the brane by having gravitational waves moving faster through the bulk than through the brane. We study the variation of the speed of light with the location of the brane and find that gravitational waves move faster through the bulk only in the cases with naked singularities.
The organization of the paper is as follows. We present in section \[sec2\] the general solution of the bulk equations of motion for gravity coupled to the dilaton field and an antisymmetric tensor of rank $q+2$, for the cases $k=0,\pm 1$, generalizing the results of [@HS] (see also [@GM]). In the next section we introduce the external $n$-brane with matter in a perfect fluid for the particular case of codimension one brane. We obtain the junction conditions and then obtain the geometry defined by the position of the $n$-brane in the bulk background. We specialise in section \[sec4\] to the five-dimensional case discussing in detail the issues of self-tuning and violation of Lorentz invariance in these backgrounds. Finally we present our conclusions and discuss some issues related to the cosmological implications of the bulk solutions.
General charged dilatonic $q$-brane background {#sec2}
==============================================
In [@kraus; @KeKi] it was found that the cosmology of a brane inside a higher dimensional bulk spacetime can be studied by considering a *static* bulk geometry with the same spatial structure, i.e., a spacetime with a constant curvature, maximally symmetric subspace of the same dimensionality as the brane. For an observer on the brane, the movement of the brane in the static geometry becomes an evolving brane Universe. This is a direct consequence of Birkhoff’s theorem in more than 4 dimensions. In [@BCG] it was explicitly shown how to map the cosmological and static metrics for the case of a five-dimensional bulk without extra matter fields. Similar results hold when there are background gauge fields [@CEG]. Notice that this is not always the case, we may have some cosmological solutions that may not be mapped to static ones in more general cases when other fields are included. In particular, in presence of a scalar field, Birkhoff’s theorem does not hold [@bgqtz]. Nevertheless, it is still possible to find some background solutions that remain static or, as we will see, that depend on the time coordinate only. Even if these solutions are no longer the most general solutions in the bulk, they are interesting on their own and we will focus on their study in this paper.
In this section, we generalize the electric black $q$-brane solution studied by Horowitz and Strominger in [@HS] for positive and constant spatial curvature ($k=1$) to the case of arbitrary dimensions and arbitrary spatial curvature $k=0,\pm 1$.[^4]
We consider the coupling of gravity to a dilaton field and an antisymmetric tensor with the following action in the Einstein frame:[^5] $$\label{generalaction}
S= \int d^{n+q+2}x \sqrt{g} \left( \alpha R - \lambda (\partial
\phi)^2 - \eta e^{-\sigma \phi} F_{q+2}^2 \right),$$ where $\phi$ is the dilaton field, $F$ is a field strength $(q+2)$ form. We have left the couplings $\alpha, \lambda, \eta, \sigma$ arbitrary. Varying the action (\[generalaction\]) yields the following equations of motion: $$\begin{aligned}
\alpha G_{\mu\nu} &=& -\frac{1}{2} \lambda(\nabla\phi)^2\,g_{\mu\nu} +
\lambda\, \nabla_{\mu}\phi\, \nabla_{\nu}\phi +
\nonumber\\&&
+\eta\,e^{-\sigma\phi}
\left( (q+2) F_\mu{}^{\lambda_1\ldots\lambda_{q+1}}
F_{\nu\lambda_1\ldots\lambda_{q+1}} - \frac{1}{2}g_{\mu\nu} F^2
\right),\qquad
\label{einstein}\\
2\,\lambda\,\nabla^2 \phi &=&-\sigma\,\eta \, e^{-\sigma\phi} F^2\,,
\label{dilaton1}\\
\nabla_{\mu_1} \left( e^{-\sigma\phi}
F^{\mu_1\mu_2\dots\mu_{q+2}}\right) &=& 0\,.\end{aligned}$$ We are looking for solution of this system for which the spacetime is a (warped) product of a $q$ dimensional space giving the dimensionality of the $q$-brane singularity ($q=0$ is simply a black hole) and an $n+2$ dimensional spacetime where the $n$-dimensional slices correspond to spaces of constant curvature. The electrically charged solution is given by $$\begin{aligned}
ds^2 &=& h_-^{\frac{-4 \lambda q (n-1)^2 b}{\alpha n (n+q) \Sigma^2}}
\left( -h_+h_-^{1-(n-1) b}dt^2 +h_+^{-1}h_-^{-1+b} dr^2 +r^2h_-^{b}
dx^2_{n,k} \right)+
\nonumber\\&&
+\, h_-^{\frac{4 \lambda (n-1)^2 b}{\alpha (n+q) \Sigma^2}} dy^2_{q}\,,
\label{eq:metric}\\
\phi&=&\frac{(n-1) \sigma b}{\Sigma^2} \ln h_-\,,
\label{eq:dil}\\
F_{try_1\ldots y_q} &=&\frac{Q \epsilon_{y_1\ldots y_q} }{r^{n}}\,,
\qquad \epsilon_{y_1\ldots y_q}=\pm 1\,,
\label{eq:F}\end{aligned}$$ where $dx^2_{n,k}$ is an $n$-dimensional spatial maximally symmetric metric of constant curvature $n(n-1)k$, $k=0,\pm 1$.[^6] The harmonic functions, $h_\pm$, depend on two constants of integration, $r_\pm$, and are given by:[^7] $$\label{eq:h+-def1}
h_+ = 1-\left(\frac{r_+}{r}\right)^{n-1}\,, \qquad
h_- = k- \left(\frac{r_- }{ r}\right)^{n-1}\,.$$ We have defined the quantities $\Sigma$ and $b$, in terms of the different parameters of the action, by the following expressions $$\begin{aligned}
\Sigma^2 &=& \sigma^2 + \frac{4 \lambda}{\alpha} \frac{q(n-1)^2}{n(n+q)}\,,
\label{eq:Sigma}\\
b &=& \frac{2\alpha n \Sigma^2}{(n-1)(\alpha n \Sigma^2+ 4(n-1)\lambda)}\,.\qquad
\label{eq:b}\end{aligned}$$ The electric charge, $Q$, of this background is related to the two constants of integration by $$\label{eq:Q}
Q^2 = \frac{4n(n-1)^2 \alpha\, \lambda
(r_+r_-)^{n-1}}{(q+2)!\,\eta\,(\alpha n \Sigma^2 +
4(n-1)\,\lambda)}\,,$$ while another combination of the constants of integration will be interpreted as the “mass” of the background. The solution with a vanishing electric charge, which will be of physical relevance concerning the problem of the cosmological constant, will be presented separately in the section \[sec:selftuning\].
Even if formally satisfying the equations of motion, this solution does not make sense physically in the regions where $h_-$ becomes negative, i.e., $r<r_-$ for $k=1$ and anywhere for $k=0,-1$, because $h_-$ appears in the solution with non-integer powers, contrary to the case of a simple Schwarzschild or Reissner-Nördstrom black hole. However, it is easy to construct a new solution that overcomes this problem:[^8] the solution will still be given by the expressions (\[eq:metric\])–(\[eq:F\]) with the quantities $\Sigma, b, Q$ still related to the parameters of the action by (\[eq:Sigma\])–(\[eq:Q\]) but the definitions (\[eq:h+-def1\]) of the functions $h_\pm$ has to be replaced by $$\label{eq:h+-def2}
h_+(r) = s(r) \left(1-\left(\frac{r_+}{r}\right)^{n-1} \right),
\qquad
h_-(r) = \left|k-\left(\frac{r_-}{r}\right)^{n-1}\right|,$$ where $$s(r)={\rm sgn} \left(k-\left(\frac{r_{-}}{r}\right)^{n-1}\right).$$
Now, it is important to observe some of the geometrical characteristics of this solution. First of all, it is not hard to find that for all $k$, $r_+$ is always a horizon, as usual. The other properties depend on the value of the constant curvature $k$:
- ***k=1:*** By computing the scalar curvature, it is possible to realize that $r=r_-$ is a scalar singularity for an arbitrary value of $b$.[^9] The background is asymptotically flat and corresponds (for $d=10$), written in the Einstein frame, to the black $q$-brane solution constructed by Horowitz and Strominger [@HS].
- ***k=0* and *k=–1:*** It is clear from the expressions for $h_-$ above, that $r=r_-$ is just a regular point, whereas $r=r_+$ remains a horizon. Furthermore, the coordinate $r$ becomes timelike in the region $r>r_{+}$ but remains spacelike for $r<r_+$, exactly the opposite of Schwarzschild black hole. The $q$-dimensional singularity at $r=0$ is then timelike. Therefore we are only interested on the static region of spacetime, and we will locate the external $n$-brane on the region $0<r<r_+$. However we would like to point out here that the region $r>r_+$ is interesting [per se]{} for cosmology. In this region $r$ becomes the time coordinate and the horizon $r_+$ is a past Cauchy horizon for this cosmological solution. Unlike the standard cosmological singularity $r=0$ becomes a timelike singularity behind the horizon, resembling the ‘white hole’ region of the Reissner-Nördstrom solution, but having a single horizon instead of two. Unlike the Schwarzschild solution, since the singularity is timelike it may be avoided by a future directed timelike curve in the region beyond the horizon.
We show in figures \[fig1\] and \[fig2\] the corresponding Penrose diagrams for these geometries illustrating the relevant regions.
A detailed study of the external $n$-brane inside the cosmological regions is beyond the scope of the present article and it is left to a future publication [@bgqtz]. In the next section we will restrict only to the static solutions, $r_\mp<r_\pm<r$ or $r<r_\mp<r_\pm$ for $k=1$ and $0<r<r_+$ for $k=0$ and $k=-1$, keeping in mind their possible relevance for cosmology as well as the self-tuning mechanism for the cosmological constant and the possible bulk violations of the 4D Lorentz invariance.
Before finishing this section a comment is in order. In the general action above we have not included a cosmological constant term in the bulk. The reason is the following. The term would be of the form $V
e^{\nu\phi}$ with $V$ some constant. However it has been shown [@wiltshire] that for $\nu\neq 0$, and for $\nu=0$ but $V >
0$, there are no solutions of the field equations consistent with the symmetries we imposed. The only possibility would be to have $\nu=0$ and $V<0$ for which some numerical solutions are known, or to restrict to the case of no dilaton couplings $b=0$ which would reduce to the case with only gauge fields and no dilatons already considered in the literature. There still remains the possibility of a nontrivial dilaton potential $V(\phi)$ with a stationary point for which there should be solutions [@wiltshire]. This possibility will not be considered here. However, it should be mentioned that, in absence of gauge fields but with a Liouville potential for the scalar field, a dilatonic domain wall ($k=0$) solution has been constructed by Cai and Zhang in [@CZ].
Codimension one brane worlds {#sec3}
============================
In order to incorporate the external $n$-brane in the bulk geometry described before we will restrict to the case of a codimension one brane world which corresponds to the case $q=0$, that is a point-like singularity. Therefore we are restricted to the case of an $n$-brane in $d=n+2$ spacetime dimensions.
To be concrete, let us define the starting action for this model. It consists of the sum of bulk action as before (setting $q=0$ and fixing the values of $\alpha$, $\lambda$ and $\eta$) plus a brane part. We have, in the Einstein frame: $$\label{actionn}
S = \frac{1}{2 \kappa_{n+2}^2} \int {d^{n+2}x \, \sqrt{g_{n+2}} \,
\left( R -\frac{1}{2}(\nabla \phi)^{2} - 2\kappa_{n+2}^2 e^{-\sigma
\phi} F_{\mu\nu}F^{\mu\nu} \right)} + S_{\rm br} + S_{\rm G.H.} \,,$$ with $$S_{\rm br} = - \int {d^{n+1}x \,\sqrt{g_{n+1}\, f^{n+1}(\phi)} \,
{\mathcal{L}}_m (\psi, \nabla\psi, g_{ab} f(\phi))}\, z(\phi) \, ,$$ as before, $\phi$ is the dilaton field, while $F_{\mu\nu}=\partial_\mu
A_\nu-\partial_\nu A_\mu$ is the field strength tensor of a gauge field, $A_\mu$; $S_{G.H.}$ is the Gibbons-Hawking term and ${\mathcal
{L}}_m$ is the lagrangian for the matter fields, $\psi$, living on the brane which we model as a perfect fluid; $\kappa_{n+2}^2$ is the effective Newton constant in $(n+2)$ dimensions. The matter on the brane couples to the bulk through gravity or [via]{} the dilaton only. The conformal coupling of the dilaton to matter is specified by the function $f(\phi)$ and we have also introduced a multiplicative coupling through the function $z(\phi)$. Varying the action (\[actionn\]) we will obtain an extra term coming from the brane source in the equations of motion for the metric and the dilaton. They are as follows ($r=\mathcal{R}(\tau)$ is the position of the brane): $$\begin{aligned}
G_{\mu\nu} &=& -\frac{1}{4}(\nabla\phi)^2\, g_{\mu\nu} +\frac{1}{2}\,
\nabla_{\mu}\phi\, \nabla_{\nu}\phi
\nonumber \\&&
+\,2\kappa_{n+2}^2\,e^{-\sigma\phi} \left( 2 F_\mu{}^\lambda
F_{\nu\lambda} - \frac{1}{2} g_{\mu\nu}F^2 \right) +\kappa_{n+2}^2
\sqrt{\frac{g_{n+1}}{g_{n+2}}} T_{ab} \delta^a_\mu\delta_\nu^b
\delta(r-{\mathcal R}(\tau))\,,
\label{eq:BraneEinstein}\\
\nabla^2 \phi & = & -2\kappa_{n+2}^2\,\sigma e^{-\sigma\phi} F^2 -
2\kappa_{n+2}^2 \sqrt{\frac{g_{n+1}}{g_{n+2}}}
\left((np-\rho)\frac{f'}{2f}+\omega_{\mathcal{L}}\rho \frac{z'}{z}
\right) \delta (r-{\mathcal R}(\tau))\,.\qquad
\label{eq:BraneDilaton}\end{aligned}$$ $T_{ab}$ is the brane stress-energy tensor coupled to the induced metric on the brane and it is given by $$\label{energytensor4d}
T^{a}{}_{b} = \frac{2}{\sqrt{g_{n+1}}}\, \frac{\delta S_{\rm
br}}{\delta g_{ac}} \, g_{cb} = {\rm diag} (-\rho, p,\ldots, p) =
f^{(n+1)/2}(\phi) z(\phi)\, {\rm diag} (-\bar{\rho}, \bar{p},\ldots,
\bar{p}) \,,$$ where $\bar{\rho}$ and $\bar{p}$ would be the energy density and pressure of the brane in absence of coupling to the dilaton, while the physical energy density and pressure coupled to the induced metric in the Einstein frame are $\rho$ and $p$. The action for the matter on the brane is expressed in terms of the energy density through the parameter $\omega_{\mathcal{L}}$:[^10] $$f^{(n+1)/2}(\phi) \, z(\phi) \, {\mathcal{L}}_m (\psi, \nabla\psi,
g_{ab} f(\phi)) = -\omega_{\mathcal{L}}\, \rho\, .$$ Notice that the energy-momentum conservation reads $$\nabla_\mu T^{\mu\nu} =0\,;$$ where $T^{\mu\nu}$ is the total $n+2$-dimensional stress-energy tensor.
Ignoring for the moment the presence of the brane, the solution of the bulk corresponds exactly to the $q=0$ case of the first section. This is given by $$\begin{aligned}
ds^2&=& -h_+ h_-^{1-(n-1)b}dt^2 +h_+^{-1} h_-^{-1+b} dr^2 +r^2 h_-^{b}
dx^2_{n,k} \,,
\\
\phi&=&\frac{(n-1) b}{\sigma} \ln h_-\,,
\\
F_{tr}&=&\frac{Q}{r^{n}}\,,
\label{eq:metric(n+2)D}\end{aligned}$$ where now the expressions for $b$ and $Q$ read $$\begin{aligned}
b &=& \frac{2 n \sigma^2}{(n-1)(n\sigma^2+ 2(n-1))} \,,
\label{eq:bsigma}\\
Q^2 &=& \frac{n(n-1)^2 (r_+r_-)^{n-1}}{2\kappa_{n+2}^2( n \sigma^2 +
2(n-1))} \,.\end{aligned}$$ And the functions $h_\pm$ are the sign amended harmonic functions (\[eq:h+-def2\]). Note that depending on the value of the dilaton to the gauge field, the parameter $b$ varies between $0$ and $2/(n-1)$. Thus, even in the cases $k=0,-1$ where $h_-$ is decreasing with $r$, the spatial warp factor $r^2h_-^b$ always remains increasing: there is no bounce.
Introducing the brane
---------------------
Let us now introduce a dynamical $n$-brane moving into the static $n+2$ bulk described in the last section. The brane will be separating two regions of the above discussed background; that is, we are gluing two slices of the metric together at the position of the brane. We will assume that our solutions possess a $\mathbb{Z}_2$ symmetry “centered” at the brane (this means essentially we change $r\rightarrow \mathcal{R}^2/r$ on the two sides of the brane and therefore identify the two sides of the spacetime in that dimension, for a detailed discussion see [@CEG; @grojean]).
In order to do this, we have to satisfy the Israel junction conditions at the brane as follows [@israel]. Consider a general boundary $X^\mu$, parametrized by the cosmic time $\tau$, $$X^\mu=(t(\tau), {\mathcal R}(\tau), x_1, \dots, x_n)\,,$$ such that the induced metric on the boundary becomes $$\label{branemetricn}
ds^2 = -d\tau^2 + a^2(\tau) dx_{n,k}^2\,,$$ with $a(\tau)$ is the scale factor on the brane, i.e. $$a(\tau) = r\, h_-^{b/2}(r)_{\bigl|r={\mathcal
R}(\tau)}$$ and the cosmic time $\tau$ is defined such that (a dot means derivative with respect to the proper time) $$\label{tpunton}
- h_+ h_-^{1-(n-1)b} \dot t^2 + h_+^{-1} h_-^{b-1}\dot {\mathcal
R}^2 = -1\,.$$ The components of the extrinsic curvature have to satisfy the junction conditions: $$\left[ K_{ab} \right]_-^+ =- \kappa^2_{n+2}\left( T_{ab} - \frac{1}{n}
g_{ab} T^c{}_c \right),$$ while the junction condition for the dilaton derives from (\[eq:BraneDilaton\]) and reads ($u$ is the unit normal to the brane) [@ScalarGravity]: $$\left[ u.\partial \phi\right]_-^+ = 2 \kappa^2_{n+2}
\left(-(np-\rho)\frac{f'}{2f}-\omega_{\mathcal{L}}\,\rho\,
\frac{z'}{z} \right).$$ Notice that for the gauge field there are no junction conditions since it couples only to the bulk and not to the brane. With a $\mathbb{Z}_2$ symmetry between the two sides of the brane the junction equations are: $$\begin{aligned}
\rho &=& \mp 2n\kappa^{-2}_{n+2} \left( \frac{1}{{\mathcal R}} +
\frac{b\,h_-'}{2\,h_-} \right) h_+^{1/2} h_-^{(1-b)/2}
\sqrt{1+h_+^{-1} h_-^{-1+b} {\dot{{\mathcal R}}}^2} \,,\qquad~
\label{eq:Jump1}\\
n p + (n-1) \rho &=&\pm 2n\kappa^{-2}_{n+2} \frac{h_+^{-1/2}
h_-^{(b-1)/2} }{\sqrt{1+h_+^{-1} h_-^{-1+b} {\dot{{\mathcal R}}}^2}}
\Biggl( {\ddot{{\mathcal R}} - \frac{(n-2)b h_-'}{2\,h_-}
{\dot{{\mathcal R}}}^2}
\nonumber\\&&
\qquad+\frac{1}{2} h_-^{1-b} h_+' + \left(\frac{1}{2} - \frac{n-1}{2}b\right) h_+
h_-^{-b} h_-' \Biggr),
\label{eq:Jump2}\\
(np-\rho)\frac{f'}{2f} +\omega_{\mathcal{L}}\rho\frac{z'}{z} &=&
\mp (n-1)\kappa^{-2}_{n+2} \frac{b}{\sigma} h_-' h_+^{1/2}
h_-^{-(1+b)/2} \sqrt{1+h_+^{-1} h_-^{-1+b} {\dot{{\mathcal R}}}^2}\,.
\label{eq:Jump3}\end{aligned}$$ Note that due to the absence of bounce in the spatial warp factor $r^2
h_-^b$, there is a simple connection between the sign of $\rho$ and the region of the spacetime cut by the $\mathbb{Z}_2$ symmetry: for positive (negative) energy density (lower (upper) signs in the previous eqs.) we keep the interior (exterior) region of the background, $r<{ \mathcal R}$ ($r>{ \mathcal R}$).
Even if quite messy, these junction equations have nice physical interpretation. Indeed a first combination of (\[eq:Jump1\])–(\[eq:Jump3\]) gives us the (non)conservation equation for the energy on the brane, $$\label{eq:conservation}
\dot{\rho} + n (\rho + p) H = -\left( (np-\rho)\frac{f'}{2f} +
\omega_{\mathcal{L}}\,\rho\, \frac{z'}{z} \right)\dot{\phi} \,,$$ where $H$ is the Hubble parameter, i.e., the time variation of $a={
\mathcal R}\, h_-^{b/2}$, the scale factor on the brane: $$\label{hubble}
H(\tau) = \frac{\dot a(\tau)}{a(\tau)} = \left( \frac{1}{{ \mathcal
R}}+\frac{b\,h_-'}{2\,h_-} \right)\, \dot{{ \mathcal R}} \,.$$ Another combination gives a Friedmann-type equation that relates the Hubble parameter to the energy density on the brane: $$\label{eq:Friedmann}
H^2 = \frac{\kappa_{n+2}^4}{4 n^2}\,\rho^2 - \left( \frac{1}{a(\tau)}+
\frac{b\,h_-'}{2\,h_-^{1+b/2}} \right)^2 h_+h_- .$$ Finally a third combination will characterize the equation of state of the matter on the brane: $p=\omega \rho$ with $$\label{eq:EqState}
\omega = \left( \frac{(n-1)\, b\, h_-'}{2n\, \sigma\, h_-} \left(
\frac{1}{{\mathcal R}} + \frac{b\, h_-'}{2\, h_-} \right)^{-1}
-\omega_{\mathcal{L}} \frac{z'}{z} +\frac{f'}{2\,f} \right)
\frac{2\,f}{n\,f'} \,.$$ In general the position of the brane will be time-dependent and so also the equation of state will be. In the interesting case of exponential couplings to the dilaton, $f=f_0\, \exp(\beta \phi)$ and $z=z_0\, \exp(\gamma \phi)$, clearly one could obtain a constant $\omega$ by tuning the parameters of the action such that $b=0$ or either by choosing a constant of integration $r_-=0$. Surprisingly when the curvature vanishes, $k=0$, the previous formula becomes: $$\label{eq:JumpW}
\omega = -\frac{2(n-1)^2\, b}{n^2\, \beta\sigma\,(2-(n-1)b)}
+\left(1-\frac{2\omega_{\mathcal{L}} \gamma}{\beta}\right) \frac{1}{n}\,,$$ describing an equation of state that remains constant. Since $b$ is related through (\[eq:bsigma\]) to the parameters that enter in the action, this expression actually fixes completely the equation of state from the original parameters of the lagrangian, and once specifying those parameters there is only one possible equation of state allowed. We may also work backwards and find which couplings in the action allow a particularly interesting equation of state (say $\omega=-1$). This is the same fine-tuning of the dilaton couplings as in the original scalar self-tuning models [@ADKS; @KSS]. We will come back to this issue in the next section.
It is easy to see that the equations (\[eq:conservation\]), (\[hubble\]) and (\[eq:Friedmann\]) reduce to the well known expressions for the case of no dilaton coupling ($b=0$) and/or no gauge field coupling ($r_+=0$).
Before describing in detail the geometry of the brane-bulk system, let us rewrite the (non)conservation eq. (\[eq:conservation\]) in a more usual form at least in absence of multiplicative coupling to the dilaton ($z=1$). Indeed this equation takes a more simpler form in the Jordan frame defined by a Weyl rescaling of the metric with the conformal coupling to the dilaton: $$d \bar{s}^2 = f(\phi)\, ds^2 \,,\quad \hbox{i.e.,} \quad d\bar{\tau}^2
= f(\phi)\, d\tau^2 \quad \hbox{and} \quad \bar{a}^2 = f(\phi)\, a^2\,.$$ The energy density and pressure coupled to this metric are precisely the quantities $\bar{\rho}$ and $\bar{p}$. Then it is easy to prove that the (non)conservation equation reads: $$\frac{d}{d\bar{\tau}} \bar{\rho} + n(\bar{\rho}+\bar{p}) \bar{H} = 0
\qquad \hbox{with}\qquad \bar{H} = \frac{1}{\bar{a}} \,
\frac{d\bar{a}}{d\bar{\tau}}\,.$$ In the presence of a multiplicative coupling, $z\neq 1$, the right hand side would not vanish, but as soon as $z=1$, our model is simply a Brans-Dicke theory with localized matter on a brane and it is well known that the usual conservation equation holds in the Jordan frame.
Geometry of the brane-bulk system
---------------------------------
Now we have to decide which part of the space-time to keep due to the $\mathbb{Z}_2$ symmetry we are requiring in order to guarantee a compact extra dimension and then finite four-dimensional Planck scale. This will be defined by the normal vector that we choose in the calculation of the extrinsic curvature and it is reflected in the sign of the energy density, as we have already explained from the junction conditions. If the normal vector points inwards, the energy density is positive whereas for outwards normal vector it is negative. So if ${\mathcal R}(\tau)$ represents the location of the brane, we can glue two interior, $r<{\mathcal R}$ (exterior, $r>{\mathcal R}$), regions by taking a positive (negative) energy density.
There will be three (two) zones (see figures \[fig1\] and \[fig2\]), defined by the black hole/0-singularity geometry, where the brane can be located for $k=1$ ($k=0,-1$), in either case, we take two exterior or interior regions. Let us analyze this point in detail.
- ***k=1* case**
\(I) $r_\mp < r_\pm < {\mathcal R(\tau)}$
.5cm a) $\rho<0$ Glue two exterior regions with no horizon or singularities in it. So extra space dimension is infinite.
.5cm b) $\rho>0$ Glue two interior regions which contain the two singularities $r=0$, $r_-$, protected by the horizon at $r_+$ if $r_+>r_-$ in which case one can naturally cut the space at the horizon and avoid in a natural way the singularities with a positive energy density on the brane. However, if $r_+<r_-$ from the brane point of view, one will see the singularity $r_-$ and not the horizon hidden behind it.
\(II) $r_- < {\mathcal R(\tau)} < r_+$
.5cm a) $\rho<0$ Glue two exterior regions which include the horizon at $r_+$ and no singularities at all.
.5cm b) $\rho>0$ Glue two interior regions which contain the two naked singularities $r=0$, $r_-$.
(II’) $r_+ < {\mathcal R(\tau)} < r_-$
.5cm a) $\rho<0$ Glue two exterior regions which include the the naked singularity at $r_-$.
.5cm b) $\rho>0$ Glue two interior regions which contain the horizon at $r_+$.
Notice that in both cases (II) and (II’) $r$ is a timelike coordinate in the interval $[r_\mp,r_\pm]$ and so the space-time there is cosmological. The singularity at $r_-$ is a null-singularity.
\(III) ${\mathcal R(\tau)} < r_\mp < r_\pm$
.5cm a) $\rho<0$ The metric becomes again static and one can glue two exterior regions with the singularity at $r_-$ and the horizon at $r_+$ that shields the singularity as long as $r_+<r_-$. However, if $r_-<r_+$ then from the point of view of the brane, one will see a naked singularity and not the horizon, located behind the singularity! In both situations, one can then cut the space at the singularity or the horizon to get a finite extra dimension.
.5cm b) $\rho>0$ Glue two interior regions which contain the naked singularity at $r=0$. One can cut the space at the singularity there.
In some cases we can see that it is possible to cut the space in such a way that naked singularities are avoided, thus getting rid of the problems described for example in [@ADKS; @FLLN]. For instance we can choose naturally the brane as described in (I)b: we then require $r_-<r_+<{\mathcal R(\tau)}$ and a positive energy density.
- ***k=0* and *k=–1* cases**
\(I) $r_+ < {\mathcal R(\tau)}$
.5cm a) $\rho<0$ The metric in this region has become time-dependent and is therefore cosmological. One can still glue two exterior regions with no horizon or singularities in it.
.5cm b) $\rho>0$ Glue two interior regions which contain the singularity at $r=0$ protected by the horizon at $r_+$. The bulk metric is again cosmological.
\(II) ${\mathcal R(\tau)} < r_+$
.5cm a) $\rho<0$ The metric becomes again static and one can glue two exterior regions keeping the horizon at $r_+$ and then cutting there the space to obtain a finite extra dimension.
.5cm b) $\rho>0$ Glue two interior static regions which contain the naked singularity at $r=0$. One can cut the space at the singularity getting a finite extra dimension.
0.3 cm There are two possibilities to obtain a finite extradimension and thus a finite Planck scale on the brane: putting the brane at ${\mathcal R}\le r_+$ and, taking a negative energy density and avoiding the singularity at $r=0$ as described in (II)a; or taking a positive energy density but dealing again with a naked singularity as described in (II)b. We will see later, that any of these choices is perfectly consistent with the positivity theorems of the energy density on our brane.
The 5 dimensional model {#sec4}
=======================
In this section we would like to describe some interesting features of the model presented in the case when we have a 3-brane representing our world. In order to do that, we just have to take $n=3$ in all the relations of the last section.
Besides simplicity and the obvious relevance of 3-branes, this will allow us to make contact with the solutions discussed in the literature for 5 dimensions. We will consider the various limits of our solution to reobtain the different systems already considered in the literature for $5$ dimensions where only gauge fields or only dilaton fields were considered.
- In the limits $r_+=b=0$ or $r_-=b=0$ both the gauge and dilaton fields vanish and we end up with the solution given in [@BCG] with vanishing bulk cosmological constant which, for $k=1$ corresponds to the usual Schwarzschild black hole solution.
- In the limit $b=0$, one obtains a solution with a vanishing dilaton field, that corresponds to the model found in [@CEG] again without bulk cosmological constant.
- The case $r_+=0$, $b=2/3$, $k=0$ corresponds to the usual scalar self-tuning geometry, as we will see below.
Static 3-brane universe
-----------------------
We will discuss now in detail the implications of the junction conditions for matter on the 3-brane, determined by the energy density and the pressure. First we will examine under which conditions the brane remains static in the bulk: ${\mathcal R(\tau)}=R_0$. The static brane solution is of particular relevance in the case of a vanishing curvature, $k=0$, since the induced metric on the brane is then 4D Poincaré invariant.
We will restrict to exponential couplings between the brane and the dilaton: $$f(\phi) = f_0\, e^{\beta\phi}\,, \qquad z(\phi) = z_0\, e^{\gamma\phi}\,.$$ Thus according to the junction equation (\[eq:JumpW\]), the equation of state for a static brane corresponds to $\omega=$const.
Remember that for $k=0$, the only region where the brane can remain static is $r<r_+$. In order to avoid the singularity problem we will deal with negative energy density and glue two exterior regions ($r>R_0$).
For a static brane, the junction equations (\[eq:Jump1\])–(\[eq:Jump3\]) simplify ($\dot{\mathcal
R}=\ddot{\mathcal R}=0$), leaving us with $$\begin{aligned}
\label{staticjunctsk0}
\frac{1}{6}\,\kappa_5^{2}\,\rho &=& -\frac{(1-b)}{R_{0}}
\left(\frac{r_-}{R_0} \right)^{1-b} \sqrt{\left( \frac{r_+}{R_0}
\right)^2-1 } \,,
\\
\frac{1}{6}\,\kappa_5^{2}\, (2+3\omega) \rho & = & \frac{1}{R_0}
\left( 2b-1 - \frac{r_+^2}{r_+^2-R_0^2} \right) \left(\frac{r_-}{R_0}
\right)^{1-b} \sqrt{\left( \frac{r_+}{R_0} \right)^2-1 } \,,\qquad
\\
\label{eq:3braneW}
\omega & = & \frac{1}{3} - \frac{2\omega_\mathcal{L}\, \gamma}{3\beta}
-\frac{4\, b}{9\beta\, (1-b)\sigma} \,.\end{aligned}$$ At this point, it should be noticed that the interpretation of one combination of the jump equations as the conservation equation is erroneous for a static brane and even if the conservation equation is trivially satisfied, we really have three independent junction conditions. [A priori]{} we have also three constants of integration $R_0,r_+$ and $r_-$. We should mention that, the combination (\[eq:3braneW\]) of the jump equations involves only parameters that appears in the action and therefore requires a kind of fine-tuning; we will discuss this issue in the next subsection. The two other equations fix the two constants of integration $r_+$ and $r_{-}$: $$r_+=R_0 \sqrt{1+\frac{1}{3(1-b)\omega}}\,, \qquad r_-=R_0 \left(
-\frac{1}{6} \kappa_5^2\, \rho R_0 \sqrt{3\omega\over(1-b)}
\right)^{1/(1-b)}.$$ And the consistency of the solution that requires $r_+>R_0$ translates into $$0\leq \omega\,.$$ Notice that since the energy density on the brane is negative, the weak energy conditions are violated anyway; in particular even if matter on the brane can have a non exotic equation of state ($0<\omega<1$), still $p+\rho<0$.[^11]
Even if of less physical relevance, a similar study can be conducted for non vanishing curvature, $k=\pm 1$. Let us consider, for instance, $k=+1$. The brane, with positive energy density, will be located in the region $r_\mp < r_\pm < R_0$. It is possible to see that in this case it is not necessary to require fine-tuning between the parameters of the action, and a possible expression for $\omega$ is the following: $$\label{omegak1}
\omega = -\frac{1}{3} \frac{R_0^2-r_-^2}{R_0^2-(1-b)r_-^2}
\left(2+\frac{r_{-}^{2}}{R_{0}^{2}-r_{-}^{2}}+
\frac{r_{+}^{2}}{R_{0}^{2}-r_{+}^{2}}\right).$$ The formula (\[omegak1\]) shows that $\omega$ is always less than zero; the physical requirement $\omega \ge -1$, moreover, is satisfied when the following condition holds $$\label{omegamu}
(1-3b)\frac{r_{-}^{2}}{R_{0}^{2}-r_{-}^{2}} \le
1-\frac{r_{+}^{2}}{R_{0}^{2}-r_{+}^{2}}\,.$$ This last expression will be interesting later on. Notice that as long as (\[omegamu\]) is satisfied, the weak energy conditions hold.
Self-tuning solutions {#sec:selftuning}
---------------------
Let us discuss the issue of the self-tuning of the cosmological constant that has attracted some interest recently [@ADKS; @KSS]. The main idea behind this proposal is that the field equations in 5-dimensions with a dilaton field allow for a solution which is 4D Poincaré invariant whatever the vacuum energy on the brane is. Furthermore there are not other maximally symmetric solutions and the brane is such that any corrections to the cosmological constant coming from matter loops on the brane can be absorbed into a shift of the dilaton field, allowing the ‘self-tuning’ of the cosmological constant. The main problem of this proposal [@ADKS; @FLLN] is the existence of naked singularities in the extra dimensions which cannot be avoided [@CEGH].
A similar proposal has been made regarding the inclusion of gauge fields instead of the dilaton [@CEG]. Again the cosmological constant can be self-tuned by adjusting the values of the charge and mass of the corresponding $AdS$ black hole solution. However the geometry of the brane-bulk system is such that the singularity of the black hole can be shielded by a horizon only for a brane with an exotic equation of state $\omega<-1$. In this subsection we would like to study the possibility of obtaining a self-tuning brane in our dilatonic backgrounds.
First we have to address an important issue regarding equation (\[eq:JumpW\]). When the gauge fields are nonvanishing, we proved that $\omega$ was completely determined by the parameters of the lagrangian, because the constant $b$ was determined by $\sigma$, the coupling of the dilaton to the gauge field, as in equation (\[eq:bsigma\]). This means that in order to get a specific equation of state we would need to tune the parameters of the action ($\gamma, \beta, \sigma$). When the gauge fields vanish i.e., $Q=0$, $b$ is an arbitrary constant of integration (only limited by $0\leq b \leq 1$) independent of the parameters of the action, however, since $r_+=0$, the junction conditions completely determine $b$ in terms of $\omega$ and once again (\[eq:JumpW\]) requires a fine-tuning between the equation of state of the matter on the brane and the dilaton couplings. This is the same fine-tuning problem as in the original scalar self-tuning models [@ADKS; @KSS].
Actually the uncharged dilatonic background solution is given by $$\begin{aligned}
ds^2&=& -h_+ h_-^{1-2b}dt^2 +h_+^{-1} h_-^{-1+b} dr^2 +r^2 h_-^{b}
dx^2_{3,k} \,,
\nonumber\\
\phi&=&\pm \sqrt{3b(1-b)}\, \ln h_- \,,
\label{eq:b=0metric}\end{aligned}$$ where the functions $h_\pm$ are now given by $$h_+= \mathop{\rm sgn} h\,, \qquad h_-= \left|h \right |, \qquad h=k-
\frac{l}{r^2}\,, \qquad k=0,\pm1\,.$$ and $l$ is an integration constant which can be positive or negative since there is no reality constraint for the electric charge anymore. The jump equations on a brane with positive energy density, for $k=0$, now read: $$\begin{aligned}
\kappa_5^2 \rho &=& 6(1-b) \frac{1}{R_0} \left(\frac{-l}{R_0}\right)^{1-b}\,,
\nonumber\\
b&=& \frac{1+3\omega}{3\omega}\,,
\nonumber\\
(3\omega-1) \frac{f'}{2f} + \omega_{\mathcal{L}} \frac{z'}{z} &=& \mp
\sqrt{\frac{b}{3(1-b)}} \,.\end{aligned}$$
A sufficient condition for $r$ to be spacelike is the choice $k=0$ and $l<0$; for simplicity we will take $l=-1$. Then we can realize that the full metric (\[eq:b=0metric\]) will be 4D Poincaré invariant under the condition that the spatial and temporal warp factors have the same dependence on $r$: $$r^{-2(1-2b)}=r^2r^{-2b}\,,
\quad \hbox{i.e.,}
\quad b=\frac23\,.$$ So, we are left with the bulk metric: $$ds^2 = \sqrt{1-\left|\frac{y}{y_0}\right|}
\left(-dt^2+dx_{3,k=0}^2\right) + dy^2 \,,$$ where the new coordinate $y$ is related to $r$ by: $r^{1/3}dr=dy$. This is just the bulk metric of the original scalar self-tuning model [@ADKS; @KSS], corresponding to the solution II of [@KSS]. It can easily be verified that for $\beta = 1/\sqrt{6}$ and $\gamma=0$ we get $\omega=-1$ which is the equation of state used in [@KSS].[^12]
We can actually go beyond the results in [@ADKS; @KSS] by noting that the other solutions with different values of $b$ will also allow for a 3-brane with a Poincaré invariant induced metric whatever the value of its vacuum energy is. These other solutions will break the 4D Poincaré invariance in the bulk, which is still safe for gauge interactions of the Standard Model living on the brane, and may lead to interesting observational consequences in gravitational waves experiments as we will discuss in the next section. Nevertheless there is no way to overcome the problem of the naked singularity in these cases since none of the self-tuning solutions with vanishing gauge field involves a horizon.
In the case of non vanishing gauge fields, even if the value of $\omega$ is again completely determined by the parameters of the lagrangian ($\beta, \gamma, \sigma $), the global geometry is such that in the case of negative energy density on the brane, there are no naked singularities present. Also $\omega$ can be in the physically allowed range, which is an improvement on the cases with only dilaton or only gauge fields.
It is easy to verify that the other maximally symmetric four-dimensional spaces, i.e. the de Sitter and anti-de Sitter geometries, cannot be obtained from our solutions.
Violation of 4D lorentz invariance
----------------------------------
It has recently been emphasized [@CEG; @cXdim] that brane world models can have an interesting effect regarding the speed of propagation of light and gravitational signals. Indeed the general solution (\[eq:metric\]) breaks the 4D Poincaré invariance in the bulk and as a consequence the speed of propagation of electromagnetic signals parallel to the brane depends on the location of the brane. It is then [a priori]{} possible to foresee signals that travel faster through the bulk than on the brane. It has been argued for a long time [@VaryingC] that faster than light propagation and/or variation of the speed of light can solve many of the cosmological puzzles (horizon problem, cosmological constant problem, [etc]{}). These ideas witness today a renewed interest in the context of brane world models [@cXdim].
The violation of 4D Lorentz invariance was extensively discussed in [@CEG], where the authors shown the correctness of the intuitive idea for which if one has a decreasing speed of gravitational waves moving away from the brane, then the brane Lorentz invariance can be recovered, in the sense that the gravitational waves prefer to move on the brane, due to the Fermat’s principle.
We can now calculate in which cases we can have a negative derivative for the speed of gravitational waves, i.e., a decreasing speed of light away of the brane such that according to Fermat’s theorem the gravitational waves will actually propagate on the brane rather than through the bulk.
Let us first examine the case of a static brane with a 4D Poincaré invariant induced metric, which requires a vanishing curvature $k=0$. From the expression (\[eq:metric(n+2)D\]) of the metric, we deduce the local speed of propagation of gravitational waves in a direction parallel to the brane: $$\label{speed0}
c^2_{\rm grav}(r)= \left(\frac{r_{+}^{2}}{r^{2}}-1\right)
\left(\frac{1}{r^{2}}\right)^{(2-3b)} r_{-}^{2(1-3b)} \,.$$ This expression is of course valid only in the region where $r$ is a spacelike coordinate. It is easy to see that, since $0<b<1$, the local speed of propagation $c^2_{\rm grav}(r)$ is always a decreasing function of $r$ in the region $0<r<r_+$, this means that this local speed of propagation will be either decreasing or increasing away from the brane in $R_{0}$ depending on the sign of the brane energy density (see figure \[fig3\]):
- Positive energy density. We keep the interior region ($r<R_0$) and thus the speed of propagation is increasing away from the brane and the gravitational waves will prefer to propagate through the bulk. Note that in this case the naked singularity at $r=0$ is not shielded by a horizon.
- Negative energy density. We keep now the exterior region ($R_0<r<r_+$), therefore in this case the gravitational waves will prefer to travel through the brane instead of the bulk and there will be no evidence of Lorentz violation.
We can then conclude that Lorentz violation would be manifest only in the case with naked singularity, similar to the situation found in [@CEG].
We can also consider the case $k=1$ (although it is less interesting since the induced metric on the brane is not Poincaré invariant). In this case the conclusion regarding the increase or decrease of the speed of gravitational waves is different. The speed of the gravitational waves as a function of the position in the bulk is given by the following expression $$\label{vel}
c_{\rm grav}^2(r)= \left(1-\frac{r_{+}^{2}}{r^{2}}\right)
\left(1-\frac{r_{-}^{2}}{r^{2}}\right)^{1-3b}r^{-2}\, ,$$ and we have a brane with positive energy density located in the region $r_+<R_0$.
The condition of decreasing speed of gravitational waves away from the brane is satisfied when $$\label{speed}
R_0^{2}+3(1-b)\frac{r_{-}^{2}r_{+}^{2}}{R_0^{2}} \le 2 r_{+}^{2}
+(2-3b)r_{-}^{2}\,.$$ The behaviour of the speed of gravitational waves is clear from the plots in figure \[fig4\]. It is very interesting to note that this condition is exactly the same as (\[omegamu\]), but with the opposite sign. The conclusion is that, with $k=1$, to have a decreasing speed of light moving away from the brane one needs to consider an exotic form of matter living on the brane, with $w \le
-1$. [Vice-versa]{}, if the matter on the brane has a standard equation of state, one must take into account possible non negligible effects of changes in the speed of gravitational waves in the visible world.
Conclusions and comments {#sec5}
========================
We have started a general treatment of higher dimensional singular geometries for which external branes of arbitrary dimension can be incorporated. There are two kinds of branes in our solutions that should not be confused. First are the $q$ dimensional bulk singularities that in the $k=1$ case are black branes and in general are ‘$q$-brane singularities’. The second brane is an external $n$-brane that is added given that the geometry factorizes (with a warp factor) between the $q$ dimensional part and the $n+2$ dimensional for which the $n$ dimensional hypersurfaces are maximally symmetric with constant curvature. We may see this, in the case $k=1$, as a brane in a background defined by a second, orthogonal, brane. In the extremal case ($r_-=r_+$) the bulk solution is supersymmetric and there may be an interesting reformulation of this configuration in terms of the AdS/CFT correspondence. For $k=0,-1$ there is no analogue to an extremal case: there are no time-like Killing vectors and there does not seem to be a supersymmetric limit.
=1Besides their intrinsic relevance, these configurations can be seen as the starting point of brane cosmology in the higher dimensional space. We have found several interesting properties of our solutions which are novel. First the $k=0,-1$ bulk solutions are to the best of our knowledge new[^13] and have very interesting geometrical structure having a natural cosmological interpretation but with a time-like singularity and a Cauchy horizon. Similar geometries have appeared in the past. The Penrose diagram of the Taub-NUT solution has the same signature change, as we have, in the Cauchy horizon, but in that case there is another horizon rather than a singularity which allows the infinite extension of the diagram [@TAUB]. A similar behaviour appears in the study of tilted Bianchi cosmologies. In these cases the Cauchy surface corresponds actually to a non-scalar singularity usually referred to as a ‘whimper’ or intermediate singularity, where the metric changes signature [@ellis]. The origin of this intermediate singularity is the fact that the fluid matter flows in a direction which is not perpendicular to the homogeneity surfaces. In our case there is no fluid in the bulk and the intermediate surface is not a singularity but a horizon. It is interesting to notice that if an observer in the region beyond the horizon extrapolates back in time she/he never finds a big-bang singularity but the horizon.
Even though a detailed study of the cosmology of these configurations is beyond our scope, we would like to point out the following interesting properties. For $k=0$ in the 5d bulk cosmological region, at late times ($t\gg r_+$ so that $h_+ \sim -1$) the three standard spatial dimensions always expand while the fifth has different behavior depending on the values of $b$, similar to the Kasner solution: For $2/3<b<1$ all $4$ spatial dimensions expand, the 5th faster than the rest. For the critical case $ b=2/3$ the four spatial dimensions expand at the same rate. In the interval $1/2<b<2/3$ three dimensions expand faster than the 5th, whereas for $b=1/2$ the 5th dimension is static and the others expand. Finally for $0<b<1/2$ the 5th dimension contracts and the others expand. All of this in the Einstein frame.
Second our solutions provide new examples of violation of 4D Lorentz invariance that may have interesting consequences regarding gravitational waves experiments and cosmology. Finally the introduction of the $n$ brane on the bulk static regions can be done in such a way that the singularities can be avoided. A substantial improvement on the situations with only gauge fields or only dilaton regarding the self tuning mechanism for the cosmological constant was achieved. It remains to be seen to what extent this is real progress in the approach to the cosmological constant problem and to what extent it is realistic to have a negative energy density brane world.
Many things remain to be explored in this subject. The possibility of a nontrivial dilaton potential with a stationary point is an open question. Also the incorporation of further antisymmetric tensors of different rank as well as the consideration of codimension larger than one brane worlds and the addition of more than one external branes are left to the future. Probably a more direct extension of the present work is the actual use of these geometries to a detailed cosmological study of this class of brane worlds.
We have made a step forward in studying the structure of a general class of brane models with properties close to what we natural expect in string theory. Their interesting properties makes them worth of further investigation, especially in the context of brane cosmology. We hope to report on progress in this direction in the near future.
We would like to thank J. Barrow, G. Gibbons, D. Mateos and P. Townsend for useful discussions. The research of F.Q. is supported by PPARC. I.Z.C. is supported by CONACyT (Mexico), Trinity College (Cambridge) and COT (Cambridge Overseas Trust). C.G. is supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC03-76SF00098, and in part by the National Science Foundation under grant PHY-95-14797. G.T. is partially supported by the European TMR project “Across the Energy Frontier" under the contract HPRN-CT-2000-0148.
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[^1]: We refrain from using the standard terminology $p$-brane to avoid confusion with the pressure $p$ in the following sections.
[^2]: For clarity we mention the distinction between the two branes involved in our construction: the $q$-brane that is electrically charged under a $q+1$ form and an $n$-brane that is coupled to the bulk through gravity only. Until section 3, we will keep $q$ and $n$ general and we will restrict to $q=0$ afterwards.
[^3]: Note however, that in string theory, black holes will carry several charges, and then, the extra singularity can be stabilized [@miriam].
[^4]: The magnetically charged case is straightforward.
[^5]: Our conventions correspond to a mostly positive lorentzian signature $(-+\ldots +)$ and the definition of the curvature in terms of the metric is such that a Euclidean sphere has positive curvature. Bulk indices will be denoted by Greek indices ($\mu,\nu\ldots$) and brane indices by Latin indices ($a,b\ldots$). The Einstein tensor in the bulk will be denoted by $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}$.
[^6]: Notice that in general, one can add a constant to the dilaton field solution, $\phi \to \phi(r) + 2 \phi_0$, by redefining the form as $F \to F
e^{\sigma \phi_{0}}$.
[^7]: Another solution has been found (in the special case of $n=3$ and $q=0$) in [@BF] and it is given by $h_+ = k-(r_+/r)^{n-1}$ and $h_- = 1-
(r_- / r)^{n-1}$.
[^8]: Notice also that it would have been possible to flip both signs in front of $(r_\pm/r)^{n-1}$ in $h_\pm$ while keeping a real electric charge, however the origin $r=0$ would be a naked singularity.
[^9]: For $b=0$, $r=r_-$ is actually another horizon as in the Reissner-Nördstrom solution.
[^10]: The value of $\omega_{\mathcal{L}}$ depends on the type of matter of the brane. Known cases correspond to $\omega_{\mathcal{L}}=-\omega$ for a time dependent scalar field [@kolb]. There are also other cases for which $\omega_{\mathcal{L}}=1$ [@H-E].
[^11]: We thank J. Cline and H. Firouzjahi for discussion on this point.
[^12]: To make the comparison with [@ADKS; @KSS] equations we have to redefine the dilaton field appropriately.
[^13]: See however [@BF; @similar] for related solutions.
|
**Embedding an Edge-colored $K(a^{(p)};\lambda,\mu )$ into a Hamiltonian Decomposition of $K(a^{(p+r)};\lambda,\mu )$**
Amin Bahmanian[^1], C. A. Rodger[^2]\
Department of Mathematics and Statistics\
Auburn University, Auburn, AL\
USA 36849-5310
Let $K(a^{(p)};\lambda,\mu )$ be a graph with $p$ parts, each part having size $a$, in which the multiplicity of each pair of vertices in the same part (in different parts) is $\lambda$ ($\mu $, respectively). In this paper we consider the following embedding problem: When can a graph decomposition of $K(a^{(p)};\lambda,\mu )$ be extended to a Hamiltonian decomposition of $K(a^{(p+r)};\lambda,\mu )$ for $r>0$? A general result is proved, which is then used to solve the embedding problem for all $r\geq \frac{\lambda}{\mu a}+\frac{p-1}{a-1}$. The problem is also solved when $r$ is as small as possible in two different senses, namely when $r=1$ and when $r=\frac{\lambda}{\mu a}-p+1$.
Keywords. Amalgamations; Detachments; Hamiltonian Decomposition; Edge-coloring; Hamiltonian Cycles; Embedding
Introduction
============
Let $G=(V,E)$ be a graph and let $H=\{H_{i}\}_{i\in I}$ be a family of graphs where $H_{i}=(V_{i},E_{i})$. We say that $G$ has an *$H$-decomposition* if $\{E_{i}:i\in I\}$ partitions $E$ and each $E_{i}$ induces an isomorphic copy of $H_{i}$. Graph decomposition in general has been studied for many classes of graphs. The decomposition of a graph into paths [@Tarsi83], cycles [@Sajna; @1] or stars [@Tarsi81] has been of special interest over the years. Of particular interest is the decomposition of a graph into Hamiltonian cycles; that is a *Hamiltonian Decomposition*. In 1892 Walecki [@L] proved the classic result that the complete graph $K_n$ is decomposable into Hamiltonian cycles if and only if $n$ is odd. Laskar and Auerbach [@LA] settled the existence of Hamiltonian decomposition of the complete multipartite graph $K_{m,\ldots,m}$. Alspach, Gavlas, and Sajna [@Alspach; @and; @Gavlas; @Sajna; @1; @Sajna; @2] collectively solved the problem of decomposing the complete graph into isomorphic cycles, but the problem remains open for different cycle lengths.
A *k-edge-coloring* of $G$ is a mapping ${\cal K}:E(G)\rightarrow C$, where $C$ is a set of $k$ *colors* (often we use $C=\{1,\ldots,k\}$), and the edges of one color form a *color class*. Another challenge is the companion embedding problem:
Let $H=\{H_{i}\}_{i\in I}$ and $H^*=\{H^*_{j}\}_{j\in J}$ be two families of graphs. Given a graph $G$ with an $H$-decomposition and a graph $G^*$ which is a supergraph of $G$ (or $G$ is a subgraph of $G^*$), under what circumstances one can extend the $H$-decomposition of $G$ into an $H^*$-decomposition of $G^*$? In other words, given an edge-coloring of $G$ (that can be considered as a decomposition when each color class induces a graph in $H$), is it possible to extend this coloring to an edge-coloring of $G^*$ so that each color class of $G^*$ induces a graph in $H^*$?
In this direction, Hilton [@H2] found necessary and sufficient conditions for a decomposition of $K_m$ to be embedded into a Hamiltonian decomposition of $K_{m+n}$, which later was generalized by Nash-Williams [@Nash87]. Hilton and Rodger [@HR] considered the embedding of Hamiltonian decompositions for complete multipartite graphs. For embedding factorizations see [@HJRW; @RW], where the connectivity of the graphs in $H^*$ is one defining property. We also note that embedding problems first were studied for latin squares by M. Hall [@MHall45]. For extensions of Hall’s theorem see [@LinRod92; @AndHil1; @AndHil2].
By the *multiplicity* of a pair of vertices $u,v$ of $G$, we mean the number of edges joining $u$ and $v$ in $G$. In this paper $K(a_1,\ldots, a_p;\lambda,\mu )$ denotes a graph with $p$ parts, the $i^{th}$ part having size $a_i$, in which multiplicity of each pair of vertices in the same part (in different parts) is $\lambda$ ($\mu $, respectively). When $a_1 = \ldots = a_p=a$, we denote $K(a_1,\ldots, a_p;\lambda,\mu )$ by $K(a^{(p)};\lambda,\mu )$. Let us say that an edge in $K(a^{(p)};\lambda,\mu )$ is pure (mixed, respectively) if its endpoints are in the same (different, respectively) part(s). If we replace each edge of $G$ by $\lambda$ multiple edges, then we denote the new graph by $\lambda G$.
The graph $K(a_1,\ldots, a_p;\lambda,\mu )$ is of particular interest to statisticians, who consider group divisible designs with two associate classes, beginning over 50 years ago with the work of Bose and Shimamoto [@BoSh]. Decompositions of $K(a_1,\ldots, a_p;\lambda,\mu )$ into cycles of length $m$ have been studied for small values of $m$ [@FuRod98; @FuRod01; @FuRodSar]. Recently, Bahmanian and Rodger have settled the existence problem completely for longest (i.e. Hamiltonian) cycles in [@BahRod1]. In this paper, we study conditions under which one can embed a decomposition of $K(a^{(p)};\lambda,\mu )$ into a Hamiltonian decomposition of $K(a^{(p+r)};\lambda,\mu )$ for $r>0$. Our proof is largely based on our results in [@BahRod1] (see Theorem \[mainthesp\] in the next section).
If $G$ is a $k$-edge-colored graph, and if $u,v\in V(G)$ then $\ell(u)$ denotes the number of loops incident with vertex $u$, $d(u)$ denotes the degree of vertex $u$ (loops are considered to contribute two to the degree of the incident vertex), and $m(u,v)$ denotes the multiplicity of pair $u$ and $v$. The subgraph of $G$ induced by the edges colored $j$ is denoted by $G(j)$, and $\omega (G)$ is the number of components of $G$.
Amalgamation and Graph Embedding
================================
*Amalgamating* a finite graph $G$ can be thought of as taking $G$, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated graph $H$. Any edge incident with an original vertex in $G$ is then incident with the corresponding new vertex in $H$, and any edge joining two vertices that are squashed together in $G$ becomes a loop on the new vertex in $H$.
A *detachment* of $H$ is, intuitively speaking, a graph $G$ obtained from $H$ by splitting some or all of its vertices into more than one vertex. That is, to each vertex $\alpha$ of $H$ there corresponds a subset $V_{\alpha}$ of $V(G)$ such that an edge joining two vertices $\alpha$ and $\beta$ in $H$ will join some element of $V_{\alpha}$ to some element of $V_{\beta}$. If $\eta$ is a function from $V(H)$ into $\mathbb {N}$ (the set of positive integers), then an *$\eta$-detachment* of $H$ is a detachment of $H$ in which each vertex $u$ of $H$ splits into $\eta(u)$ vertices. For a more precise definition of amalgamation and detachment, we refer the reader to [@BahRod1].
Since two graphs $G$ and $H$ related in the above manner have an obvious bijection between the edges, an edge-coloring of $G$ or $H$, naturally induces an edge-coloring on the other graph. Hence an amalgamation of a graph with colored edges is a graph with colored edges.
The technique of vertex amalgamation, which was developed in the 1980s by Rodger and Hilton, has proved to be very powerful in decomposing of various classes of graphs. For a survey about the method of amalgamation and embedding partial edge-colorings we refer the reader to [@AndRod1]. In [@BahRod1], the authors proved a general detachment theorem for multigraphs. For the purpose of this paper we use a very special case of this theorem as follows:
\[mainthesp\] Let $H$ be a $k$-edge-colored graph all of whose color classes are connected, and let $\eta$ be a function from $V(H)$ into $\mathbb{N}$ such that for each $v \in V(H)$ $\eta (v) = 1$ implies $\ell_H (v) = 0$, $d_{H(j)}(v)/\eta(v)$ is an even integer for $1\leq j\leq k$, $\binom{\eta(v)}{2}$ divides $\ell_H(v)$, and $\eta(v)\eta(w)$ divides $m_{H}(v,w)$ for each $w\in V(H)\backslash\{v\}$. Then there exists a loopless $\eta$-detachment $G$ of $H$ in which each $v\in V(H)$ is detached into $v_1,\ldots, v_{\eta(v)}$, all of whose color classes are connected, and for $v\in V(H)$
- $m_G(v_i, v_{i'}) = \ell_H(v)/\binom {\eta(v)}{2} $ for $1\leq i<i'\leq \eta(v)$ if $\eta (v) \geq 2$,
- $m_G(v_i, w_{i'}) = m_H(v, w)/(\eta (v) \eta (w)) $ for $w\in V(H)\backslash\{v\}$, $1\leq i\leq \eta(v)$ and $1\leq i'\leq \eta(w)$, and
- $d_{G(j)}(v_i) = d_{H(j)}(v)/\eta (v)$ for $1\leq i\leq \eta(v)$ and $1\leq j\leq k$.
Here is our main result:
\[embed1th\] Let $G=K(a^{(p)};\lambda,\mu )$ with $a>1$, $\lambda\geq 0$, $\mu \geq 1$, $\lambda\neq \mu $, $r\geq 1$, and let $\omega_j=\omega\big(G(j)\big)$. For $1\leq j\leq k$, define $$\label{congcond}
s_j\equiv \omega_j \pmod r \mbox{ with }1\leq s_j\leq r,$$ and suppose $$\label{ineqsumcond}
\sum_{j=1}^k s_j\geq kr-\mu a^2\binom{r}{2}.$$ Then a $k$-edge-coloring of $G$ can be embedded into a Hamiltonian decomposition of $G^*=K(a^{(p+r)};\lambda,\mu )$ if and only if:
1. $k=\big(\lambda(a-1)+\mu a(p+r-1)\big)/2$,
2. $\lambda\leq \mu a(p+r-1)$,
3. Every component of $G(j)$ is a path (possibly of length $0$) for $1\leq j\leq k$, and
4. $\omega_j\leq ar$ for $1\leq j\leq k$.
By Theorem 4.3 in [@BahRod1], for $K(a^{(p+r)};\lambda,\mu )$ to be Hamiltonian decomposable, conditions (i) and (ii) are necessary and sufficient. (Condition (i) follows since $k$ must be $d_{G^*}(v)/2$. Condition (ii) follows since each Hamiltonian cycle must use at least $p+r$ mixed edges, so there must be sufficiently many mixed edges for all pure edges to be used.) For $1\leq j\leq k$, for $G(j)$ to be extendable to a Hamiltonian cycle in $K(a^{(p+r)};\lambda,\mu )$, the degree of each vertex has to be at most 2, and thus every component must be a path. Moreover, since each new vertex can link together two disjoint paths, the number of components of every color class can not exceed the number of new vertices, $ar$. This proves the necessity of (i)–(iv).
Let $G=(V,E)$, and let $u$ be a vertex distinct from vertices in $V$. Define the new graph $G_1=(V_1,E_1)$ with $V_1=V\cup\{u\}$ by adding to $G$ the vertex $u$ incident with $\mu a^2\binom{r}{2}$ loops, and adding $\mu ar$ edges between $u$ and each vertex $v$ in $V$ (see Figure \[figure:G\_1G\_2\]). Note that for each $v\in V$, $d_{G_1}(v)=\lambda(a-1)+\mu a(p-1)+\mu ar=\lambda(a-1)+\mu a(p+r-1)=2k$.
Now we extend the $k$-edge-coloring of $G$ to a $(k+1)$-edge-coloring of $G_1$ as follows:
1. Each edge in $G$ has the same color as it does in $G_1$,
2. For every $v\in V$, color the $\mu ar$ edges between $v$ and $u$ so that $d_{G_1(j)}(v)=2$ for $1\leq j\leq k$. Since $d_{G(j)}(v)\leq 2$ for $1\leq j\leq k$, and since $d_{G_1}(v)=2k$, this can be done. Notice that for every component of $G(j)$ (which is a path), exactly two edges (from end points of the path) are connected to $u$; so at this point $d_{G_1(j)}(u)=2\omega_j$ for $1\leq j\leq k$.
3. For $1\leq j\leq k$ color $r-s_j$ ($\geq 0$) loops with $j$. This coloring of loops can be done, since by condition (2) of the theorem we have: $$\begin{aligned}
\sum_{j=1}^k s_j\geq kr-\mu a^2\binom{r}{2} & \iff & \sum_{j=1}^k r-\sum_{j=1}^k s_j\leq \mu a^2\binom{r}{2} \\
& \iff & \sum_{j=1}^k (r-s_j)\leq \mu a^2\binom{r}{2}=\ell_{G_1}(u).\end{aligned}$$ Moreover we color the remaining $\sum_{j=1}^k s_j-kr+\mu a^2\binom{r}{2}$ ($\geq 0$) loops with the new color $k+1$. Thus for $1\leq j\leq k$, $$d_{G_1(j)}(u)=2\omega_j+2(r-s_j)=2r+2(\omega_j-s_j),$$ and $d_{G_1(k+1)}(u)=2\Big(\sum_{j=1}^k s_j-kr+\mu a^2\binom{r}{2}\Big)$. By (1) $d_{G_1(j)}(u)$ is an even multiple of $r$ for $1\leq j\leq k$. Now to show that $d_{G_1(k+1)}(u)$ is an even multiple of $r$, first we show that $\sum_{j=1}^{k}\omega_j=\mu a^2pr/2$. $$\begin{aligned}
\sum \nolimits_{j=1}^k \omega_j & = & \sum \nolimits_{j=1}^k (pa-|E(G(j))|) \\
& = & kpa- |E|\\
&= & pa\big(\lambda(a-1)+\mu a(p+r-1)\big)/2-pa\big(\lambda(a-1)+\mu a(p-1)\big)/2 \\
&= & \mu a^2pr/2.
\end{aligned}$$ Notice that $\mu a(p+r-1)$ is even, since otherwise, in particular $a$ would be odd, so $k$ would not be an integer. Thus, $$\begin{aligned}
d_{G_1(k+1)}(u) & \equiv & 2\sum \nolimits_{j=1}^k \omega_j+\mu a^2r(r-1) \\
& \equiv & \mu a^2pr+\mu a^2r(r-1)\\
&\equiv & \mu a^2r(p+r-1)\\
&\equiv & 0 \pmod {2r}.
\end{aligned}$$
Let $b_1,\ldots,b_{k+1}$ be even integers such that $d_{G_1(j)}(u)=b_jr$ for $1\leq j\leq k+1$. Note that for $1\leq j\leq k$, we have $$b_j/2=1+\frac{\omega_j-s_j}{r}\leq 1+\lfloor \frac{ar-1}{r}\rfloor\leq 1+(a-1)=a.$$
Since each component of $G(j)$ is joined to $u$ in $G_1(j)$, each color class of $G_1$ is connected. Let $\eta$ be a function from $V_1$ into $\mathbb N$ such that $\eta(v)=1$ for each $v\in V$, and $\eta(u)=r$. Now by Theorem \[mainthesp\], there exists an $\eta$-detachment $G_2$ of $G_1$, all of whose color classes are connected, (see Figure \[figure:G\_1G\_2\]) in which $u$ is detached into $r$ new vertices $u_1,\ldots,u_r$ such that:
- $m_{G_2}(u_i,u_{i'})=\mu a^2\binom{r}{2}/\binom{r}{2}=\mu a^2,$ for $1\leq i<i'\leq r$;
- $m_{G_2}(u_i,v)=\mu ar/r=\mu a$ for each $v\in V$ and each $i$, $1\leq i \leq r$;
- $d_{G_2(j)}(u_i)=b_jr/r=b_j$ for $1\leq i \leq r$ and $1\leq j \leq k+1$.
We observe that $d_{G_2}(u_i)=ap(\mu a)+(r-1)\mu a^2=\mu a^2(p+r-1)$ for $1\leq i\leq r$, and is even. Note that by (c), $d_{G_2(j)}(u_i)=d_{G_2(j)}(u_{i'})$ and is even for $1\leq i\leq i' \leq r$, and we know that $d_{G_2}(u_i)\leq 2ka$ for $1\leq i \leq r$. Therefore, since $G(k+1)$ is an even graph, (so it has a 2-factorization), we can recolor each 2-factor of color class $k+1$ with a color $j$, $1\leq j\leq k$ such that $d_{G_2(j)}(u_i)\leq 2a$. We let $b'_1,\ldots,b'_{k}$ be even integers such that in the resulting edge-coloring of $G_2$ obtained from recoloring the color class $k+1$, $d_{G_2(j)}(u)=b'_jr$ for $1\leq j\leq k$.
Now we define the new graph $G_3$ by adding $\lambda\binom{a}{2}$ loops on every vertex $u_i$ in $G_2$, for $1\leq i \leq r$ (see Figure \[figure:G\_3\]).
We extend the $k$-edge-coloring of $G_2$ to a $k$-edge-coloring of $G_3$ such that:
1. Each edge in $G_2$ has the same color at it does in $G_3$,
2. For $1\leq i\leq r$ and $1\leq j\leq k$, there are $a-b'_j/2$ ($\geq 0$) loops incident with $u_i$ colored $j$. This is possible, for the following reason: $$\begin{aligned}
\sum_{j=1}^k (a-b'_j/2) & = & ka-\frac{1}{2}\sum_{j=1}^k d_{G_2(j)}(u_1) \\
& = & ka- \frac{1}{2} d_{G_2}(u_1) \\
&= & ka-\frac{1}{2}\mu a^2(p+r-1)\\
&= & \frac{a}{2}\big( \lambda(a-1)+\mu a(p+r-1) \big)-\frac{1}{2}\mu a^2(p+r-1)\\
&= & \frac{a}{2} \lambda(a-1)= \ell_{G_3}(u_1).
\end{aligned}$$
Since for $1\leq j\leq k$, $G_2(j)$ is a connected spanning subgraph of $G_3(j)$, $G_3(j)$ is also connected. Let $\eta'$ be a function from $V_3$ into $\mathbb N$ such that $\eta'(v)=1$ for each $v\in V$, and $\eta'(u_i)=a$ for $1\leq i\leq r$. Now by Theorem \[mainthesp\], there exists an $\eta'$-detachment $G_4$ of $G_3$, all of whose color classes are connected, in which $u_i$ is detached into $a$ new vertices $u_{i1},\ldots,u_{ia}$ for $1\leq i\leq r$ such that:
- $m_{G_4}(u_{ij},u_{ij'})=\lambda\binom{a}{2}/\binom{a}{2}=\lambda$ for $1\leq i\leq r$ and $1\leq j<j'\leq a$;
- $m_{G_4}(u_{ij},u_{i'j'})=\mu a^2/a^2=\mu $ for $1\leq i<i'\leq r$ and $1\leq j<j'\leq a$;
- $m_{G_4}(u_{ij},v)=\mu a/a=\mu $ for each $v\in V$ and for $1\leq i \leq r$; and
- $d_{G_4(j)}(u_{ii'})=2a/a=2$ for $1\leq i \leq r$, $1\leq i' \leq a$.
Therefore $G_4=K(a^{(p+r)};\lambda,\mu )$, and each color class in $G_4$ is a Hamiltonian cycle, so the proof is complete.
A natural perspective of this embedding problem is to keep $a, p, \lambda$ and $\mu$ fixed, and ask for which values of $r$ the embedding is possible. The following result completely settles this question for all $r\geq \frac{\lambda(a-1)+\mu a(p-1)}{\mu a(a-1)}$.
\[embed1thasym\] Let $G=K(a^{(p)};\lambda,\mu )$ with $a>1$, $\lambda\geq 0$, $\mu \geq 1$, $\lambda\neq \mu $, and $$\label{rlargewrtp}
r\geq \frac{\lambda(a-1)+\mu a(p-1)}{\mu a(a-1)}.$$ Then a $k$-edge-coloring of $G$ can be embedded into a Hamiltonian decomposition of $K(a^{(p+r)};\lambda,\mu )$ if and only if of are satisfied.
It is enough to show that (\[rlargewrtp\]) implies (\[ineqsumcond\]). Since $s_j\geq 1$ for $1\leq j\leq k$, $\sum_{j=1}^k s_j \geq k$. Thus, if we show that $k\geq kr-\mu a^2\binom{r}{2}$, we are done. This is true by the following sequence of equivalences: $$\begin{aligned}
k(r-1)\leq \mu a^2\binom{r}{2} & \iff & \\
(r-1)\big(\lambda (a-1)+\mu a (p+r-1)\big) \leq \mu a^2 r(r-1) & \iff & \\
\lambda (a-1) \leq \mu a(ar-p-r+1)=\mu a\big( r(a-1)-(p-1) \big) & \iff & \\
\lambda (a-1)/(\mu a) \leq r(a-1)-(p-1)& \iff & \\
r\geq \frac{\lambda(a-1)+\mu a(p-1)}{\mu a(a-1)}.
\end{aligned}$$
Another immediate corollary of Theorem \[embed1th\] is the following complete solution to the embedding problem when $r=1$:
\[embedcorr=1\] Let $G=K(a^{(p)};\lambda,\mu )$ with $a>1$, $\lambda\geq 0$, $\mu \geq 1$, $\lambda\neq \mu $. Then a $k$-edge-coloring of $G$ can be embedded into a Hamiltonian decomposition of $K(a^{(p+1)};\lambda,\mu )$ if and only if:
1. $k=\big(\lambda(a-1)+\mu ap\big)/2$,
2. $\lambda\leq \mu ap$,
3. Every component of $G(j)$ is a path (possibly of length $0$) for $1\leq j\leq k$, and
4. $\omega_j\leq a$ for $1\leq j\leq k$.
Since $r=1$, we have $s_1=\ldots=s_k=1$, so $k=\sum_{j=1}^k s_j= k-\mu a^2\binom{1}{2}=k$, and thus condition (\[ineqsumcond\]) of Theorem \[embed1th\] is satisfied.
Whenever $\lambda \leq \mu a$ and $p\leq a$, the embedding problem is completely solved for all values of $r\geq 1$.
Condition \[rlargewrtp\] can be rewritten as $r\geq \frac{\lambda}{\mu a}+\frac{p-1}{a-1}$. Since we are assuming that $\lambda \leq \mu a$ and $p\leq a$, we have $\frac{\lambda}{\mu a}+\frac{p-1}{a-1}\leq 2$. Therefore the result follows from Theorem \[embed1thasym\] and Corollary \[embedcorr=1\].
The following result completely settles the embedding problem for the smallest value of $r$ in another sense, namely with respect to the inequality (ii) of Theorem \[embed1th\]; so it settles the case where $\lambda=\mu a(p+r-1)$, or equivalently where $r=\frac{\lambda}{\mu a}-p+1$. The proof is similar to that of Theorem \[embed1th\], so only an outline of the proof is provided, the details being omitted. The proof of the necessity of condition (ii) of Theorem \[embed1th\] shows that, in a Hamiltonian decomposition of $K(a^{(p)};\lambda a(p+r-1),\lambda)$, each Hamiltonian cycle contains exactly $a-1$ pure edges from each part, and exactly $p+r$ mixed edges.
Let $a>1$ and $r,\mu \geq 1$. A $k$-edge-coloring of $G=K(a^{(p)};\mu a(p+r-1),\mu )$ can be embedded into a Hamiltonian decomposition of $G^*=K(a^{(p+r)};\mu a(p+r-1),\mu )$ if and only if:
- $k=\mu a^2(p+r-1)/2$,
- Every component of $G(j)$ is a path (possibly of length $0$) for $1\leq j\leq k$,
- $G(j)$ has exactly $a-1$ pure edges from each part, and at most $p-1$ mixed edges for $1\leq j \leq k$, and
- $\omega_j\leq r$ for $1\leq j \leq k$.
The necessity of (i)–(iii) follows as described in Theorem \[embed1th\]. Let $m_j$ be the number of mixed edges in $G(j)$. To extend each component $P$ of $G(j)$ to a Hamiltonian cycle in $G^*$, two mixed edges have to join $P$ to the new vertices, and since each Hamiltonian cycle in $G^*$ contains exactly $p+r$ mixed edges, we have that $$\label{1eqa}
m_j+2\omega_j\leq p+r.$$ Since $G(j)$ is a collection of paths, for $1\leq j\leq k$, we have $|V(G(j))|=|E(G(j))|+\omega_j$. Therefore $ap=m_j+p(a-1)+\omega_j$ and thus $$\label{2eqa}
m_j+\omega_j=p.$$ Combining (\[1eqa\]) and (\[2eqa\]) implies (iv).
To prove sufficiency, we define the graph $G_1$ as it is defined in Theorem \[embed1th\]. We extend the $k$-edge-coloring of $G$ to a $k$-edge-coloring of $G_1$ such that $d_{G_1(j)}(v)=2$ for each $v\in V$ and $1\leq j\leq k$. This is possible by the same argument as in Theorem \[embed1th\]. At this point $d_{G_1(j)}(u)=2\omega_j\leq 2r$ for $1\leq j\leq k$. So we can color the loops incident with $u$ such that $d_{G_1(j)}(u)=2r$ for $1\leq j\leq k$, simply by assigning the color $j$ to $r-\omega_j$ loops.
Now we detach the vertex $u$ into $r$ new vertices $u_1,\ldots,u_r$ to obtain the new graph $G_2$ (as we did in the proof of Theorem \[embed1th\]). Note that $d_{G_2(j)}(u_i)=2r/r=2$ for each $i$, $1\leq i \leq r$ and each $j$, $1\leq j \leq k$. Now we define the new graph $G_3$ by adding $a-1$ loops of color $j$, $1\leq j\leq k$, on every vertex $u_i$ in $G_2$, for each $i$, $1\leq i \leq r$. So we have $d_{G_3(j)}(u_i)=2a$. Using Theorem \[mainthesp\], detach each vertex $u_i$ into $a$ new vertices $u_{i1},\ldots,u_{ia}$ for $1\leq i\leq r$, to obtain the new graph $G_4$ in which, $G_4(j)$ is connected and $d_{G_4(j)}(u_{ii'})=2a/a=2$ for $1\leq j\leq k$, $1\leq i \leq r$, $1\leq i' \leq a$. This completes the proof.
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[^1]: mzb0004@auburn.edu
[^2]: rodgec1@auburn.edu
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abstract: 'We describe a time evolution algorithm for quantum spin chains whose Hamiltonians are composed of an infinite uniform left and right bulk part, and an arbitrary finite region in between. The left and right bulk parts are allowed to be different from each other. The algorithm is based on the time-dependent variational principle (TDVP) of matrix product states, and the system Hamiltonian needs to be representable as a matrix product operator. The algorithm is inversion-free and very simple to adapt from an existing TDVP code for finite systems. The importance of working in the projective Hilbert space is highlighted. We study the transverse-field Ising model as an illustrative example. Two features of quantum dynamics are examined: finite speed of information spreading and thermalization from a local quench. We also offer a derivation of TDVP directly from symplectic geometry.'
author:
- Yantao Wu
bibliography:
- 'abc.bib'
title: 'Time-dependent variational principle of mixed matrix product states in the thermodynamic limit'
---
Introduction
============
Over the last two decades, research in quantum dynamics has benefited greatly from numerical algorithms that can simulate accurately the real-time dynamics of many-body quantum systems. For one-dimensional systems, two time evolution algorithms, both based on matrix product states (MPS), have proved reliable: the time evolving block decimation (TEBD) method [@tebd] and the time-dependent variational principle (TDVP) algorithm [@tdvp; @finite_TDVP]. For translationally invariant systems, both methods can generalize to the thermodynamic limit: the iTEBD [@itebd] and the iTDVP [@iTDVP; @Tangent_space], eliminating the undesirable finite-size effects and reducing the complexity dependence of the system size from linear to constant. Based on locality [@Locality], one expects that for systems composed of uniform left and right bulk parts and finite impurities in between, the time evolution algorithms should also have an efficient thermodynamic version. While it is not clear to us how this can be done for TEBD, a TDVP-based method to deal with such cases has been put forth in [@nonuniform_TDVP].
After [@nonuniform_TDVP] was published, tangent space methods of MPS have developed significantly [@finite_TDVP; @VUMPS; @iTDVP; @Tangent_space]. It is thus worthwhile to revisit the problem and apply these development. In this paper, we greatly simplify the algorithm in [@nonuniform_TDVP] and improve it in many ways. While [@nonuniform_TDVP] only treats nearest-neighbor interactions, we will be able to treat any Hamiltonian that can be written as a matrix product operator (MPO). [@nonuniform_TDVP] also uses inverses of matrices conditioned by the MPS Schmidt coefficients, which can be very small. This leads to the paradoxical situation where the larger the bond dimension of the MPS is and hence the better the variational approximation, the more numerically unstable the algorithm becomes. The algorithm described below will be completely inversion-free. [@nonuniform_TDVP] considers only the Hamiltonians whose left and right bulk parts are the same, and the quenches which only change the finite region of impurities. We will allow the left and right bulks to be different and the quenches to change the bulk parts. In addition, we give a much more detailed derivation of the final TDVP equation.
The core idea of TDVP is very simple. The states representable by MPSs with a given bond dimension form a submanifold, ${\mathcal{H}_\text{MPS}}$, of the entire Hilbert space [@Geometry_MPS]. For a state, ${| \Psi(t) \rangle}$, at time $t$, the time evolution governed by its Hamiltonian $\hat{H}$ leads the state out of ${\mathcal{H}_\text{MPS}}$, i.e. $\hat{H}{| \Psi(t) \rangle}$ is not in the tangent space, $T_{{| \Psi(t) \rangle}}{\mathcal{H}_\text{MPS}}$, of ${\mathcal{H}_\text{MPS}}$ at ${| \Psi(t) \rangle}$. For the time evolution to stay in ${\mathcal{H}_\text{MPS}}$, the TDVP mandates to approximate $\hat{H}{| \Psi(t) \rangle}$ as its [*orthogonal projection*]{} on $T_{{| \Psi(t) \rangle}}{\mathcal{H}_\text{MPS}}$ in the integration of the time evolution. One then chooses a small time step, and integrates the projected $\hat{H} {| \Psi(t) \rangle}$ to obtain a trajectory in ${\mathcal{H}_\text{MPS}}$ which hopefully approximates the true quantum dynamics. What not obvious about this TDVP prescription is that the resulting dynamics in ${\mathcal{H}_\text{MPS}}$ is also symplectic, just as the Schr[ö]{}dinger evolution in the full Hilbert space. In the Appendix \[app:symp\], we offer a derivation of TDVP directly from symplectic geometry, which makes the symplectic properties of TDVP apparent.
The technical difficulty in applying TDVP to MPSs comes from the fact that there is a lot of gauge freedom in an MPS, i.e. the same quantum state can be represented by two MPSs with very different matrix elements. This means that the time evolution of the quantum state does not uniquely specify how the matrix elements of an MPS should evolve. One thus needs to specify a gauge choice for the MPS and its tangent vector, which we will do below.
This paper is organized as follows. In Sec. \[sec:system\], we describe the system of interest and its MPS approximation. We will examine very carefully the gauge freedom of the MPS. In Sec. \[sec:tangent\], we review some facts about the tangent space of ${\mathcal{H}_\text{MPS}}$ and provide a gauge choice for the tangent vector. In Sec. \[sec:ortho\], we construct the orthogonal projection of $\hat{H}{| \Psi \rangle}$. In order for this construction to be possible in the thermodynamic limit, we work in the projective Hilbert space. In Sec. \[sec:integration\], we give an integration scheme to obtain the TDVP dynamics. In Sec. \[sec:Ising\], we study the transverse-field Ising model as an example. Two features of quantum dynamics are examined: finite speed of information spreading and thermalization from a local quench. In Sec. \[sec:discussion\], we discuss future work and conclude.
The system of interest, its MPS approximation, and gauge freedom {#sec:system}
================================================================
We consider an infinite quantum spin chain with a local Hilbert space of dimension $d$ on each site. The system has an infinite left and right bulk part, and a finite region of impurities with length $n_W$ in between. Let the Hamiltonian $\hat{H}$ be written as an infinite MPO with four-index MPO elements $W^{ss'}_{ab}$ with $a,b=1,\cdots, d_W$ and $s, s' = 1,\cdots,d$, where $d_W$ is the bond dimension of the MPO: $$\begin{split}
\hat{H} &= \sum_{{{\bf s}},{{\bf s}}'}(... W_{{[i-1]}}^{s_{i-1}s'_{i-1}} W_{{[i]}}^{s_is'_i} W_{{[i+1]}}^{s_{i+1}s'_{i+1}}...){| {{\bf s}} \rangle}{\langle {{\bf s}}'|}
\\
&=\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$W_A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$W_A$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$W_1$};
\draw (6,1.5) -- (6.5,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (1.5,2) -- (1.5,2.5);
\draw (3.5,1) -- (3.5,.5);
\draw (3.5,2) -- (3.5,2.5);
\draw (5.5,1) -- (5.5,.5);
\draw (5.5,2) -- (5.5,2.5);
\draw (7.10,1.5) node[scale=1.25] (X) {\dots};
\draw (7.6,1.5) -- (8.1,1.5);
\draw[rounded corners] (8.1,2) rectangle (9.25,1);
\draw (8.75,1.5) node (X) {$W_{n_W}$};
\draw (9.25,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$W_Z$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$W_Z$};
\draw (13.25,1.5) -- (13.75,1.5);
\draw (8.75,1) -- (8.75,.5);
\draw (8.75,2) -- (8.75,2.5);
\draw (10.75,1) -- (10.75,.5);
\draw (10.75,2) -- (10.75,2.5);
\draw (12.75,1) -- (12.75,.5);
\draw (12.75,2) -- (12.75,2.5);
\end{tikzpicture} \dots
\end{split}$$ where $W_{[i]} = W_A$ for all lattice sites $i < 1$ and $W_{[i]} = W_Z$ for all $i > n_W$, and $W_{[i]}$ are arbitrary for $i = 1,\cdots, n_W$. In the following, for notational conciseness, we drop the physical index $s$ on the tensors in an MPS or an MPO when confusion does not arise. Based on locality principles like the Lieb-Robinson bound [@Lieb-Robinson], we assume that the MPS approximating the time-evolved quantum states has the form\
$$\begin{split}
|\Psi(A; &B^i; Z)\rangle = \sum_{{{\bf s}}}(... A_{{[i-1]}}^{s_{i-1}} A_{{[i]}}^{s_i} A_{{[i+1]}}^{s_{i+1}}...){| {{\bf s}} \rangle}
\\
&=\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1$};
\draw (6,1.5) -- (6.5,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.10,1.5) node[scale=1.25] (X) {\dots};
\draw (7.75,1.5) -- (8.25,1.5);
\draw[rounded corners] (8.25,2) rectangle (9.25,1);
\draw (8.75,1.5) node (X) {$B^n$};
\draw (9.25,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$Z$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z$};
\draw (13.25,1.5) -- (13.75,1.5);
\draw (8.75,1) -- (8.75,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\end{split}
\label{eq:MPS}$$ where $n$, the number of inhomogeneous tensors $B^i$, needs to be larger than $n_W$. We require $A_{{[i]}} = A$ for all $i < 1$, and $A_{{[i]}} = Z$ for all $i > n$. The tensors $A_{{[i]}}$ on lattice sites $1$ to $n$ are denoted as $B^i$ and are allowed to change arbitrarily, except restrained by the bond dimension $D$. In the following analysis, in order for the variational manifold to be well-defined, we fix the bond dimension of the MPS to a given value. Here we note that as the local information spreads with real-time dynamics in a spin chain, in order for the MPS approximation to remain accurate, $n$ needs to increase with time. As shown in Sec. \[sec:integration\], it is very easy to dynamically expand $n$. For now, we take it to be a fixed number.
Gauge freedom
-------------
Eq. \[eq:MPS\] defines the variational manifold used to describe the time evolution of the system. $A$, $B^1, \cdots, B^n$, $Z$ are all complex tensors of dimension $d\times D\times D$, constituting the manifold of variational coefficients that we have access to: $${\mathcal{M}_\text{MPS}}= {\mathbb{C}}^{d\times D\times D} \times {\mathbb{C}}^{d\times D\times D} \times \prod_{i=1}^n {\mathbb{C}}^{d\times D\times D}.$$ The variational manifold of quantum states is then $${\mathcal{H}_\text{MPS}}= \{{| \Psi(A; B^i; Z) \rangle} | (A; B^i; Z) \in {\mathcal{M}_\text{MPS}}\}.$$ The (complex) dimension of ${\mathcal{M}_\text{MPS}}$ is much larger than that of ${\mathcal{H}_\text{MPS}}$, because of the gauge symmetries in an MPS. For example, with a $D\times D$ invertible matrix $X$, the tensors $A' = X^{-1}AX, B'^1 = X^{-1}B^1, B'^{i\not=1}=B^{i\not=1}, Z' = Z$ constitute a different point in ${\mathcal{M}_\text{MPS}}$, yet they give the same state in ${\mathcal{H}_\text{MPS}}$. In fact, it will turn out in Sec. \[sec:ortho\] that it is necessary to work in the projective space of ${\mathcal{H}_\text{MPS}}$: $${{{\bf P}}\mathcal{H}_\text{MPS}}= {\mathcal{H}_\text{MPS}}/{\mathbb{C}},$$ which has more gauge freedom than ${\mathcal{H}_\text{MPS}}$, e.g. with any complex number $\alpha$, the tensors $A' = \alpha A, B'_i =B_{i}, Z' = Z$ give the same point in ${{{\bf P}}\mathcal{H}_\text{MPS}}$.
To quantify the MPS gauge freedom, we need to find a group $G$ acting on ${\mathcal{M}_\text{MPS}}$, so that ${\mathcal{M}_\text{MPS}}/G \cong {{{\bf P}}\mathcal{H}_\text{MPS}}$, where ${\mathcal{M}_\text{MPS}}/G$ is the set of orbits of the group action. Thus, $G$ has to be large enough so that one element in ${{{\bf P}}\mathcal{H}_\text{MPS}}$ is in only one orbit, and it has to be small enough so that the group action is free [@Quotient]. One finds that $G$ is $$G = {\mathbb{C}}_A \times {\mathbb{C}}_Z \times (\prod_{i=1}^{n+1} {\text{GL}}({\mathbb{C}}; D))/{\mathbb{C}}_{B^1}$$ where ${\mathbb{C}}_A$, ${\mathbb{C}}_Z$, are ${\mathbb{C}}_{B^1}$ are groups of scalar multiplication on tensor $A$, $Z$, and $B^1$, each with complex dimension one. ${\text{GL}}({\mathbb{C}};D)$ is the multiplicative group of complex matrices of dimension $D \times D$. The complex dimension of $G$ is the number of the complex equations that one can impose in the gauge choice of tangent vectors to ${{{\bf P}}\mathcal{H}_\text{MPS}}$. It is equal to $$\dim_{\mathbb{C}}G = 1 + 1 + (n+1)D^2 - 1 = 2D^2 + (n-1)D^2 + 1.$$ We view these gauge symmetries as assets, and will “spend” them as we see fit to simplify the tangent vectors to ${{{\bf P}}\mathcal{H}_\text{MPS}}$ in the following sections.
Mixed canonical form of MPS
---------------------------
The gauge freedom of an MPS can be exploited to bring the MPS in a convenient form. For a entirely uniform MPS, as in the standard practice, one can write it in the mixed canonical form [@Tangent_space]: $$\begin{aligned}
{| \Psi(A) \rangle} &=
\dots \begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,0) -- (1,0);
\draw[rounded corners] (1,0.5) rectangle (2,-0.5);
\draw (1.5,0) node (X) {$A$};
\draw (2,0) -- (3,0);
\draw[rounded corners] (3,0.5) rectangle (4,-0.5);
\draw (3.5,0) node {$A$};
\draw (4,0) -- (5,0);
\draw[rounded corners] (5,0.5) rectangle (6,-0.5);
\draw (5.5,0) node {$A$};
\draw (6,0) -- (9,0);
\draw[rounded corners] (9,0.5) rectangle (10,-0.5);
\draw (9.5,0) node {$A$};
\draw (10,0) -- (11,0);
\draw[rounded corners] (11,0.5) rectangle (12,-0.5);
\draw (11.5,0) node {$A$};
\draw (12,0) -- (12.5,0);
\draw (1.5,-.5) -- (1.5,-1); \draw (3.5,-.5) -- (3.5,-1); \draw (5.5,-.5) -- (5.5,-1); \draw (11.5,-.5) -- (11.5,-1);
\draw (9.5,-.5) -- (9.5,-1);
\draw (5,1) node {$\phantom{X}$};
\end{tikzpicture} \dots \\
&=
\dots \begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,0) -- (1,0);
\draw[rounded corners] (1,0.5) rectangle (2,-0.5);
\draw (1.5,0) node (X) {$A_L$};
\draw (2,0) -- (3,0);
\draw[rounded corners] (3,0.5) rectangle (4,-0.5);
\draw (3.5,0) node {$A_L$};
\draw (4,0) -- (5,0);
\draw[rounded corners] (5,0.5) rectangle (6,-0.5);
\draw (5.5,0) node {$A_C$};
\draw (6,0) -- (9,0);
\draw[rounded corners] (9,0.5) rectangle (10,-0.5);
\draw (9.5,0) node {$A_R$};
\draw (10,0) -- (11,0);
\draw[rounded corners] (11,0.5) rectangle (12,-0.5);
\draw (11.5,0) node {$A_R$};
\draw (12,0) -- (12.5,0);
\draw (1.5,-.5) -- (1.5,-1); \draw (3.5,-.5) -- (3.5,-1); \draw (5.5,-.5) -- (5.5,-1); \draw (11.5,-.5) -- (11.5,-1);
\draw (9.5,-.5) -- (9.5,-1);
\draw (5,1) node {$\phantom{X}$};
\end{tikzpicture} \dots \\
&= \dots \begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,0) -- (1,0);
\draw[rounded corners] (1,0.5) rectangle (2,-0.5);
\draw (1.5,0) node (X) {$A_L$};
\draw (2,0) -- (3,0);
\draw[rounded corners] (3,0.5) rectangle (4,-0.5);
\draw (3.5,0) node {$A_L$};
\draw (4,0) -- (5,0);
\draw[rounded corners] (5,0.5) rectangle (6,-0.5);
\draw (5.5,0) node {$A_L$};
\draw (6,0) -- (7,0);
\draw(7.5,0) circle (0.5);
\draw (8,0) -- (9,0);
\draw (7.5,0) node {$C_A$};
\draw[rounded corners] (9,0.5) rectangle (10,-0.5);
\draw (9.5,0) node {$A_R$};
\draw (10,0) -- (11,0);
\draw[rounded corners] (11,0.5) rectangle (12,-0.5);
\draw (11.5,0) node {$A_R$};
\draw (12,0) -- (12.5,0);
\draw (1.5,-.5) -- (1.5,-1); \draw (3.5,-.5) -- (3.5,-1); \draw (5.5,-.5) -- (5.5,-1); \draw (11.5,-.5) -- (11.5,-1);
\draw (9.5,-.5) -- (9.5,-1);
\draw (5,1) node {$\phantom{X}$};
\end{tikzpicture} \dots \end{aligned}$$ The tensors $\{A_L, A_R, A_C, C_A\}$ satisfy the following relations: $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A_L$};
\draw (1.5,-1.5) node {$\bar{A}_L$};
\draw (2,1.5) -- (2.5,1.5); \draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\end{tikzpicture}
\hspace{10mm}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0.5,1.5) -- (1,1.5); \draw (0.5,-1.5) -- (1,-1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A_R$};
\draw (1.5,-1.5) node {$\bar{A}_R$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (1,-1.5) edge[out=0,in=0] (1,1.5);
\end{tikzpicture}
\label{eq:canonical}$$ and $$\begin{split}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node (X) {$A_C$};
\draw (1,-1) -- (1,-1.5);
\draw (1.5,-0.5) -- (2,-0.5);
\end{tikzpicture}
&=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (2.5,0) -- (3,0);
\draw[rounded corners] (3,0.5) rectangle (4,-0.5);
\draw (3.5,0) node (X) {$A_L$};
\draw (3.5,-0.5) -- (3.5,-1);
\draw (4,0) -- (5,0);
\draw (5.5,0) circle (0.5);
\draw (5.5,0) node {$C_A$};
\draw (6,0) -- (6.5,0);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0.5,0) -- (1,0);
\draw (1.5,0) circle (0.5);
\draw (1.5,0) node (X){$C_A$};
\draw (2,0) -- (3,0);
\draw[rounded corners] (3,0.5) rectangle (4,-0.5);
\draw (3.5,0) node {$A_R$};
\draw (3.5,-0.5) -- (3.5,-1);
\draw (4,0) -- (4.5,0);
\end{tikzpicture}
\end{split}.
\label{eq:AC}$$ The tensors $A_L$ and $A_R$ are respectively called the [*left and right canonical forms*]{} of $A$. $A_C$ is called the [*center site tensor*]{}, and $C_A$ the [*bond matrix*]{}. When the tensors do not have uniformity at all, similar left and right canonical tensors can be found that satisfy Eq. \[eq:canonical\] [@finite_TDVP]. The mixed-canonical form is the key to inversion-free TDVP algorithms [@Tangent_space]. Motivated by this, we also write the MPS in Eq. \[eq:MPS\] into the mixed-canonical form: $$\begin{split}
|\Psi(&A; B^i; Z)\rangle
\\
&=\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1$};
\draw (6,1.5) -- (7,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$B^2$};
\draw (8,1.5) -- (8.5,1.5);
\draw (9.10,1.5) node[scale=1.25] (X) {\dots};
\draw (9.75,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$B^n$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z$};
\draw (13.25,1.5) -- (14.25,1.5);
\draw[rounded corners] (14.25,2) rectangle (15.25,1);
\draw (14.75,1.5) node (X) {$Z$};
\draw (15.25,1.5) -- (15.75,1.5);
\draw (14.75,1) -- (14.75,.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.5,1) -- (7.5,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\\
&=\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A_L$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A_C$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1_R$};
\draw (6,1.5) -- (7,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$B^2_R$};
\draw (8,1.5) -- (8.5,1.5);
\draw (9.10,1.5) node[scale=1.25] (X) {\dots};
\draw (9.75,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$B^n_R$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z_R$};
\draw (13.25,1.5) -- (14.25,1.5);
\draw[rounded corners] (14.25,2) rectangle (15.25,1);
\draw (14.75,1.5) node (X) {$Z_R$};
\draw (15.25,1.5) -- (15.75,1.5);
\draw (14.75,1) -- (14.75,.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.5,1) -- (7.5,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\\
&=\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A_L$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A_L$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1_L$};
\draw (6,1.5) -- (7,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$B^2_C$};
\draw (8,1.5) -- (8.5,1.5);
\draw (9.10,1.5) node[scale=1.25] (X) {\dots};
\draw (9.75,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$B^n_R$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z_R$};
\draw (13.25,1.5) -- (14.25,1.5);
\draw[rounded corners] (14.25,2) rectangle (15.25,1);
\draw (14.75,1.5) node (X) {$Z_R$};
\draw (15.25,1.5) -- (15.75,1.5);
\draw (14.75,1) -- (14.75,.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.5,1) -- (7.5,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\\
&=\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A_L$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A_L$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1_L$};
\draw (6,1.5) -- (7,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$B^2_L$};
\draw (8,1.5) -- (8.5,1.5);
\draw (9.10,1.5) node[scale=1.25] (X) {\dots};
\draw (9.75,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$B^n_L$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z_C$};
\draw (13.25,1.5) -- (14.25,1.5);
\draw[rounded corners] (14.25,2) rectangle (15.25,1);
\draw (14.75,1.5) node (X) {$Z_R$};
\draw (15.25,1.5) -- (15.75,1.5);
\draw (14.75,1) -- (14.75,.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.5,1) -- (7.5,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\end{split}
\label{eq:mixed_MPS}$$ Here $\{A_L, A_R, A_C\}$ and $\{Z_L, Z_R, Z_C\}$ are respectively the mixed canonical tensors of a uniform MPS made of $A$ and $Z$, and satisfy Eq. \[eq:canonical\]. $B^1_L, \cdots, B^{n-1}_L$ and $B^2_R, \cdots, B^n_R$ also respectively satisfy the left and right canonical relations in Eq. \[eq:canonical\]. However, $B_L^n$ and $B_R^1$ do not satisfy any canonical relation, because bringing them into canonical forms will destroy the uniformity of the $Z$ and $A$ matrices. This, however, as we will see, is not an essential difficulty.
Before we proceed, we note that Eq. \[eq:canonical\] is an eigen-relation for the transfer operators ${E_{A_L}}$ and ${E_{A_R}}$ defined as $${E_{A_L}}=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5, 1.5) -- (1,1.5);
\draw (0.5, -1.5) -- (1,-1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A_L$};
\draw (1.5,0) node(X) {};
\draw (1.5,-1.5) node {$\bar{A}_L$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
\hspace{10mm}
{E_{A_R}}=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5, 1.5) -- (1,1.5);
\draw (0.5, -1.5) -- (1,-1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A_R$};
\draw (1.5,-1.5) node {$\bar{A}_R$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}.
\label{eq:E}$$ In fact, Eq. \[eq:canonical\] is the eigen-relation for the non-degenerate leading eigenvalue of the transfer operators, which is 1 for a normalized uniform MPS [@Tangent_space]. This is very important, because it means that if one propagates an arbitrary boundary tensor from left through infinitely many ${E_{A_L}}$, only the leading left-eigvector of ${E_{A_L}}$ survives, which is a two-index delta tensor. The analogous fact is true for ${E_{A_R}}$, too.
The tangent space of matrix product states and gauge choices {#sec:tangent}
============================================================
We now analyze the tangent space to ${{{\bf P}}\mathcal{H}_\text{MPS}}$, following [@Tangent_space]. The tangent space of ${{{\bf P}}\mathcal{H}_\text{MPS}}$ can be obtained from the tangent space of ${\mathcal{H}_\text{MPS}}$ by identifying tangent vectors different by multiples of ${| \Psi \rangle}$. Therefore, we will still work with tangent vectors to ${\mathcal{H}_\text{MPS}}$ knowing that we can add arbitrary multiples of ${| \Psi \rangle}$ to the tangent vector whenever needed.
At ${| \Psi(A; B^i; Z) \rangle}$, the tangent vectors to ${\mathcal{H}_\text{MPS}}$ result from infinitesimal changes on the tensor elements: $a \equiv \delta A$, $b^i\equiv \delta B^i$, and $z \equiv \delta Z$, and are given by $$\begin{split}
{| \Phi(&a; b^i; z) \rangle} \equiv {| \Psi(A+a; B^i+b^i; Z+z) \rangle} - {| \Psi(A; B^i; Z) \rangle}
\\
&=\sum_{i=-\infty}^{0} \dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$a$};
\draw (4,1.5) -- (5,1.5);
\draw (3.5,0.2) node {$i$};
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$A$};
\draw (6,1.5) -- (7,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$A$};
\draw (8,1.5) -- (8.5,1.5);
\draw (9.10,1.5) node[scale=1.25] (X) {\dots};
\draw (9.75,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$B^n$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z$};
\draw (13.25,1.5) -- (13.75,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.5,1) -- (7.5,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\\
&+\sum_{i=1}^{n} \dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1$};
\draw (6,1.5) -- (6.5,1.5);
\draw (7.1,1.5) node[scale=1.25] (X) {\dots};
\draw (7.75,1.5) -- (8.25,1.5);
\draw[rounded corners] (8.25,2) rectangle (9.25,1);
\draw (8.75,1.5) node (X) {$b^i$};
\draw (8.75,0.2) node {$i$};
\draw (9.25,1.5) -- (9.75,1.5);
\draw (10.35,1.5) node[scale=1.25] (X) {\dots};
\draw (11,1.5) -- (11.5,1.5);
\draw[rounded corners] (11.5,2) rectangle (12.5,1);
\draw (12.00,1.5) node (X) {$B^n$};
\draw (12.5,1.5) -- (13.5,1.5);
\draw[rounded corners] (13.5,2) rectangle (14.5,1);
\draw (14,1.5) node (X) {$Z$};
\draw (14.5,1.5) -- (15,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (8.75,1) -- (8.75,.5);
\draw (12.,1) -- (12.,.5);
\draw (14.,1) -- (14.,.5);
\end{tikzpicture} \dots
\\
&+\sum_{i=n+1}^\infty
\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1$};
\draw (6,1.5) -- (6.5,1.5);
\draw (7.10,1.5) node[scale=1.25] (X) {\dots};
\draw (7.75,1.5) -- (8.25,1.5);
\draw[rounded corners] (8.25,2) rectangle (9.25,1);
\draw (8.75,1.5) node (X) {$Z$};
\draw (9.25,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$z$};
\draw (10.75,0.2) node (X) {$i$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z$};
\draw (13.25,1.5) -- (13.75,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (8.75,1) -- (8.75,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\\
&=\sum_{i=-\infty}^{0} \dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A_L$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$a_L$};
\draw (4,1.5) -- (5,1.5);
\draw (3.5,0.2) node {$i$};
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$A_R$};
\draw (6,1.5) -- (7,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$A_R$};
\draw (8,1.5) -- (8.5,1.5);
\draw (9.10,1.5) node[scale=1.25] (X) {\dots};
\draw (9.75,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$B^n_R$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z_R$};
\draw (13.25,1.5) -- (13.75,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (7.5,1) -- (7.5,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\\
&+\sum_{i=1}^{n} \dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A_L$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A_L$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1_L$};
\draw (6,1.5) -- (6.5,1.5);
\draw (7.1,1.5) node[scale=1.25] (X) {\dots};
\draw (7.75,1.5) -- (8.25,1.5);
\draw[rounded corners] (8.25,2) rectangle (9.25,1);
\draw (8.75,1.5) node (X) {$b^i_L$};
\draw (8.75,0.2) node {$i$};
\draw (9.25,1.5) -- (9.75,1.5);
\draw (10.35,1.5) node[scale=1.25] (X) {\dots};
\draw (11,1.5) -- (11.5,1.5);
\draw[rounded corners] (11.5,2) rectangle (12.5,1);
\draw (12.00,1.5) node (X) {$B^n_R$};
\draw (12.5,1.5) -- (13.5,1.5);
\draw[rounded corners] (13.5,2) rectangle (14.5,1);
\draw (14,1.5) node (X) {$Z_R$};
\draw (14.5,1.5) -- (15,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (8.75,1) -- (8.75,.5);
\draw (12.,1) -- (12.,.5);
\draw (14.,1) -- (14.,.5);
\end{tikzpicture} \dots
\\
&+\sum_{i=n+1}^\infty
\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A_L$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A_L$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^1_L$};
\draw (6,1.5) -- (6.5,1.5);
\draw (7.10,1.5) node[scale=1.25] (X) {\dots};
\draw (7.75,1.5) -- (8.25,1.5);
\draw[rounded corners] (8.25,2) rectangle (9.25,1);
\draw (8.75,1.5) node (X) {$Z_L$};
\draw (9.25,1.5) -- (10.25,1.5);
\draw[rounded corners] (10.25,2) rectangle (11.25,1);
\draw (10.75,1.5) node (X) {$z_R$};
\draw (10.75,0.2) node (X) {$i$};
\draw (11.25,1.5) -- (12.25,1.5);
\draw[rounded corners] (12.25,2) rectangle (13.25,1);
\draw (12.75,1.5) node (X) {$Z_R$};
\draw (13.25,1.5) -- (13.75,1.5);
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (8.75,1) -- (8.75,.5);
\draw (10.75,1) -- (10.75,.5);
\draw (12.75,1) -- (12.75,.5);
\end{tikzpicture} \dots
\end{split}
\label{eq:Phi}$$ where we have also written $\Phi$ in the mixed canonical form. The meaning of the subscripts on $a_L$, $b_L^i$, and $z_R$ will become clear in a moment.
Gauge choices of the tangent vectors
------------------------------------
Due to the gauge freedom, parameters $a_L, b_L^i$, and $z_R$ are redundant in describing a tangent vector to ${{{\bf P}}\mathcal{H}_\text{MPS}}$, which poses a problem to computing the projection of $\hat{H}{| \Psi \rangle}$. We now spend the gauge symmetries contained in $G$ to fix these redundancies. Out of the $2D^2+(n-1)D^2 + 1$ gauge symmetries of ${{{\bf P}}\mathcal{H}_\text{MPS}}$, we impose at once $2D^2+(n-1)D^2$ restraints on $a_L, b_L^i$, and $z_R$: $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (3,1);
\draw[rounded corners] (1,-1) rectangle (3,-2);
\draw (2.0,1) -- (2.0,-1);
\draw (2.0,1.5) node {$a_L(b^i_L)$};
\draw (1.5,0) node(X) {};
\draw (2.0,-1.5) node {$\bar{A}_L(\bar{B}^i_L)$};
\draw (3,1.5) -- (3.5,1.5); \draw (3,-1.5) -- (3.5,-1.5);
\end{tikzpicture}
=
0
\hspace{10mm}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0.5,1.5) -- (1,1.5); \draw (0.5,-1.5) -- (1,-1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$z_R$};
\draw (1.5,0) node(X) {};
\draw (1.5,-1.5) node {$\bar{Z}_R$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
=0
\label{eq:tangent_canonical}$$ where the $i$ above only goes from $1$ to $n-1$. We still have one last symmetry to spend, which we reserve for $b_L^n$ until Eq. \[eq:bn\_L\]. Eq. \[eq:tangent\_canonical\] can be explicitly satisfied by giving $a_L, b_L^i,$ and $z_R$ an effective parametrization: $$\begin{split}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (-0.3,-0.5) -- (0.20,-0.5);
\draw[rounded corners] (0.2,0) rectangle (1.8,-1);
\draw (1,-0.5) node (X) {$a_L(b_L^i)$};
\draw (1,-1) -- (1,-1.5);
\draw (1.8,-0.5) -- (2.3,-0.5);
\end{tikzpicture}
&=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (1.9,0) -- (2.4,0);
\draw[rounded corners] (2.4,0.5) rectangle (4.4,-0.5);
\draw (3.4,0) node (X) {$V_{A_L(B_L^i)}$};
\draw (3.4,-0.5) -- (3.4,-1);
\draw (4.4,0) -- (5.4,0);
\draw[rounded corners] (5.4,0.5) rectangle (7.0,-0.5);
\draw (6.2,0) node {$X_{A(B^i)}$};
\draw (7,0) -- (7.5,0);
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node (X) {$z_R$};
\draw (1,-1) -- (1,-1.5);
\draw (1.5,-0.5) -- (2,-0.5);
\end{tikzpicture}
&=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0.5,0) -- (1,0);
\draw[rounded corners] (1,0.5) rectangle (2,-0.5);
\draw (1.5,0) node (X){$X_Z$};
\draw (2,0) -- (3,0);
\draw[rounded corners] (3,0.5) rectangle (4,-0.5);
\draw (3.5,0) node {$V_{Z_R}$};
\draw (3.5,-0.5) -- (3.5,-1);
\draw (4,0) -- (4.5,0);
\end{tikzpicture}
\end{split}
\label{eq:XY}$$ where the right (left) index of $V_{A_L}(V_{Z_R})$ has dimension $D(d-1)$. $V_{A_L}$ is determined by requiring its column vectors be orthonormal among themselves and orthogonal to those of $A_L$: $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$V_{A_L}$};
\draw (1.5,-1.5) node {$\bar{V}_{A_L}$};
\draw (2,1.5) -- (2.5,1.5); \draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\end{tikzpicture}
\hspace{10mm}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A_C$};
\draw (1.5,-1.5) node {$\bar{V}_{A_L}$};
\draw (2,1.5) -- (2.5,1.5); \draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A_L$};
\draw (1.5,-1.5) node {$\bar{V}_{A_L}$};
\draw (2,1.5) -- (2.5,1.5); \draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
=0 .
\label{eq:V}$$ $V_{B_L^i}$ are similarly determined for $i=1,\cdots, n-1$, and $V_{Z_R}$ is determined from a right version of Eq. \[eq:V\].
The Gram matrix of the tangent vector
-------------------------------------
Using Eq. \[eq:canonical\] and \[eq:Phi\]-\[eq:V\] we now compute the inner product ${\langle \Phi|\Phi \rangle}$, also known as the Gram matrix, which is needed later for computing the orthogonal projection of $\hat{H}{| \Psi \rangle}$: $$\begin{split}
{\langle \Phi(&\bar{X}_A;\bar{X}_{B^i};\bar{X}_Z;\bar{b}^{n}_L)|\Phi({X}_A;{X}_{B^i};{X}_Z;{b}_L^n) \rangle}
\\
&= \sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_A$};
\draw (1.5,-1.5) node {$\bar{X}_A$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,-1.5) -- (2.5,-1.5);
\draw[rounded corners] (2.5,2) rectangle (4.5,-2);
\draw (3.5,0) node {$({E_{A_R}})^m$};
\draw (4.5,1.5) -- (5.0,1.5);
\draw (4.5,-1.5) -- (5.0,-1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw[rounded corners] (5,-1) rectangle (6,-2);
\draw (5.5,1) -- (5.5,-1);
\draw (5.5,1.5) node {$B^1_R$};
\draw (5.5,0) node(X) {};
\draw (5.5,-1.5) node {$\bar{B}^1_R$};
\draw (6,-1.5) edge[out=0,in=0] (6,1.5);
\end{tikzpicture}
+ \sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$B^n_L$};
\draw (1.5,-1.5) node {$\bar{B}^n_L$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,-1.5) -- (2.5,-1.5);
\draw[rounded corners] (2.5,2) rectangle (4.5,-2);
\draw (3.5,0) node {$({E_{Z_L}})^m$};
\draw (4.5,1.5) -- (5.0,1.5);
\draw (4.5,-1.5) -- (5.0,-1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw[rounded corners] (5,-1) rectangle (6,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (5.5,1.5) node {$X_Z$};
\draw (5.5,0) node(X) {};
\draw (5.5,-1.5) node {$\bar{X}_Z$};
\draw (6,-1.5) edge[out=0,in=0] (6,1.5);
\end{tikzpicture}
\\
&+\sum_{i=1}^{n-1}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_{B^i}$};
\draw (1.5,-1.5) node {$\bar{X}_{B^i}$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
+
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$b^n_{L}$};
\draw (1.5,-1.5) node {$\bar{b}^n_{L}$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\draw (1.5,1) -- (1.5,-1);
\end{tikzpicture}.
\end{split}$$ To simplify ${\langle \Phi|\Phi \rangle}$ further, we explicitly split out the contribution of ${E_{A_R}}$ from its leading eigenspace: $${E_{A_R}}=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0.5,-1.5) edge[out=0,in=0] (0.5,1.5);
\draw (2.0,0) circle (.5);
\draw (2,0) node {$l_{A_R}$};
\draw (3,1.5) edge[out=180,in=90] (2,0.5);
\draw (3,-1.5) edge[out=180,in=270] (2,-0.5);
\end{tikzpicture}
+
\tilde{E}_{A_R}$$ where $l_{A_R}$ is the leading left-eigenvector of ${E_{A_R}}$, and $\tilde{E}_{A_R}$ is the contribution from the sub-leading eigenspace of ${E_{A_R}}$. Then, $$\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1.5,1.5) -- (2.5,1.5);
\draw (1.5,-1.5) -- (2.5,-1.5);
\draw[rounded corners] (2.5,2) rectangle (4.5,-2);
\draw (3.5,0) node {$({E_{A_R}})^m$};
\draw (4.5,1.5) -- (5.5,1.5);
\draw (4.5,-1.5) -- (5.5,-1.5);
\end{tikzpicture}
=
\sum_{m=0}^\infty \begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,-1.5) edge[out=0,in=0] (0.5,1.5);
\draw (2.0,0) circle (.5);
\draw (2,0) node {$l_{A_R}$};
\draw (3,1.5) edge[out=180,in=90] (2,0.5);
\draw (3,-1.5) edge[out=180,in=270] (2,-0.5);
\end{tikzpicture}
+
\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1.5,1.5) -- (2.5,1.5);
\draw (1.5,-1.5) -- (2.5,-1.5);
\draw[rounded corners] (2.5,2) rectangle (4.5,-2);
\draw (3.5,0) node {$(\tilde{E}_{A_R})^m$};
\draw (4.5,1.5) -- (5.5,1.5);
\draw (4.5,-1.5) -- (5.5,-1.5);
\end{tikzpicture}.
\label{eq:split}$$ This splitting is useful because $\tilde{E}_{A_R}$ has a spectral radius less than one, and the second term on the right-hand side of Eq. \[eq:split\] converges. We now have $$\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_A$};
\draw (1.5,-1.5) node {$\bar{X}_A$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,-1.5) -- (2.5,-1.5);
\draw[rounded corners] (2.5,2) rectangle (4.5,-2);
\draw (3.5,0) node {$({E_{A_R}})^m$};
\draw (4.5,1.5) -- (5.0,1.5);
\draw (4.5,-1.5) -- (5.0,-1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw[rounded corners] (5,-1) rectangle (6,-2);
\draw (5.5,1) -- (5.5,-1);
\draw (5.5,1.5) node {$B^1_R$};
\draw (5.5,0) node(X) {};
\draw (5.5,-1.5) node {$\bar{B}^1_R$};
\draw (6,-1.5) edge[out=0,in=0] (6,1.5);
\end{tikzpicture}
=
\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_A$};
\draw (1.5,-1.5) node {$\bar{X}_A$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
+F_A
\label{eq:FA}$$ where $F_A$ is a finite number. Here we have used the normalization of the state: $${\langle \Psi|\Psi \rangle} =
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (4.,0) circle (.5);
\draw (4.,0) node {$l_{A_R}$};
\draw (5,1.5) edge[out=180,in=90] (4,0.5);
\draw (5,-1.5) edge[out=180,in=270] (4,-0.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw[rounded corners] (5,-1) rectangle (6,-2);
\draw (5.5,1) -- (5.5,-1);
\draw (5.5,1.5) node {$B^1_R$};
\draw (5.5,0) node(X) {};
\draw (5.5,-1.5) node {$\bar{B}^1_R$};
\draw (6,-1.5) edge[out=0,in=0] (6,1.5);
\end{tikzpicture} = 1.
\label{eq:normalization}$$ An relation analogous to Eq. \[eq:FA\] holds for the $Z$ tensors, too. This gives the final form of the Gram matrix: $$\begin{split}
{\langle \Phi(&\bar{X}_A;\bar{X}_{B^i};\bar{X}_Z;\bar{b}^{n}_L)|\Phi({X}_A;{X}_{B^i};{X}_Z;{b}_L^n) \rangle}
\\
&=
\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_A$};
\draw (1.5,-1.5) node {$\bar{X}_A$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
+
\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_Z$};
\draw (1.5,-1.5) node {$\bar{X}_Z$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
+
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$b^n_{L}$};
\draw (1.5,-1.5) node {$\bar{b}^n_{L}$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\draw (1.5,1) -- (1.5,-1);
\end{tikzpicture}
\\
&+\sum_{i=1}^{n-1}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1.5) node {$X_{B^i}$};
\draw (1.5,-1.5) node {$\bar{X}_{B^i}$};
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\end{tikzpicture}
+ F_A(X_A) + F_Z(X_Z).
\end{split}
\label{eq:Gram}$$ The Gram matrix is thus essentially diagonal. As we will see, the finite terms $F_A$ and $F_Z$ drop when computing the orthogonal projection of $\hat{H}{| \Psi \rangle}$.
Orthogonal projection of $\hat{H}{| \Psi \rangle}$ {#sec:ortho}
===================================================
We now determine the $\Phi({X}_A;{X}_{B^i};{X}_Z;{b}^{n}_L)$ that is the orthogonal projection of ${\hat{H}{| \Psi \rangle}}$ on the tangent space at ${| \Psi \rangle}$. It is given by the solution to the minimization problem $$\min_{X_A,X_{B^i},X_Z,b^n_L}{\left\lVert{\hat{H}{| \Psi \rangle}}- \Phi({X}_A;{X}_{B^i};{X}_Z;{b}^{n}_L)\right\rVert}_2^2.$$ $X_A$ is determined by $$\frac{\partial{\langle \Phi|\Phi \rangle}}{\partial \bar{X}_A} =
\frac{\partial F_A}{\partial \bar{X}_A} +
\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0.0,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node (X) {$X_A$};
\draw (1.5,-0.5) -- (2.0,-0.5);
\end{tikzpicture}
= \frac{\partial{\langle \Phi | \hat{H}|\Psi \rangle}}{\partial \bar{X}_A}.
\label{eq:X_A}$$ Here, $$\begin{split}
&\frac{\partial{\langle \Phi | \hat{H}|\Psi \rangle}}{\partial \bar{X}_A}
\\
& =\sum_{m=0}^\infty \cdots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (2.5,3.5) -- (3,3.5);
\draw (2.5,1.5) -- (3,1.5);
\draw (2.5,-0.5) -- (3,-0.5);
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {${E^{[W]}_{A_L}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw[rounded corners] (8.2, 4) rectangle (10.2,-1);
\draw (9.2,1.5) node {$({E^{[W]}_{A_R}})^m$};
\draw (10.2,3.5) -- (11.2,3.5);
\draw (10.2,1.5) -- (11.2,1.5);
\draw (10.2,-.5) -- (11.2,-0.5);
\draw[rounded corners] (11.2, 4) rectangle (13.2,-1);
\draw (12.25,1.5) node {$\displaystyle\prod_{i=1}^n E_{B^i_R}^{{[W]}}$};
\draw (13.2,3.5) -- (14.2,3.5);
\draw (13.2,1.5) -- (14.2,1.5);
\draw (13.2,-.5) -- (14.2,-0.5);
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw[rounded corners] (14.2, 4) rectangle (15.4,-1);
\draw (14.8,1.5) node {${E^{[W]}_{Z_R}}$};
\draw (15.4,3.5) -- (15.9,3.5);
\draw (15.4,1.5) -- (15.9,1.5);
\draw (15.4,-.5) -- (15.9,-.5);
\end{tikzpicture}\cdots
\end{split}
\label{eq:long}$$ where we have defined the MPO transfer matrices $${E^{[W]}_{A_L}}=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5, 1.5) -- (1,1.5);
\draw (0.5, -1.5) -- (1,-1.5);
\draw (0.5, 0) -- (1,0);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,0.5);
\draw (1.5,1.5) node {$A_L$};
\draw[rounded corners] (1,-0.5) rectangle (2,0.5);
\draw (1.5,-1) -- (1.5,-0.5);
\draw (1.5,0) node(X) {$W_A$};
\draw (1.5,-1.5) node {$\bar{A}_L$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,0) -- (2.5,0);
\draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
\hspace{10mm}
{E^{[W]}_{Z_R}}=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5, 1.5) -- (1,1.5);
\draw (0.5, -1.5) -- (1,-1.5);
\draw (0.5, 0) -- (1,0);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,0.5);
\draw (1.5,1.5) node {$Z_R$};
\draw[rounded corners] (1,-0.5) rectangle (2,0.5);
\draw (1.5,-1) -- (1.5,-0.5);
\draw (1.5,0) node(X) {$W_Z$};
\draw (1.5,-1.5) node {$\bar{Z}_R$};
\draw (2,1.5) -- (2.5,1.5);
\draw (2,0) -- (2.5,0);
\draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture},
\label{eq:E}$$ and others analogously. Before we proceed, we need some facts about the MPO transfer matrices.
Note that for a uniform MPS of tensor $A$ with $m$ sites, ${\langle \Psi|\hat{H}|\Psi \rangle} \sim ({E^{[W]}_{A_L}})^m$ up to some unimportant boundary terms. The extensivity of energy thus requires that $({E^{[W]}_{A_L}})^m$ be asymptotically linear in $m$. This can only happen if the leading eigenvalue of ${E^{[W]}_{A_L}}$ equals one and is defective. In fact, for a typical MPO, the leading eigenvalue of ${E^{[W]}_{A_L}}$ is indeed one with algebraic multiplicity two and geometric multiplicity one [@VUMPS], i.e. ${E^{[W]}_{A_L}}$ has one eigenvector and one generalized eigenvector in the leading eigenspace. This behavior can be attributed to the Schur form (lower triangular form) of the $W$ matrix of an MPO [@Schur; @VUMPS], on which we give a review in Appendix \[app:schur\]. We denote the left eigenvector and generalized eigenvector of ${E^{[W]}_{A_L}}$ by ${(I_1|}$ and ${(L_A^{{[W]}}|}$, and the right eigenvector and generalized eigenvector of ${E^{[W]}_{A_R}}$ by ${|I_{d_W})}$ and ${|R_A^{{[W]}})}$. The ${(L_A^{{[W]}}|}$ and ${|R_A^{{[W]}})}$ can be efficiently computed by an algorithm given in the Appendix of [@VUMPS]. (They are known as quasi-fixed points there.) We analogously define ${(L_Z^{{[W]}}|}$ and ${|R_Z^{{[W]}})}$.
As the left boundary tensor propagates through infinitely many ${E^{[W]}_{A_L}}$ to meet the center site $A_C$ in Eq. \[eq:long\], only the leading eigenspace survives. The same applies to the right side. Thus, $$\frac{\partial{\langle \Phi | \hat{H}|\Psi \rangle}}{\partial \bar{X}_A} = \left[{(L_A^{{[W]}}|} + \alpha {(I_1|}\right] E_C \left[{|R_Z^{{[W]}})} + \beta {|I_{d_W})}\right],
\label{eq:E_C}$$ where $$E_C \equiv
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (4.7,3.5) -- (5.2,3.5);
\draw (4.7,1.5) -- (5.2,1.5);
\draw (4.7,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw[rounded corners] (8.2, 4) rectangle (10.2,-1);
\draw (9.2,1.5) node {$({E^{[W]}_{A_R}})^m$};
\draw (10.2,3.5) -- (11.2,3.5);
\draw (10.2,1.5) -- (11.2,1.5);
\draw (10.2,-.5) -- (11.2,-0.5);
\draw[rounded corners] (11.2, 4) rectangle (13.2,-1);
\draw (12.25,1.5) node {$\displaystyle\prod_{i=1}^n E_{B^i_R}^{{[W]}}$};
\draw (13.2,3.5) -- (13.7,3.5);
\draw (13.2,1.5) -- (13.7,1.5);
\draw (13.2,-.5) -- (13.7,-0.5);
\draw (5,1.5) node (X) {$\phantom{X}$};,
\end{tikzpicture}.$$ Here, $\alpha$ and $\beta$ are two complex numbers. They occur because every time ${(L_A^{{[W]}}|}$ passes through ${E^{[W]}_{A_L}}$, there arises a new term of ${(I_1|}$: $ {(L_A^{{[W]}}|} {E^{[W]}_{A_L}}= {(L_A^{{[W]}}|} + e {(I_1|}$, where $e$ is the energy density of the chain [@VUMPS]. Their values, however, do not matter because of the following lemmas.
${(I_1|}E_C = 0$. (This lemma, and others below, are based on the Schur form of the MPO. See Appendix \[app:schur\] for a discussion of their proofs.) \[lem:1\]
${(L_A^{{[W]}}|}E_C{|I_{d_W})} = 0$. \[lem:2\]
Thus, $$\begin{split}
\frac{\partial{\langle \Phi | \hat{H}|\Psi \rangle}}{\partial \bar{X}_A}
& =\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw[rounded corners] (8.2, 4) rectangle (10.2,-1);
\draw (9.2,1.5) node {$({E^{[W]}_{A_R}})^m$};
\draw (10.2,3.5) -- (11.2,3.5);
\draw (10.2,1.5) -- (11.2,1.5);
\draw (10.2,-.5) -- (11.2,-0.5);
\draw[rounded corners] (11.2, 4) rectangle (13.2,-1);
\draw (12.25,1.5) node {$\displaystyle\prod_{i=1}^n E_{B^i_R}^{{[W]}}$};
\draw (13.2,3.5) -- (14.2,3.5);
\draw (13.2,1.5) -- (14.2,1.5);
\draw (13.2,-.5) -- (14.2,-0.5);
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw[rounded corners] (14.2, 4) rectangle (15.4,-1);
\draw (14.8,1.5) node {$R_Z^{{[W]}}$};
\end{tikzpicture}.
\end{split}$$ As with $E_{A_R}$, we split out of ${E^{[W]}_{A_R}}$ the term associated with the leading eigenspace. To do this, we need the following lemma in linear algebra.
Let $E$ be a matrix with leading eigenvalue one, according to which there is one eigenvector and one generalized eigenvector. Let ${(v_1|}$ be the left generalized eigenvector, ${(v_2|}$ the left eigenvector, ${|u_1)}$ the right eigenvector, and ${|u_2)}$ the right generalized eigenvector. Then, for an integer $m > 0$, $$E^m = {|u_1)}{(v_1|} + m {|u_1)}{(v_2|} + {|u_2)}{(v_2|} + \tilde{E}^m,$$ where $\tilde{E}$ is the contribution to $E$ from the sub-leading eigenspace. \[lem:3\]
When applying Lemma \[lem:3\] to ${E^{[W]}_{A_R}}$, the contribution associated with the ${|u_1)} = {|I_{d_W})}$ drops because of the following lemma.
$$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw[rounded corners] (8.2, 4) rectangle (9.4,-1);
\draw (8.8,1.5) node {$I_{d_W}$};
\end{tikzpicture} = 0.$$
\[lem:4\]
Thus, we have $$\begin{split}
&\frac{\partial{\langle \Phi | \hat{H}|\Psi \rangle}}{\partial \bar{X}_A}
=\sum_{m=0}^\infty
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw[rounded corners] (8.2, 4) rectangle (9.4,-1);
\draw (8.8,1.5) node {$R_{A}^{{[W]}}$};
\end{tikzpicture}
\\
& \hspace{5mm}+
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw[rounded corners] (8.2, 4) rectangle (11.2,-1);
\draw (9.7,1.5) node {$\displaystyle\sum_{m=0}^\infty (\tilde{E}^{[W]}_{A_R})^m$};
\draw (11.2,3.5) -- (12.2,3.5);
\draw (11.2,1.5) -- (12.2,1.5);
\draw (11.2,-.5) -- (12.2,-0.5);
\draw[rounded corners] (12.2, 4) rectangle (14.2,-1);
\draw (13.25,1.5) node {$\displaystyle\prod_{i=1}^n E_{B^i_R}^{{[W]}}$};
\draw (14.2,3.5) -- (15.2,3.5);
\draw (14.2,1.5) -- (15.2,1.5);
\draw (14.2,-.5) -- (15.2,-0.5);
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw[rounded corners] (15.2, 4) rectangle (16.4,-1);
\draw (15.8,1.5) node {$R_Z^{{[W]}}$};
\end{tikzpicture},
\end{split}
\label{eq:jj}$$ where we have made use of the following lemma.
$$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$l_{A_R}^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2, 4) rectangle (7.2,-1);
\draw (6.25,1.5) node {$\displaystyle\prod_{i=1}^n E_{B^i_R}^{{[W]}}$};
\draw (7.2,3.5) -- (8.2,3.5);
\draw (7.2,1.5) -- (8.2,1.5);
\draw (7.2,-0.5) -- (8.2,-0.5);
\draw[rounded corners] (8.2, 4) rectangle (9.4,-1);
\draw (8.8,1.5) node {$R_{Z}^{{[W]}}$};
\end{tikzpicture} = 1,$$
where $l_{A_R}^{{[W]}}$ is the left eigenvector of ${E^{[W]}_{A_R}}$. \[lem:5\]
Note that the second term of Eq. \[eq:jj\] converges. Now substitute Eq. \[eq:jj\] into Eq. \[eq:X\_A\], and divide the equation by $\sum_{m=0}^\infty 1$. The finite terms drop, and we obtain $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (-0.25,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node (X) {$X_A$};
\draw (1.5,-0.5) -- (2.25,-0.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2,4) rectangle (6.2,3);
\draw (5.7,3.5) node {$A_C$};
\draw (5.7,3.0) -- (5.7,2.0);
\draw[rounded corners] (5.2,2.0) rectangle (6.2,1);
\draw (5.7,1.5) node(X) {$W_A$};
\draw (5.7,1.0) -- (5.7,-0.);
\draw[rounded corners] (5.2,0.0) rectangle (6.2,-1);
\draw (5.7,-0.5) node {$\bar{V}_{A_L}$};
\draw (6.2,1.5) -- (7.7,1.5);
\draw (6.2,3.5) -- (7.7,3.5);
\draw (6.2,-.5) -- (6.7,-.5);
edge\draw (6.7,-0.5) edge[out=0,in=0] (6.7,-1.5);
\draw (7.7,-0.5) edge[out=180,in=180] (7.7,-1.5);
\draw (7.7,1.5) -- (8.7,1.5);
\draw (7.7,3.5) -- (8.7,3.5);
\draw (7.7,-.5) -- (8.7,-.5);
\draw[rounded corners] (8.7, 4) rectangle (9.9,-1);
\draw (9.3,1.5) node {$R_{A}^{{[W]}}$};
\end{tikzpicture}.
\label{eq:X_A_final}$$ Analogously, we have $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (-0.25,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node (X) {$X_Z$};
\draw (1.5,-0.5) -- (2.25,-0.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_Z^{{[W]}}$};
\draw (4.2,3.5) -- (6.7,3.5);
\draw (4.2,1.5) -- (6.7,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
edge\draw (5.2,-0.5) edge[out=0,in=0] (5.2,-1.5);
\draw (6.2,-.5) -- (6.7,-.5);
\draw (6.2,-0.5) edge[out=180,in=180] (6.2,-1.5);
\draw[rounded corners] (6.7,4) rectangle (7.7,3);
\draw (7.2,3.5) node {$Z_C$};
\draw (7.2,3.0) -- (7.2,2.0);
\draw[rounded corners] (6.7,2.0) rectangle (7.7,1);
\draw (7.2,1.5) node(X) {$W_Z$};
\draw (7.2,1.0) -- (7.2,-0.);
\draw[rounded corners] (6.7,0.0) rectangle (7.7,-1);
\draw (7.2,-0.5) node {$\bar{V}_{Z_R}$};
\draw (7.7,1.5) -- (8.2,1.5);
\draw (7.7,3.5) -- (8.2,3.5);
\draw (7.7,-.5) -- (8.2,-.5);
\draw (7.7,1.5) -- (8.7,1.5);
\draw (7.7,3.5) -- (8.7,3.5);
\draw (7.7,-.5) -- (8.7,-.5);
\draw[rounded corners] (8.7, 4) rectangle (9.9,-1);
\draw (9.3,1.5) node {$R_{Z}^{{[W]}}$};
\end{tikzpicture},
\label{eq:X_Z_final}$$ and for $i = 1, \cdots, n-1$, $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (-0.25,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node (X) {$X_{B^i}$};
\draw (1.5,-0.5) -- (2.25,-0.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2, 4) rectangle (7.2,-1);
\draw (6.25,1.5) node {$\displaystyle\prod_{j=1}^{i-1} E_{B^j_L}^{{[W]}}$};
\draw (7.2,3.5) -- (8.2,3.5);
\draw (7.2,1.5) -- (8.2,1.5);
\draw (7.2,-.5) -- (8.2,-0.5);
\draw[rounded corners] (8.2,4) rectangle (9.2,3);
\draw (8.7,3.5) node {$B^i_C$};
\draw (8.7,3.0) -- (8.7,2.0);
\draw[rounded corners] (8.2,2.0) rectangle (9.2,1);
\draw (8.7,1.5) node {$W_i$};
\draw (8.7,1.0) -- (8.7,-0.);
\draw[rounded corners] (8.2,0.0) rectangle (9.2,-1);
\draw (8.7,-0.5) node {$\bar{V}_{B^i_L}$};
\draw (9.2,1.5) -- (11.7,1.5);
\draw (9.2,3.5) -- (11.7,3.5);
\draw (9.2,-.5) -- (9.7,-.5);
edge\draw (9.7,-0.5) edge[out=0,in=0] (9.7,-1.5);
\draw (10.7,-0.5) edge[out=180,in=180] (10.7,-1.5);
\draw (10.7,-.5) -- (11.7,-.5);
\draw[rounded corners] (11.7, 4) rectangle (14.3,-1);
\draw (13.05,1.5) node {$\displaystyle\prod_{j=i+1}^n E_{B^j_R}^{{[W]}}$};
\draw (14.3,3.5) -- (15.3,3.5);
\draw (14.3,1.5) -- (15.3,1.5);
\draw (14.3,-.5) -- (15.3,-0.5);
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw[rounded corners] (15.3, 4) rectangle (16.5,-1);
\draw (15.9,1.5) node {$R_Z^{{[W]}}$};
\end{tikzpicture}.
\label{eq:X_B_final}$$ We now determine $b_L^n$, which is given by $$\begin{split}
\frac{\partial{\langle \Phi|\Phi \rangle}}{\partial \bar{b}_L^n} &=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node {$b_L^n$};
\draw (1,-0.8) node(X) {};
\draw (1,-1) -- (1,-1.5);
\draw (1.5,-0.5) -- (2,-0.5);
\end{tikzpicture}
= \frac{\partial{\langle \Phi | \hat{H}|\Psi \rangle}}{\partial \bar{b}_L^n}
\\
&=
\left[{(L_A^{{[W]}}|} + \alpha {(I_1|}\right]
E_D
\left[{|R_Z^{{[W]}})} + \beta {|I_{d_W})}\right],
\end{split}
\label{eq:b_n}$$ where $$E_D \equiv
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2, 4) rectangle (7.2,-1);
\draw (6.25,1.5) node {$\displaystyle\prod_{i=1}^{n-1} E_{B^i_L}^{{[W]}}$};
\draw (7.2,3.5) -- (8.2,3.5);
\draw (7.2,1.5) -- (8.2,1.5);
\draw (7.2,-.5) -- (7.7,-0.5);
edge\draw (7.7,-0.5) edge[out=0,in=0] (7.7,-1.5);
\draw[rounded corners] (8.2,4) rectangle (9.2,3);
\draw (8.7,3.5) node {$B^n_C$};
\draw (8.7,3.0) -- (8.7,2.0);
\draw[rounded corners] (8.2,2.0) rectangle (9.2,1);
\draw (8.7,1.5) node {$W_n$};
\draw (8.7,1.0) -- (8.7,-1.5);
\draw (9.2,3.5) -- (10.2,3.5);
\draw (9.2,1.5) -- (10.2,1.5);
\draw (9.7,-.5) -- (10.2,-0.5);
\draw (9.7,-0.5) edge[out=180,in=180] (9.7,-1.5);
\end{tikzpicture}.$$ Here the $\alpha$ and $\beta$ are the same as in Eq. \[eq:E\_C\]. Two lemmas are now in order:
${(I_1|}E_D{|I_{d_W})} = 0$. \[lem:6\]
$ {(I_1|}E_D{|R_{Z}^{{[W]}})} =
{(L_A^{{[W]}}|}E_D{|I_{d_W})} = B_C^n.
$ \[lem:7\]
Thus, $$b_L^n =
{(L_A^{{[W]}}|} E_D {|R_Z^{{[W]}})} + (\alpha+\beta)B_C^n.
\label{eq:bn_L}$$ But note that $b^n_L = (\alpha+\beta)B_C^n$ gives a contribution of $(\alpha+\beta){| \Psi \rangle}$ to ${| \Phi \rangle}$, which can be dropped in the projective space. Also recall that we still have one gauge symmetry to spend, which we use to write $b_L^n = {(L_A^{{[W]}}|} E_D {|R_Z^{{[W]}})}$. Thus, finally, we have $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.60},scale=.4]
\draw (0,-0.5) -- (0.5,-0.5);
\draw[rounded corners] (0.5,0) rectangle (1.5,-1);
\draw (1,-0.5) node {$b_L^n$};
\draw (1,-0.8) node(X) {};
\draw (1,-1) -- (1,-1.5);
\draw (1.5,-0.5) -- (2,-0.5);
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (5,1.5) node (X) {$\phantom{X}$};,
\draw[rounded corners] (3, 4) rectangle (4.2,-1);
\draw (3.6,1.5) node {$L_A^{{[W]}}$};
\draw (4.2,3.5) -- (5.2,3.5);
\draw (4.2,1.5) -- (5.2,1.5);
\draw (4.2,-0.5) -- (5.2,-0.5);
\draw[rounded corners] (5.2, 4) rectangle (7.2,-1);
\draw (6.25,1.5) node {$\displaystyle\prod_{i=1}^{n-1} E_{B^i_L}^{{[W]}}$};
\draw (7.2,3.5) -- (8.2,3.5);
\draw (7.2,1.5) -- (8.2,1.5);
\draw (7.2,-.5) -- (7.7,-0.5);
edge\draw (7.7,-0.5) edge[out=0,in=0] (7.7,-1.5);
\draw[rounded corners] (8.2,4) rectangle (9.2,3);
\draw (8.7,3.5) node {$B^n_C$};
\draw (8.7,3.0) -- (8.7,2.0);
\draw[rounded corners] (8.2,2.0) rectangle (9.2,1);
\draw (8.7,1.5) node {$W_n$};
\draw (8.7,1.0) -- (8.7,-1.5);
\draw (9.2,3.5) -- (10.2,3.5);
\draw (9.2,1.5) -- (10.2,1.5);
\draw (9.7,-.5) -- (10.2,-0.5);
\draw (9.7,-0.5) edge[out=180,in=180] (9.7,-1.5);
\draw[rounded corners] (10.2, 4) rectangle (11.4,-1);
\draw (10.8,1.5) node {$R_Z^{{[W]}}$};
\end{tikzpicture}.
\label{eq:bn_final}$$ We can now put Eq. \[eq:X\_A\_final\]-\[eq:X\_B\_final\] and Eq. \[eq:bn\_final\] back into Eq. \[eq:Phi\] to obtain ${| \Phi \rangle}_{H} = \text{Proj}_{T{{{\bf P}}\mathcal{H}_\text{MPS}}} {\hat{H}{| \Psi \rangle}}$.
While the derivation leading to ${| \Phi \rangle}_H$ is quite long, the result is intuitively clear. The $A$ and $Z$ matrices evolve as if they are in an entirely uniform MPS, by the iTDVP algorithm in [@Tangent_space; @iTDVP]. The $B$ matrices evolve by the same finite TDVP algorithm in [@finite_TDVP] except under the influence of the boundary tensors ${(L_A^{{[W]}}|}$ and ${|R_Z^{{[W]}})}$. The only thing unclear is how to patch these three time evolutions together, which we explain in the next section.
Integration scheme {#sec:integration}
==================
Here we explain how to evolve ${| \Psi \rangle}$ to $e^{\delta t \hat{H}} {| \Psi \rangle}$ using ${| \Phi \rangle}_H$.
MPO tensor $W_A$, $W_1, \cdots, W_{n_W}$, $W_Z$; MPS tensor $\{A_L, A_R, C_A, A_C\}$, $\{Z_L, Z_R, C_Z, Z_C\}$, $B^1_C, B^2_R, \cdots, B^n_R$; $L_A^{{[W]}}$, $R_Z^{{[W]}}$; time step $\delta t$ MPS tensor $\{A_L, A_R, C_A, A_C\}$, $\{Z_L, Z_R, C_Z, Z_C\}$, $B^1_C, B^2_R,\cdots,B^n_R$; $L_A^{{[W]}}$, $R_Z^{{[W]}}$ {$A_L,A_R,C_A,A_C$} $\gets$ iTDVP($W_A$,$A_L,A_R,C_A,A_C,\delta t$) Compute $L_A^{{[W]}}$ with $A_L$ and $W_A$ {$B^1_L,\cdots,B^{n-1}_L,B^{n}_C$} $\gets$ right sweep of finite-size TDVP($B^1_C,B^2_R,\cdots,B^{n}_R$,$L_A^{{[W]}}$,$R_Z^{{[W]}}$,$\delta t/2$) {$Z_L,Z_R,C_Z,Z_C$} $\gets$ iTDVP($W_Z$,$Z_L,Z_R,C_Z,Z_C,\delta t$) Compute $R_Z^{{[W]}}$ with $Z_R$ and $W_Z$ {$B^1_C,B^2_R,\cdots,B^{n}_R$} $\gets$ left sweep of finite-size TDVP($W_1,\cdots,W_n$,$B^1_L,\cdots,B^{n-1}_L,B^{n}_C$,$L_A^{{[W]}}$,$R_Z^{{[W]}}$,$\delta t/2$)
\[tab:mixed-iTDVP\]
In iTDVP, one first puts the center site $A_C$ at left infinity. Then one exponentiates the terms in ${| \Phi \rangle}_H$, one by one from left to right, to sequentially act on the current state. As the algorithm sweeps from left infinity to site $0$, the effect of the left boundary tensor decays away and the $A_C$ and $C_A$ tensors converge to their respective limits. The iTDVP algorithm in [@iTDVP] finds these limits without doing the actual sweep, thus is very efficient. However, there is something very peculiar about the sweeping process: in obtaining $\{A_C(t+\delta t), C_A(t+\delta(t))\}$ from $\{A_C(t), C_A(t)\}$, when the action of one term in ${| \Phi \rangle}_H$ is completed, one ends up with $C_A(t)$ instead of $C_A(t+\delta t)$ as the bond matrix. (One step of the sweep consist of two half-steps, and $C_A(t+\delta t)$ is obtained after the first half-step.) See page 35 of [@Tangent_space] or Table 1 of [@iTDVP] for the details. This peculiar fact is the key to patch the iTDVP and the finite TDVP algorithms, which we now do.
Suppose that at time $t$, we have a mixed iMPS centered at $B_1^C(t)$: $$\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (3.4,1);
\draw (2.2,1.5) node (X) {$A_L(t)$};
\draw (2.2,1) -- (2.2,.5);
\draw (3.4,1.5) -- (10.0,1.5);
\draw[rounded corners] (10.0,2) rectangle (11.8,1);
\draw (10.9,1.5) node {$B_C^1(t)$};
\draw (10.9,1.0) -- (10.9,0.5);
\draw (11.8,1.5) -- (12.3,1.5);
\draw (12.8,1.5) node[scale=1.25] (X) {\dots};
\draw (13.3,1.5) -- (13.8,1.5);
\draw[rounded corners] (13.8,2) rectangle (15.6,1);
\draw (14.7,1.5) node {$B_R^{n}(t)$};
\draw (14.7,1.0) -- (14.7,0.5);
\draw (15.6,1.5) -- (16.6,1.5);
\draw[rounded corners] (16.6,2) rectangle (18.4,1);
\draw (17.5,1.5) node {$Z_R(t)$};
\draw (17.5,1.0) -- (17.5,0.5);
\draw (18.4,1.5) -- (18.9,1.5);
\end{tikzpicture} \dots$$ To make the MPS centered at $A_C(t)$ at left infinity, one needs to borrow a $C_A(t)$ from $B^1_C(t)$, so that one has $$\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (3.4,1);
\draw (2.2,1.5) node (X) {$A_R(t)$};
\draw (2.2,1) -- (2.2,.5);
\draw (3.4,1.5) -- (7.2,1.5);
\draw[rounded corners] (7.2,2) rectangle (9.0,1);
\draw (8.1,1.5) node {$C_A^{-1}(t)$};
\draw (9.0,1.5) -- (10.0,1.5);
\draw[rounded corners] (10.0,2) rectangle (11.8,1);
\draw (10.9,1.5) node {$B_C^1(t)$};
\draw (10.9,1.0) -- (10.9,0.5);
\draw (11.8,1.5) -- (12.3,1.5);
\draw (12.8,1.5) node[scale=1.25] (X) {\dots};
\draw (13.3,1.5) -- (13.8,1.5);
\draw[rounded corners] (13.8,2) rectangle (15.6,1);
\draw (14.7,1.5) node {$B_R^{n}(t)$};
\draw (14.7,1.0) -- (14.7,0.5);
\draw (15.6,1.5) -- (16.6,1.5);
\draw[rounded corners] (16.6,2) rectangle (18.4,1);
\draw (17.5,1.5) node {$Z_R(t)$};
\draw (17.5,1.0) -- (17.5,0.5);
\draw (18.4,1.5) -- (18.9,1.5);
\end{tikzpicture} \dots$$ One then performs iTDVP on $A$ for $\delta t$ to arrive at $$\dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (3.4,1);
\draw (2.2,1.5) node (X) {$A_L(t+\delta t)$};
\draw (2.2,1) -- (2.2,.5);
\draw (3.4,1.5) -- (4.4,1.5);
\draw[rounded corners] (4.4,2) rectangle (6.2,1);
\draw (5.3,1.5) node {$C_A(t)$};
\draw (6.2,1.5) -- (7.2,1.5);
\draw[rounded corners] (7.2,2) rectangle (9.0,1);
\draw (8.1,1.5) node {$C_A^{-1}(t)$};
\draw (9.0,1.5) -- (10.0,1.5);
\draw[rounded corners] (10.0,2) rectangle (11.8,1);
\draw (10.9,1.5) node {$B_C^1(t)$};
\draw (10.9,1.0) -- (10.9,0.5);
\draw (11.8,1.5) -- (12.3,1.5);
\draw (12.8,1.5) node[scale=1.25] (X) {\dots};
\draw (13.3,1.5) -- (13.8,1.5);
\draw[rounded corners] (13.8,2) rectangle (15.6,1);
\draw (14.7,1.5) node {$B_R^{n}(t)$};
\draw (14.7,1.0) -- (14.7,0.5);
\draw (15.6,1.5) -- (16.6,1.5);
\draw[rounded corners] (16.6,2) rectangle (18.4,1);
\draw (17.5,1.5) node {$Z_R(t)$};
\draw (17.5,1.0) -- (17.5,0.5);
\draw (18.4,1.5) -- (18.9,1.5);
\end{tikzpicture} \dots$$ Thus, the bond matrix $C_A(t)$ cancels, and one next carries out the right sweep of finite TDVP on $B$ for $\delta t/2$ with boundary tensors ${(L_{A(t+\delta t)}^{{[W]}}|}$ and ${|R_{Z(t)}^{{[W]}})}$. Then one does iTDVP on $Z$ for $\delta t$ and sweeps on $B$ leftward for $\delta t/2$ with boundary tensors ${(L_{A(t+\delta t)}^{{[W]}}|}$ and ${|R_{Z(t+\delta t)}^{{[W]}})}$. This completes the mixed-iTDVP for one step of $\delta t$. For a pseudocode, see Table \[tab:mixed-iTDVP\]. We call this algorithm [*mixed-iTDVP*]{}. Globally, mixed-iTDVP is second order in $\delta t$ if $A$ and $Z$ are eigenstates of the bulk Hamiltonian on the left and right. It is first order in $\delta t$ if $A$ and $Z$ evolve non-trivially. To dynamically expand $n$, simply upgrade some number of $A$ and $Z$ matrices to be part of $B$. The algorithm can also be used to find the ground state when the evolution is in imaginary time. When the time step is infinite, the algorithm reduces to the conventional one-site density matrix renormalization group [@DMRG]. When the time step approaches 0, however, the imaginary time-evolution algorithm has the benefit of ensuring finding the global energy minimum, as long as the initial state has non-zero overlap with the ground state.
Example: transverse-field Ising model {#sec:Ising}
=====================================
As an illustrative example, we study the quantum dynamics of the transverse-field Ising chain: $$\hat{H} = -\sum_{i}\hat{\sigma}^z_i \hat\sigma^z_{i+1} + h_x \sum_{i} \hat{\sigma}^x_i + h_z \hat\sigma^z_{i_0}.
\label{eq:Ising}$$ The system is exactly solvable when $h_z = 0$ and is critical at $h_x = 1$ [@Kogut]. At criticality, the dispersion relation becomes linear $$E({{\bf k}}) = v_s {|{{\bf k}}|},$$ giving a characteristic sound velocity $v_s = 2$ [@Kogut]. We focus on quenching the local Hamiltonian $h_z \hat{\sigma}_{i0}^z$ in the following. When the quench is local, we observe that the entanglement entropy does not grow linearly with time, at least for the examples that we study below. This means that one can study the quantum dynamics for long times with a relatively small bond dimension, well into the regime of thermalization. We use superscript $0$ to denote the pre-quenched Hamiltonian $\hat H^0$, and $1$ to denote the post-quenched Hamiltonian $\hat H^1$.
speed of information spreading
------------------------------
We first consider local quantum quenches due to $h_z\hat\sigma_{i_0}^z$, where we place $i_0$ in the middle of the inhomogeneous region $B$. For the quench with $h_x^0=h_x^1=1.05$, $h_z^0 = 0$, and $h_z^1 = 1.0$, we measure the time dependence of ${\langle \sigma^x_i \rangle}$ on the whole chain, shown in Fig. \[fig:sigma\_x\].
![ ${\langle \sigma^x_i \rangle}$ as a function of time, represented both in a curve plot and a contour plot, for the quench $h_x^0=h_x^1=1.05$, $h_z^0 = 0$, and $h_z^1 = 1.0$. The computation is done for $\delta t = 0.005$, $D = 10$. The same computation is done for $D =20$, too, and the result is well-converged with $D$. []{data-label="fig:sigma_x"}](sx)
![ ${\langle \sigma^x_i \rangle}$ as a function of time, represented both in a curve plot and a contour plot, for the quench $h_x^0=h_x^1=1.05$, $h_z^0 = 0$, and $h_z^1 = 1.0$. The computation is done for $\delta t = 0.005$, $D = 10$. The same computation is done for $D =20$, too, and the result is well-converged with $D$. []{data-label="fig:sigma_x"}](export)
Note that there exists a very sharp wave-front as the information of the local quench spreads. The slope of the wave-front can be computed to give the speed of information spreading, $v_w$. More specifically, we do a linear fit of the function $i_\text{ridge}(t)$, which equals the site of the left-most local maximum of ${\langle \sigma^x_i \rangle}$ at time $t$, and take the slope of the linear fit as the slope of the wave-front. For quenches over a broad range of parameters, $v_w$ are all very close to $v_s$, as shown in Table \[tab:vw\].
[0.48]{}
$h^1_z$ $h_x$ $D$ $v_w$ $R^2$
--------- ------- ----- --------- ----------
0.1 1.01 20 1.93(2) 0.99984
0.1 1.05 20 2(0) 1
0.1 1.5 10 1.96(2) 0.999819
1.0 1.01 10 1.95(2) 0.999749
1.0 1.05 10 1.96(2) 0.999835
1.0 1.5 10 1.95(2) 0.999749
: Velocity of the wave-front in local quenches. The number in the parenthesis is the uncertainty of the fit on the last digit. $R^2$ is the $R$-square of the linear fit.[]{data-label="tab:vw"}
\
[0.48]{}
$h^0_z$ $h_x$ $D$ $v_w$ $R^2$
--------- ------- ----- --------- ----------
0.1 1.05 20 1.96(2) 0.9998
0.1 1.5 10 1.95(2) 0.999785
1.0 1.05 10 1.94(2) 0.99979
1.0 1.5 10 1.93(2) 0.999817
: Velocity of the wave-front in local quenches. The number in the parenthesis is the uncertainty of the fit on the last digit. $R^2$ is the $R$-square of the linear fit.[]{data-label="tab:vw"}
Because of the discrete nature of $i$, $i_\text{ridge}(t)$ can be ambiguous up to $\pm 1$. This contributes to the slight non-linearity of $r_\text{ridge}(t)$, indicated by $R^2 < 1$. We note that for the fit with a perfect $R^2$, i.e. 1, $v_w$ is exactly 2. This raises the curious question as to how $v_w$ is related with $v_s$ in general. A reasonable conjecture seems to be $$\lim_{h_z \rightarrow 0} \lim_{h_x \rightarrow 1} v_w = v_s.$$ Also, what causes the closeness of $v_w$ to $v_s$ off criticality shown Table \[tab:vw\]? Incidentally, we recently proposed an effective method to numerically compute $v_s$ in a general way with Monte Carlo Renormalization Group [@MCRG_diag], so this question may be pursued, at least numerically, in the future.
thermalization
--------------
Another feature observed in the quench in Fig. \[fig:sigma\_x\] is that the ground state of $\hat{H}^0$ evolves under $\hat{H}^1$ to approach the ground state of $\hat{H}^1$. For example, for the quench in Fig. \[fig:sigma\_x\], one can compute the difference between ${\langle \sigma_x^i \rangle}(t)$ and ${\langle \sigma_x^i \rangle}_{\text{GS1}}$, where the latter is the mean magnetization in the $x$-direction in the ground state of the post-quench Hamiltonian $\hat{H}^1$. This is shown in Fig. \[fig:thermo\].
![ Left: ${\langle \sigma^x_i \rangle}$ in the ground state of $\hat{H}^1$. Right: ${\langle \sigma^x_i \rangle}(t) - {\langle \sigma^x_i \rangle}_\text{GS1}.$ The quench is the same as in Fig. \[fig:sigma\_x\]. []{data-label="fig:thermo"}](x)
![ Left: ${\langle \sigma^x_i \rangle}$ in the ground state of $\hat{H}^1$. Right: ${\langle \sigma^x_i \rangle}(t) - {\langle \sigma^x_i \rangle}_\text{GS1}.$ The quench is the same as in Fig. \[fig:sigma\_x\]. []{data-label="fig:thermo"}](delta)
Here we note that as $t \rightarrow \infty$, ${\langle \sigma^x_i \rangle}(t)$ approaches ${\langle \sigma^x_i \rangle}_\text{GS1}$ point-wise. The same phenomenon is observed for ${\langle \sigma^y_i \rangle}$ and ${\langle \sigma^z_i \rangle}$, as well. This suggests that $$\lim_{t\rightarrow \infty} {| \Psi(t) \rangle} = {| \Psi_1 \rangle}$$ where ${| \Psi(0) \rangle}$ is the ground state of $\hat{H}^0$, and ${| \Psi_1 \rangle}$ is the ground state of $\hat{H}^1$. This belongs to the case of [*strong thermalization*]{} first discussed in [@Thermalization], where an initial state evolves into a thermal state. We find this phenomenon very interesting, and do not yet know how general it is. A more detailed study is deferred to future work.
Discussion {#sec:discussion}
==========
In this paper, we gave a detailed derivation of the TDVP equation for mixed infinite MPSs. The result is a simple combination of the finite TDVP and infinite TDVP algorithms, both of which are inversion-free. The method was applied to local quenches of the transverse-field Ising model, and interesting phenomenon were found, which calls for future work. We also expect future work on the algorithmic side. For example, we note that the mixed infinite MPS is very similar to the variational ansatz of the elementary excitations [@Tangent_space] of a translationally invariant system: $${| \Psi_k \rangle} =\sum_{x} e^{ikx} \dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A$};
\draw (4,1.5) -- (5,1.5);
\draw[rounded corners] (5,2) rectangle (6,1);
\draw (5.5,1.5) node {$B^x$};
\draw (5.5,0.25) node {$x$};
\draw (1.5,1) -- (1.5,.5);
\draw (3.5,1) -- (3.5,.5);
\draw (5.5,1) -- (5.5,.5);
\draw (6,1.5) -- (7.0,1.5);
\draw[rounded corners] (7,2) rectangle (8,1);
\draw (7.5,1.5) node (X) {$Z$};
\draw (8,1.5) -- (9,1.5);
\draw[rounded corners] (9,2) rectangle (10,1);
\draw (9.5,1.5) node (X) {$Z$};
\draw (10,1.5) -- (10.5,1.5);
\draw (7.5,1) -- (7.5,.5);
\draw (9.5,1) -- (9.5,.5);
\end{tikzpicture} \dots
\label{eq:MPS}$$ We thus hope that the current method can help develop a time-evolution algorithm for the elementary excitations.
The code is based on ITensor [@ITensor] (version 3, C`++`), and is available upon request. The author is grateful for the help received on the ITensor Support Q&A. He is grateful for mentorship from his advisor Roberto Car at Princeton. He acknowledges support from the DOE Award DE-SC0017865.
Appendix
========
Symplectic derivation of TDVP {#app:symp}
-----------------------------
The derivations [@tdvp] of TDVP in the literature have been based on a variational principle, hence the name. This has the benefit of not needing differential geometry, but buries the symplectic structure of TDVP under the heavy calculations in the derivation. Here we give a derivation directly from symplectic geometry, which is quite elegant and may be preferable to a person who knows some basic differential geometry. We assume knowledge of basic differential geometry at the level of chapter 5 and 8 of [@Geometry].
Let ${\mathcal{H}}$ be a complex vector space with (complex) dimension $m$. ${\mathcal{H}}$ can also be viewed as a real manifold with real dimension $2m$, and thus with a tangent space $T_\Psi {\mathcal{H}}$ at $\Psi \in {\mathcal{H}}$ of real dimension $2m$. $T_\Psi {\mathcal{H}}$ can be complexified to give $(T_\Psi {\mathcal{H}})^{\mathbb{C}}$ which has complex dimension $2m$. Let $J$ be a linear complex structure on $(T_\Psi {\mathcal{H}})^{\mathbb{C}}$. $J^2=1$ and have two eigenvalues $i$ and $-i$, each with an eigenspace of complex dimension $m$. $(T_\Psi{\mathcal{H}})^{\mathbb{C}}$ can then be written as a direct sum of the eigenspaces of $J$: $(T_\Psi{\mathcal{H}})^{\mathbb{C}}= (T_\Psi {\mathcal{H}})^+ \oplus(T_\Psi {\mathcal{H}})^-$, where $J(T_\Psi {\mathcal{H}})^+ = i(T_\Psi {\mathcal{H}})^+$ and $J(T_\Psi {\mathcal{H}})^-=-i(T_\Psi {\mathcal{H}})^-$. Note that $\dim_{\mathbb{C}}(T_\Psi {\mathcal{H}})^+ = m = \dim_{\mathbb{C}}{\mathcal{H}}$, and a linear isomorphism can be established: $(T_\Psi {\mathcal{H}})^+ \cong {\mathcal{H}}$. This allows one to extend the inner product of ${\mathcal{H}}$ to $(T_\Psi {\mathcal{H}})^+$: $$I(X, Y) \equiv {\langle X|Y \rangle}, \hspace{5mm} \forall X,Y \in (T_\Psi {\mathcal{H}})^+ \cong {\mathcal{H}}.$$ Note that we do not define an inner product on $(T_\Psi {\mathcal{H}})^-$. $I$ allows a definition of a metric $g$ on $(T_\Psi {\mathcal{H}})^{\mathbb{C}}$: $\forall X,Y \in (T_\Psi {\mathcal{H}})^+$, $$\begin{split}
g(\bar{Y}, X) &= I(Y, X),
\\
g(Y, X) &= 0,
\\
g(\bar{Y}, \bar{X}) &= 0.
\end{split}$$ This $g$ is known as the Hermitian metric. It is such that $g(JX, JY) = g(X, Y)$ for all $X, Y \in (T_\Psi {\mathcal{H}})^{\mathbb{C}}$. $g$ defines a two-form $\Omega$: $$\Omega(X, Y) = g(JX, Y), \hspace{5mm} \forall X,Y \in (T_\Psi {\mathcal{H}})^{\mathbb{C}}.$$ (It is not hard to show $\Omega(X, Y) = -\Omega(Y,X)$.) Because vector spaces are “flat”, $g$ does not change from point to point, thus $d\Omega = 0$. This means $\Omega$ is symplectic. A manifold with a compatible complex structure $J$, Hermitian structure $I$, Riemannian structure $g$, and symplectic structure $\Omega$ is known as a K[ä]{}hler manifold. We have essentially shown that any complex vector space with an inner product is K[ä]{}hler.
Let $\xi, \eta, \chi, \phi \in (T_\Psi {\mathcal{H}})^+$. $\Omega$ and $I$ are connected by the following: $$\begin{split}
\Omega(\chi + \bar\phi, \xi + \bar\eta) &= g(J(\chi + \bar\phi), \xi+\bar\eta)
\\
&= g(i\chi-i\bar\phi, \xi+\bar\eta)
\\
&=g(i\chi,\bar\eta) + g(-i\bar\phi,\xi)
\\
&=I(\eta,i\chi) + I(i\phi,\xi).
\end{split}$$
On ${\mathcal{H}}$, for a Hamiltonian operator $\hat{H}$, consider the Hamiltonian flow of the Hamiltonian function $H: \Psi \in {\mathcal{H}}\mapsto {\langle \Psi|\hat{H}|\Psi \rangle}$. For $\xi, \eta$ infinitesimal: $$\begin{split}
dH(\xi+\bar\eta)|_\Psi &= {\langle \Psi+\eta|\hat{H}|\Psi+\chi \rangle} - {\langle \Psi|\hat{H}|\Psi \rangle}
\\
&= I(\eta, \hat{H}\Psi) + I(\Psi, \hat{H} \xi)
\\
& = I(\eta, \hat{H}\Psi) + I(\hat{H}\Psi, \xi)
\\
&= \Omega(X_H, \xi+\bar\eta)
\end{split}$$ where $X_H$ is the Hamiltonian flow of $H$: $$X_H = -i\hat{H} \Psi + \overline{-i\hat{H}\Psi}.$$ This is nothing but the Schr[ö]{}dinger flow. Thus, the Schr[ö]{}dinger dynamics can be viewed as a symplectic flow of the Hamiltonian function $H(\Psi)$.
Now let $M$ be a submanifold of ${\mathcal{H}}$. Does $H$ induce a symplectic Schr[ö]{}dinger flow on $M$? Yes! Let the inclusion function from $M$ to ${\mathcal{H}}$ be denoted as $${\text{inc}}: M \to {\mathcal{H}}, \hspace{5mm} {\text{inc}}: \Psi\in M \mapsto \Psi \in {\mathcal{H}}.$$ Both the Hamiltonian function and the symplectic form have a restriction on $M$: $$H_M = H\circ {\text{inc}}, \hspace{5mm} \Omega_M = {\text{inc}}^* \Omega.$$ Because the exterior differentiation $d$ and the pullback ${\text{inc}}^*$ commutes, $d\Omega_M = 0$, and thus $M$ is also symplectic. We now look for the Hamiltonian flow $X_{H_M}$ associated with $H_M$ on $M$. For all $\xi, \eta \in (T_\Psi M)^+$, we look for $X_{H_M} \in (T_\Psi M)^{\mathbb{C}}$ such that $\Omega_M(X_{H_M}, \xi+\bar\eta) = dH_M(\xi+\bar\eta)|_\Psi$. $$\begin{split}
dH_M(\xi+\bar\eta)|_\Psi &= dH({\text{inc}}_*(\xi+\bar\eta))|_\Psi
\\
&= dH(\xi+\bar\eta)|_\Psi
\\
&= I(\eta, \hat{H}\Psi) + I(\hat{H}\Psi, \xi).
\end{split}$$ Now here is the key, because $\xi, \eta$ are both only in $(T_\Psi M
)^+$, $\hat{H}\Psi$ can be replaced with its orthogonal projection on $(T_\Psi M)^{\mathbb{C}}$, $\text{Proj} \hat{H}\Psi$: $$\begin{split}
dH_M(\xi+\bar\eta) & = I(\eta, \text{Proj}\hat{H}\Psi) + I(\text{Proj}\hat{H}\Psi, \xi)
\\
&= \Omega(X_{H_M}, \xi+\bar\eta)
\end{split}$$ where $X_{H_M}$ is the Hamiltonian flow of $H_M$ on $M$: $$X_{H_M} = -i\text{Proj}\hat{H} \Psi + \overline{-i\text{Proj}\hat{H}\Psi}.$$ This gives the TDVP dynamics on $M$ and the dynamics is symplectic.
Schur form of MPO {#app:schur}
-----------------
As discussed in the main text, the $W$ matrix of an MPO is lower-triangular, known as the Schur form. For example, in terms of the operator-valued matrices $\hat{W}_{ab} = \sum_{ss'}W^{ss'}_{ab}{| s \rangle}{\langle s'|}$, the $W$ matrix of the transverse-field Ising Hamiltonian (when $h_z = 0$) in Eq. \[eq:Ising\] can be expressed as, $$\hat{W}=
\begin{bmatrix}
{\openone}&0&0\\
-\hat \sigma^z&0&0\\
h_x\hat \sigma^x&\hat \sigma^z&{\openone}\end{bmatrix}
\label{eq:TFI_MPO}$$ where $\hat \sigma^x$ and $\hat \sigma^z$ are the Pauli matrices. To us, the important features of $\hat{W}$ are that $\hat{W}$ is lower triangular and that $\hat{W}_{11} = \hat{W}_{d_Wd_W} = {\openone}$. This means that the dominant left-eigenvector ${(I_1|}$ of ${E^{[W]}_{A_L}}$ and right-eigenvector ${|I_{d_W})}$ of ${E^{[W]}_{Z_R}}$ are $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (3,4) rectangle (4,-1);
\draw (3.5,1.5) node[scale=1.5] {$I_1$};
\draw (4,-0.5) -- (5,-0.5);
\draw (4,1.5) -- (5,1.5);
\draw (5.5,1.5) node[scale=1.5] {$a$};
\draw (4,3.5) -- (5,3.5);
\draw (5,1.5) node (X) {$\phantom{X}$};
\end{tikzpicture}
=
\delta_{a1}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1) edge[out=180,in=180] (1,4);
\end{tikzpicture},
\hspace{8mm}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (5,4) rectangle (6.5,-1);
\draw (5.75,1.5) node[scale=1.5] {$I_{d_W}$};
\draw (4,-0.5) -- (5,-0.5);
\draw (4,1.5) -- (5,1.5);
\draw (3.5,1.5) node[scale=1.5] {$a$};
\draw (4,3.5) -- (5,3.5);
\draw (5,1.5) node (X) {$\phantom{X}$};
\end{tikzpicture}
=
\delta_{ad_W}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1) edge[out=0,in=0] (1,4);
\end{tikzpicture}.$$ In addition, the generalized eigenvector ${(L_A^{{[W]}}|}$ and ${|R_Z^{{[W]}})}$ satisfy the following relation [@VUMPS]: $$\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (2,4) rectangle (4,-1);
\draw (3.0,1.5) node[scale=1.5] {$L^{{[W]}}_{A}$};
\draw (4,-0.5) -- (5,-0.5);
\draw (4,1.5) -- (5,1.5);
\draw (5.7,1.5) node[scale=1.5] {$d_W$};
\draw (4,3.5) -- (5,3.5);
\draw (5,1.5) node (X) {$\phantom{X}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1) edge[out=180,in=180] (1,4);
\end{tikzpicture},
\hspace{8mm}
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw[rounded corners] (5,4) rectangle (7,-1);
\draw (6,1.5) node[scale=1.5] {$R^{{[W]}}_{Z}$};
\draw (4,-0.5) -- (5,-0.5);
\draw (4,1.5) -- (5,1.5);
\draw (3.5,1.5) node[scale=1.5] {$1$};
\draw (4,3.5) -- (5,3.5);
\draw (5,1.5) node (X) {$\phantom{X}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=0.6},scale=.4]
\draw (1,-1) edge[out=0,in=0] (1,4);
\end{tikzpicture}.$$ We now discuss the proofs of the lemmas in the main text.
Lemma \[lem:1\]: Because ${(I_1|}$ is non-zero only when its middle index is one, $W_A$ only contributes a ${\openone}$ to ${(I_1|}E_C$. Thus, ${(I_1|}E_C = 0$ by Eq. \[eq:V\].
Lemma \[lem:2\]: Because ${|I_{d_W})}$ is non-zero only when its middle index is $d_W$, and that the only non-zero element in the $d_W$ column of $W$ is $W_{d_Wd_W}$, the ${(L_A^{{[W]}}|}$ contributes only as ${(I_{d_W}|}$. This makes ${(L_A^{{[W]}}|}E_C{|I_{d_W})} = 0$ by Eq. \[eq:V\].
Lemma \[lem:3\]: This is proved by putting $E$ into its Jordan canonical form.
Lemma \[lem:4\]: Similar to Lemma \[lem:2\].
Lemma \[lem:5\]: Because of the Schur form, ${(l_{A_R}^{{[W]}}|}$ is non-zero only when its middle index is 1, and is equal to ${(l_{A_R}|}$ in that case. Then this lemma reduces to Eq. \[eq:normalization\].
Lemma \[lem:6\]: Because of the Schur form, ${(I_1|}E_D$ is only non-zero when its middle index is 1, but ${|I_{d_W})}$ is only non-zero when its middle index is $d_W$. This makes the whole thing zero.
Lemma \[lem:7\]: Similar to Lemma \[lem:2\], ${(L_A^{{[W]}}|} ({|R_Z^{{[W]}})})$ contributes only as ${(I_{d_W}|} ({|I_1)})$ and $W_n$ contributes only as ${\openone}$. Thus, the whole expression reduces to $B_C^n$.
|
---
abstract: 'Contraction hierarchies are a simple hierarchical routing technique that has proved extremely efficient for static road networks. We explain how to generalize them to networks with time-dependent edge weights. This is the first hierarchical speedup technique for time-dependent routing that allows bidirectional query algorithms.'
author:
- |
Veit Batz, Robert Geisberger and Peter Sanders\
Universität Karlsruhe (TH), 76128 Karlsruhe, Germany\
bibliography:
- 'hwy.bib'
title: |
Time Dependent Contraction Hierarchies\
— Basic Algorithmic Ideas[^1]
---
Introduction
============
This technical note explains how contraction hierarchies (CHs) can be generalized to allow time-dependent edge weights. We assume familiarity with CHs [@GSS08b; @Gei08]. Like many of the most successful speedup techniques for routing in road networks, the CH query-algorithm uses *bidirectional* search. This is a challenge since bidirectional searching in a time-dependent network requires knowing the arrival time[^2] which is what we want to compute in the first place.
Due to the difficulty of bidirectional routing, the first promising approaches to fast routing used goal directed rather than hierarchical routing and accepted suboptimal routes [@NDLS08]. SHARC routing [@BD08] was specifically developed to encode hierarchical information into a goal-directed framework allowing unidirectional search and recently was generalized to exact time-dependent routing [@Del08]. Schultes [@Sch08] gives a way to make queries in static networks unidirectional but this approach does not directly yield a time-dependent approach.
Preliminaries {#s:preliminaries}
=============
There are classical results on time-dependent route planning [@CH66] that show that a simple generalization of Dijkstra’s unidirectional algorithm works for time-dependent networks $G=(V,E)$ if the objective function is travel time and a cost function $f:\real\rightarrow\real$ has the *FIFO-property*: $\forall \tau<\tau'\gilt \tau+f(\tau)\leq \tau'+f(\tau')$, i.e., there is no overtaking. We focus on this case and further assume that the travel time functions are representable by a piece-wise linear function. However, all our algorithms view travel-time functions (TTFs) as an abstract data type with a small number of operations, basically evaluation, chaining (operation ${\mathbf{*}}$ computes a time-dependent function for a sequence of edges) and minimum computations. Also note, that the format used in public transportation with lists of departure times and arrival times can also be represented in this way. The basic primitives can be implemented in such a way that evaluation at a point in time takes logarithmic time[^3] and the other operations take time linear in the number of line segments representing the inputs.
It seems that any exact time-dependent preprocessing technique needs a basic ingredient that computes travel times not only for a point in time a travel time *profile* but for an entire *time-interval*. An easy way to implement this profile queury a generalization of Dijkstra’s algorithm to profiles [@KS93]. Tentative distances then become TTFs. Adding edge weights is replaced by chaining TTFs and taking the minimum takes the minimum of TTFs. Unfortunately, the algorithm looses its label-setting property. However, the performance as a label-correcting algorithm seems to be good in important practical cases.
Construction {#s:construction}
============
The most expensive preprocessing phase of static CHs orders the nodes by importance. For a first version we propose to adopt the *static* algorithm for the time-dependent CHs (TCHs). This is based on the assumption that averaged over the planning period, the importance of a node is not heavily affected by its exact traffic pattern.
The second stage of CH-preprocessing – contraction – is in principle easy to adapt to time-dependence: we *contract* the nodes of the graph in the order computed previously. When contracting node $v\in
V'$, we are given a current (time-dependent) overlay graph $G'=(V',E')$. For every combination of incoming edge $(u,v)\in E'$ and outgoing edge $(v,w)\in E'$ we have to decide whether the path $\seq{u,v,w}$ may be a shortest path at any point in time. If so, we have to insert the shortcut $(u,w)$ into the next overlay graph $G''=(V'\setminus\set{v}, E'')$. The weight function of this shortcut can be computed by chaining the weight functions of its constituents. Later, we only need to consider shortcuts during time intervals when they may represent a shortest path.[^4] The required information can be computed by running profile-Dijkstra from each node $u$ with $(u,v)\in E'$. The shortcut is needed for $w$ if $c((u,v){\mathbf{*}}(v,w))<d(u,w)$ at any point in time.
Query {#s:query}
=====
The basic static query algorithm for CHs consists of a forward search in an upward graph ${G_{\uparrow}}=(V,{E_{\uparrow}})$ and a backward search in a downward graph ${G_{\downarrow}}$. Wherever, these searches meet, we have a candidate for a shortest path. The shortest such candidate is a shortest path.
Since the departure time is known, the forward search is easy to generalize. In particular, the only overhead compared to the static case is that we have to evaluate each relaxed edge for one point in time. In our experience with a plain time-dependent Dijkstra, this means a small constant factor overhead in practice.
The most easy way to adapt the backward search is to explore *all* nodes that can *reach* $t$ in ${G_{\downarrow}}$. Experiments for static CHs [@Gei08] indicate that this search space is only a small constant factor larger than the search space that takes edge weights into account. During this exploration we mark all edges connecting nodes that can reach $t$. Let ${E_{\mathrm{marked}}}$ denote the set of marked edges.
Now, we can perform an $s$–$t$-query by a forward search from $s$ in $(V, {E_{\uparrow}}\cup{E_{\mathrm{marked}}})$.
The above algorithm is correct.
(Outline) This immediately follows from the properties of TCHs. The detailed proof is analogous to the proof in [@Gei08]. Roughly, the properties of TCHs imply that there must be a shortest path $P$ in the TCH that consists of two segments: One using only eges in ${G_{\uparrow}}$ leading to a peak node $v_p$ and one connecting $v_p$ to $t$ in ${G_{\downarrow}}$. Since all edges of $P$ are in the search space of our forward search, this path or some other shortest path will be found.
Refinements {#ss:crefinements}
===========
Node Ordering {#ss:orefinements}
-------------
Note that there are many ways to adapt the node ordering to take time-dependence into account without resorting to full-fledged time-dependent processing. For example, we can take the average travel time of an edge or look at a sample of departure times and base our priority for node-ordering on the entire sample.
Contraction {#ss:crefinements}
-----------
The main difficulty in constructing TCHs is that the the complexities of time-dependent edge weights and tentative distances grows with progressive contraction and with the diameter of the profile-Dijkstra searches. One way to counter this is to use approximations. With some care, this can be done without compromising the exactness of queries. In particular, we propose to compute piece-wise linear approximations that are always within a factor $1+\epsilon$ from the true travel time.
First, during a local search, we can replace tentative distances with less complex upper bounds on the tentative distance. The worst that can happen is that we introduce additional shortcuts. The hope is that for sufficiently good approximations of the true tentative distance, the number of superfluous shortcuts will be small. The intuition behind this is that if traffic changes the shortest path at all, it is unlikely that the travel time difference is tiny.
For shortcuts that are actually introduced, we compute both upper and lower bounds. For comparing a shortcut $a$ with a witness $b$, we compare a lower bound for $a$ with an upper bound for $b$. Once the (approximate) TCH is computed, we have a choice whether we want to condense it into an exact TCH (i.e., for all shortcuts introduced, we compute there exact edge cost functions) or we later modify the query to compute exact shortest paths using approximate TCHs (ATCH). Note that the complexity of the functions affects the space requirements but has little influence on the cost of evaluation and thus on the query time.
Query {#ss:qrefinements}
-----
We can prune the forward search by marking all nodes $v$ in the backward search space with a lower bound $\ell(v)$ on the travel time to $t$. Note that this information can be gathered with a static Dijkstra algorithm that is likely to be faster than time-dependent Dijkstra. Furthermore, we compute an upper bound $U$ for the travel time from $s$ to $t$ using any static routing technique, unpacking of the statically optimal path $P$, and time-dependent evaluation of $P$. Now, during forward search, if $d(s,v)+\ell(v) > U$ we do not need to continue the search.
There are various ways to compute better upper and lower bounds. Assume we have computed a lower bound $L$ on the total travel time using search in a static graph. Using $U$, $L$ and the departure time, we know a time window $W$ for the arrival time. For computing the lower bounds $\ell(v)$ we can then perform a variation of Dijkstras algorithm that computes minimum travel times over a time interval. If the time interval is small, this might be fast.
#### Exact Routing in ATCHs (Outline) {#exact-routing-in-atchs-outline .unnumbered}
We modify our query algorithm to compute a graph that contains all edges that *might* be in the shortest path tree using upper and lower bounds in a conservative way. Then, using the pruning techniques from above, we remove all parts of this graph that cannot be part of a shortest path from $s$ to $t$ at a given departure time. Then, we unpack all surviving edges. Hopefully, the resulting graph will mostly consist of a small number of partially overlapping paths from $s$ to $t$. Finally, we perform an exact forward search from $s$ in the unpacked graph.
Conclusions {#s:conclusions}
===========
We have developed algorithmic ideas for time dependent routing using CHs. Now experiments have to show whether already the most basic approach or some of its refinements yields a good exact query algorithm for road networks or public transportation. If problems show up, it is likely that the density of the graph or the complexity of shortcuts gets out of hands in the later stages of contraction. From the experience with static routing [@BDSSSW08], it is likely that such problems could be mitigated using a combination with goal directed techniques, e.g., arc-flags. Again from [@BDSSSW08] it could be expected that at least this combination will outperform SHARC [@Del08].
For commercial applications, approximate queries are not a big problem. In this case, many simplifications suggest themselves where we can simply use approximations of time dependent functions that are neither upper nor lower bounds and where we only introduce shortcuts that bring significant improvements.
[^1]: Partially supported by DFG grant SA 933/4-1 and a Google Research Award
[^2]: Wlog we assume that a query specifies source, destination and departure time.
[^3]: Actually our implementation uses a bucketing heuristics that takes constant time on average.
[^4]: Although this can be viewed as a violation of the FIFO-property, we do not get a problem when appliying time-dependent Dijkstra – it never makes sense to wait for a shortcut to become valid since this would not result in a shortest connection.
|
---
abstract: 'We investigate the gravitational field of static perfect-fluid in the presence of electric field. We adopt the equation of state $p(r)=-\rho(r)/3$ for the fluid in order to consider the closed ($S_3$) or the open ($H_3$) background spatial topology. Depending on the scales of the mass, spatial-curvature and charge parameters ($K$, $R_0$, $Q$), there are several types of solutions in $S_3$ and $H_3$ classes. Out of them, the most interesting solution is the Reisner-Norström type of black hole. Due to the electric field, there are two horizons in the geometry. There exists a curvature singularity inside the inner horizon as usual. In addition, there exists a naked singularity at the antipodal point in $S_3$ outside the outer horizon due to the fluid. Both of the singularities can be accessed only by radial null rays.'
author:
- Inyong Cho
title: Fluid Black Holes with Electric Field
---
Introduction
============
The spatial topology of the Universe is one of the unresolved problems in cosmology. From the recent cosmic microwave background radiation data, the density fraction of the curvature is estimated as $\Omega_k = 0.000 \pm 0.005$ ($95\%$, [*Planck*]{} TT+lowP+lensing+BAO) [@Ade:2015xua]. Because of the observational error, it is not possible to determine the spatial topology from the data at the current stage. Some other efforts have been made in the inflation models in the closed/open universe [@Ellis:2002we; @Ellis:2003qz; @Labrana:2013oca; @Bucher:1994gb; @White:2014aua]. The investigation of primordial density perturbation shows that the peculiar predictions of those models are beyond the resolution of the current observational data. Therefore, one needs to consider other ways in order to catch an idea of the background spatial topology, for example, the investigation of the gravitating localized objects in different topologies.
The pure closed/open ($S_3/H_3$) spatial topology is achieved by a constant matter field with the equation of state, $$\begin{aligned}
\label{eom0}
p=-\frac{1}{3}\rho = {\rm constant},\end{aligned}$$ where $\rho>0$ for $S_3$ and $\rho<0$ for $H_3$. The resulting metric is well known as $$\begin{aligned}
\label{metric0}
ds^2 = \mp dt^2 +\frac{dr^2}{1-kr^2/R_0^2} +r^2d\Omega_2^2,\end{aligned}$$ where $k=+1/-1$ represents $S_3/H_3$, and $\rho =\pm 3/(8\pi R_0^2)$. For $S_3$, the ranges of the radial coordinate, $0 \leq r \leq R_0$ and $r \geq R_0$, are considered separately. (We shall call the former $S_3$-I and the latter $S_3$-II.) For $S_3$-II, we take $g_{00}=+1$ to consider only one time coordinate.
The metric is the only solution to the Einstein’s equation with the matter of Eq. . There is no additional mass term unlike in vacuum which admits the flat Minkowski space as the massless limit of the Schwarzschild spacetime. In order to achieve a nontrivial structure such as a black hole in $S_3/H_3$, other type of matter than Eq. needs to be introduced. Then, the $S_3/H_3$ nature will be exposed only at some place of space while a nontrivial geometry is formed elsewhere.
For the nontrivial geometrical structure that admits the inherent $S_3$/$H_3$ topology, the static fluid configuration with the equation of state $p(r) = -\rho(r)/3$ was recently studied in Ref. [@Cho:2016kpf]. It was found that there are a black-hole solution ($S_3$-I, $S_3$-II, $H_3$), a nonstatic cosmological solution ($S_3$-II, $H_3$), and a singular static solution ($H_3$). The nontrivial geometries of these three types of solutions are sourced by fluid. At some region of space, the signature of the $S_3/H_3$ topology appears (near the equator for $S_3$-I, near the center for $S_3$-II, and at the asymptotic region for $H_3$). In this sense, we interpret the nontrivial geometrical configuration as a gravitating object formed in the $S_3/H_3$ background spatial topology. This object can be considered as a large fluid object which is produced in a global universe, or a local compact object which is produced in a local $S_3$/$H_3$ space.
In this paper, we consider the same static fluid in Ref. [@Cho:2016kpf] with the electric field in spherical symmetry. If there is only the electric field, the spacetime is described by the Reisner-Norström solution. If we add the constant matter of Eq. to the electric field, there is no consistent static solution to the Einstein’s equation. Therefore, as in Ref. [@Cho:2016kpf] we consider the fluid of $p(r) = -\rho(r)/3$. The mixture of electric field and fluid form the geometry, and we expect that the $S_3/H_3$ topology due to fluid unveils at some region of space. When the electric field is turned off, the system reduces to the fluid-only case investigated in Ref. [@Cho:2016kpf]. There are some other works on the gravitating solutions for static fluids (see e.g., Refs. [@Bekenstein:1971ej; @Sorkin:1981wd; @Pesci:2006sb; @Semiz:2008ny; @Lake:2002bq; @Bronnikov:2008ia; @Cho:2017nhx]).
This paper consists as following. In Sec. II, we introduce the model and field equations. In Sec. III, we classify the solutions and discuss the spacetime structure. In Sec. IV, we discuss the geodesic motions. In Sec. V, we study the stability of the solutions. In Sec. VI, we conclude.
Model and field equations
=========================
We consider the electric field and the perfect fluid in static state. The static metric ansatz for spherical symmetry is given by $$\begin{aligned}
\label{metricfgr}
ds^2 = -f(r)dt^2 +g(r)dr^2 +r^2d\Omega_2^2.\end{aligned}$$ The energy-momentum tensor for the fluid is given by $$\begin{aligned}
\label{emF}
T^\mu_\nu = {\rm diag}[-\rho(r), p(r),p(r),p(r)],\end{aligned}$$ and we consider the equation of state which meets the $S_3/H_3$ boundary condition, $$\begin{aligned}
\label{eos}
p(r)=-\frac{1}{3}\rho(r) .\end{aligned}$$ The field-strength tensor for the electric field is given by $$\begin{aligned}
\label{Fmunu}
{\cal F}_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu .\end{aligned}$$ We consider the static electric field only, then the vector potential is given by $$\begin{aligned}
A_\mu = [A_0(r),0,0,0].\end{aligned}$$ Then the nonvanishing components of ${\cal F}_{\mu\nu}$ in Eq. are $$\begin{aligned}
\label{Ftr}
{\cal F}_{01} = -{\cal F}_{10} = E(r) = [f(r)A_0(r)]',\end{aligned}$$ where $E(r)$ is the electric field, and the prime denotes the derivative with respect to $r$. The energy-momentum tensor for the electric field is given by $$\begin{aligned}
\label{emE}
{\cal T}^\mu_\nu
= {\cal F}^{\mu\alpha}{\cal F}_{\nu\alpha}
-\frac{1}{4}\delta^\mu_\nu {\cal F}_{\alpha\beta}{\cal F}^{\alpha\beta}
= \frac{E^2(r)}{2f(r)g(r)} {\rm diag}(-1,-1,1,1).\end{aligned}$$ With the metric and the energy-momentum tensors and , the nonvanishing components of the Einstein’s equation, $G^\mu_\nu = 8\pi(T^\mu_\nu +{\cal T}^\mu_\nu$), are $$\begin{aligned}
G^0_0 &= -\frac1{r^2} + \frac{1}{r^2 g} - \frac{g'}{r g^2}
= - 8\pi \left[ \rho(r) +\frac{E^2(r)}{2f(r)g(r)} \right] ,\label{G00} \\
G^1_1 &= -\frac1{r^2} + \frac{1}{r^2 g}+ \frac{f'}{rfg}
= 8\pi \left[ p(r) -\frac{E^2(r)}{2f(r)g(r)} \right] , \label{G11} \\
G^2_2 &= G^3_3 = \frac{f'}{2rfg} - \frac{f'^2}{4f^2 g} - \frac{g'}{2rg^2} -
\frac{f' g'}{4f g^2} + \frac{f''}{2fg}
= 8\pi \left[ p(r) +\frac{E^2(r)}{2f(r)g(r)} \right] . \label{G22}\end{aligned}$$ Since the fluid and the electric field are minimally coupled only thorough gravity, the conservation of the energy-momentum tensor is satisfied individually, $\nabla_\mu T^{\mu\nu} = 0$ and $\nabla_\mu {\cal T}^{\mu\nu} = 0$, which provide the field equations, $$\begin{aligned}
\label{TEeqn}
\rho' +\frac{f'}{f}\rho =0,
\qquad
\frac{3E^2}{fg} \left( \frac{E'}{E} -\frac{f'}{2f} -\frac{g'}{2g} +\frac{2}{r} \right) =0.\end{aligned}$$ These field equations give solutions for fluid and electric field in terms of the gravitational field, $$\begin{aligned}
\label{solrhoE}
\rho(r) = {\rm constant} \times f(r),
\qquad
E(r) = {\rm constant} \times \frac{\sqrt{f(r)g(r)}}{r^2}.\end{aligned}$$
Classification of solutions
===========================
With the solutions in Eq. and the equation of state , the Einstein equations - are solved, $$\begin{aligned}
\rho(r) &= -\frac{3}{8\pi\alpha} \left\{ 1\mp \frac{2\alpha |\beta|}{r}
\left[ \beta(r^2+\alpha) \right]^{1/2}
+\frac{Q^2}{3}\left( \frac{1}{\alpha} +\frac{1}{2r^2} \right) \right\}, \label{rho}\\
f(r) &= \frac{\rho(r)}{\rho_c}, \qquad
g^{-1}(r) = -\frac{8\pi}{3} (r^2+\alpha) \rho(r), \label{g} \\
E(r) &= \frac{Q}{3r^2\left[ \beta(r^2+\alpha) \right]^{1/2}}, \label{E}\end{aligned}$$ where, $Q$ is the electric charge, $\alpha$ and $\beta$ are integration constants, and $\rho_c = -9\beta/(8\pi)$. The above solutions reduce to those of the fluid-only solutions in Ref. [@Cho:2016kpf] when $Q=0$, and to the Reisner-Norström (RN) solution when $\alpha \to\infty$ and $\beta \to 0$ with $\alpha\beta = {\rm finite} = M^{2/3}$.
In order to catch the idea of the spatial topology, we transform the radial coordinate $r$ to $\chi$, and use the metric $$\begin{aligned}
\label{metric2}
ds^2 = -f(\chi)dt^2 +g(\chi)d\chi^2 + R_0^2b^2(\chi)d\Omega_2^2,\end{aligned}$$ where $b(\chi)$ is introduced in the subsections below. We introduced a new parameter $R_0\equiv \sqrt{|\alpha|}$ which is related with the curvature. In addition, we introduce another parameter $K \equiv 2R_0^2|\beta|^{3/2}$ interpreted as a mass parameter analogous to the fluid-only black hole investigated Ref. [@Cho:2016kpf]. Depending on the signatures of $\alpha$ and $\beta$, the solutions are classified into three categories. Two of them meet the $S_3$ boundary condition, and the other does the $H_3$ condition. The classes are summarized in Table I. When both of the parameter $K$ and the charge $Q$ are turned off, the metric reduces to that of the pure $S_3$/$H_3$ in Eq.
Class $\rho(\chi)$ $f(\chi)$ $g(\chi)$
----------------- ------------------------------------------------------------------------------------------------------ ------------------------------------------------------------- -------------------------------------------
$S_3$-I $\quad$ $\qquad \frac{3}{8\pi R_0^2} \left[ 1- K \cot\chi -\frac{Q^2}{6R_0^2} (1-\cot^2\chi) \right] \qquad$ $\qquad \frac{\rho(\chi)}{\rho_c}, \quad (\rho_c>0) \qquad$ $\qquad \frac{3}{8\pi \rho(\chi)} \qquad$
$S_3$-II $\frac{3}{8\pi R_0^2} \left[ 1 \mp K \tanh\chi -\frac{Q^2}{6R_0^2} (1+\tanh^2\chi) \right]$ $\frac{\rho(\chi)}{\rho_c}, \quad (\rho_c<0)$ $-\frac{3}{8\pi \rho(\chi)}$
$H_3$ $-\frac{3}{8\pi R_0^2} \left[ 1 \mp K \coth\chi +\frac{Q^2}{6R_0^2} (1+\coth^2\chi) \right]$ $\frac{\rho(\chi)}{\rho_c}, \quad (\rho_c<0)$ $-\frac{3}{8\pi \rho(\chi)}$
: Classification of solutions. The signature of $\rho_c$ is chosen so that $f(\chi)g(\chi)>0$.
$S_3$-I
-------
This is the case of $\alpha<0$ and $\beta<0$. The transformation is performed by $$\label{rS3I}
r = R_0b(\chi) = R_0\sin\chi
\quad (0\leq \chi \leq \pi,\; 0\leq r \leq R_0).$$ Note that for a given value of $r$, $\chi$ is double valued. The metric becomes $$\label{metricS3I}
ds^2 = -\frac{3}{8\pi R_0^2\rho_c} \left[ 1- K \cot\chi
-\frac{Q^2}{6R_0^2} (1-\cot^2\chi) \right] dt^2
+\frac{R_0^2}{1- K \cot\chi
-(Q^2/6R_0^2) (1-\cot^2\chi)} d\chi^2
+R_0^2\sin^2\chi d\Omega_2^2.$$ Here, $\rho_c>0$. This solution states that the fluid with the electric field strength in Eq. closes the space in a finite region $ 0\leq r \leq R_0$. We believe that the fluid is responsible for this closure since the same phenomenon occurs even in the fluid-only case in Ref. [@Cho:2016kpf]. Both of the Ricci scalar and the Kretschmann scalar diverge at $\chi=0$ and $\pi$, i.e., there exist curvature singularities at both poles. For the pure fluid case ($Q=0$) investigated in Ref. [@Cho:2016kpf], the background $S_3$ topology is exposed at the boundary around the equator ($\chi \approx\pi/2$, i.e., $r\approx R_0$), $$\begin{aligned}
ds_3^2 \approx R_0^2d\chi^2 +R_0^2\sin^2\chi d\Omega_2^2.\end{aligned}$$ With the electric field, however, there is a charge correction, $$\begin{aligned}
ds_3^2 \approx \frac{R_0^2}{1-Q^2/(6R_0^2)}d\chi^2 +R_0^2\sin^2\chi d\Omega_2^2.\end{aligned}$$ The location of the horizon is found from $g_{\chi\chi}^{-1}=0$, $$\chi_h = \chi_\pm \equiv \cot^{-1} \left( \frac{3KR_0^2 \mp \sqrt{J_1}}{Q^2} \right),
\quad\mbox{where}\quad
J_1=9K^2R_0^4 -6Q^2R_0^2+Q^4.$$ Depending on the existence of the horizon, there are two types of solutions. (See Fig. 1 for the graphical view of the metric function.)
\(i) [**RN black-hole type solution**]{}: If $J_1>0$, there exist two horizons at $\chi_h=\chi_\pm$, which coalesce when $J_1=0$. This solution mimics the Reisner-Nordström geometry of the charge black hole. The spacetime is regular at $\chi<\chi_-$ and $\chi>\chi_+$. The singularity at the north pole ($\chi=0$) is inside the inner horizon, and is not accessible by the timelike observers as in the RN black hole. The singularity at the south pole ($\chi=\pi$) is naked, but is not accessible either by the timelike observers as in the fluid black hole investigated in Ref. [@Cho:2016kpf]. The geodesics are studied in the next section.
\(ii) [**Naked singular solution**]{}: If $J_1<0$, there is no horizon. Both singularities are naked, but neither of them are accessible.
{width="30.00000%"}
$S_3$-II
--------
This is the case of $\alpha<0$, $\beta>0$. The transformation is performed by $$\label{rS3II}
r=R_0b(\chi) =R_0\cosh\chi
\quad ( -\infty < \chi < \infty,\; r\geq R_0),$$ Again, $\chi$ is double-valued for a given value of $r$. The metric becomes $$\label{metricS3II}
ds^2 = -\frac{3}{8\pi R_0^2\rho_c} \left[ 1 \ominus\oplus K \tanh\chi
-\frac{Q^2}{6R_0^2} (1+\tanh^2\chi) \right] dt^2
+\frac{R_0^2}{-\left[ 1 \ominus\oplus K \tanh\chi
-(Q^2/6R_0^2) (1+\tanh^2\chi) \right]} d\chi^2
+R_0^2\cosh^2\chi d\Omega_2^2.$$ Here, $\rho_c<0$. The fluid curves the space in a flipped way to the $S_3$-I case; the space is confined in the open region $r\geq R_0$. The curvature is finite everywhere. The location of the horizon is $$\chi_h = \chi_\pm \equiv \tanh^{-1} \left( \frac{\ominus\oplus 3KR_0^2 \pm \sqrt{J_2}}{Q^2} \right),
\quad\mbox{where}\quad
J_2=9K^2R_0^4 +6Q^2R_0^2 -Q^4.$$ (The $\pm$ roots are valid for both $\ominus$ and $\oplus$.) There are four types of solutions. (See Fig. 2.) Two of them are black-hole type solutions (Schwarzschild and Reisner-Nordström types) without a singularity, and the others are regular and nonstatic solutions.
Let us consider the $\ominus$ solution.
If $J_2>0$, there are three types of solutions.
\(i) [**RN black-hole type solution**]{}: For $Q^2 \geq 3(1+K)R_0^2$, there are two horizons at $\chi_\pm$ and this is the RN black-hole type.
\(ii) [**Schwarzschild black-hole type solution**]{}: For $3(1-K)R_0^2 < Q^2 < 3(1+K)R_0^2$, there exists only one horizon. Inside the horizon (the trapped region), $f(\chi), g(\chi) <0$ and $\rho>0$. The spacetime is nonstatic in the trapped region, and static outside. The structure is similar to that of the Schwarzschild black hole.
\(iii) [**Nonstatic solution**]{}: For $Q^2 \leq 3(1-K)R_0^2$, there is no horizon and the spacetime is nonstatic everywhere. This type of solution is special for $S_3$-II. This is analogous to the solution in Eq. describing the region $r\geq R_0$ in which the roles of the temporal and the radial coordinates are exchanged.
If $J_2<0$, there is one type of solution.
\(iv) [**Regular solution**]{}: The spacetime is regular everywhere while $\rho<0$.
For the $\oplus$ solution, the situation is the same with the $\ominus$ solution with $\chi \to -\chi$. Therefore, out of four types (i)-(iv), the only change is in (ii). Now, the region of $\chi < \chi_h$ is static, and the region of $\chi > \chi_h$ is nonstatic.
{width="30.00000%"}
$H_3$
-----
This is the case of $\alpha>0$, $\beta>0$. The transformation is performed by $$\label{rH3}
r= R_0b(\chi) =R_0\sinh\chi \quad (\chi \geq 0,\; r \geq 0),$$ and the metric becomes $$\label{metricH3}
ds^2 = -\frac{3}{8\pi R_0^2(-\rho_c)} \left[ 1 \ominus\oplus K \coth\chi
+\frac{Q^2}{6R_0^2} (1+\coth^2\chi) \right] dt^2
+\frac{R_0^2}{1 \ominus\oplus K \coth\chi +(Q^2/6R_0^2) (1+\coth^2\chi)} d\chi^2
+R_0^2\sinh^2\chi d\Omega_2^2.$$ Here, $\rho_c<0$. The curvature diverges at $\chi=0$. The location of the horizon is $$\chi_h = \chi_\pm \equiv \coth^{-1} \left( \frac{\oplus\ominus 3KR_0^2 \mp \sqrt{J_3}}{Q^2} \right),
\quad\mbox{where}\quad
J_3=9K^2R_0^4 -6Q^2R_0^2 -Q^4.$$ The solutions are classified as below. (See Fig. 3.)
For the $\ominus$ solution in Eq. , there are three types of solutions for $J_3>0$.
\(i) [**RN black-hole type solution**]{}: For $3(K-1)R_0^2 < Q^2 < 3KR_0^2$, there are two horizons at $\chi_\pm$ and this is the RN black-hole type.
\(ii) [**dS-type solution**]{}: For $Q^2 \leq 3(K-1)R_0^2 $, there is only one horizon at $\chi_+$. The spacetime is static inside the horizon, and nonstatic outside. This is a de Sitter-like solution. This solution is achieved when the electric charge $Q$ is small. When $Q=0$, this corresponds to the cosmological solution of the fluid-only case in Ref. [@Cho:2016kpf] for which the spacetime is nonstatic everywhere. It was interpreted as a universe expanding from an initial singularity. For the present case, however, the horizon is formed due to the electric field inside which the spacetime is static.
\(iii) [**Naked singular solution**]{}: For $Q^2 \geq 3KR_0^2$, the solution is static everywhere, but with a singularity at the center.
For the $\oplus$ solution in Eq. , or for $J_3<0$, there is no horizon, and the solution is singular static like (iii).
{width="30.00000%"}
Gauss’ Law
----------
Let us discuss the Gauss’ law in the $\chi$ coordinate. The field-strength tensor ${\cal F}_{\mu\nu}$ in Eq. in the $r$ coordinate with the components is transformed to ${\cal F}'_{\mu\nu}$ in the $\chi$ coordinate with the nonzero components, $$\begin{aligned}
\label{Ftchi}
{\cal F}'_{t\chi} = -{\cal F}'_{\chi t} = E(\chi)
= \frac{Q}{3|\beta|^{1/2}R_0^2b^2(\chi)} .\end{aligned}$$ The electric flux is then $$\begin{aligned}
\Phi_E= \oint E\sqrt{g^{(2)}}d^2x
= \iint \frac{Q}{3|\beta|^{1/2}R_0^2b^2(\chi)}
\times R_0^2b^2(\chi) \sin\theta d\theta d\phi
= \frac{4\pi Q}{3|\beta|^{1/2}}
= \frac{4\pi Q}{\sqrt{8\pi |\rho_c|}},\end{aligned}$$ where we used the relation $\rho_c = -9\beta/(8\pi)$. Compared with the Gauss’ law in flat space, there is a correction due to fluid by the factor $\sqrt{8\pi |\rho_c|}$.
Mass
----
In this section, let us discuss the mass of the black-hole solutions. For the fluid-only case in Ref. [@Cho:2016kpf], it was investigated that the horizon structure of the fluid black hole is similar to that of the Schwarzschild black hole. The parameters are related with the Schwarzschild mass $M$ as $$\begin{aligned}
\label{KM}
K = \left( \frac{R_0^2}{4M^2} -1 \right)^{-1/2},
\quad
\left( -\frac{R_0^2}{4M^2} +1 \right)^{-1/2},
\quad
\left( \frac{R_0^2}{4M^2} +1 \right)^{-1/2},\end{aligned}$$ for the type $S_3$-I, $S_3$-II, and $H_3$, respectively. For the $S_3$-I type, there is an upper limit in the mass, $M \to R_0/2$ as $ K \to \infty$. In this limit, the horizon approaches the equator of $S_3$, $\chi_h = \cot^{-1}(1/K) \to \pi/2$.
Other than the Schwarzschild mass, it is interesting to consider the Misner-Sharp mass ${\cal M}$ which can be used for black-hole thermodynamics [@Misner:1964je]. We evaluate ${\cal M}$ in this work. When the metric is given by $$\begin{aligned}
\label{metricMS}
ds^2 = h_{ab} dx^adx^b +r^2(x)d\Omega_2^2,\end{aligned}$$ where $a,b=0,1$, the Misner-Sharp mass is defined as $$\begin{aligned}
\label{MS}
{\cal M} = \frac{1}{2} (1-h^{ab} \partial_a r \partial_b r).\end{aligned}$$ In the $\chi$ coordinate, we have $r=R_0b(\chi)$ and Eq. becomes $$\begin{aligned}
{\cal M}(\chi) = -\frac{4\pi R_0^3}{3s} \rho(\chi) b(\chi) [b'(\chi)]^2 +\frac{R_0}{2}b(\chi),\end{aligned}$$ where $s$ is the signature of $\rho_c$ ($s=+1$ for $S_3$-I, and $s=-1$ for the others). The mass depends on the radial coordinate $\chi$.
For the fluid-only case ($Q=0$), the mass is still $\chi$ dependent, while one has ${\cal M}_{\rm Sch} = M$ for the ordinary Schwarzschild black hole. For the fluid black-hole solutions, one can show with the aid of Eq. that the Misner-Sharp mass evaluated on the horizon coincides with the Schwarzschild mass, ${\cal M}(\chi_h) = M$. This indicates that the horizon structure of the fluid black hole is the same with that of the Schwarzschild black hole.
For the ordinary RN black hole, the Misner-Sharp mass is given by ${\cal M}_{\rm RN} =M-Q^2/(2r) = M-Q^2/[2R_0b(\chi)]$. For the RN black-hole type solutions obtained in this work ($Q\neq 0$), keeping the mass relation of $K$ in Eq. , the Misner-Sharp mass evaluated on the horizons does not coincide with that of the ordinary RN black hole, ${\cal M}(\chi_\pm) \neq {\cal M}_{\rm RN}(\chi_\pm)$.
Although the horizon structure of the fluid black hole ($Q=0$) is the same with that of the ordinary one, the thermodynamics must be very different because the off-horizon structure is very different. We shall study the thermodynamics using the Misner-Sharp mass in a separate work including the charged case.
Geodesics
=========
In this section, we discuss the geodesics of the solutions. We focus mainly on the black-hole solutions. For simplicity, we define a function, $$\begin{aligned}
\label{Fchi}
F(\chi) \equiv \frac{8\pi R_0^2}{3s}\rho(\chi).\end{aligned}$$ The geodesic equations become $$\begin{aligned}
&\mbox{$t$-eq. : }\quad
\frac{1}{F(\chi)} \frac{d}{d\lambda} \left[ F(\chi) \frac{dt}{d\lambda}\right]
=0 ,\label{S3teq}\\
&\mbox{$\phi$-eq. : }\quad
\frac{1}{b^2(\chi)} \frac{d}{d\lambda} \left[ b^2(\chi) \frac{d\phi}{d\lambda}\right]
=0.\label{S3phieq}\end{aligned}$$ From Eqs. and , we denote the conserved quantities $E$ (energy) and $L$ (angular momentum) as $$\begin{aligned}
E \equiv F(\chi) \frac{dt}{d\lambda} = {\rm constant},
\qquad
L \equiv b^2(\chi) \frac{d\phi}{d\lambda} = {\rm constant}.\end{aligned}$$ The $\chi$-equation can be derived from the metric as $$\begin{aligned}
\label{chieq}
g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = - \varepsilon,\end{aligned}$$ where $\varepsilon = 0,1$ for null and timelike geodesics, individually. On the $\theta=\pi/2$ plane, Eq. becomes $$\begin{aligned}
\frac{1}{2} \left( \frac{d\chi}{d\lambda} \right)^2 + V(\chi) = \frac{3E^2}{16\pi R_0^4|\rho_c|} \equiv \tilde{E}^2,\end{aligned}$$ where the effective potential is given by $$\begin{aligned}
V(\chi) = \frac{1}{2} F(\chi) \left[ \frac{L^2}{b^2(\chi)} +\frac{\varepsilon}{R_0^2} \right].\end{aligned}$$ We summarize $V(\chi)$ in Table II. The effective potential $V(\chi)$ of the black-hole type solutions is plotted in Fig. 4-6.
For the RN black-hole type solution of $S_3$-I, the singularities at both poles are not accessible except by the radial null geodesic. For the fluid-only case in Ref. [@Cho:2016kpf], the one at the north pole inside the horizon was accessible since the inner geometry was similar to that of the Schwarzschild black hole. However, for the present case, it is not because the inner geometry is similar to that of the charged black hole. The nonaccessibility to the naked singularity at the south pole is similar to the fluid-only case. The geodesic observer starting from the outer static region falls into the inner static region passing the intermediate nonstatic region. Afterwards, the observer bounces back to the nonstatic region and then enters the outer static region. This later motion after the bounce proceeds in the other copy of the spacetime accompanied in the usual RN geometry. The geodesic as a whole is an oscillatory orbit in the infinite tower of the RN spacetime.
For $S_3$-II, the RN black-hole type solution, when the energy level ($\tilde E$) is low, the oscillatory orbit is similar to that of $S_3$-I. When the energy level is increased, the geodesic observer can reach the inner static region behind the inner horizon. When the energy level is high enough, the geodesic observer can escape to the asymptotic infinity in the static region. The Schwarzschild black-hole type solution has the similar geodesic structure to that of the usual Schwarzschild black hole. When the energy level is low, all the geodesic motions fall into the black hole. However, $V(\chi)$ approaches a constant value as $\chi \to -\infty$.
For $H_3$, the singularity at the center is not accessible except by the radial null geodesic, which is different from the fluid-only case. Similarly to the $S_3$-I, it is due the electric charge. When the energy level is low, the geodesic motion is oscillatory as in $S_3$-I. When the energy level is high, the geodesic observer can reach the asymptotic infinity. Another interesting solution is dS-type. For this solution, the geodesics escape from the static region crossing the de Sitter-like horizon and reach asymptotic infinity. This is different from the pure de Sitter space in which there can be a stable geodesic motion inside the horizon.
Class $F(\chi)$ $V(\chi)$
----------------- ------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------
$S_3$-I $\quad$ $\qquad 1- K \cot\chi -(Q^2/6R_0^2) (1-\cot^2\chi) \qquad$ $\qquad \frac{1}{2} [1- K \cot\chi -(Q^2/6R_0^2) (1-\cot^2\chi)] \left( \frac{L^2}{\sin^2\chi} +\frac{\varepsilon}{R_0^2} \right) \qquad$
$S_3$-II $-1 \pm K \tanh\chi +(Q^2/6R_0^2) (1+\tanh^2\chi) $ $\frac{1}{2} [-1 \pm K \tanh\chi +(Q^2/6R_0^2) (1+\tanh^2\chi)] \left( \frac{L^2}{\cosh^2\chi} +\frac{\varepsilon}{R_0^2} \right)$
$H_3$ $1 \mp K \coth\chi +(Q^2/6R_0^2) (1+\coth^2\chi)$ $\frac{1}{2} [1 \mp K \coth\chi +(Q^2/6R_0^2) (1+\coth^2\chi)] \left( \frac{L^2}{\sinh^2\chi} +\frac{\varepsilon}{R_0^2} \right)$
: Effective potential $V(\chi)$
{width="30.00000%"}
{width="30.00000%"} {width="30.00000%"}\
(i) (ii)\
{width="30.00000%"} {width="30.00000%"}\
(i) (ii)
Stability
=========
In this section, we study the stability of the solutions. We introduce linear spherical scalar perturbations with the metric ansatz, $$\begin{aligned}
ds^2 = -f(t,\chi)dt^2 + g(t,\chi) d\chi^2 + R_0^2 b^2(\chi) d\Omega_2^2.\end{aligned}$$ The metric perturbations are introduced as $$\begin{aligned}
f(t,\chi) &= f_0(\chi) + \epsilon f_1(t,\chi), \label{p1}\\
g(t,\chi) &= R_0^2 \big[ g_0(\chi) + \epsilon g_1(t,\chi) \big], \label{p2}\end{aligned}$$ where $\epsilon$ is a small parameter, and the subscript $0$ stands for the background solutions obtained in Sec. III. Using the function $F(\chi) = 8\pi R_0^2\rho_0(\chi)/3s$ defined in Eq. , where $\rho_0(\chi)$ is the background solution in Table I, we have $$\begin{aligned}
f_0(\chi) &= \frac{\rho_0(\chi)}{\rho_c} = \frac{3s}{8\pi R_0^2\rho_c}F(\chi), \\
g_0(\chi) &= \frac{1}{F(\chi)}.\end{aligned}$$ The contravariant form of the energy-momentum tensor for fluid is written as $$\begin{aligned}
T^{\mu\nu} = (\rho +p)u^\mu u^\nu + pg^{\mu\nu},\end{aligned}$$ with the velocity four-vector $$\begin{aligned}
u^\mu = \big[ u^0(t,\chi),u^1(t,\chi),0,0 \big].\end{aligned}$$ For the fluid at hand, $p=-\rho/3$, the perturbations for the energy density and the four-velocity are introduced by $$\begin{aligned}
\rho(t,\chi) &= \rho_0(\chi) +\epsilon \rho_1(t,\chi), \label{p3}\\
u^0(t,\chi) &= u_0^0(\chi) +\epsilon u_1^0(t,\chi), \label{p4}\\
u^1(t,\chi) &= u_0^1(\chi) +\epsilon u_1^1(t,\chi). \label{p5}\end{aligned}$$ We have $u_0^1(\chi)=0$ for the comoving background fluid. From the normalization $u^\mu u_\mu=-1$, we have $u_0^0(\chi)=1/\sqrt{f_0(\chi)}$ and $u_1^0(t,\chi)=-f_1u_0^0/(2f_0) = -f_1/(2f_0^{3/2})$.
For the electric field, we introduce the simplest perturbation along the radial direction only, by which there is no magnetic field induced by the perturbation, $$\begin{aligned}
\label{p6}
{\cal F}'_{t\chi} = -{\cal F}'_{\chi t} = E(t,\chi) = E_0(\chi) + \epsilon E_1(t,\chi),\end{aligned}$$ where $E_0(\chi)$ is given in Eq. .
Now we apply the perturbations , , and -, and expand the field equations in the first order of $\epsilon$. From the $(0,1)$ component of the Einstein’s equation, we get $$\begin{aligned}
u_1^1(t,\chi) = -\sqrt{\frac{2\pi R_0^2\rho_c}{3}} \frac{\dot{g_1}b'F}{s^2b\sqrt{F}}.\end{aligned}$$ Therefore, the perturbations of the four-vector, $u_1^0$ and $u_1^1$ in Eqs. and , are expressed by the background functions and the metric perturbations. There are seven equations in total for four perturbations, $f_1$, $g_1$, $\rho_1$ and $E_1$; three from Einstein’s equation, two from $\nabla_\mu T^{\mu\nu} =0$ , and two from $\nabla_\mu {\cal T}^{\mu\nu} =0$. Four of them are independent equations. After manipulating equations with $$\begin{aligned}
f_1(t,\chi) &= e^{i\omega t} \psi(\chi),\\
g_1(t,\chi) &= e^{i\omega t} \varphi(\chi),\end{aligned}$$ the equation for $\varphi(\chi)$ is decoupled as $$\begin{aligned}
\label{PE1}
-F^2\varphi''
-\left[ 3FF' + F^2 \left( 3\frac{b''}{b'} +s\frac{b}{b'} \right) \right] \varphi'
+\left[ \frac{\omega^2}{\sigma} -2FF''
-FF'\left( 4\frac{b''}{b'}-\frac{b'}{b} -s\frac{b}{b'} \right)
-2F^2 \left( \frac{b'''}{b'} -\frac{b'^2}{b^2} +s \frac{bb''}{b'^2} -s \right)
\right] \varphi =0,\end{aligned}$$ where $\sigma \equiv 1/(8\pi R_0^4\rho_c s) = 1/(8\pi R_0^4|\rho_c|) >0$ for all classes. The coefficients of the above equation depend only on the background functions $F(\chi)$ and $b(\chi)$.
By transforming the radial coordinate and the amplitude function as $$\begin{aligned}
z = \int^\chi_0 \frac{d\chi}{\sqrt{2}F(\chi)}, \qquad
\Phi(z) = N \frac{F(\chi)b'(\chi)}{z} \varphi(\chi),\end{aligned}$$ where $N$ is a normalization constant, we get the perturbation equation in the nonrelativistic Schrödinger-type, $$\begin{aligned}
\label{PE2}
\left[ -\frac{1}{2}\frac{d^2}{dz^2} - \frac{1}{z}\frac{d}{dz}
+U(z) \right] \Phi(z) = -\frac{\omega^2}{\sigma} \Phi(z)
=-8\pi R_0^4|\rho_c| \omega^2 \Phi(z) \equiv \Omega \Phi(z).\end{aligned}$$ The potential is given by $$\begin{aligned}
U[z(\chi)] = F^2 \left[ -\frac{F''}{F} +\left( \frac{F'}{F} \right)^2
+\frac{F'}{F} \left( \frac{b''}{b'} +2\frac{b'}{b} +4s \right)
+2 \left( \frac{b''}{b'} \right)^2 +s
\right],\end{aligned}$$ where we used $sb/b' = -b''/b'$, $b'''/b'=-s$, and $b''/b=-s$. Since there always exists a positive eigenvalue $\Omega$ for any type of potential $U$, i.e., $\omega^2 <0$, this system is [*unconditionally unstable*]{}.
The stability story is very similar to the fluid-only case. When perturbations are introduced to the static fluid, the fluid becomes time dependent, which drives the Universe to undergo the Friedmann expansion. This type of instability does not necessarily mean that the black-hole structure is destroyed. Instead, the instability indicates that the background universe undergoes expansion while the black-hole structure sustains.
When the perturbation of the electric field is considered, the instability can be related with the destruction of the black-hole structure. It is known that the Cauchy (inner) horizon of the charged black hole is unstable to form a singularity [@Gursel:1979zza]. The perturbation introduced in this work may develop such an instability in the RN black-hole type solution.
Conclusions
===========
We investigated the gravitational field of static fluid plus electric field. Both of the fluid and the electric field are the sources of the gravitational field, but the way to curve the spacetime is a bit different from each other. By adopting the equation of state $p(r) = -\rho(r)/3$, the fluid is responsible for the topology of the background space. The spatial topology can be either closed ($S_3$) or open ($H_3$). Such a nature of the spatial topology is not observed everywhere. Instead, the signature of the background spatial topology appears at some place of the spacetime.
Based on the background topology, there exist various types of solutions in three classes which we named as $S_3$-I, $S_3$-II, and $H_3$. Interesting classes are $S_3$-I and $H_3$ although the class $S_3$-II has most varieties in solution. The most interesting solutions are the black-hole solutions. Due to the presence of the electric field, the black-hole geometry mimics that of the Reisner-Norström spacetime. This type of black hole exists in both $S_3$ and $H_3$ spaces. (There exists also a Schwarzschild-type black hole in $S_3$-II.) The central singularity inside the black hole of this type of solution is due to the electric source as well as the fluid source. There is a naked singularity in $S_3$-I at the antipodal point which is not accessible except by the radial null rays. The formation of this singularity is caused by the fluid. The geodesics of the Reisner-Norström black-hole type solution exhibit the oscillatory orbit in the infinite tower of the spacetime encountered in the usual Reisner-Norström geometry.
All the solutions obtained in this paper are unconditionally unstable. This is not surprising because the stability story is similar to the fluid-only case in Ref. [@Cho:2016kpf]. The reason of the instability is that the static fluid becomes unstable (time dependent) with small perturbations and drives the background geometry to the Friedmann expansion. In addition, there is an electric field for which it is well known that the pure charged black-hole solution (Reisner-Norström geometry) is unstable under perturbations.
The solutions investigated in this paper are useful in studying the magnetic monopole in the closed/open space, which is under investigation currently. Usually, the outside geometry of the magnetic monopole is the same with that of the charged black hole (Reisner-Norström geometry) [@Gibbons:1990um; @Cho:1975uz; @Bais:1975gu; @Yasskin:1975ag; @Cordero:1976jc]. Since we obtained the charged black-hole solution in $S_3$/$H_3$ with the aid of fluid, it is very interesting to investigate the magnetic monopole in the presence of fluid. It may give rise to insight about the monopole in the closed/open space. The asymptotic geometry of this type of the gauge monopole is worth while to investigate and will be very interesting to compare with the usual monopole geometry. In addition, the removal of the singularity is also a very interesting issue. For the usual case, the monopole field removes the singularity of the charged solution. For this case, however, the formation of the singularity is caused not only by the electric charge, but also by the fluid. It is interesting to see if the monopole field can regularize the singular behavior of the fluid.
The author is grateful to Hyeong-Chan Kim and Gungwon Kang for useful discussions. This work was supported by the grant from the National Research Foundation funded by the Korean government, No. NRF-2017R1A2B4010738.
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abstract: 'We revisit qubit-qutrit quantum systems under collective dephasing and answer some of the questions which have not been asked and addressed so far in the literature. In particular, we examine the possibilities of non-trivial phenomena of [*time-invariant*]{} entanglement and [*freezing*]{} dynamics of entanglement for this dimension of Hilbert space. Interestingly, we find that for qubit-qutrit systems both of these peculiar features coexist, that is, we observe not only time-invariant entanglement for certain quantum states but we find also find evidence that many quantum states freeze their entanglement after decaying for some time. To our knowledge, the existance of both these phenomena for one dimension of Hilbert space is not found so far. All previous studies suggest that if there is freezing dynamics of entanglement, then there is no time-invariant entanglement and vice versa. In addition, we study local quantum uncertainity and other correlations for certain families of states and discuss the interesting dynamics. Our study is an extension of similar studies for qubit-qubit systems, qubit-qutrit, and multipartite quantum systems.'
author:
- Mazhar Ali
title: 'Qubit-Qutrit ($2 \otimes 3$) quantum systems: An investigation of some quantum correlations under collective dephasing'
---
Introduction {#Sec:intro}
============
Quantum correlations have their role in potential applications in quantum information theory. This includes remote state preparation [@Dakic-Nat-2012], entanglement distribution [@Streltsov-PRL108-2012; @Chuan-PRL109-2012], transmission of correlations [@Streltsov-PRL111-2013], and quantum metrology [@Modi-PRX1-2011] to name a few. This utilization of quantum correlations is already enough motivation to study, characterize and quantify them. There are classical correlations which have no quantum in it. Quantum correlations are difficult to characterize and quantify and there are several different techniques to capture them. Entanglement, quantum discord and local quantum uncertainity are kind of quantum correlations. Even there are different measures to compute entanglement and quantum discord. Nevertheless, these correlations have attracted lot of interest and considerable efforts have been devoted to develop a theory of these correlations [@Horodecki-RMP-2009; @gtreview; @Chiara-RPP2018]. The advancement in experimental setups during last couple of decades, enabled us to work for realistic realizations of quantum devices utilizing quantum correlations. Due to unavoidable interactions of delicate quantum systems with their environment, it is essential to simulate the effects of noisy environments on quantum correlations. Such investigations are already an active area of research [@Aolita-review] and several authors have studied decoherence effects on quantum correlations for both bipartite and multipartite systems [@lifetime; @Aolita-PRL100-2008; @bipartitedec; @Band-PRA72-2005; @lowerbounds; @Lastra-PRA75-2007; @Guehne-PRA78-2008; @Lopez-PRL101-2008; @Ali-work; @Weinstein-PRA85-2012; @Ali-JPB-2014; @Ali-2015; @Ali-2016].
There are several types of experimental setups to test the ideas of quantum information. One of the technological advanced setup is to trap the ions/atoms and perform quantum computations by logic gates, measurements etc. In these experiments, the typical noise is caused by intensity fluctuations of electromagnetic fields which leads to collective dephasing process. This process degrades quantum correlations and there are already many investigations of the effects of collective dephasing on entanglement for bipartite and multipartite quantum systems [@Yu-CD-2002; @AJ-JMO-2007; @Li-EPJD-2007; @Song-PRA80-2009; @Ali-PRA81-2010; @Karpat-PLA375-2011; @Liu-arXiv; @Carnio-PRL-2015; @Carnio-NJP-2016; @Ali-IJQI-2017; @Ali-EPJD-2017]. It has been reported in these studies that collective dephasing process offers not only the expected exponential decay of entanglement but also the abrupt end of entanglement (sudden death of entanglement). In addition to these two dynamical behavior, some of the recent studies demonstrated that there are two other types of non-trivial dynamics of entanglement present/observed under collective dephasing. First, there is so called [*time-invariant*]{} entanglement [@Karpat-PLA375-2011; @Liu-arXiv; @Ali-EPJD-2017]. Time-invariant entanglement does not necessarily mean that the quantum states live in decoherence free subspaces (DFS). In fact the quantum states may change at every instance whereas their entanglement remain constant throughout the dynamical process. This feature was first observed for qubit-qutrit systems [@Karpat-PLA375-2011] and then later on observed for qubit-qubit systems as well [@Liu-arXiv]. Recently, we have investigated time-invariant phenomenon for genuine entanglement of three and four qubits and explicitly observed this phenomenon [@Ali-EPJD-2017]. The second non-trivial feature of entanglement decay is called [*freezing*]{} dynamics of entanglement [@Carnio-PRL-2015; @Carnio-NJP-2016; @Ali-IJQI-2017]. It was shown that a specific two qubits state may first decay upto some numerical value before suddenly stop decaying and maintain this stationary entanglement [@Carnio-PRL-2015; @Carnio-NJP-2016]. Recently, we have explored freezing dynamics for various genuine multipartite specific states of three and four qubits, including random states and found evidence for it [@Ali-IJQI-2017]. More recently, we have explored the possibility of either time-invariant entanglement or freezing dynamics for qutrit-qutrit ($3 \otimes 3$) systems [@Ali-MPLA-2019]. We found no evidence for time-invariant entanglement, however we observed the exclusive evidence for freezing dynamics of entanglement [@Ali-MPLA-2019]. We have noticed that in all previous studies on time-invariant entanglement and freezing dynamics for a given Hilbert space, there is either time-invariant entanglement or freezing dynamics behavior. We have not found so far these two features occuring together for one dimension of Hilbert space. Interestingly, for qubit-qutrit systems we find both these features present. As we show below, there are certain states which exibit either time-invariant entanglement or sudden death of entanglement but never freezing dynamics. On the other hand, some other quantum states exhibit either freezing dynamics or sudden death but never time-invariant dynamics. However, we get both peculiar features for Hilbert space of dimension $6$.
The two other quantum correlations which we study in this work are quantum discord and local quantum uncertainity. Quantum discord may be defined as the difference between quantum mutual information and classical correlations [@Ollivier-PRL88-2001; @Vedral-et-al; @Luo-PRA77-2008]. Quantum discord may be nonzero even for separable states and have applications in quantum information. Due to complicated minimization process, the computation of discord is not an easy task and analytical results are known only for some restricted families of states. For $2 \otimes d$ quantum systems, analytical results for quantum discord are known for a specfic family of states [@Ali-JPA43-2010] and the general procedure to calculate discord is also worked out [@Rau-2017]. The dynamics of quantum discord under decoherence has been studied [@Werlang-work] and is found to be more robust than quantum entanglement. In this work, we also study dynamics of quantum discord and classical correlation under collective dephasing for a specfic family of states. The other quantum correlation which we study in this work is recently proposed, known as local quantum uncertainity [@Girolami-PRL110-2013]. This measure is based on idea of skew information and it is discord type correlation [@Modi-RMP84-2012]. Recently, the effects of decoherence on discord-like measures including local quantum uncertainity has been studied [@Bera-RPP81-2018; @Karpat-CJP96-2018; @Slaoui-work]. Here in this work, we study local quantum uncertainity for several families of quantum states under collective dephasing. We find that in situations where entanglement exhibits time-invariant feature, local quantum uncertainity first keep on increasing to a specfic value and then exhibit freezing dynamics after long time. In instances, where there is entanglement sudden death, local quantum uncertainity first decays, then increase and finally tend to freeze in the long time. On the other hand, in situations where entanglement exhibits freezing dynamics, local quantum uncertainity first decays very slowly to a value and then decays abruptly and finally tend to exhibit freezing dynamics as well. Finally, we examine the random pure states and calculate their entanglement at infinity. We find that more than half random states main their entanglement at infinity and hence all other correlations as well under collective dephasing.
This paper is organized as follows. In section \[Sec:Model\], we briefly discuss our model of interest and obtain the most general solution for an arbitrary initial density matrix. In section \[Sec: QCs\], we review the idea of entanglement for qubit-qutrit systems and describe the method to compute negativity for an arbitrary initial quantum state. We also briefly examine the concept of quantum discord and how to compute it for an arbitrary bipartite state. We also briefly review local quantum uncertainity and how to compute it for any state for $2 \otimes d$ quantum systems. In section \[Sec: Res\], we provide our main results for various initial states. Finally, we conclude our work in section \[Sec: Cc\].
Collective dephasing for qubit-qutrit systems {#Sec:Model}
=============================================
Our physical model consists of a qubit and a qutrit (one two-level atom and one three-level atom for an example) $A$ and $B$ that are coupled to a noisy environment, collectively. The qutrit as an atom, can be realized with well known “V”-type energy level configuration in which the transition among excited levels is forbidden. This means that first excited state will decay to ground level only and similarly the second excited level will also decay to ground level. The atoms are sufficiently far apart and they do not interact with each other, so that we can treat them as independent. The collective dephasing refers to coupling of atoms to the same noisy environment, which can be stochastic magnetic fields $B(t)$. There are at least two approaches to write a Hamiltonian for such physical situations. First, the Hamiltonian could be time independent, like in case of a qubit $H = \hbar \omega/2 \, \sigma_z$ with $\omega$ as energy splitting between excited states of atom. One can write a unitary propagator $U(t) = \exp(- i H t/\hbar)$. As there are fluctuations in magnetic field strength, the integration over it will induce a probability distribution $p(w)$ of characteristic energy splitting. The time evolution of atom can be written as an integral over $p(\omega)$ and unitary evolution, i.e., $\rho(t) = \int p(\omega) U(t) \rho(0) U(t)^\dagger \, d\omega$. The form of $p(\omega)$ will determine the nature of noise. Another approach, which we have taken in this work and most of the work in literature is to take the Hamiltonian as time dependent and embed the fluctuations of magnetic field in stochastic function $B(t)$, which already includes the information about characteristic function and so that the ensemble average over it introduce the decay parameter $\Gamma$. Both approaches are equivalent and generates the same dynamics. However, we point out, to our knowledge the present work and recent works are restricted to a very specific orientation of magnetic field and the theory of a general description of magnetic fields in any arbitrary directions is still not worked out. The Hamiltonian of the quantum system (with $\hbar = 1$) can be written as [@Karpat-PLA375-2011] $$\begin{aligned}
H(t) = - \frac{\mu}{2} \, \big[ \, B(t) (\sigma_z^A + \sigma_z^B) \, \big] \, , \label{Eq:Ham} \end{aligned}$$ where $\mu$ is gyromagnetic ratio and $\sigma_z^A$ is standard Pauli matrix for qubit and $\sigma_z^B$ is the dephasing operator for qutrit $B$. The stochastic magnetic fields refer to statistically independent classical Markov processes satisfying the conditions $$\begin{aligned}
\langle B(t) \, B(t')\rangle &=& \frac{\Gamma}{\mu^2} \, \delta(t-t') \,, \nonumber \\
\langle B(t)\rangle &=& 0 \, ,\end{aligned}$$ with $\langle \cdots \rangle$ as ensemble time average and $\Gamma$ denote the phase-damping rate for collective dephasing.
Let $|2\rangle$, $|1\rangle$, and $|0\rangle$ be the first excited state, second excited, and ground state of the qutrit, respectively. We choose the computational basis $ \{ \, |0,0\rangle$, $|0,1\rangle$, $|0,2\rangle \, |1,0\rangle$, $|1,1\rangle$, $|1,2\rangle \, \}$, where we have dropped the subscripts $A$ and $B$ with the understanding that first basis represents qubit $A$ and second qutrit $B$. Also the notation $|0 \rangle \otimes |0\rangle = |0 \, 0 \rangle$ has been adopted for simplicity. The time-dependent density matrix for the system is obtained by taking ensemble average over the noisy field, i.e., $\rho(t) = \langle\rho_{st}(t)\rangle$, where $\rho_{st}(t) = U(t) \rho(0) U^\dagger(t)$ and $U(t) = \exp[-\mathrm{i} \int_0^t \, dt' \, H(t')]$. The most general solution of $\rho(t)$ under the assumption that the system is not initially correlated with environment is given as
$$\begin{aligned}
\rho(t) = \left(
\begin{array}{llllll}
\rho_{11} & \xi \, \rho_{12} & \xi^4 \, \rho_{13} & \xi^4 \, \rho_{14} & \xi^9 \, \rho_{15} & \xi^{16} \,\rho_{16} \\
\xi \, \rho_{21} & \rho_{22} & \xi \, \rho_{23} & \xi \, \rho_{24} & \xi^4 \, \rho_{25} & \xi^9 \, \rho_{26} \\
\xi^4 \, \rho_{31} & \xi \, \rho_{32} & \rho_{33} & \rho_{34} & \xi \, \rho_{35} & \xi^4 \, \rho_{36} \\
\xi^4 \, \rho_{41} & \xi \, \rho_{42} & \rho_{43} & \rho_{44} & \xi \, \rho_{45} & \xi^4 \, \rho_{46} \\
\xi^9 \, \rho_{51} & \xi^4 \, \rho_{52} & \xi \, \rho_{53} & \xi \, \rho_{54} & \rho_{55} & \xi \, \rho_{56} \\
\xi^{16} \, \rho_{61} & \xi^9 \, \rho_{62} & \xi^4 \, \rho_{63} & \xi^4 \, \rho_{64} & \xi \, \rho_{65} & \rho_{66}
\end{array}
\right) \,,\label{Eq:MF}\end{aligned}$$
where $\xi = e^{- \Gamma t/8}$. We note that decoherence free subspaces (DFS) [@Yu-CD-2002] do appear in this system as a common characteristic of collective dephasing. Another interesting property of the dynamics is the fact that all initially zero matrix elements remain zero.
Entanglement, quantum discord and local quantum uncertainity for $2 \otimes 3$ quantum systems {#Sec: QCs}
==============================================================================================
In this section, we briefly review the correlations, which we study in this work for qubit-qutrit systems. In subsection \[SS:ent\], we briefly review entanglement and a computable measures of entanglement. In subsection \[SS:dis\], we review the quantum discord and how to compute it for any bipartite quantum state. In subsection \[SS:lq\], we discuss local quantum uncertainity and how to compute it for a given state in $2 \otimes d$ quantum systems.
Quantum entanglement {#SS:ent}
--------------------
The question of quantum entanglement for qubit-qubit $(2 \otimes 2)$ quantum systems and qubit-qutrit $(2 \otimes 3)$ quantum systems has been solved. It is well known that for bipartite quantum systems, if the partial transpose with respect of any one of the subsystem has at least one negative eigenvalue then the quantum state is entangled or NPT [@Peres-PRL-1996]. Whereas if the partial transposed matrix has all positive eigenvalues (PPT), then entanglement/separability depends upon the dimension of Hilbert space. The PPT states for $2 \otimes 2$ and $2 \otimes 3$ are separable (not entangled), whereas for larger dimensions of Hilbert space, there may exist PPT-entangled states (also called bound entangled states) [@Horodecki-RMP-2009]. Hence, for a given density matrix of qubit-qutrit system, one can easily find the eigenvalues of partially transposed matrix (partial transpose can be taken with respect to any subsystem). It is not hard to look for possible negative eigenvalues. The sum of absolute values of all possible negative eigenvalues is defined as a legitimate measure of quantum entanglement, namely [*negativity*]{} [@Vidal-PRA65-2002]. Hence, negativity is defined as $$N (\rho) = 2 \, \bigg( \, \sum_i \, | \eta_i| \, \bigg)\, ,$$ where $\eta_i$ are possible negative eigenvalues and multiplication with $2$ is for normalization so that for maximally entangled states, this measure should have numerical value of $1$. For specific quantum states, this definition is sufficient to compute and study the dynamics of negativity. For random states, it is more easy to use [*entanglement monotone*]{}, which is based on PPT-mixtures idea [@Bastian-PRL106-2011] and very easy to compute numerical value of entanglement for any density matrix. The description of semi-definite programming (SDP) and genuine negativity is described in details in Ref.[@Bastian-PRL106-2011]. We denote this measure by $E(\rho)$ in this paper. For bipartite systems, this monotone is equivalent to [*negativity*]{}.
Quantum Discord {#SS:dis}
---------------
Quantum discord is one of the measure of quantum correlations which are captured using von Neumann entropy. This measure has been intensively investigated in previous 18 years in various contexts and many studies focused on the quantification of this measure for various dimensions of Hilbert space. The literature on this measure is so extensive that it is not possible to cite each of them, so we only provide fundamental references. We discuss the main ideas very briefly to compute quantum discord for a given bipartite quantum state. Any bipartite state may have both quantum and classical correlations, which are jointly captured by quantum mutual information. In particular, if $\rho^{AB}$ denotes the density operator of a composite bipartite system $AB$, and $\rho^A$ ($\rho^B$) the density operator of part $A$ ($B$), respectively, then the quantum mutual information is defined as [@Groisman-PRA72-2005] $$\begin{aligned}
\mathcal{I} (\rho^{AB}) = S (\rho^A) + S (\rho^B) - S(\rho^{AB})\, , \label{Eq:QMI}\end{aligned}$$ where $S(\rho) = - \mathrm{tr} \, ( \rho \, \log_2 \rho )$ is the von Neumann entropy. We take all logarithms base $2$ in this work. Quantum mutual information may be written as a sum of classical correlation $\mathcal{C}(\rho^{AB})$ and quantum discord $\mathcal{Q} (\rho^{AB})$, that is, $\mathcal{I} (\rho^{AB}) = \mathcal{C} (\rho^{AB}) + \mathcal{Q} (\rho^{AB})$ [@Ollivier-PRL88-2001; @Vedral-et-al; @Luo-PRA77-2008]. Quantum discord can be positive in separable mixed states (that is, with no entanglement).
Quantum discord can be quantified [@Ollivier-PRL88-2001] via von Neumann type measurements which consist of one-dimensional projectors that sum to the identity operator. Let the projection operators $\{ A_k\}$ describe a von Neumann measurement for subsystem $A$ only, then the conditional density operator $\rho_k$ associated with the measurement result $k$ is $$\begin{aligned}
\rho_k = \frac{1}{p_k} (A_k \otimes \mathbb{I}_B) \, \rho \, (A_k \otimes \mathbb{I}_B) \,,\end{aligned}$$ where the probability $p_k$ equals $\mathrm{tr} [(A_k \otimes \mathbb{I}_B) \, \rho \, (A_k \otimes \mathbb{I}_B)]$. The quantum conditional entropy with respect to this measurement is given by [@Luo-PRA77-2008] $$\begin{aligned}
S (\rho | \{A_k\}) := \sum_k p_k \, S(\rho_k) \, ,
\label{Eq:QCE}\end{aligned}$$ and the associated quantum mutual information of this measurement is defined as $$\begin{aligned}
\mathcal{I} (\rho|\{A_k\}) := S (\rho^B) - S(\rho|\{A_k\}) \, . \label{Eq:QMIM} \end{aligned}$$ A measure of the resulting classical correlations is provided [@Ollivier-PRL88-2001; @Vedral-et-al; @Luo-PRA77-2008] by $$\begin{aligned}
\mathcal{C}(\rho) := \sup_{\{A_k\}} \, \mathcal{I} (\rho|\{A_k\}) \, . \label{Eq:CC} \end{aligned}$$ The obstacle to computing quantum discord lies in this complicated maximization procedure for calculating the classical correlation because the maximization is to be done over all possible von Neumann measurements of $A$. Once $\mathcal{C}(\rho)$ is in hand, quantum discord is simply obtained by subtracting it from the quantum mutual information, $$\begin{aligned}
\mathcal{Q}(\rho) := \mathcal{I}(\rho) - \mathcal{C}(\rho) \, .\end{aligned}$$ This maximization process is not easy in general and analytical results for quantum discard are only known for very specific quantum states. In this work, we have been only able to calculate it for only one family of quantum states for $2 \otimes 3$ quantum system.
Local quantum uncertainity {#SS:lq}
--------------------------
First of all we briefly review the concept of local quantum uncertainity (LQ). This is a measure of quantum correlations which has been defined for $2 \otimes d$ quantum systems [@Girolami-PRL110-2013]. It is a quantum discord type measure and we will see in the results below that for certain quantum states, quantum discord and local quantum uncertainity captures precisely same correlations and are equal to each other, whereas for some other states, they are different measures. It is defined as the minimum skew information which is obtained via local measurement on qubit part only. This measure has the advantage that there is no need for complicated minimization over parameters related with measurement operations. This measure is defined as $$LQ(\rho) \equiv \, \min_{K_A} \, \mathcal{I} (\rho , K_A \otimes \mathbb{I}_B) \,,$$ where $K_A$ is some local observable on subsystem $A$, and $\mathcal{I}$ is the skew information of the density operator $\rho$, defined as $$\mathcal{I} (\rho , K_A \otimes \mathbb{I}_B) \, = \,- \frac{1}{2} \, \rm{Tr} ( \, [ \, \sqrt{\rho}, \, K_A \otimes \mathbb{I}_B ]^2 \, ) \,.$$ It has been shown [@Girolami-PRL110-2013] that for $2 \otimes d$ quantum systems, the compact formula for local quantum uncertainity is given as $$LQ(\rho) = 1 - \rm{max} \, \{ \lambda_1 \,, \lambda_2 \, , \lambda_3 \, \}\,,$$ where $\lambda_i$ are the eigenvalues of $3 \times 3$ matrix $\mathcal{M}$, whose matrix elements are calculated by relationship $$m_{ij} \equiv \rm{Tr} \, \{ \, \sqrt{\rho} \, (\sigma_i \otimes \mathbb{I}_B) \, \sqrt{\rho} \, (\sigma_j \otimes \mathbb{I}_B) \, \}\,,$$ where $i,j = 1,2,3$ and $\sigma_i$ are the standard Pauli matrices.
Main results {#Sec: Res}
============
In this section, we will present our main results for various families of quantum states.
### Two parameter class of states
The class of quantum states with two real parameters $\alpha$ and $\gamma$ in a $2 \otimes d$ quantum system [@Chi-JPA36-2003] is given as $$\begin{aligned}
\rho_{\alpha, \gamma} =& \alpha \, \sum_{i = 0}^{1} \sum_{j = 2}^{d-1} \, | i\, j \rangle\langle i \, j| + \beta \, ( | \phi^+ \rangle\langle \phi^+|
+ | \phi^- \rangle\langle \phi^-| + | \psi^+ \rangle\langle \psi^+| \, ) \nonumber \\& + \gamma \, | \psi^- \rangle\langle \psi^-| \,,
\label{Eq:rhoag}\end{aligned}$$ where $\{ \, |i \, j \rangle : \, i = 0, \, 1, j = 0, \, 1, \, \ldots \, , \, d-1 \, \}$ is an orthonormal basis for $2 \otimes d$ quantum system and $$\begin{aligned}
| \, \phi^\pm \rangle &=& \frac{1}{\sqrt{2}} \, ( \, |0\, 0\rangle \pm | 1\,1 \rangle \, ) \\
| \, \psi^\pm \rangle &=& \frac{1}{\sqrt{2}} \, (\, |0\,1 \rangle \pm |1\,0 \rangle ) \, , \end{aligned}$$ and the parameter $\beta$ is dependent on $\alpha$ and $\gamma$ by the unit trace condition, $$\begin{aligned}
2 \, ( d - 2) \alpha + 3 \, \beta + \gamma = 1 \, .\end{aligned}$$ From Eq. (\[Eq:rhoag\]) one can easily obtain the range of parameters as $ 0 \leq \alpha \leq 1/(2 (d-2))$ and $0 \leq \gamma \leq 1$. We note that the states of the form $\rho_{0, \gamma}$ are equivalent to Werner states [@Wer-PRA89] in a $2 \otimes 2$ quantum systems. Moreover, the states $\rho_{\alpha, \gamma}$ have the property that their PPT (positive partial transpose) region is always separable [@Chi-JPA36-2003]. It is also known that an arbitrary quantum state $\rho$ in $2 \otimes d$ can be transformed to $\rho_{\alpha, \gamma}$ with the help of local operations and classical communication (LOCC).
We have already calculated quantum discord, classical correlation and entanglement for this family in an earlier work [@Ali-JPA43-2010]. Here we simply extend the previous results for collective dephasing (an additional parameter $\Gamma t$). It turns out that classical correlations for this family of states does not depend on decay parameter and are constant in time. The expression for classical correlations is given as $$\begin{aligned}
\mathcal{C}(\rho_{\alpha,\gamma}) = - (3 \, \beta + \gamma) \, \log(\frac{3 \, \beta + \gamma }{2}) + 2 \, \beta \, \log(2 \, \beta)
+ (\beta + \gamma) \log(\beta + \gamma) \, .
\label{Eq:cc23} \end{aligned}$$ The quantum discord is calculated using the standard procedure discussed in previous section and is given as $$\begin{aligned}
\mathcal{Q}(\rho_{\alpha,\gamma})(t) =& 1 - 2 \, \alpha -2 \, \beta -(\beta + \gamma) \, \log(\beta + \gamma) + \frac{\beta + \gamma
+ \xi \, (\beta-\gamma)}{2} \, \log\bigg(\frac{\beta + \gamma + \xi \, (\beta - \gamma)}{2}\bigg)
\nonumber \\& + \frac{\beta + \gamma - \xi \, (\beta - \gamma)}{2} \, \log\bigg(\frac{\beta + \gamma - \xi \, (\beta - \gamma)}{2}\bigg) \,.
\label{Eq:qd23}\end{aligned}$$ We can see that as $t \to \infty$, $\xi \to 0$, and $\mathcal{Q}(\rho_{\alpha,\gamma})(\infty) = 1 - 2 \, \alpha - 3 \, \beta - \gamma = 0$ as expected.
The local quantum uncertainity for this family of state turns out to be $$\begin{aligned}
LQ(\rho_{\alpha,\gamma})(t) = 1 - 2 \, \alpha - 2 \, \beta - \bigg[\sqrt{\beta (1 + \xi) + \gamma (1 - \xi)} \, \sqrt{\beta \, (1 - \xi)
+ \gamma \, (1 + \xi)} \bigg]\,.
\label{Eq:lqab1}\end{aligned}$$ We note the similarity between local uncertainity Eq.(\[Eq:lqab1\]) and quantum discord Eq.(\[Eq:qd23\]). Indeed, it turns out that for $t = 0$, and for the initial states (i) $\alpha = \beta = 0$ and $\gamma = 1$, (ii) $\alpha = \gamma = 0$, and $\beta = 1/3$, (iii) $\gamma = 0$, and (iv) $ \beta = 0$, local quantum uncertainity and quantum discord turns out be exactly equal as can be checked easily. However, for more general cases with $\alpha, \beta, \gamma \neq 0$, and under collective dephasing, both measured are different as will shown below.
The negativity for this family of states is straight forward to calculate and is given as $$N (\rho_{\alpha, \gamma})(t) = \rm{max} \big[ \, 0 \, , \xi \, (\gamma - \beta) - 2 \, \beta \big] \,.$$ It is easy to see that for $\beta = 0$, the states decay asymptotically and entanglement is lost only at infinity, whereas for $ \beta \neq 0$, negativity is lost at $$\Gamma t = 8 \, \log \frac{\gamma - \beta}{ 2 \, \beta} \, .
\label{Eq:Nabe}$$
We plot entanglement, discord, classical correlation, and local quantum uncertainity for state $\rho_{\alpha, \gamma}(t)$ in Figure (\[Fig:1\]). We have taken specific values of $\alpha = 0.1$, $\beta = 0.1$, and $\gamma = 0.5$. Quantum discord $ \mathcal{Q} (\rho_{\alpha, \gamma})(t)$ plotted as solid line decays slowly as well as negativity (dashed line) and local quantum uncertainity (big dashed line). Classical correlation (dashed orange line) is constant in time with fixed initial value. Negativity ends at $\Gamma t \approx 5.54$ (not shown in Figure \[Fig:1\]), whereas quantum discord becomes zero at infinity. Local quantum uncertainity and quantum discord become zero at the same time as expected.
Let us take another set of initial values with $\alpha = 0.12$, $\beta = 0.12$, and $\gamma = 0.4$ for state $\rho_{\alpha, \gamma}(t)$. Figure (\[Fig:2\]) depicts entanglement (dashed line), classical correlation (thick dashed orange line), quantum discord (solid line) and local quantum uncertainity (big dashed line) for this set of values against decay parameter $\Gamma t$. As we have reduced the fraction of maximally entangled state ($\gamma$) and increased the noisy components $\alpha$ and $\beta$ slighly, nevertheless, the resulting dynamics is interesting and different than earlier case. The numerical values of all correlations are lower than the earlier case. This fact is understandable as we have reduced the fraction of $\gamma$, so maximally entangled state feeded almost all correlations in $\rho_{\alpha, \gamma}$. Another main difference is vanishing of entanglement at $\Gamma t \approx 1.233$ so called sudden death of entanglement. Classical correlation is constant as mentioned earlier. Quantum discord and local quantum uncertainity decaying slowly as expected and both becoming zero only at infinity.
### Search for freezing dynamics of entanglement
It has already been shown explicitly [@Karpat-PLA375-2011] that certain qubit-qutrit entangled states exhibit time-invariant entanglement feature under collective dephasing. However, the question of freezing dynamics of entanglement has not been explored so far. Therefore, we look for such possibility encouraged by the existance of decohence free subspaces where entangled states can reside. Of course, alone the presence of such decoherence free spaces do not guarantee that either time-invariant entanglement or freezing dynamics must occur. In fact all previous studies suggest that for all other dimensions of Hilbert space studied so far, either time-invariant entanglement appear or freezing dynamics. To our knowledge, these both possibilities have never been observed for any single dimension of Hilbert space. Interestingly, as we will demonstrate that qubit-qutrit systems offer all kind of dynamical features of entanglement, that is, entanglement sudden death, asymptotic decay of entanglement, time-invariant entanglement, and freezing dynamics of entanglement under collective dephasing.
Let us define a single parameter class of states, which are mixture of entangled states residing in decoherence free subspace and states which decay. The states are defined as $$\rho_\alpha = \alpha \, |\psi_3 \rangle\langle \psi_3| + (1 - \alpha) \, |\psi_2 \rangle\langle \psi_2| \,,$$ where $0 \leq \alpha \leq 1$, the maximally entangled state $|\psi_2 \rangle$ is defined as $$|\psi_2 \rangle = \frac{1}{\sqrt{2}} \, (|0 \, 1\rangle + |1 \, 2 \rangle)\,,$$ and another maximally entangled state $|\psi_3 \rangle$ is defined as $$|\psi_3 \rangle = \frac{1}{\sqrt{2}} \, (|0 \, 2\rangle + |1 \, 0 \rangle)\, .$$ In this mixture $|\psi_2 \rangle$ decays, whereas $|\psi_3 \rangle$ lives in decoherence free subspace. Therefore the time evolution of this state can be written as $$\rho_\alpha (t) = \alpha \, \rho_3 + (1 - \alpha) \, \rho_2 (t) \, .$$ There are only two possible negative eigenvalues for the partial transpose of this state, namely $$\begin{aligned}
v_1(\alpha) = \frac{1}{4} \, \bigg[ (1-\alpha) - \sqrt{(1-\alpha)^2 + 4 \, \alpha^2} \bigg] \nonumber \\
v_2 (\alpha,\,\xi) = \frac{1}{4} \, \bigg[ \alpha - \sqrt{\alpha^2 + 4 \, \xi^{18} \, (1- \alpha)^2} \bigg] \,.\end{aligned}$$ Negativity for these states can be written as $$N_\alpha = 2 \, \big[ \, {\rm max} ( 0, \, - v_1(\alpha) ) + {\rm max} (0, \, - v_2(\alpha)) \, \big]\, .$$ It is obvious that $v_1(\alpha)$ does not depend on decay parameter and this value is negative for any $\alpha > 0$. The other eigenvalue $v_2(\alpha, \xi)$ is also negative for any $\alpha > 0$ at the start ($\Gamma t = 0$), however, as decoherence is turned on, this value quickly becomes positive. So we can see very clearly that all states with $ 0 < \alpha < 1$ must exhibit freezing dynamics of entanglement.
Figure (\[Fig:3\]) shows negativity plotted against decay parameter $\Gamma t$ for various choices of parameter $\alpha$. It is clear that all initial amounts of entanglement determined by choice of $\alpha$ decay as evident from $v_2(\alpha)$ until it becomes zero and hence the residual entanglement in decoherence free subspace becomes dominant as dictated by $v_1(\alpha)$. Hence this family of states exhibit freezing dynamics of entanglement such that quantum states changes with time but its entanglement is locked in time (stationary). It is interesting to note that for qubit-qutrit systems, time-invariant entanglement and freezing dynamics exist. We have not found this coincidence in any other dimension of Hilbert space so far.
It is straight forward to calculate local quantum uncertainity for $\rho_\alpha$ which is given as $$LQ_\alpha = 1 - \frac{1}{2} \, \sqrt{\alpha (1-\alpha)} \,.$$ This value is symmetric about $\alpha = 0.5$ as expected because all correlations must be symmetric about this value. For time-evolved state, local quantum uncertainity is given as $$LQ_\alpha(t) = 1 - \lambda_\alpha(t) \,,$$ where $\lambda_\alpha(t) = \rm{max} [w_{11}(t), w_{33}(t)]$, and $$w_{11}(t) = w_{22}(t) = \frac{\sqrt{\alpha} \, \bigg[ \sqrt{(1-\alpha)(1 - \xi^9)} + \sqrt{(1 - \alpha)(1 + \xi^9)} \bigg]}{2 \sqrt{2}} \, ,$$ and $$w_{33}(t) = (1-\alpha) \, \sqrt{1 - \xi^{18}} \, ,$$ where $w_{ii}(t)$ are the eigenvalues of the symmetric $3 \times 3$ matrix.
In Figure (\[Fig:4\]) we plot the local quantum uncertainity against decay parameter $\Gamma t$ for same values of parameter $\alpha$ as in Figure (\[Fig:3\]). As can be seen that just like entanglement freezing, the local quantum uncertainity initially decays to some value and then also tend to freezing dynamics of local uncertainity. At $\Gamma t = \infty$, the stationary value of local quantum uncertainity is given as $$LQ_\alpha (\infty) = 1 - \rm{max} \bigg[ (1-\alpha),\, \sqrt{\alpha (1-\alpha)/2} \bigg]\, ,$$ which is abvious a nonzero value.
### A review on time-invariant entanglement for qubit-qutrit systems
As we have noticed in all earlier reports of time-invariant entanglement, the quantum state exhibiting this interesting phenomenon must be a mixture of two entangled states and one of the state must reside in decoherence free subspace. However, we have seen above that if we mix state $|\psi_3 \rangle$ and $|\psi_2 \rangle$, we do not obsereve any time-invariant entanglement rather freezing dynamics of entanglement. So this suggests that we must look for some other entangled state to be mixed with $|\psi_3 \rangle$. One of such state is $$|\psi_1 \rangle = \frac{1}{\sqrt{2}} \, (|0 \, 0\rangle + |1 \, 2 \rangle)\, .$$ Actually the first report of time-invariant entanglement for qubit-qutrit systems [@Karpat-PLA375-2011] took a state which was mixture of these two type of states. To generalize this observation for more general states, first let us consider the states, $$\tilde{\rho}_\alpha = \alpha \, |\psi_1 \rangle\langle \psi_1| + \frac{1 - \alpha}{6} \, \mathbb{I}_6 \,,$$ where $\mathbb{I}_6$ is $6 \times 6$ identity matrix and $ 0 \leq \alpha \leq 1$. Such states are called isotropic states and they are NPT for $1/4 < \alpha \leq 1$, and hence entangled. To avoid confusion, we differentiate these state by taking tilde over $\rho_\alpha$. This could have been avoided by calling single parameter any other name than $\alpha$, however we preferred to keep it like that. We can now define two parameter family of states, which are mixture of isotropic states and $|\psi_3\rangle$, given as $$\rho_{\alpha,\beta} = \beta \, |\psi_3\rangle \langle \psi_3 | + (1 - \beta) \, \tilde{\rho}_\alpha \, ,$$ where $ 0 \leq \beta \leq 1$. Entanglement properties for this family of states are quite interesting. The partial transpose with respect to subsystem $A$ have maximum two possible negative eigenvalues and the rest of $4$ eigenvalues are definitely positive for the given range of parameters $\alpha$ and $\beta$. The $2$ possible negative eigenvalues are such that when one is positive, then other is negative and vice versa. They are never negative at the same time. The time evolution of these states can be written as $$\rho_{\alpha,\beta}(t) = \beta \, |\psi_3\rangle \langle \psi_3 | + (1 - \beta) \, \tilde{\rho}_\alpha(t) \, .$$ Hence $\tilde{\rho}_\alpha(t)$ decays, whereas $|\psi_3\rangle$ remain dynamically invariant as it lives in DFS. Now there is an additional parameter $\Gamma t$ involved in the density matrix. The two possible negative eigenvalues of partially transposed matrix are given as $$\begin{aligned}
x_1 &=& \frac{1}{6} \, \big[ 1 + 2 \, \alpha (1-\beta) - 4 \, \beta \, \big] \nonumber \\
x_2 &=& \frac{1}{6} \, \big[ 1 + 2 \, \beta - \alpha \, (1-\beta)(1 + 3 \, \xi^{16} ) \, \big]\, .
\label{Eq:NEVs}\end{aligned}$$
As we have mentioned earlier that these two eigenvalues can not be negative at the same time. We also observe that one of the eigenvalue $x_1$ does not depend upon $\xi$, so if this eigenvalue is negative then as the other cannot be negative so this necessary means time-invariant entanglement. On the other hand if $x_1$ is positive then $x_2$ must be negative. However $x_2$ depends on $\xi$ and it not difficult to see that $x_2$ can become positive in a finite time, leading to finite time end of entanglement. As long as $\beta > 1/2$, $x_2$ is positive for all ranges of $\alpha$, hence we can get time-invariant entanglement, whereas for other values we would get sudden death of entanglement. Negativity for these state is given as $$N_{\alpha\,, \beta} = 2 \, \big[ \, {\rm max} \big( 0, \, - x_1(\alpha,\beta) \big) + {\rm max} \big(0, \, - x_2(\alpha , \beta, \xi) \big) \, \big]\, .$$
In Figure (\[Fig:5\]), we plot negativity against parameter $\Gamma t$ for four different set of values of $\alpha$ and $\beta$. We see that for $\beta > 1/2$, that is, for $\alpha = 0.4$, $\beta = 0.7$ (red thick dashed line) and $\alpha = 0.5$, $\beta = 0.8$ (blue thick dashed line), we get time-invariant entanglement on the one hand and for other range, $\alpha = 0.9$, $\beta = 0.2$ (solid line), and $\alpha = 0.8$, $\beta = 0.3$ (thin dashed line), we see end of negativity at finite times.
We have also calculated local quantum uncertainity for $\rho_{\alpha, \, \beta}(t)$. Following the procedure mentioned in previous section, we get a diagonal matrix and hence the eigenvalues of the resulting $3 \times 3$ matrix. It is simple to pick the maximum eigenvalue for given set of parameters. In Figure (\[Fig:6\]), we plot $LQ_{\alpha, \, \beta}$ against parameter $\Gamma t$ for same set of values for $\alpha$ and $\beta$ as in Figure (\[Fig:5\]). We observe quite interesting dynamics for local quantum uncertainity as compared with earlier cases. First we see that for two instances where we get time-invariant entanglement, the local quantum uncertainity first increases and then tends to freeze to a specfic positive value. Intuitively one can understand the freezing behavior of local quantum uncertainity as due to stationary correlations (not decaying due to decoherence subspace) in state $|\psi_3 \rangle$. However, it is not intuitive why these correlations first increase before becoming stationary. For the other two instances, where we get sudden death of entanglement, that is, for $(\alpha = 0.8, \, \beta = 0.3)$ (thin dashed line) local quantum uncertainity first decays for a short time and then once again increase and then tends to freeze to a constant value. Whereas for $(\alpha = 0.9, \, \beta = 0.2)$ (solid line), local quantum uncertainity first decreases for a short time, then increase to a value and then once again decays to another value and then finally exhibits freezing dynamics. As we mentioned, the freezing part of correlations can be explain easily whereas other parts of dynamics are counterintuitive.
### Comparison with dynamics of random states
In order to compare dynamics of quantum states with generic states, we have generated $100$ random pure states. A state vector for qubit-qutrit systems, randomly distributed according to the Haar measure can be generated in the following way [@Toth-CPC-2008]: First, we generate a vector such that both the real and the imaginary parts of the vector elements are Gaussian distributed random numbers with a zero mean and unit variance. Second we normalize the vector. It is easy to prove that the random vectors obtained this way are equally distributed on the unit sphere [@Toth-CPC-2008]. Note that the random pure states, which we generate in the global Hilbert space of dimension $6$, so the unit sphere is not the Bloch ball.
After generation of $100$ random pure states, we find their time-evolved density matrices interacting with collective dephasing and compute negativity using PPT-mixture package [@Bastian-PRL106-2011], for each state against parameter $\Gamma t$. From this data we can also obtain an error estimate to indicate the reliability of the measure. This can, for instance, be defined as a confidence interval [@Ali-JPB-2014] $$\begin{aligned}
CI = \mu \, \pm \, \sqrt{\delta} \, ,
\label{interval}\end{aligned}$$ where $\mu$ stands for mean value and $\delta$ for variance of quantity being measured. Note, however, that this is not a confidence interval in the mathematical sense.
In Figure (\[Fig:7\]), we plot entanglement monotone (negativity) $E(\rho)$ for random pure states against parameter $\Gamma t$. The thick dashed (blue) line presents the mean value of entanglement, whereas thick dashed-dotted (red) lines represent confidence interval $CI$ with top line as sum of mean value and variance, where as below thick dashed-dotted line with difference of mean value and variance. As we can see that many states tend to exhibit freezing dynamics of entanglement (about $57\%$) where as many exhibit sudden death of entanglement (about $43\%$).
Finally we analyze the asymptotic states by taking $\xi = 0$ in time-evolved density matrices for random states. We then compute entanglement monotone (negativity) for these states and as mentioned earlier about $57\%$ of them are found to be entangled. In Figure (\[Fig:8\]) we show bar graph for random states at infinity against number of random states. It is obvious that all entangled states will be having nonzero local quantum uncertainity as well.
Conclusions {#Sec: Cc}
===========
We have studied dynamics of quantum correlations of qubit-qutrit systems under Markovian collective dephasing. We have investigated some aspects of this simple system not studied before. In particular, we have studied two non-trivial features of entanglement dynamics, namely, time-invariant entanglement and freezing dynamics of entanglement. All previous studies on these two features of entanglement dynamics for bipartite as well as for multipartite quantum systems gave the impression that we could not have both features available for one specific quantum system under collective dephasing. The reason for this impression was the observation that for qubit-qubit systems we detected time-invariant entanglement whereas we did not find any freezing dynamics of entanglement under same collective dephasing model [@Liu-arXiv]. We did find freezing dynamics for qubit-qubit systems however for more general directions of magnetic fields [@Carnio-PRL-2015] instead of specific z-direction where we have only time-invariant feature available. For three qubits, we found evidence for freezing dynamics of genuine entanglement whereas we found no evidence for time-invariant entanglement [@Ali-EPJD-2017]. On the other hand, for four qubits, we found no evidence for freezing dynamics of entanglement but we do found time-invariant entanglement [@Ali-EPJD-2017]. More recently, we examined qutrit-qutrit quantum systems where we found freezing dynamics of entanglement but no time-invariant entanglement [@Ali-MPLA-2019]. There is no concrete mathematical arguments for mutual exclusiveness of these features for any specific Hilbert space. Contrary to earlier impression, for qubit-qutrit quantum systems, we found time-invariant entanglement as well as freezing dynamics entanglement. The future investigations might shed more light on relationship between these possibilities and dimensions of subsystems if there is any such relationship. In addition, we have studied dynamics of quantum discord for a specific family of quantum states and local quantum uncertainity for several families of states. We have seen that for some states quantum discord and local quantum uncertainity decay asymptotically and become zero only at infinity. For these states only classical correlations remain constant and do not decay. For other states which exhibit freezing dynamics of entanglement, local quantum uncertainity also tends to exhibit freezing dynamics. For quantum states which exhibit time-invariant entanglement, local quantum uncertainity first increase to a specific value and then become stationary at nonzero values. For same states which exhibit sudden death of entanglement, local quantum uncertainity first decay for short time, then increase for some time and finally reach a nonzero stationary value. Finally we have compared the dynamics of specific states with generic states by generating random pure states. We have seen that most of random pure states under collective dephasing exhibit freezing dynamics of entanglement and maintain this nonzero value even at infinity. Some random pure states do become separable at finite time. Another future avenue would be to explore more general $d \otimes N$ quantum systems for $d \neq N$ to find more examples.
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---
abstract: 'Luminous compact blue galaxies (LCBGs) are a diverse class of galaxies characterized by high luminosities, blue colors, and high surface brightnesses. Residing at the high luminosity, high mass end of the blue sequence, LCBGs sit at the critical juncture of galaxies that are evolving from the blue to the red sequence. Yet we do not understand what drives the evolution of LCBGs, nor how they will evolve. Based on single-dish [HI]{} observations, we know that they have a diverse range of properties. LCBGs are [HI]{}-rich with M$_{HI}=$10$^{9-10.5}~$M$_\odot$, have moderate M$_{dyn} =$10$^{10-12}~$M$_\odot$, and 80% have gas depletion timescales less than 3 Gyr. These properties are consistent with LCBGs evolving into low-mass spirals or high mass dwarf ellipticals or dwarf irregulars. However, LCBGs do not follow the Tully-Fisher relation, nor can most evolve onto it, implying that many LCBGs are not smoothly rotating, virialized systems. GMRT and VLA [HI]{} maps confirm this conclusion revealing signatures of recent interactions and dynamically hot components in some local LCBGs, consistent with the formation of a thick disk or spheroid. Such signatures and the high incidence of close companions around LCBGs suggest that star formation in local LCBGs is likely triggered by interactions. The dynamical masses and apparent spheroid formation in LCBGs combined with previous results from optical spectroscopy are consistent with virial heating being the primary mechanism for quenching star formation in these galaxies.'
author:
- 'D.J. Pisano'
- 'C.A. Garland'
- Katie Rabidoux
- Spencer Wolfe
- 'R. Guzmán'
- 'J. Pérez-Gallego'
- 'F.J. Castander'
title: 'The Evolution of Luminous Compact Blue Galaxies: Disks or Spheroids?'
---
[address=[WVU Dept. of Physics, P.O. Box 6315, Morgantown, WV 26506, USA]{},altaddress=[djpisano@mail.wvu.edu]{} ]{}
[ address=[Natural Sciences Department, Castleton State College, Catleton, VT, USA]{} ]{}
[ address=[WVU Dept. of Physics, P.O. Box 6315, Morgantown, WV 26506, USA]{} ]{}
[ address=[WVU Dept. of Physics, P.O. Box 6315, Morgantown, WV 26506, USA]{} ]{}
[ address=[Dept. of Astronomy, Univ. of Florida, Gainesville, FL, USA]{} ]{}
[ address=[Dept. of Astronomy, Univ. of Florida, Gainesville, FL, USA]{} ]{}
[ address=[Insitut de Ciéncies de l’Espai, Barcelona, Spain]{} ]{}
Introduction
============
Galaxies have evolved dramatically since a redshift of one with the appearance of a red sequence [@bell04] and an order of magnitude decrease in the cosmic star formation rate [@madau98]. Yet the physical mechanisms that drive this evolution, this quenching of star formation in luminous galaxies, are poorly understood. Luminous compact blue galaxies (LCBGs) reside at the high mass tip of the blue sequence, M$_\star \sim$3$\times$10$^{10}~$M$_\odot$ [@kauffmann03], yet because of their prolific star formation they are poised to evolve off the blue sequence in the near future. As such, LCBGs sit at the critical juncture for understanding how galaxies evolve from the blue to the red sequence. We are conducting a multi-wavelength study of local LCBGs to better understand some of the processes that can trigger and quench their intense star formation and how LCBGs may subsequently evolve.
LCBGs are defined as those galaxies with $M_B\le$$-$ 18.5 mag, $SB_e(B)\le$ 21 mag arcsec$^{-2}$ (equivalent to $r_{eff}\le$ 4 kpc), and $B-V\le$ 0.6 mag [@werk04; @garland04]. They represent a small, extreme subset of the blue sequence and are not a distinct population. While having similar colors and surface brightness as blue compact dwarfs, LCBGs are significantly brighter and more metal-rich [@kunth00; @werk04]. We report here on single-dish and interferometric [HI]{} observations of local LCBGs (D$\le$200 Mpc) selected from the SDSS.
Single-Dish Results
===================
We have collected single-dish [HI]{} data for 163 local LCBGs selected from SDSS. Of these about 40% are original observations with Arecibo, the Green Bank Telescope (GBT), or Nançay, while the remainder are published archival data. Of the 63 LCBGs we targeted for original observations, 94% were detected with a 5$\sigma$ M$_{HI}$ detection limit over 140 km s $^{-1}$ of 2.5$\times$10$^8~$M$_\odot$, so there are very few [HI]{}-deficient LCBGs. Using the integrated flux of the [HI]{} line, we find that local LCBGs are [HI]{}-rich with M$_{HI}$=10$^{9-10.5}~$M$_\odot$, similar to the Milky Way or M 31. Using the [HI]{} linewidth with their optical sizes and inclinations, we are able to derive an enclosed dynamical mass for LCBGs assuming that they are rotating systems [@garland04]. We find that local LCBGs have M$_{dyn} =$10$^{10-12}~$M$_\odot$, intermediate between the LMC and the Milky Way and close to the maximal halo mass of $\sim$10$^{12}~$M$_\odot$ from [@cattaneo06] for quenching of star formation by virial heating. Combining M$_{HI}$ with infrared or radio continuum measurements of the star formation rate, we find that 80% of local LCBGs have a gas depletion timescale, $\tau_{HI}$=M$_{HI}$/SFR, less than 3 Gyr. About half of the sample have a close, optically-bright companion [@garland04], a higher rate than for field galaxies in general [@james08; @pisano02], yet there appears to be no significant difference in the properties of isolated and un-isolated LCBGs (within the single-dish beam). Overall, these properties are consistent with local LCBGs following a range of evolutionary paths becoming massive dwarf elliptical, dwarf irregular, or low-mass spiral galaxies depending on their exact properties.
While single-dish [HI]{} observations are a powerful tool for constraining the nature and evolution of LCBGs, their inherently poor angular resolution means we are not getting a complete picture. For example, despite the presence of large amounts of [HI]{} and signatures of normal rotation, LCBGs do not follow the Tully-Fisher relation for normal spiral galaxies, nor can they evolve onto it. Therefore, in order to better understand the kinematics of LCBGs, it is essential to have resolved maps of them from an interferometer.
Interferometer Results
======================
![[HI]{} kinematics of three local LCBGs. From top to bottom they are SDSS1507+5511, SDSS0119+1452, and Mrk 325. Left column: Velocity fields with contours every $\sim$10 km s$^{-1}$, Right column: Velocity dispersion maps with contours every $\sim$5 km s $^{-1}$. The synthesized beam is in the lower left of each panel. \[HIfig\]](figure1.eps){width=".9\textwidth"}
We have used the GMRT and the VLA to map the [HI]{} in 18 local LCBGs with synthesized beams of $\sim$10$^{\prime\prime}$–60$^{\prime\prime}$; some examples are shown in Figure \[HIfig\]. These data allow us to probe the internal kinematics of local LCBG revealing signatures of interactions and give us a more robust measure of their dynamical masses. These data can also reveal optically-faint, but gas-rich companions that may be responsible for triggering star formation in LCBGs. Many optically-isolated LCBGs have gas-rich companions and others have signatures of recent interactions implying a much higher incidence of interactions than is derived solely from optical data.
While many LCBGs have quiescent, rotating disks, those shown in Figure \[HIfig\] have velocity dispersions comparable to the rotation velocities of the galaxy. This is similar to what is seen in high redshift star-forming galaxies [@shapiro08] and is consistent with the formation of a dynamically hot component, such as a bulge or thick disk, as a result of a minor merger. Similar features are also seen in the optical velocity fields of the ionized gas in local LCBGs [@perez-gallego09; @perez-gallego09b]. Although a number of LCBGs have disturbed velocity fields, their [HI]{} linewidths still trace the gravitational potential of the galaxy, therefore the derived dynamical masses are still reasonably estimates of the halo mass. The masses combined with the ongoing spheroid formation is consistent with star formation being quenched by virial heating in local LCBGs [@cattaneo06].
Conclusions
===========
LCBGs reside at the high mass tip of the blue sequence and are poised to evolve onto the red sequence in the near future. In addition to their diverse optical properties, our observations of local LCBGs reveal a diverse range of [HI]{} properties. However, the vast majority of LCBGs are [HI]{}-rich, have moderate M$_{dyn}$, and have short gas depletion timescales. LCBGs have properties consistent with evolving into low mass spiral galaxies or high mass dwarf elliptical or dwarf irregular galaxies. As such, we believe that LCBGs are a common stage in the evolution of galaxies and not a distinct class of galaxy. Our resolved maps of local LCBGs reveal that some have signatures of ongoing bulge or thick disk formation, probably due to a recent minor merger. Combined with the higher incidence of companions compared to field galaxies, this implies that the star formation in LCBGs is triggered by interactions. Finally, optical spectroscopy [@perez-gallego09; @perez-gallego09b] reveals that of 22 LCBGs studied only one has a low-luminosity AGN and only six have signatures of supernova-driven winds. The high velocity dispersions and large halo masses seen in these galaxies, combined with the spectroscopic results, are consistent with virial heating being the dominant quenching mechanism in local LCBGs [@cattaneo06].
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---
abstract: 'Weyl semimetals are a class of topological materials that exhibit a bulk Hall effect due to time-reversal symmetry breaking. We show that for the idealized semi-infinite case, the Casimir force between two identical Weyl semimetals is repulsive at short range and attractive at long range. Considering plates of finite thickness, we can reduce the size of the long-range attraction even making it repulsive for all distances when thin enough. In the thin-film limit, we study the appearance of an attractive Casimir force at shorter distances due to the longitudinal conductivity. Magnetic field, thickness, and chemical potential provide tunable nobs for this effect, controlling the Casimir force: whether it is attractive or repulsive, the magnitude of the effect, and the positions and existence of a trap and antitrap.'
author:
- 'Justin H. Wilson'
- 'Andrew A. Allocca'
- Victor Galitski
bibliography:
- 'arxivreferences.bib'
title: Repulsive Casimir force between Weyl semimetals
---
In 1948, Casimir [@Casimir1948b] showed that quantum fluctuations in the electromagnetic field cause a force between two perfectly conducting, electrically neutral objects. This has since been extended to other materials [@Lifshitz:1956vv; @Bordag:2001tt]. Throughout this time, Casimir repulsion between two materials in vacuum has been a long sought after phenomenon [@Dzyaloshinskii1961; @Boyer1974]. There are principally four categories in which repulsion can be achieved: (i) modifying the dielectric of the intervening medium [@Dzyaloshinskii1961; @Kenneth2002; @Munday2009], (ii) pairing a dielectric object and a permeable object [@Boyer1974] (such as with metamaterials [@Leonhardt2007; @*Rosa2009]), (iii) using different geometries [@Boyer1968; @Fulling2007; @Levin2010], and (iv) breaking time-reversal symmetry [@Tse2012; @Grushin2011]. In this paper, we are concerned with Casimir repulsion in identical time-reversal broken systems. Specifically, we will study how Weyl semimetals with time-reversal symmetry breaking can exhibit Casimir repulsion. The key ingredient to Casimir repulsion in this paper is the existence of a nonzero bulk Hall conductance $\sigma_{xy}\neq 0$, $\sigma_{xy}=-\sigma_{yx}$ [@Hosur2013].
It is a general theorem that mirror symmetric objects without time-reversal symmetry breaking can only attract one another with the Casimir effect [@Kenneth2006]. This is understood with the Lifshitz formula [@Lifshitz:1956vv] where if we have two materials characterized by the two reflection matrices $R_1$ and $R_2$ and separated by a distance $a$ in a parallel plate geometry, we have $$\begin{aligned}
E_c(a) = \hbar\int \, \frac{d^2 k}{(2\pi)^2} \int \frac{d\omega}{2\pi}\, \operatorname{tr}\log [\mathbb I - R_1 R_2 e^{-2 q_z a}],\label{eq:Lifshitz}\end{aligned}$$ where the trace is a matrix trace and $q_z = \sqrt{\omega^2 + k^2}$. This integral generally yields an attractive force; however, if we break time reversal symmetry, obtaining antisymmetric off-diagonal terms in the reflection matrix $R_{xy}=-R_{yx}$ there is the possibility of Casimir repulsion [@Klimchitskaya2009]. One candidate is a two-dimensional Hall material [@Tse2012], and similarly, another is a topological insulator where the surface states have been gapped by a magnetic field [@Grushin2011; @Rodriguez-Lopez2014]. A Hall conductance does not guarantee repulsion; longitudinal conductance can overwhelm any repulsion from the Hall effect (although the magnetic field can lead to interesting transitions [@Allocca2014]), and a Hall effect that is too strong can suppress Kerr rotation and hence lead to attraction. The latter case is an interesting phenomenon where “more” of a repulsive material can lead to attraction.
![The setup we will consider here is two Weyl semimetals separated by a distance $a$ in vacuum and with distance between Weyl cones $2\mathbf b$ in $k$ space (split in the $z$ direction). \[fig:Picture-of-setup\]](plots/Weylfinal){width="6cm"}
The material we are interested in is marginal in both the case of longitudinal conductance and in an overwhelming Hall effect: Weyl semimetals [@Hosur2013] with the Casimir setup seen in Fig. \[fig:Picture-of-setup\] and the resulting normalized Casimir pressure seen in Figs. \[fig:SemiInfiniteBulkHall\]–\[fig:thinfilmresults\]. These materials have linearly dispersive band structures characterized by Weyl nodes with different chiralities and characterized by a chiral anomaly [@Aji2012]. Clean Weyl semimetals at zero temperature have a zero dc longitudinal conductivity and optical conductivity $\operatorname{Re}[\sigma_{xx}]\propto \omega$ [@Hosur2012]. Additionally, they exhibit a bulk Hall effect exemplified in the dc limit by an axionic field theory [@Zyuzin2012] where in addition to the Maxwell action, we have $$\begin{aligned}
\label{eq:axionic-action}
S_{A} = \frac{e^2}{32 \pi^2\hbar c} \int d^3 r\, dt \, \theta(\mathbf r, t) \epsilon^{\mu \nu \alpha \beta} F_{\mu \nu} F_{\alpha \beta},\end{aligned}$$ where $\theta(\mathbf r, t) = 2\mathbf b \cdot \mathbf r - 2 b_0 t$ and $2 \mathbf b$ is the distance between Weyl nodes in $\mathbf k$ space whereas $2 b_0$ is their energy offset, $e$ is the charge of an electron, $\hbar$ is Planck’s constant, $c$ is the speed of light, $F_{\mu \nu}$ is the electromagnetic field strength tensor, and $\epsilon^{\mu \nu \alpha \beta}$ is the fully antisymmetric four-tensor. Inversion symmetry breaking Weyl semimetals, on the other hand, do not exhibit a dc Hall effect [@Halasz2012] and therefore will not see the effects described in this paper. The electrodynamics of this were investigated in Ref. [@Grushin2012] where the authors even comment on the possibility for a repulsive Casimir effect.
The marginal nature of Weyl semimetals makes them prime candidates for tuning the Casimir force between attractive and repulsive regimes. In constructed Weyl semimetals made of heterostructures of normal and topological insulators [@Burkov2011] an external magnetic field can control the Hall effect [@Chen2013a] and hence the repulsive effects. Additionally, some of the first materials that have been predicted were pyrochlore iridates [@Wan2011; @*Chen2012; @*Witczak-Krempa2012]; these could also see a repulsion tunable with carrier doping or an additional magnetic field.
In a real material and experiment at finite temperatures, disorder and interactions should be taken into account, and in Weyl semimetals they lead to a finite dc conductivity [@Hosur2012; @Burkov2011a; @Burkov2011]. We simulate this effect in the latter part of this paper by raising the chemical potential in our clean system, leading to intraband transitions that contribute to the longitudinal conductivity (in the dc limit these are singular contributions). To begin, we consider two semi-infinite slabs of a Weyl semimetal ($z<0$ and $z>a$ to be precise), neglecting all frequency dependence to the conductivities by assuming the electromagnetic response is captured by Eq. . The result is just a material that is solely a bulk Hall material with current responses given by the Hall conductivity $\sigma_{xy} = e^2 b/2\pi^2 \hbar$. This response can be encoded in the dielectric function so that $\epsilon_{xx}=\epsilon_{yy}=\epsilon_{zz}=1$ and $\epsilon_{xy}=-\epsilon_{yx}=i\sigma_{xy}/\omega$. With this set up, if an incident wave $\mathbf k$ hits such a material it will break up into two different polarizations in the material $\mathbf k^{\pm}$ that satisfies $k_x^\pm=k_x$, $k_y^\pm=k_y$, and $(k_z^\pm)^2 = k_z(k_z\pm \sigma_{xy}/c)$. Additionally, the two elliptical polarizations $\mathbf D_\pm = \epsilon(\omega) \mathbf E_\pm$ are $\mathbf D_\pm \propto \tfrac{\omega}{c k^{\pm}} (k_z \pm \sigma_{xy}/c) \mathbf{e}_1 \mp i k_z^\pm \mathbf{e}_2$ where $\mathbf{e}_2$ is perpendicular to the plane of incidence and $\mathbf{e}_1= \mathbf{e}_2 \times \mathbf{k}^\pm$. Notice that for $k_z<\sigma_{xy}/c$, one of the polarizations is evanescent.
The incident and reflected polarizations can be broken up into transverse electric (TE) and transverse magnetic (TM) modes, and the reflection matrix $R(\omega,\mathbf k)$ just connects incident to reflected $(E^{\mathrm{TM}}_r, E^{\mathrm{TE}}_r)^T = R(\omega, \mathbf k)(E^{\mathrm{TM}}_0, E^{\mathrm{TE}}_0)^T$. As shown in the Lifshitz formula Eq. , we need the imaginary frequency reflection matrix. If we let $\omega\rightarrow i \omega$ and define $q_z^2 = \omega^2/c^2+k_x^2+k_y^2$, the reflection matrix for a semi-infinite slab of this bulk Hall material is $$\begin{aligned}
R_\infty(i q_z) = \frac1{ \sigma_{xy}/c}\begin{pmatrix}
Q_- - \sigma_{xy}/c & - Q_+ +2 q_z \\
Q_+ -2 q_z & Q_- - \sigma_{xy}/c
\end{pmatrix},\end{aligned}$$ where we have defined for brevity $Q_{\pm} = \sqrt{2q_z(\sqrt{q_z^2+\sigma_{xy}^2/c^2} \pm q_z)}$ (the real frequency version of $R_\infty$ is found in the Supplemental Material [@supplement]). Inspecting $R_\infty(i q_z)$, we see that the reflection matrix only depends on the ratio $c q_z/\sigma_{xy}$. This dependence has implications for the Casimir force. After changing variables to solely $q_z$, we can inspect the Casimir pressure $P_c(a) = - \partial E_c/\partial a$, and we have an expression $P_c = \frac{2 \hbar c}{(2\pi)^2} \int d q_z\, q_z^3 \; g\!\left[\tfrac{q_z}{\sigma_{xy}/c}, 2 q_z a \right], \label{eq:semi-inf-force}$ with a function $g(\tfrac{q_z}{\sigma_{xy}/c}, 2 q_z a)$ written out in the Supplemental Material [@supplement] for completeness. If we then change variables to $x = 2 a q_z$ and normalize by Casimir’s original result for perfect conductors $P_0 = - \frac{\hbar c\pi^2}{240 a^4}$ [@Casimir1948b], we can write the equation for the pressure as $P_c/P_0 = f(\sigma_{xy} a/c)$.
![(Color online) The normalized Casimir force between two semi-infinite bulk Hall materials. Repulsion is seen for $\sigma_{xy} a/c \lesssim 4.00$. $P_0$ is the distant-dependent ideal Casimir force [@Casimir1948b]. For $\sigma_{xy}a/c\rightarrow\infty$, $P_c/P_0\rightarrow 1$.[]{data-label="fig:SemiInfiniteBulkHall"}](plots/SemiInfiniteBulkHall.pdf){width="8.6cm"}
With this formulation, we plot normalized force $P_c/P_0$ as a function of $\sigma_{xy} a /c$ obtaining the single function seen in Fig. \[fig:SemiInfiniteBulkHall\]. We see that for $\sigma_{xy} a/c \lesssim 4.00$ we obtain repulsion whereas for $\sigma_{xy} a/c \gtrsim 4.00$ we obtain attraction. Thus, these similar materials trap each other at a fixed distance simply dependent on the Hall conductivity, $a_{\mathrm{Trap}} \approx \frac{4.00}{\sigma_{xy}/c}$. If we insert the value of $\sigma_{xy}=e^2 b/2\pi^2\hbar$ into this expression, we find $a_{\mathrm{Trap}} \approx 860/b$. This means that if $1/b\sim O(\unit{nm})$, then $a_{\mathrm{Trap}}\sim O(\unit{\mu m})$ quite reasonable.
As the distance between the materials gets long, $P_c/P_0 \rightarrow 1$. This behavior is markedly different from the thin film Hall case obtained by Tse and MacDonald in Ref. [@Tse2012] and Rodriguez-Lopez and Grushin Ref. [@Rodriguez-Lopez2014]. They found a small \[two-dimensional (2D)\] quantum Hall effect implies a quantized and repulsive Casimir force at long distances. In our case, we get attraction at long distances for a bulk Hall material independent of the magnitude of the Hall effect. To resolve this seeming inconsistency, imagine a finite thickness of the bulk Hall material of thickness $d$, then the two-dimensional conductivity $\sigma_{xy} d$ diverges as $d\rightarrow\infty$, and in the case of a 2D quantum Hall plate with infinite Hall conductivity, the Casimir effect is attractive and approaches $P_0$.
To make this argument more precise, one can actually find the reflection matrix for a bulk Hall system of thickness $d$ and the result is (derivation of $R_d$ depends only on the axionic action Eq. and can be found in the Supplemental Material [@supplement], calculated for real frequencies) $$\begin{aligned}
R_d(i q_z) & = \begin{pmatrix}
R_{xx} & R_{xy} \\ - R_{xy} & R_{xx}
\end{pmatrix},\end{aligned}$$ with
$$\begin{aligned}
R_{xx} & = -\tfrac12\tfrac{\sigma_{xy}}{c}( Q_- \sinh Q_+ d +\tfrac{\sigma_{xy}}{c}\cosh Q_+ d - Q_+ \sin Q_- d -\tfrac{\sigma_{xy}}{c}\cos Q_- d)/D, \\
R_{xy} & = -\tfrac12 \tfrac{\sigma_{xy}}{c}( Q_+ \sinh Q_+ d + 2 q_z \cosh Q_+ d - Q_- \sin Q_- d - 2q_z \cos Q_- d)/D,\end{aligned}$$
where $$\begin{aligned}
D = (Q_+^2 +\tfrac12\tfrac{\sigma_{xy}^2}{c^2}) \cosh Q_+ d + (2 q_z Q_+ + \tfrac{\sigma_{xy}}{c} Q_-) \sinh Q_+ d + (Q_-^2 - \tfrac12 \tfrac{\sigma_{xy}^2}{c^2}) \cos Q_- d + (2q_z Q_- - \tfrac{\sigma_{xy}}{c} Q_+)\sin Q_- d.\end{aligned}$$
It can be shown that $\lim_{d\rightarrow \infty} R_d(iq_z) = R_\infty(i q_z)$.
Similarly, in the limit of $d \rightarrow 0$, if we keep $\sigma_{xy}^{2\mathrm D} = \sigma_{xy} d$ constant, we obtain what was found in Ref. [@Tse2012], $$\begin{aligned}
\lim_{d\rightarrow 0} R_d(i q_z) = \frac1{1 + (\sigma_{xy}^{2\mathrm D}/2c)^2 } \begin{pmatrix}
-(\sigma_{xy}^{2\mathrm D}/2c)^2 & -\sigma_{xy}^{2\mathrm D}/2c \\
\sigma_{xy}^{2\mathrm D}/2c & -(\sigma_{xy}^{2\mathrm D}/2c)^2
\end{pmatrix}.\end{aligned}$$ For the rest of our discussion, define $R_0(i q_z) = \lim_{d\rightarrow 0} R_d(i q_z)$ with $\sigma_{xy}^{2\mathrm D} = \sigma_{xy} d$ held constant.
With the correct limits identified, we first notice that we can write $R_d$ as a function of only two variables $R_d = R_d(c q_z/\sigma_{xy}, \sigma_{xy}^{2\mathrm D}/c)$. Thus, by similar arguments to what we had for the semi-infinite case, the Casimir pressure $P_c = P_0 f(\sigma_{xy} a/c, \sigma_{xy}^{2\mathrm D}/c)$.
The limiting cases can be understood now by considering first Eq. . The exponential constrains $q_z \sim 1/a$ and since technically the thin-film limit is $\lim_{q_z d\rightarrow 0} R_d(i q_z) = R_0(i q_z)$, we have that $d/a \rightarrow 0$. In other words, the thin film limit is applicable when we are considering $d \ll a$. The opposite limit is just when $q_z d \rightarrow \infty$, and by similar arguments, that means $d \gg a$ is when the semi-infinite case applies. Both limits leave $\sigma_{xy} a/c$ and $\sigma_{xy} d/c$ unaffected (although in the thin film case $\sigma_{xy} a$ drops out whereas in the semi-infinite case $\sigma_{xy} d\rightarrow \infty$ has the same limit as $q_z d \rightarrow \infty$).
The thin film limit can be evaluated exactly [@Tse2012] and has the form $P_c^{\mathrm{TF}} = P_0 \frac{90}{\pi^4} \operatorname{Re}\{ \operatorname{Li}_4\left[(\sigma_{xy}^{2\mathrm D}/c)^2/(\sigma_{xy}^{2\mathrm D}/c+2 i)^2 \right]\}$ where $\operatorname{Li}_4$ is the polylogarithm of degree 4. Note that this function has a minimum value of $P_c^{\mathrm{TF}} \approx -0.117 P_0$ representing how repulsive we can get. For large enough $\sigma_{xy}^{2\mathrm D}/c$, the force does become attractive—corresponding roughly to when $(\sigma_{xy}^{2\mathrm D}/c)^2>\sigma_{xy}^{2\mathrm D}/c$ (i.e. when Kerr rotation is suppressed).
![(Color online) A plot of the normalized Casimir force for various thicknesses of a bulk Hall material (idealized Weyl semimetal). It begins slightly repulsive for small $\sigma_{xy} d/c$, and as this increases, it becomes more repulsive until it reaches the maximum for a thin film material (the dashed line) at which point it increases to the semi-infinite limit. $P_0 = - \frac{\hbar c\pi^2}{240 a^4}$, and $\sigma_{xy}=e^2 b/2\pi^2 \hbar$ is the bulk Hall response. Figure \[fig:thinfilmresults\] takes into account more material properties. \[fig:ThickBulkHallmaterial\]](plots/ThickBulkHallMaterial.pdf){width="8.6cm"}
The cross-over between these regimes can be seen in Fig. \[fig:ThickBulkHallmaterial\]. As $\sigma_{xy}d/c$ is increased, the Casimir energy approaches the semi-infinite case. However, for any finite $d$, each curve asymptotically approaches its thin-film value (and never goes lower than the minimum value represented by the dashed horizontal line in Fig. \[fig:ThickBulkHallmaterial\]). This not only clearly connects our case to the previously known thin-film result, but also provides a theoretical justification for considering a thin-film limit $d\ll a$ with a two-dimensional conductance $\sigma_{\mu\nu} d$.
Until now the plates have been idealized. Using the thin-film limit illustrated above as a reference allows us to easily consider some of the effects of taking into account the full frequency response of the material. Thus, we pick a $\sigma_{xy} d$ that is reasonably in the repulsive regime (for all distances) in order to analyze the effects of including some of the lowest-order frequency dependence into the conductivities. We will mainly be interested in the effects of virtual vacuum transitions that are low in energy, which correspond to plates that are far apart from one another. Thus, we will use the low-energy chiral Hamiltonian for a pair of Weyl nodes, $$\begin{aligned}
H_W = \pm \hbar v_{\mathrm F} \bm \sigma \cdot (\mathbf k \pm \mathbf b),\end{aligned}$$ where $v_{\mathrm F}$ is the Fermi velocity and $\mathbf b$ is the position of the of Weyl node in $\mathbf k$ space. The exact band structure will be important as the plates get closer although weighting will still be larger on the lower-energy modes.
To the conductivities, we fix $k_z$ and calculate two-dimensional conductivities using the Kubo-Greenwood formulation (see the Supplemental Material [@supplement] for details) then integrate the resulting expression over $k_z$ with a symmetric cutoff $\sigma_{\mu \nu}(i \omega) = \int_{-\Lambda}^\Lambda \frac{d k_z}{2 \pi} \tilde\sigma_{\mu \nu}(i \omega;k_z)$ [@Goswami2013] where $\tilde\sigma_{\mu\nu}(i\omega;k_z)$ is the two-dimensional conductivity with $k_z$ fixed. We evaluate this at an imaginary frequency to aid the Casimir calculations.
We perform this procedure at finite chemical potential $\mu$ and throw out terms that go to zero when the cutoff $\Lambda\rightarrow \infty$. This yields the conductivities, $$\begin{gathered}
\sigma_{xx}(i\omega) = \frac{e^2}{12 \pi^2 \hbar v_{\mathrm F}} \left[ \tfrac 53 \omega + 2 \omega \log \left( \tfrac{v_{\mathrm F} \Lambda}{\omega} \right) \right. \\ \left.+ 4 \tfrac{\mu^2}{\hbar^2\omega} - \omega \log \left(1 + \tfrac{4 \mu^2}{\hbar^2\omega^2} \right)\right],\end{gathered}$$ and $\sigma_{xy}(i \omega) = \frac{e^2 b}{2 \pi^2 \hbar}$ is unchanged at this order. Due to the linear dispersion of the Weyl nodes, we have a logarithmic cutoff dependence. Note that rotating to real frequencies we get the correct result for two Weyl nodes for $\operatorname{Re}[\sigma_{xx}(\omega)]$ [@Hosur2012], and a result with the appropriate logarithmic divergence for $\operatorname{Im}[\sigma_{xx}(\omega)]$ [@Rosenstein2013]. This can be understood in terms of charge renormalization due to the band structure, but for ease of our purposes we let $\Lambda \sim 1/a_0$ where $a_0$ is the lattice spacing. For our plots we choose a lattice spacing of $a_0=\unit[1]{nm}$, a thickness of $d=\unit[20]{nm}$, $b=0.3 (2\pi/a_0)$, $\Lambda=2\pi/a_0$, $v_{\mathrm{F}} = \unit[6\times 10^5]{m/s}$, and $\mu=0$ unless its the parameter we are varying.
Now, one can use one of two equivalent ways of calculating the Casimir energy: the reflection matrix as given in Ref. [@Tse2012], or using a microscopic analysis to find the photon dressed RPA current-current correlators [@supplement]. In order to avoid an unphysical negative $\sigma_{xx}(i\omega)$ as well as for consistency, we cutoff the photon energies in the Lifshitz formula to run from $0$ to $\Lambda$—an approximation valid for $a\gg \frac{c}{v_{\mathrm{F}}} \Lambda^{-1}$.
First, we see that we get an anti-trap for these at approximately $\unit[650]{nm}$, and if we increase $b$ as in Fig. \[fig:WeylVaryb\] (with, say, an applied magnetic field), it not only moves closer to zero separation, but the overall repulsive behavior can be enhanced. On the other hand, if we increase $v_{\mathrm F}$ as we see in Fig. \[fig:WeylVaryvf\], we see the region of attraction is suppressed, but the overall repulsive behavior at long distances is maintained. Modifying $\Lambda$ will have effects similar to modifying $v_{\mathrm F}$, but since it appears logarithmically, it would need to change by orders of magnitude to give appreciable changes (a simple plot for this is provided in the Supplemental Material [@supplement] but is not relevant for the discussion here). This antitrap effect is occurring at short distances when higher order band effects also play a role, but any other effects will contribute to the longitudinal conductivity in such a way that an anti-trap will appear.
Interestingly, when we introduce a finite chemical potential as we see in Fig. \[fig:WeylVarymu\], in addition to the anti-trap we get at shorter distances, we start to see a trap at much longer distances appear. This is not surprising since at zero frequency there is a divergent longitudinal conductivity. Thus, we know that at long distances, the Casimir force must be attractive, but by modifying the Hall effect, we have an intermediate regime of repulsion. A similar effect would occur if we took finite temperatures or disorder corrections to the longitudinal conductivities.
Considering the form of the conductance in terms of the fine-structure constant $\sigma_{xy}d/c = \alpha \frac{2bd}\pi$, we see that $b d$ controls the strength of the repulsion in the thin-film limit. Without longitudinal conductance, the repulsive regime roughly corresponds to when $(\sigma_{xy}d/c)^2 \lesssim \sigma_{xy}d/c$ or equivalently $ \frac{2bd}\pi\lesssim \frac1{\alpha}$. The longitudinal conductance introduces $v_\mathrm{F}$ into the scheme, relevant photons have $\omega \approx c/a$, and thus it becomes important for $\sigma_{xx} d/c \sim \alpha \frac{c}{v_\mathrm{F}} \frac{d}{a} \gtrsim O(1)$ (neglecting constants) which both emphasizes that $v_\mathrm{F}$ controls the longitudinal conductance’s contribution to the Casimir effect and that the term is suppressed at longer distances.
We have shown here how Weyl semimetals can exhibit a tunable repulsive Casimir force (with, for instance, magnetic-field tuning $\mathbf b$) and how it can depend on the thickness of the material. In the thin-film limit, we showed how the semimetallic nature of these materials can work to create attraction at shorter distance scales and how a finite longitudinal conductivity will create long-distance attraction along with repulsion at intermediate distances. Recently the first experimental observation of Weyl semimetals [@Huang2015; @Zhang2015] provided optimism that these theoretical materials could be a reality. The marginal nature of these materials could be useful for controlling the Casimir force between attractive and repulsive regimes.
This work was supported by the DOE-BES (Grant No. DESC0001911) (A.A.A. and V.G.), the JQI-PFC (J.H.W.), and the Simons Foundation. We thank Liang Wu and Mehdi Kargarian for discussions.
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Supplementary materials {#supplementary-materials .unnumbered}
=======================
Axionic Electrodynamics
=======================
In the main text, we mention the axionic term which appears in the action alongside the usual Maxwell action $$\begin{aligned}
\label{eq:axionic-actionsup}
S_{A} = \frac{e^2}{32 \pi^2\hbar c} \int d^3 r\, dt \, \theta(\mathbf r, t) \epsilon^{\mu \nu \alpha \beta} F_{\mu \nu} F_{\alpha \beta},\end{aligned}$$ where $e$ is the electric charge, $\hbar$ is Planck’s constant, $c$ is the speed of light, $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the electromagnetic field tensor, $\epsilon^{\mu \nu \alpha \beta}$ is the fully antisymmetric 4-tensor, and $\theta(\mathbf r, t) = 2 \mathbf b \cdot \mathbf r - 2 b_0 t$ is the axionic field.
For our purposes, we will set $\mathbf b = b \hat{\mathbf z}$ and $b_0 = 0$. If we only apply $S_{\text{A}}$ for $z>0$, there is no resulting surface current (i.e. this is the surface without Fermi arcs), and the current response is $$\begin{aligned}
j^x(\mathbf r) & = \frac{e^2 b}{2 \pi^2} E^y(\mathbf r), \\
j^y(\mathbf r) & = - \frac{e^2 b}{2 \pi^2} E^x(\mathbf r).\end{aligned}$$
Now, we solve Maxwell’s equations in the bulk Hall system after taking the Fourier transform $$\begin{aligned}
\mathbf k \cdot \mathbf E & = -i \sigma_{xy} \hat{\mathbf z} \cdot \mathbf B, \\
\mathbf k \cdot \mathbf B & = \omega \mathbf B, \\
\mathbf k \times \mathbf E & = \omega \mathbf B, \\
\mathbf k \times \mathbf B & = i \sigma_{xy} \hat{\mathbf z} \times \mathbf E - \omega \mathbf E.\end{aligned}$$ One can define a frequency-dependent dielectric permitivity $\epsilon(\omega)$ to be $$\begin{aligned}
\epsilon(\omega) & = \begin{pmatrix}
1 & i \sigma_{xy}/\omega & 0 \\
- i \sigma_{xy}/\omega & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}\end{aligned}$$ in which case, Maxwell’s equations can be recast as a single equation for the electric displacement $\mathbf D = \epsilon(\omega) \mathbf E$, $$\begin{aligned}
\label{eq:Maxwell-displacement-fieldsup}
[\mathbf k \otimes \mathbf k - k^2 \mathbb I] \epsilon^{-1}(\omega) \mathbf D = - \omega^2 \mathbf D.\end{aligned}$$ The determinant of this matrix equation yields the frequencies that a wave vector $\mathbf k$ can have. In our case, we obtain $$\begin{aligned}
\label{eq:Bulk-weyl-dispersionsup}
\omega_{\pm}^2 = k^2 + \tfrac12 \sigma_{xy}^2 \pm \sigma_{xy} \sqrt{ k_z^2 + \tfrac14 \sigma_{xy}^2}.\end{aligned}$$ There is a polarization associated with each of these frequencies which we can obtain from Eq. using $\mathbf k \cdot \mathbf D = 0$ (no free charge). We choose $\mathbf e_1 = \hat{\mathbf y} \times \hat{\mathbf k}$ and $\mathbf e_2= \hat{\mathbf y}$ as our basis for the polarizations (assuming $k_y = 0$ without loss of generality). The resulting (unnormalized) polarizations are $$\begin{aligned}
\mathbf D_{1,2} = \frac{\omega_{\pm}}{k}\left(\sqrt{k_z^2 + \tfrac14 \sigma_{xy}^2} \mp \tfrac12 \sigma_{xy} \right) \mathbf e_1 \pm i k_z \mathbf e_2.\end{aligned}$$ Notice that these polarizations are elliptical.
To find the reflection of a wave off a half space filled with this material, $k_x$, $k_y$, and $\omega$ must remain the same on either side of the material, but $k_z$ can change, and matching both sides of the dispersion Eq. , we obtain simply that for an incident wave with wave-vector $\mathbf q = (q_x, q_y, q_z)$, the transmitted wave has $$\begin{aligned}
(k_z^{\pm})^2 = q_z(q_z \pm \sigma_{xy}). \label{eq:kz-transmitsup}\end{aligned}$$ Each of these can be associated with a (unnormalized) polarization as (assuming $q_y = 0$ without loss of generality) $$\begin{aligned}
\mathbf D_{\pm}= \tfrac{\omega}{k^{\pm}} (q_z \pm \sigma_{xy}) \mathbf{e}_1 \mp i k_z^\pm \mathbf{e}_2.\end{aligned}$$ At this point, we note some interesting electromagnetic properties appearing here. Eq. implies that this material is birefringent, and for an incident wave (at any angle) with $q_z< |\sigma_{xy}|$, only one (elliptical) polarization even propagates into the material while the other is an evanescent wave – independent of the angle of incidence.
Reflection coefficient of semi-infinite bulk
--------------------------------------------
Now, to obtain the reflection matrix, we call our incident wave $\mathbf E_0$ with wave-vector $\mathbf q$, our reflected wave $\mathbf E_r$ with wave-vector $\mathbf q_r =(q_x,q_y,-q_z)$, as well as $\mathbf E_{\pm} = \epsilon^{-1}(\omega) \mathbf D_\pm$ with wave vectors $\mathbf k_\pm$ for the two polarizations it is transmitted into. The relevant Maxwell equations at the interface between vacuum and the bulk Hall material are then given by $$\begin{aligned}
\begin{split}
(\mathbf E_0 + \mathbf E_r - \mathbf E_+ - \mathbf E_-) \times \hat{\mathbf z} & = 0, \\
(\mathbf q \times \mathbf E_0 + \mathbf q_r \times \mathbf E_r - \mathbf k_+ \times \mathbf E_+ - \mathbf k_- \times \mathbf E_-) \times \hat{\mathbf z} & = 0.
\end{split} \label{eq:matching-conditionssup}\end{aligned}$$ We can break up the polarization of the incident and reflected waves into transverse electric (TE) and transverse magnetic (TM), and the reflection matrix is defined such that $$\begin{aligned}
\begin{pmatrix}
E^{\mathrm{TM}}_r \\ E^{\mathrm{TE}}_r
\end{pmatrix} & = R(\omega, \mathbf q)\begin{pmatrix}
E^{\mathrm{TM}}_0 \\ E^{\mathrm{TE}}_0
\end{pmatrix}.\end{aligned}$$ Solving for this matrix, we obtain $$\begin{aligned}
R_\infty(\omega, \mathbf q) = \frac1{ \sigma_{xy}} \begin{pmatrix}
\sigma_{xy} + k_z^- - k_z^+ & i (2 q_z - k_z^- - k_z^+) \\
-i (2 q_z - k_z^- - k_z^+) & \sigma_{xy} + k_z^- - k_z^+
\end{pmatrix}.\end{aligned}$$ This can then be rotated to imaginary frequencies and the result is in the main text.
Reflection coefficient of thickness $d$ sample
----------------------------------------------
If we have a material of thickness $d$, then we need additional matching conditions due to Maxwell’s equations at the other interface. This requires restricting our action Eq. to $0<z<d$. To solve this, we just need to add another set of matching conditions. In addition to the incident $\mathbf E_0$ and reflected $\mathbf E_r$ waves, we now have forward moving Weyl polarizations $\mathbf E_\pm^{\uparrow}$ with $\mathbf k_\pm^\uparrow = (k_x,k_y,k_z^\pm)$, backwards moving Weyl polarizations $\mathbf E_\pm^{\downarrow}$ with $\mathbf k_\pm^\downarrow = (k_x,k_y, -k_z^\pm)$, and a transmitted wave $\mathbf E_t$ with wave-vector the same as the transmitted $\mathbf k$.
The resulting matching conditions are $$\begin{aligned}
\begin{split}
(\mathbf E_0 + \mathbf E_r - \mathbf E^\uparrow_+ - \mathbf E^\uparrow_- - \mathbf E^\downarrow_+ - \mathbf E^\downarrow_- )\times \hat{\mathbf z} & = 0, \\
(\mathbf k \times \mathbf E_0 + \mathbf k_r \times \mathbf E_r - \mathbf k^\uparrow_+ \times \mathbf E^\uparrow_+ - \mathbf k^\uparrow_- \times \mathbf E^\uparrow_- - \mathbf k^\downarrow_+ \times \mathbf E^\downarrow_+ - \mathbf k^\downarrow_- \times \mathbf E^\downarrow_-)\times \hat{\mathbf z} & = 0, \\
(\mathbf E^\uparrow_+ e^{i k_z^+ d} + \mathbf E^\uparrow_- e^{i k_z^- d}+ \mathbf E^\downarrow_+ e^{-i k_z^+ d} + \mathbf E^\downarrow_- e^{-i k_z^- d} - \mathbf E_t e^{i k_z d})\times \hat{\mathbf z} & = 0,\\
(\mathbf k_+^\uparrow \times \mathbf E^\uparrow_+ e^{i k_z^+ d} + \mathbf k_-^\uparrow \times \mathbf E^\uparrow_- e^{i k_z^- d} \phantom{+ \mathbf k_-^\downarrow \times \mathbf E^\downarrow_- e^{-i k_z^- d} - \mathbf k \times \mathbf E_t e^{i k_z d})\times \hat{\mathbf z}} \\+ \mathbf k_+^\downarrow \times \mathbf E^\downarrow_+ e^{-i k_z^+ d} + \mathbf k_-^\downarrow \times \mathbf E^\downarrow_- e^{-i k_z^- d} - \mathbf k \times \mathbf E_t e^{i k_z d})\times \hat{\mathbf z} & = 0.
\end{split}\end{aligned}$$ These equations can still be solved and the result is $$\begin{aligned}
R_d(\omega, \mathbf q) = \begin{pmatrix}
R_{xx} & R_{xy} \\ -R_{xy} & R_{xx}
\end{pmatrix},\end{aligned}$$ where $$\begin{aligned}
R_{xx} & = \sigma_{xy} \{\sin (k_z^- d) [i k_z^+ \cos (k_z^+ d)+\sigma_{xy} \sin (d
k_z^+)]-i k_z^- \cos (k_z^- d) \sin (k_z^+ d)\}/D, \\
R_{xy} & = \sigma_{xy}
\{k_z^- \cos (k_z^- d) \sin (k_z^+ d)+\sin (k_z^- d) [k_z^+
\cos (k_z^+ d)-2 i k_z \sin (k_z^+ d)]\}/D, \\
D = & [2 i k_z^-
\cos (k_z^- d)+(2 k_z-\sigma_{xy} ) \sin (k_z^- d)] [2 i k_z^+ \cos
(k_z^+ d)+(2 k_z+\sigma_{xy} ) \sin (k_z^+ d)].\end{aligned}$$ And again, this can be rotated to imaginary frequencies to obtain the result in the main text.
And as stated in the text, the various limits (semi-infinite to thin film limits) apply to this reflection matrix.
Calculating the conductivities
==============================
In order to calculate the conductivities in the clean limit we consider the Hamiltonian near a Weyl node $$\begin{aligned}
H_W = \pm \hbar v_{\mathrm{F}} \bm \sigma \cdot (\mathbf k \pm \mathbf b).\end{aligned}$$ For simplicity we set $\hbar = 1 = v_{\mathrm{F}}$ unless otherwise specified. We consider a pair of these nodes, calculating the quantities separately for each node and adding them together (which will just introduce a factor of two for both $\sigma_{xx}$ and $\sigma_{xy}$).
To find the conductivities, we find a complete basis of states which are easily found by diagonalizing the two-by-two matrix $H_W$ in momentum space (label them $\ket{f_{k\pm}}$). The relevant matrix elements are then (choosing the negative sign for the Hamiltonian) $$\begin{aligned}
\braket{f_{k+}|\sigma_x|f_{k-}} & = \frac{(k_z - b)k_x + i \epsilon k_y}{\epsilon \sqrt{\epsilon^2 - (k_z-b)^2} }, \\
\braket{f_{k+}|\sigma_y|f_{k-}} & = \frac{(k_z - b)k_y - i \epsilon k_x}{\epsilon \sqrt{\epsilon^2 - (k_z-b)^2} }, \\
\braket{f_{k\pm}|\sigma_x|f_{k\pm}} & = \pm \frac{k_x}{\epsilon},\end{aligned}$$ where $\epsilon= \sqrt{k_x^2 + k_y^2 + (k_z-b)^2} $.
We then fix $k_z$ and use the Kubo-Greenwood formula for the intra- and inter-band transitions separately to obtain two-dimensional conductivities. Thus, $$\begin{aligned}
\tilde\sigma_{\mu \nu}^{\mathrm{inter}}(i\omega;k_z)= \frac{e^2}{i}\sum_{\mathbf k, \gamma\neq \gamma'} \frac{n_{\mathbf k \gamma} - n_{\mathbf k \gamma'}}{\epsilon_{\mathbf k \gamma }- \epsilon_{\mathbf k \gamma'}} \frac{\braket{f_{\mathbf k \gamma} | j_{ \mu} | f_{\mathbf k \gamma'}}\braket{f_{\mathbf k \gamma'} | j_{ \nu} | f_{\mathbf k \gamma}}}{i \omega + \epsilon_{\mathbf k \gamma} - \epsilon_{\mathbf k \gamma'}},\end{aligned}$$ where $n_{\mathbf k \gamma}$ is the occupation in that band of that momentum and $j_\mu = \sigma_\mu$ are the single particle current operators. For intra-band quantities with $n_{\mathbf k \pm} = \theta(\mu \mp \epsilon_{\mathbf k})$ $$\begin{aligned}
\tilde\sigma_{\mu \nu}^{\mathrm{intra}}(i\omega;k_z)= - \frac{e^2}{i}\sum_{\mathbf k } \delta(\mu - \epsilon_{\mathbf k,+}) \frac{\braket{f_{\mathbf k +} | j_{ \mu} | f_{\mathbf k +}}\braket{f_{\mathbf k +} | j_{ \nu} | f_{\mathbf k +}}}{i \omega },\end{aligned}$$ assuming only the upper-band for simplicity and without loss of generality (due to particle-hole symmetry).
Adding these contributions together at finite chemical potential yields [@Tse2012] $$\begin{aligned}
\tilde\sigma_{xx}( i \omega; k_z) &= \frac{e^2}{4\pi} \left[\left(1 - \frac{4 (k_z - b)^2}{\omega^2} \right) \frac{i}4 \log\left( \frac{2\Delta - i \omega}{2\Delta + i\omega}\right) + \frac{\Delta}{ \omega}\right], \\
\tilde\sigma_{xy}( i\omega; k_z ) & = \frac{e^2}{2\pi} \left[ \frac{k_z - b}{\omega} \frac{i}2 \log\left( \frac{2\Delta - i \omega}{2\Delta + i\omega}\right)\right],\end{aligned}$$ where $\Delta = \max\{|k_z - b|,|\mu|\}$. With these quantities we can then use the cutoff procedure explained in [@Goswami2013] $$\begin{aligned}
\sigma_{\mu \nu}(i\omega) = \int_{-\Lambda}^\Lambda \frac{d k_z}{2\pi} \tilde \sigma_{\mu \nu}(i\omega; k_z).\end{aligned}$$ Through which we obtain (throwing away terms that go to zero as the cutoff increases to infinity and multiplying by two for the two nodes and bringing back in the constants $\hbar$ and $v_{\mathrm{F}}$) $$\begin{aligned}
\sigma_{xx}(i\omega) & = \frac{e^2}{12 \pi^2 \hbar v_{\mathrm F}} \left[ \tfrac 53 \omega + 2 \omega \log \left( \tfrac{v_{\mathrm F} \Lambda}{\omega} \right) + 4 \tfrac{\mu^2}{\hbar^2\omega} - \omega \log \left(1 + \tfrac{4 \mu^2}{\hbar^2\omega^2} \right)\right], \\
\sigma_{xy}(i \omega) & = \frac{e^2 b}{2 \pi^2 \hbar}.\end{aligned}$$ These quantites are what we use in the next section as input for the Casimir Force. \[sec:conductivitysup\]
Casimir Force calculation
=========================
Differentiating the Casimir energy
----------------------------------
In the main text, we find the Casimir pressure by taking the derivative of the energy $P_c = - \partial E_c/\partial a$. This leads to the expression for force found in the text for two semi-infinite Weyl plates $$\begin{aligned}
P_c = \frac{2 \hbar c}{(2\pi)^2} \int d q_z\, q_z^3 \; g\!\left[\tfrac{q_z}{\sigma_{xy}/c}, 2 q_z a \right], \label{eq:semi-inf-forcesup}\end{aligned}$$ and the function $g(u,v)$ is defined by $$\begin{aligned}
g(u,v) = -4 \frac{ R_{xx}(u)^2 - R_{xy}^2(u) - [R_{xx}(u)^2 + R_{xy}^2(u)]^2 e^{-v}}{2[R_{xx}(u)^2 - R_{xy}^2(u)] - [R_{xx}(u)^2 + R_{xy}^2(u)]^2 e^{-v}},\end{aligned}$$ where $R_{xx}(u)$ and $R_{xy}(u)$ are the matrix elements of $R_\infty(ic q_z/\sigma_{xy})$ (Eq. (3) in the main text), $$\begin{aligned}
R_{xx}(u) & =\sqrt{2u(\sqrt{u^2+1}-1)} - 1, \\
R_{xy}(u) & = 2u - \sqrt{2u(\sqrt{u^2+1}+1)}.\end{aligned}$$
Two-dimensional plates calculation
----------------------------------
Another well-known approach completely equivalent to the Lifshitz formula comes directly from quantum field theory.
To put it briefly, if we have a conductivity like we calculated above and write it as a response function $\Pi(i\omega)$ such that $$\begin{aligned}
\sigma(i\omega) &= -\Pi(i\omega)/\omega\end{aligned}$$ And calculate the RPA response function considering photons “skimming” along the surface of our material $$\begin{aligned}
\widetilde{\Pi}(i\omega,q) &= \left[\mathbb I - \Pi(i\omega) D(i\omega,q,0) \right]^{-1} \Pi(i\omega),\end{aligned}$$ with the photon propagator $$\begin{aligned}
D(i\omega,q,z) &= \begin{pmatrix} \frac{\sqrt{q^2+\omega^2}}{2\omega^2} & 0 \\ 0 & \frac{1}{2\sqrt{q^2+\omega^2}}\end{pmatrix} e^{-\sqrt{q^2+\omega^2}|z|},\end{aligned}$$ then the Casimir energy takes the form $$\begin{aligned}
E_c(a) &= \frac1\pi \int \frac{d^2q}{(2\pi)^2} \int _0^{\infty} d\omega \operatorname{tr}\log\left[\mathbb I - \widetilde{\Pi}_1(i\omega,q)D(i\omega,q,a)\widetilde{\Pi}_2(i\omega,q)D(i\omega,q,a) \right],\end{aligned}$$ For our particular case of the conductivities introduced in Section \[sec:conductivitysup\], we need to also cutoff the frequencies in this integral to go from $0$ to $v_{\mathrm F}\Lambda$ for consistency. We expect higher energy virtual photons to not play a large role.
We also show the result we obtain from varying the cutoff in Fig. \[fig:cutoff-varysup\]. Note that it does not affect the effect much unless it varies by orders of magnitude.
![The Casimir energy for the conductivities as defined in the main text with $b= \unit[0.3 (2\pi)]{nm^{-1}}$, $v_{\mathrm{F}} = \unit[6\times 10^5]{m/s}$, and $\mu=0.$ The cutoff is varied here.[]{data-label="fig:cutoff-varysup"}](plots/WeylVaryLambda.pdf){width="8cm"}
|
---
author:
- |
Z. Bern,[^1] A. De Freitas$^*$\
Department of Physics and Astronomy\
UCLA, Los Angeles, CA 90095-1547
- |
L. Dixon[^2]\
Stanford Linear Accelerator Center\
Stanford University\
Stanford, CA 94309
- |
A. Ghinculov$^*$ and H.L. Wong$^*$\
Department of Physics and Astronomy\
UCLA, Los Angeles, CA 90095-1547
title: 'QCD and QED Corrections to Light-by-Light Scattering'
---
Introduction {#IntroSection}
============
Light-by-light scattering is one of the most fundamental processes in QED. Theoretically, it proceeds at leading order, $\Ord(\alpha^4)$, via one-loop box diagrams containing charged particles. At center-of-mass energies $\sqrt{s}$ far below the mass of the electron, the process is described by the Euler-Heisenberg effective Lagrangian [@EulerHeisenberg], and the cross section rises rapidly with energy, $\sigma \propto s^3/m_e^8$. The cross section peaks at $\sqrt{s} \approx 3 m_e$ [@KarplusNeuman], then begins to fall rapidly, $\sigma \propto 1/s$ at fixed angles for $s \gg
m_e^2$ [@Akhiezer]. At still higher energies, similar thresholds are crossed for the muon, tau, and light quarks – or rather, the light hadrons. (See .) The final significant Standard Model thresholds reached are those of the $W$ boson [@JT; @GPRone] and the top quark.
The direct experimental evidence for $\ggtogg$ scattering is still scant, particularly for energies above a few GeV. At optical (electron volt) energies, at least two experiments have been proposed to detect the Euler-Heisenberg interaction via birefringence of the vacuum using laser photons in a magnetic field [@EHExpt]. The Crystal Ball experiment at the SPEAR storage ring had some unpublished evidence for $\ggtogg$ scattering at $\sqrt{s}$ of order several MeV, after subtracting beam-off $\gamma$ ray backgrounds [@Godfrey]. More extensive data are available for the process of Delbrück scattering, in which two of the four photons are supplied by the Coulomb field of a nucleus. Several experiments have studied this process, for incident photon energies ranging from a few MeV to about a GeV. Higher order Coulomb ($Z\alpha$) corrections are often important in the comparison with theory [@Delbruck]. Light-by-light scattering via an electron loop is also tested quite precisely, if indirectly, by the measurement of the anomalous magnetic moment of the electron [@electrongm2], as well as that of the muon [@BNLgm2] (the latter is even sensitive to muonic and hadronic light-by-light scattering).
At much higher energies, the small size of the Standard Model light-by-light scattering cross section provides a potential window to new physics. By backscattering a laser pulse off an intense, high energy electron beam [@GammaBackscatter], it is possible to create $\gamma\gamma$ collisions with $\sqrt{s}$ of order 100–1000 GeV, high levels of initial state polarization, and luminosities of order tens of fb$^{-1}$ per year. It has been shown that the $\ggtogg$ process at high energies is a sensitive probe [@Hooman] of theories with large extra dimensions [@ADD], for example. Despite this interest in light-by-light scattering, the process has to date only been calculated to leading order (one loop), except in the low energy limit, where the two-loop corrections to the Euler-Heisenberg Lagrangian are known [@TwoLoopEulerHeisenberg].
In this paper, we present the two-loop corrections to $\ggtogg$ in the ultra-relativistic regime where the kinematic invariants $s,t,u$ are much greater than the (squared) charged fermion masses. This regime is relevant for two ranges of center of mass energy in the Standard Model:
- For $m_c \ll \sqrt{s} \lsim 2 m_W$ (and neglecting the tiny bottom quark contribution), the QCD corrections from attaching a gluon line to the quark box are most important. These give rise to finite $\Ord(\alpha^4\alpha_s)$ corrections to the cross section.
- For $m_e \ll \sqrt{s} \lsim 2 m_\mu \approx 2 m_\pi$, the dominant corrections are QED corrections from attaching a photon line to the electron box. There are also QED corrections from inserting a fermion loop onto an external leg; however, these are cancelled completely if the theory is renormalized at zero momentum transfer, i.e. by using $\alpha \equiv \alpha(0) = 1/137.036\ldots$ as the coupling constant. Then the two-loop QED corrections become identical to those of QCD, up to an overall constant.
The Feynman diagrams for $\ggtogg$ at two loops are a small subset of those required for gluon-gluon scattering, $gg \to gg$. The interference of these two loop amplitudes with the $gg \to gg$ tree amplitudes is an essential ingredient for obtaining the next-to-next-to-leading order QCD corrections to jet production at hadron colliders. This interference was recently evaluated in a tour de force calculation by Glover, Oleari and Tejeda-Yeomans [@GOTYgggg]. In the light-by-light case, the tree amplitudes vanish. Thus the [*next*]{}-to-leading order corrections require a different interference, of two-loop amplitudes with one-loop amplitudes. Instead of evaluating this interference directly, we have computed the two-loop $\ggtogg$ amplitudes in a helicity basis. Thus polarized as well as unpolarized cross sections can be obtained, information which is quite useful in the photon case, because of the high initial state photon polarizations that are possible with backscattering.
Rather than calculating the Feynman diagrams for the two-loop $\ggtogg$ helicity amplitudes, we employed a unitarity- or cut-based technique [@CutBased; @BRY; @AllplusTwo] to generate the required loop momentum integrals. These integrals were then evaluated using techniques recently developed [@IntegralsSV; @IntegralsAGO; @IntegralsTalks] to handle double box and related integrals where all internal lines are massless. The restriction to massless internal lines limits the validity of our results to the ultrarelativistic region where all kinematic invariants are much greater than the relevant fermion masses.
Remarkably, some of the helicity amplitudes stay quite simple, even at two loops. Some of this simplicity can be understood via unitarity and supersymmetry Ward identities [@SWI].
This paper is organized as follows. In section 2 we present analytic results for the two-loop $\ggtogg$ helicity amplitudes. (The more complicated formulae are relegated to an appendix.) In section 3 we give numerical results for the QCD and QED corrections. In section 4 we present our conclusions.
The Two-Loop Amplitudes {#AmplitudesSection}
=======================
The one-loop Feynman diagrams for photon-photon scattering via a charged fermion consist of six box diagrams, which are related by permutations of the external photons to the diagram shown in a. At two loops, there are $6 \cdot 10 = 60$ nonvanishing Feynman diagrams, corresponding to connecting any two sides of the box with a photon propagator (or a gluon propagator, in the case that the charged fermion is a quark). Up to permutations, these diagrams are depicted in b. Diagrams consisting of two separate charged fermion triangles connected by a single photon or gluon, as illustrated in c, vanish by Furry’s theorem [@Furry] or simple group theory.
We did not evaluate the Feynman diagrams directly. Instead we computed the unitarity cuts in various channels, working to all orders in the dimensional regularization parameter $\epsilon = (4-D)/2$. Essentially we followed the approach first employed at two loops for the special cases of $N=4$ supersymmetric amplitudes [@BRY] and the pure gluon four-point amplitude with all plus helicities [@AllplusTwo]. These amplitudes were simple enough that a compact expression for the integrand could be given. The fermion loop contributions with all plus helicities are about as simple [@TLS]. However, for the generic helicity configuration, the integrands become quite complicated.
We have used general integral reduction algorithms developed for the all-massless planar four-point topologies [@IntegralsSV; @IntegralsAGO], in order to reduce the loop integrals to a minimal basis of master integrals. Some mild extensions of these techniques are required in order to incorporate polarization vectors for photons of definite helicity [@ggggpaper]. We then expand the master integrals in a Laurent series in $\e$, which begins at order $1/\e^4$. Many of the master integral Laurent expansions quoted in refs. [@IntegralsSV; @IntegralsAGO] are in terms of Nielsen functions [@NielsenRef], usually denoted by $S_{n,p}(x)$, with $n+p
\leq 4$. However, using various identities [@NielsenIds] the results can be expressed in terms of the polylogarithms [@Lewin], $$\begin{aligned}
\li{n}(x) &=& \sum_{i=1}^\infty { x^i \over i^n }
= \int_0^x {dt \over t} \li{{n-1}}(t),\nonumber \\
\li2(x) &=& -\int_0^x {dt \over t} \ln(1-t) \,,
\label{PolyLogDef}\end{aligned}$$ with $n=2,3,4$.
It is reassuring that all of the poles in $\e$ cancel for each helicity amplitude in the QCD case. In QCD, two loops is the first order at which $\alpha_s$ appears in the $\ggtogg$ amplitude; therefore there can be no ultraviolet divergence. Any infrared divergence would have to be cancelled by real gluon radiation; but the process $\ggtogg g$ is forbidden by group theory.
In addition to infrared finiteness, the calculational framework was also tested by gauge invariance: replacing a photon polarization vector by its momentum vector produces a vanishing result. Finally, the same computer programs for evaluating the cuts and reducing the integrals were also used to compute the fermion-loop contribution to the two-loop $gg \to gg$ amplitudes in the helicity formalism and the ’t Hooft-Veltman dimensional regularization scheme [@ggggpaper]. The interference of the two-loop $gg \to gg$ helicity amplitudes with the tree amplitudes, after summing over all external helicities and colors and accounting for the different scheme used, is in complete agreement with the calculation using conventional dimensional regularization [@GOTYgggg].
The QED case requires exactly the same set of two-loop diagrams as in QCD, up to an overall factor. In addition, there are external fermion bubble insertions of the form shown in d. In dimensional regularization with massless fermions, these diagrams would vanish by virtue of containing scale-free integrals. This vanishing represents a cancellation of ultraviolet and infrared divergences. However, if one renormalizes QED in the conventional on-shell scheme, to avoid infrared divergences one should retain a fermion mass in the external bubbles. Now the bubble integral is nonzero and ultraviolet-divergent. But this divergence, and indeed the entire integral, is exactly cancelled by the on-shell-scheme counterterm, precisely because the external leg is a real, on-shell photon.
Hence the on-shell-renormalized two-loop QED-corrected amplitude is the same as the two-loop QCD-corrected amplitude, up to overall coupling constant factors. In the on-shell scheme, the coupling constant should of course be set to $\alpha \equiv \alpha(0) = 1/137.036\ldots$. This value should be used for all the QED couplings associated with the real, external photons — that is, for all couplings [*except*]{} the extra one associated with the virtual photon in each two-loop graph. The typical virtuality of the extra photon is not zero, but of order $s$, assuming that the kinematic invariants $s,t,u$ are all comparable in magnitude and much larger than the squared fermion mass, $m_f^2$. Thus a running coupling $\alpha(\mu)$ with $\mu\approx\sqrt{s}$ should be used for the virtual photon insertion; whether it should be the $\MSbar$ running coupling, or that defined via the photon propagator at momentum transfer $\mu$, is theoretically indistinguishable at this order. For $m_e \ll \mu < m_\mu$, the latter coupling at one-loop order is $$\alpha(\mu)\ =\ { \alpha \over
1 - {\alpha\over3\pi} \Bigl[ \ln\Bigl({\mu^2 \over m_e^2}\Bigr)
- {5\over3} \Bigr] } \,.
\label{RunningAlpha}$$ For $\mu \gsim 2 m_\pi$, the running of $\alpha(\mu)$ receives hadronic corrections as well, which are best evaluated via a dispersion relation using the $e^+e^- \to $ hadrons data. In any event, the precise $\alpha(\mu)$ used makes very little difference, since it only appears in an $\Ord(\alpha)$ correction to a process with a rather tiny cross section.
For the QCD corrections, we use the $\MSbar$ running coupling $\alpha_s(\mu)$, again with $\mu = \sqrt{s}$, keeping in mind that our calculation is only to leading order in $\alpha_s$.
We consider the process $$\gamma(k_1,\lambda_1) + \gamma(k_2,\lambda_2)
\ \to \ \gamma(k_3,\lambda_3) + \gamma(k_4,\lambda_4),
\label{KinematicsDef}$$ where $k_i$ and $\lambda_i$ are the photon momenta and helicities. In terms of the center-of-mass energy $\sqrt{s}$ and scattering angle $\theta$, the Mandelstam variables are $s = (k_1+k_2)^2$, $t = (k_1-k_4)^2 = -s/2 \times (1-\cos\theta)$, and $u = (k_1-k_3)^2 = -s/2 \times (1+\cos\theta)$, with $s>0$, $t<0$, $u<0$. Parity, time-reversal invariance, and Bose symmetry imply that there are only four independent helicity amplitudes, $M_{--++}$, $M_{-+++}$, $M_{++++}$, and $M_{+--+}$. Actually, crossing symmetry relates $M_{++++}$ and $M_{+--+}$. However, representing the two-loop amplitudes in a crossing-symmetric fashion in terms of master integrals would lead to more cumbersome formulae, so we shall present all four amplitudes instead. We adopt the overall phase convention of refs. [@JT; @GPR]. Then formula (9) in ref. [@GPR] can be applied to our helicity amplitudes in order to obtain the $\ggtogg$ differential cross section for generic circular and transverse photon polarizations (Stokes parameters).
The one-loop helicity amplitudes due to a fermion of mass $m_f$ in the loop take a very simple form in the ultra-relativistic limit, $\{s,t,u\} \gg m_f^2$ [@Akhiezer; @KarplusNeuman; @GvdB]. We write $$\cm^\oneloop_{\lambda_1\lambda_2\lambda_3\lambda_4}
= 8 \, N Q^4 \, \alpha^2 \,
M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1)} \,,
\label{OneLoopDecomp}$$ where $N$ is the fermion color factor (3 for quarks, 1 for leptons), and $Q$ is the fermion charge in units of $e$. The functions $M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1)}$ are given by $$\begin{aligned}
M_{--++}^{(1)} &=&
1
\,, \nonumber \\
M_{-+++}^{(1)} &=&
1,
\nonumber \\
M_{++++}^{(1)} &=&
- {1\over2} {t^2+u^2\over s^2}
\Bigl[ \ln^2\Bigl({t\over u}\Bigr) + \pi^2 \Bigr]
- {t-u\over s} \ln\Bigl({t\over u}\Bigr) - 1
\,, \nonumber \\
M_{+--+}^{(1)} &=&
- {1\over2} {t^2+s^2\over u^2}
\ln^2\Bigl(-{t\over s}\Bigr)
- {t-s\over u} \ln\Bigl(-{t\over s}\Bigr) - 1 \nonumber \\
&& \null \hskip 2 cm
- i \pi \biggl[ {t^2+s^2\over u^2} \ln\Bigl(-{t\over s}\Bigr)
+ {t-s\over u} \biggr]
\,.
\label{OneLoopFunctions}\end{aligned}$$
The QCD- and QED-corrected two-loop amplitudes are $$\begin{aligned}
\cm^{\twoloop,\,{\rm QCD}}_{\lambda_1\lambda_2\lambda_3\lambda_4}
&=& 4 \, (N^2-1) Q^4 \, \alpha^2 \, {\alpha_s(\mu) \over \pi}
\, M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(2)} \,, \nonumber\\
\cm^{\twoloop,\,{\rm QED}}_{\lambda_1\lambda_2\lambda_3\lambda_4}
&=& 8 \, N Q^6 \, \alpha^2 \, {\alpha(\mu) \over \pi}
\, M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(2)} \,,
\label{TwoLoopDecomp}\end{aligned}$$ where the explicit values of the $M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(2)}$ are given in Appendix \[AmplitudeAppendix\].
Note from that $M_{--++}$ remains remarkably simple even at two loops — it is just a constant, independent of $s$, $t$ and $u$! This simplicity is actually predictable. The lack of an imaginary part for $M_{--++}^{(2)}$ (as for $M_{--++}^{(1)}$ and $M_{-+++}^{(1)}$) can be deduced by considering the possible unitarity cuts of the amplitude. Let $f$ be a massless fermion. Then a supersymmetry Ward Identity [@SWI] shows that the tree-level amplitudes for $\gamma\gamma f \bar{f}$ and $\gamma\gamma\gamma f \bar{f}$ vanish if all the photons have the same helicity (when considered as outgoing particles). The same result is true if one (or more) photons are replaced by gluons with the same helicity. Assigning opposite (outgoing) helicities to intermediate particles on opposite sides of a cut, it is easy to see that all two- or three-particle cuts vanish for the all-outgoing-plus-helicity two-loop amplitude $M_{--++}^{(2)}$. This argument assumes that a $D=4$ helicity assignment is valid for the intermediate particles, a condition which is justified for the processes considered here by the absence of infrared or ultraviolet divergences. Thus $M_{--++}^{(2)}$ must be a dimensionless rational function. Crossing symmetry implies that it is totally symmetric in $s,t,u$. Imposing at most single poles in $s,t,u$, and using $s+t+u=0$, one finds that $M_{--++}^{(2)}$ must actually be a constant.
shows that $M_{-+++}^{(2)}$ is also fairly simple, containing only logarithms, and not $\li2$, $\li3$ or $\li4$ functions. Its two-particle cuts in $D=4$ are the product of two rational functions (just like the cuts of a one-loop amplitude), which may partially account for this simplicity; however, the three-particle cuts do not similarly simplify at a glance.
Interestingly, up to the overall normalization these amplitudes are identical to the subleading color contributions for the $gg \rightarrow
\gamma \gamma$ amplitudes presented in ref. [@gggamgamPaper]. (In that paper all particles are taken to be outgoing, which corresponds to flipping the helicity labels for particles 1 and 2.) These amplitudes are relevant for improved estimates of the di-photon background to production of a light (mass $<$ 140 GeV) Higgs boson at the Large Hadron Collider [@HiggsBkgdPaper].
It is instructive to quote the values of the helicity amplitudes in various limits. The values of the one-loop functions for $90^\circ$ scattering ($t=-s/2$, $u=-s/2$) are $$\begin{aligned}
M_{--++}^{(1)}(90^\circ) &=& 1 \,, \nonumber\\
M_{-+++}^{(1)}(90^\circ) &=& 1 \,, \nonumber\\
M_{++++}^{(1)}(90^\circ)
&\approx& -3.46740 \,, \nonumber\\
M_{+--+}^{(1)}(90^\circ)
&\approx& -0.12169 + 0.46574 \pi i\,,
\hbox{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\label{M1RightAngle}\end{aligned}$$ while the two-loop functions are $$\begin{aligned}
M_{--++}^{(2)}(90^\circ) &=& -1.5, \nonumber \\
M_{-+++}^{(2)}(90^\circ)
&\approx& 0.17770 - 0.23287 \pi i \,, \nonumber \\
M_{++++}^{(2)}(90^\circ)
&\approx& 1.24077 + 1.00717 \pi i \,, \nonumber \\
M_{+--+}^{(2)}(90^\circ)
&\approx& 0.01445 + 0.36840 \pi i \,.
\hbox{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
\label{M2RightAngle}\end{aligned}$$ Note that except for the first, the two-loop functions are smaller than their one-loop counterparts.
In the small-angle limit $|t| \ll s$, the values of the one-loop amplitudes, up to corrections suppressed by powers of $t/s$, are $$\begin{aligned}
M_{--++}^{(1)} &\sim&
1
\,, \nonumber \\
M_{-+++}^{(1)} &\sim&
1
\,, \nonumber \\
M_{++++}^{(1)} &\sim&
- {1\over2} X^2 - X - {\pi^2 \over 2} - 1
\,,\nonumber \\
M_{+--+}^{(1)} &\sim&
- {1\over2} X^2 - X - 1 - i \pi ( X + 1 )
\,,
\hbox{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\nonumber \\
M_{+-+-}^{(1)} &\sim&
0
\,,\nonumber \\
\label{M1smallt}\end{aligned}$$ while the two-loop amplitudes are, $$\begin{aligned}
M_{--++}^{(2)} &\sim &
-{3\over2}
\, , \nonumber \\
M_{-+++}^{(2)} &\sim &
{1\over4} X^2 + {1\over2} X
+ {\pi^2\over8} - {1\over4}
+ i \pi \Bigl( {1\over4} X + {1\over4} \Bigr)
\, , \nonumber \\
M_{++++}^{(2)} &\sim &
- {1\over24} X^4 - {1\over12} X^3
- \Bigl( {\pi^2\over12} + {1\over2} \Bigr) X^2
- \Bigl( {5\over12} \pi^2 + {1\over2} \Bigr) X
+ {7\over360} \pi^4 - \zeta_3 - {7\over12} \pi^2 - {1\over4}
\nonumber \\ &&\null \hskip1cm
+ i \pi \Bigl( {1\over2} X^2 + {1\over2} X
+ 2 \zeta_3 + {\pi^2\over6} - {13\over4} \Bigr)
\,, \nonumber \\
M_{+--+}^{(2)} &\sim &
- {1\over24} X^4 - {1\over12} X^3
+ \Bigl( {\pi^2\over6} - {1\over2} \Bigr) X^2
+ \Bigl( {5\over6} \pi^2 - {1\over2} \Bigr) X
+ {11\over180} \pi^4 - \zeta_3 + {5\over12} \pi^2 - {1\over4}
\nonumber \\ && \null \hskip1cm
+ i \pi \Bigl( -{1\over6} X^3 - {3\over4} X^2 - {3\over2} X
- 2 \zeta_3 + {11\over4} \Bigr)
\,, \nonumber \\
M_{+-+-}^{(2)} &\sim&
-{1\over2} \, ,
\label{M2smallt}\end{aligned}$$ where $X \equiv -\ln(-s/t)$. Notice that the small-angle scattering amplitudes where both photons flip their helicity have no logarithmic enhancements at one- or two-loops; those with one helicity flip have no logs at one loop but a factor of $X^2$ at two loops; and those with no helicity flips pick up one factor of $X^2$ for each loop. For small angles these contributions are actually power suppressed compared to $t$-channel vector exchange diagrams which first appear at three loops [@LeadingLogs].
Next we give the differential cross sections in terms of the above amplitudes. The leading-order $\ggtogg$ unpolarized differential cross section for a single fermion flavor is given by $${d\sigma^{\rm LO} \over d\cos\theta}
= N^2 Q^8 \, {\alpha^4 \over 2 \pi \, s} \Bigl[
| M_{--++}^{(1)} |^2 + 4 \, | M_{-+++}^{(1)} |^2
+ | M_{++++}^{(1)} |^2 + | M_{+--+}^{(1)} |^2 + | M_{+-+-}^{(1)} |^2
\Bigr] \,,
\label{LeadingCrossSection}$$ where $$M_{+-+-}^{(L)}(s,t,u) = M_{+--+}^{(L)}(s,u,t) \,.
\label{mppmBySymmetry}$$ The QCD- and QED-corrected unpolarized differential cross sections for a single fermion flavor are given by $$\begin{aligned}
{d\sigma^{\rm QCD} \over d\cos\theta}
&=& {d\sigma^{\rm LO} \over d\cos\theta}
+ {d\sigma^{\alpha_s} \over d\cos\theta} \,,
\nonumber \\
{d\sigma^{\rm QED} \over d\cos\theta}
&=& {d\sigma^{\rm LO} \over d\cos\theta}
+ {d\sigma^{\alpha} \over d\cos\theta} \,,
\label{QCDCrossSection}\end{aligned}$$ where $$\begin{aligned}
{d\sigma^{\alpha_s} \over d\cos\theta}
&= & N (N^2-1) Q^8 \,
{\alpha^4 \, \alpha_s(\mu) \over 2 \pi^2 \, s} I^{(2,1)} \,, \nonumber\\
{d\sigma^{\alpha} \over d\cos\theta}
&=& N^2 Q^{10} \,
{\alpha^4 \, \alpha(\mu) \over \pi^2 \, s} I^{(2,1)} \,,
\label{alphasCrossSection}\end{aligned}$$ and $$\begin{aligned}
I^{(2,1)} &\equiv&
\Re \Bigl[
M_{--++}^{(2)} M_{--++}^{(1)\ *}
+ 4 \, M_{-+++}^{(2)} M_{-+++}^{(1)\ *}
+ M_{++++}^{(2)} M_{++++}^{(1)\ *}
\nonumber \\ && \null \hskip 2 cm
+ M_{+--+}^{(2)} M_{+--+}^{(1)\ *}
+ M_{+-+-}^{(2)} M_{+-+-}^{(1)\ *} \Bigr] \,.
\label{I21}\end{aligned}$$ The cross sections for circularly polarized photons can be constructed easily from by reweighting the products of $M$’s. For arbitrary initial photon polarizations, linear as well as circular, eqs. (9)–(15) of ref. [@GPR] can be used, after making the replacement (for the QCD case) $$F_{\lambda_1\lambda_2\lambda_3\lambda_4} \to
\cm_{\lambda_1\lambda_2\lambda_3\lambda_4}
= 8 \, N Q^4 \, \alpha^2 \, M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1)}
+ 4 \, (N^2-1) Q^4 \, \alpha^2 \, {\alpha_s(\mu) \over \pi}
\, M_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(2)} \,.
\label{Freplace}$$ (Note that our definitions of $t$ and $u$ are reversed with respect to ref. [@GPR].)
Numerical Results
=================
The QCD $K$ factor is conventionally defined as the ratio of next-to-leading to leading-order cross sections, $$K \equiv { d\sigma^{\rm QCD}/d\cos\theta
\over d\sigma^{\rm LO}/d\cos\theta } \,.
\label{KDef}$$ In a region between quark thresholds, and below the $W$ mass, we can write $$K = 1 + {\alpha_s(\mu) \over \pi}
{ \sum_i (N_i^2-1) Q_i^4 \over \sum_i N_i Q_i^4 } \, k,
\label{KEqn}$$ where $$k \equiv
{ \sum_{ \{ \lambda \} } \Re \Bigl[ M_{ \{ \lambda \} }^{(2)}
M_{ \{ \lambda \} }^{(1)\ *} \Bigr]
\over \sum_{ \{ \lambda \} } | M_{ \{ \lambda \} }^{(1)} |^2 } \,.
\label{kDef}$$ The label $i$ runs over the number of quarks and leptons with masses much less than $\sqrt{s}$; $N_i=3$ for quarks, $N_i=1$ for leptons; and $Q_i$ is the fermion charge. For energies between $m_b$ and $m_W$, for example, $(\sum_i (N_i^2-1) Q_i^4 ) / ( \sum_i N_i Q_i^4 )
= 70/87$. For energies above $m_W$, one should add the $W$ loop contributions to $\cm^\oneloop_{\{ \lambda \} }$. This dilutes the QCD corrections considerably, since the $W$ loops quickly dominate [@JT; @GPRone] (see ), and they have no QCD corrections.
In , $k$ is plotted as a function of the center-of-mass scattering angle $\theta$. Due to Bose symmetry, only the forward region $\cos\theta > 0$ has to be plotted. We also plot the corresponding curves for fully circularly polarized initial photons, but summing over final-state helicities, $$k_{\lambda_1\lambda_2} \equiv
{ \sum_{\lambda_3,\lambda_4}
\Re \Bigl[ M_{ \{ \lambda \} }^{(2)} M_{ \{ \lambda \} }^{(1)\ *} \Bigr]
\over \sum_{\lambda_3,\lambda_4} | M_{ \{ \lambda \} }^{(1)} |^2 } \,.
\label{klambdaDef}$$ Taking into account CP invariance, there are two independent cases, $++$ and $+-$. Although the QCD corrections are of order $\alpha_s/\pi$ for these two cases, they have opposite signs, and in the unpolarized cross section there is a large cancellation. The QCD corrections to the unpolarized cross section are only of order $0.1 \times \alpha_s/\pi$ for central scattering angles, becoming as large as $\alpha_s/\pi$ only for $|\cos\theta| \geq 0.95$.
displays the QCD corrections to the cross section in the 10–100 GeV region where quark boxes are important, and the quarks can be taken to be approximately massless. The QED corrections can be neglected in this region as they are a factor of 8 or so smaller. The two independent initial helicity configurations are shown, after integrating the differential cross section with an acceptance cut of $30^\circ < \theta < 150^\circ$, and multiplying by $s$ as in . The QCD corrections were computed by multiplying the leading order one-loop result, which contains the full fermion (and $W$) mass dependence, by the $K$ factor formula (\[KEqn\]) evaluated for massless fermions in both numerator and denominator (which are of course integrated separately over $\theta$). We can apply even across the $b$ quark threshold, simply because the $b$ quark’s small charge of $-1/3$ lends it a tiny contribution to the $\ggtogg$ cross section. (Nevertheless, we interpolate $( \sum_i (N_i^2-1) Q_i^4 ) / (\sum_i N_i Q_i^4 )$ from $272/345 \approx 0.788$ to $70/87 \approx 0.805$ across the threshold.) We see that the QCD corrections are numerically rather small for central scattering angles in this entire energy range. The small size of the correction explicitly demonstrates the reliability of the leading order prediction.
The QED corrections can also be extracted from . In the region between $m_e$ and $m_\mu$, for example, the corrections to the unpolarized cross section are $${ d\sigma^{\rm QED} \over d\cos\theta } =
{ d\sigma^{\rm LO} \over d\cos\theta } \times \biggl[ 1
+ 2 {\alpha(\mu) \over \pi} \, k \biggr] \,.
\label{QEDKfactor}$$ For the same angular cuts ($30^\circ < \theta < 150^\circ$) as in the QCD case, this amounts to only a $0.07\%$ decrease in the cross section for $++$ initial helicities, and a $0.35\%$ increase for $+-$ initial helicities. It will be a true challenge to measure the $\ggtogg$ reaction to this level of precision.
Conclusions
===========
In this paper we presented the two-loop QCD and QED corrections to light-by-light scattering by fermion loops in the ultrarelativistic limit where all kinematic invariants are much greater than the relevant fermion masses. These corrections reliably give the leading Standard Model corrections for most energies below 100 GeV. The corrections are quite small numerically, showing that the leading order computations are robust.
Some of the helicity amplitudes remain quite simple even at two loops, as a consequence of unitarity and a supersymmetry Ward identity for tree amplitudes.
To extend our results to regions where the kinematic invariants are comparable to the masses in the loops, the technology for computing two-loop double box integrals should first be extended to include massive internal lines, which seems feasible. Probably the most important application would then be to compute the electroweak corrections to the $W$ box contribution to $\ggtogg$, since that contribution dominates at high energies, where new physics contributions are most likely to be found.
We thank Hooman Davoudiasl and George Gounaris for useful conversations.
Two-loop helicity Amplitudes {#AmplitudeAppendix}
============================
The explicit expressions for the two-loop amplitudes appearing in are $$M_{--++}^{(2)} =
- {3\over 2 }\,,
\hskip 12. cm \null
\label{FppppSL}$$ $$\begin{aligned}
M_{-+++}^{(2)} & = &
{1\over 8} \biggl[ { x^2 + 1\over y^2 } ((X + i \pi)^2 + \pi^2 )
+ {1\over 2} (x^2 + y^2) ((X-Y)^2 + \pi^2) \nonumber \\
&& \hskip 4 cm \null
- 4 \biggl({1\over y} - x \biggr) (X + i \pi) \biggr]
+ \Bigl\{t \leftrightarrow u \Bigr\} \,,
\hskip 2.8 cm \null
\label{FmpppSL}\end{aligned}$$ $$\begin{aligned}
M_{++++}^{(2)} &=&
- 2 x^2 \biggl[ \li4(-x) + \li4(-y)
- (X + i\pi) \Bigl( \li3(-x) + \li3(-y) \Bigr) \nonumber \\
&& \hskip1.0cm
+ {1\over12} X^4 - {1\over3} X^3 Y + {\pi^2 \over 12} X Y
- {4 \over 90} \pi^4
+ i {\pi\over6} X \Bigl( X^2 - 3 X Y + \pi^2 \Bigr) \biggr] \nonumber \\
&& \null
-(x-y) \Bigl( \li4(-x/y) - {\pi^2\over6} \li2(-x) \Bigr) \nonumber\\
&& \null
- x \biggl[ 2 \li3(-x) - \li3(-x/y) - 3 \zeta_3
- 2 (X + i\pi) \li2(-x) \nonumber \\
&& \hskip1.0cm
+ (X-Y) ( \li2(-x/y) + X^2 )
+ {1\over 12} ( 5 (X-Y) + 18 i \pi) ((X-Y)^2 + \pi^2) \nonumber \\
&& \hskip1.0cm
- {2\over 3} X (X^2 + \pi^2) - i \pi (Y^2 + \pi^2) \biggr]
\nonumber \\
&& \null
+ {1\over 4} { 1 - 2x^2 \over y^2 } ((X + i\pi)^2 + \pi^2)
- {1\over 8} ( 2 x y + 3 ) ((X-Y)^2 + \pi^2)
+ { \pi^2 \over 12} \nonumber \\
&& \null
+ \biggl( {1\over 2 y} + x \biggr) (X + i\pi) - {1\over 4}
+ \Bigl\{ t \leftrightarrow u \Bigr\} \,,
\label{FmmppSL}\end{aligned}$$ $$\begin{aligned}
M_{+--+}^{(2)} &=&
- 2 {x^2+1 \over y^2} \biggl[
\li4(-x/y) - \li4(-y)
+ {1\over2} (X - 2 Y - i\pi) ( \li3(-x) - \zeta_3 ) \nonumber \\
&& \hskip2.0cm
+ {1\over 24} ( X^4 + 2 i \pi X^3 - 4 X Y^3 + Y^4
+ 2 \pi^2 Y^2 ) + {7\over 360} \pi^4 \biggr] \nonumber \\
&& \null
- 2 {x-1\over y} \biggl[ \li4(-x) - \zeta_4
- {1\over 2} (X + i\pi) (\li3(-x) - \zeta_3 ) \nonumber \\
&& \hskip2.0cm
+ {\pi^2\over 6} \Bigl( \li2(-x) - {\pi^2\over6}
- {1\over2} X^2 \Bigr)
- {1\over 48} X^4 \biggr] \nonumber \\
&& \null
+ \biggl(2 {x\over y} - 1\biggr)
\biggl[ \li3(-x) - (X + i\pi) \li2(-x)
+ \zeta_3 - {1\over 6} X^3
- {\pi^2\over 3} (X + Y) \biggr] \nonumber \\
&& \null
+ 2 \biggl(2 {x\over y} + 1 \biggr)
\biggl[ \li3(-y) + (Y + i\pi) \li2(-x) - \zeta_3
+ {1\over 4} X ( 2 Y^2 + \pi^2 ) \nonumber \\
&& \hskip2.5cm
- {1\over 8} X^2 (X + 3 i\pi) \biggr]
- {1\over 4} ( 2 x^2 - y^2 ) ((X-Y)^2 + \pi^2) \nonumber \\
&& \null
- {1\over 4} \Bigl(3 + 2 {x\over y^2} \Bigr) ((X + i\pi)^2 + \pi^2)
- {2-y^2 \over 4 x^2} ((Y + i\pi)^2 + \pi^2)
+ {\pi^2\over 6} \nonumber \\
&& \null
+ {1\over 2} ( 2 x + y^2 ) \biggl[ {1\over y} (X + i\pi)
+ {1\over x} (Y + i\pi) \biggr]
- {1\over 2} \,.
\label{FmpmpSL}\end{aligned}$$ Here $$x \equiv {t\over s} \,, \quad y \equiv {u\over s} \,, \quad
X \equiv \ln\Bigl(-{t\over s}\Bigr) \,, \quad
Y \equiv \ln\Bigl(-{u\over s}\Bigr) \,.
\label{XYdef}$$
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[^1]: Research supported by the US Department of Energy under grant DE-FG03-91ER40662.
[^2]: Research supported by the US Department of Energy under grant DE-AC03-76SF00515.
|
---
abstract: 'Load balancing is a widely accepted technique for performance optimization of scientific applications on parallel architectures. Indeed, balanced applications do not waste processor cycles on waiting at points of synchronization and data exchange, maximizing this way the utilization of processors. In this paper, we challenge the universality of the load-balancing approach to optimization of the performance of parallel applications. First, we formulate conditions that should be satisfied by the performance profile of an application in order for the application to achieve its best performance via load balancing. Then we use a real-life scientific application, MPDATA, to demonstrate that its performance profile on a modern parallel architecture, Intel Xeon Phi, significantly deviates from these conditions. Based on this observation, we propose a method of performance optimization of scientific applications through load imbalancing. We also propose an algorithm that finds the optimal, possibly imbalanced, configuration of a data parallel application on a set of homogeneous processors. This algorithm uses functional performance models of the application to find the partitioning that minimizes its computation time but not necessarily balances the load of the processors. We show how to apply this algorithm to optimization of MPDATA on Intel Xeon Phi. Experimental results demonstrate that the performance of this carefully optimized load-balanced application can be further improved by 15% using the proposed load-imbalancing optimization.'
author:
- |
Alexey Lastovetsky\
\
\
Lukasz Szustak\
\
\
Roman Wyrzykowski\
\
bibliography:
- 'Lastovetsky\_Szustak\_sc15.bib'
title: 'Model-based optimization of MPDATA on Intel Xeon Phi through load imbalancing'
---
Introduction {#sec:intro}
============
Load balancing is a widely accepted technique for optimization of the computation performance of scientific applications on parallel architectures. Indeed, the intuition suggests that unlike unbalanced applications, the balanced ones do not waste processor cycles on waiting at points of synchronization and data exchange, maximizing this way the utilization of the processors.
In this paper, we challenge the universality of the load-balancing approach to optimization of the computation performance of parallel applications. First, we try to understand the limitations of the load-balancing approach. We formulate conditions that should be satisfied by the performance profile of an application in order for the application to achieve its best performance via load balancing.
Then we use a real-life scientific application to demonstrate that its performance profile on a modern parallel architecture does not satisfy these conditions. The application we use implements the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA), which is one of the major parts of the dynamic core of the EULAG geophysical model [@SMO06]. EULAG (Eulerian/semi-Lagrangian fluid solver) is an established numerical model developed for simulating thermo-fluid flows across a wide range of scales and physical scenarios [@PIO12; @SMO90]. In particular, it can be used in numerical weather prediction (NWP), simulation of urban flows, areas of turbulence, ocean currents, etc. This solver, originally developed for conventional HPC systems, is currently being re-written for modern HPC platforms. In particular, MPDATA has been recently re-written and optimized for execution on an Intel Xeon Phi coprocessor [@SZU13; @SZU15].
In our experiments, we observe significant deviations of the MPDATA performance profile from the conditions required for applicability of the load-balancing techniques. Based on this observation, we propose a general method of performance optimization of scientific applications through load imbalancing as well as an algorithm that finds the optimal, possibly imbalanced, configuration of a data parallel application on a set of homogeneous processors. This algorithm uses functional performance models of the application [@twamley2005; @ijhpca2007] to find the partitioning that minimizes its computation time but not necessarily balances the load of the processors. Finally, we apply this algorithm to optimization of MPDATA on Intel Xeon Phi. Experimental results demonstrate that the performance of this carefully optimized load-balanced application can be further improved by 15% using the proposed load-imbalancing method.
The contributions of the work presented in the paper are as follows:
- Formulation of the conditions that should be satisfied to guarantee that load balancing will minimize the computation time of parallel application.
- Building the performance profile of a real-life scientific application on a modern HPC platform and demonstration of its significant deviation from the conditions that guarantee that load balancing be a safe technique for performance optimization.
- A new optimization method that uses the performance profile for optimization of the application through its imbalancing.
- A partitioning algorithm finding the optimal and generally speaking uneven distribution of computations of an application between homogeneous processing units using its functional performance model.
- Application of the proposed partitioning algorithm to optimization of MPDATA on Intel Xeon Phi, resulting in further acceleration of this carefully optimized load-balanced application by up to 15%.
The rest of the paper is structured as follows. Section \[sec:per\] overviews load-balancing techniques and formulates the conditions when these techniques would minimize the computation time of parallel applications. Section \[sec:opt\] analyzes the performance profile of MPDATA on Intel Xeon Phi and introduces the new approach to minimization of the computation time through load imbalancing. Section \[sec:model\] introduces a partitioning algorithm for (uneven) distribution of computations between homogeneous processors, minimizing the computation time of the application. Section \[sec:app\] applies this algorithm to optimization of MPDATA on Xeon Phi. Section \[sec:result\] presents experimental results, and Section \[sec:con\] concludes the paper.
Load balancing and performance {#sec:per}
==============================
In this section we overview load-balancing techniques used for optimization of the performance of parallel scientific applications on both homogeneous and heterogeneous platforms. We also formulate the conditions when application of these techniques will optimize the computation performance.
Load balancing is a widely accepted and commonly used approach to performance optimization of scientific applications on parallel architectures. Load balancing algorithms can be classified as static or dynamic. Static algorithms (for example, those based on data partitioning) [@Fatica2009; @Yang2010; @Ogata2008; @ijhpca2007] require a priori information about the parallel application and platform. This information can be gathered either at compile-time or runtime. Static algorithms are also known as predicting-the-future because they rely on accurate performance models as input to predict the future execution of the application. Static algorithms are particularly useful for applications where data locality is important because they do not require data redistribution. However, these algorithms are unable to balance on non-dedicated platforms, where load changes with time. Dynamic algorithms (such as task scheduling and work stealing) [@Linderman2008; @Augonnet2009; @QuintanaOrti2009] balance the load by moving fine-grained tasks between processors during the calculation. Dynamic algorithms do not require a priori information about execution but may incur significant communication overhead due to data migration. Dynamic algorithms often use static partitioning for their initial step due to its provably near-optimal communication cost, bounded tiny load imbalance, and lesser scheduling overhead [@Song2012].
Whatever load balancing algorithm is used, the goal is always to minimize the computation time of the application. The intuition behind the assumption that balancing the application improves its performance is the following: a balanced application does not waste processor cycles on waiting at points of synchronization and data exchange, maximizing this way the utilization of the processors. Is this assumption always true? To answer this question, let us formulate the assumption in a mathematical form. Consider an application, the computation performance of which can be modeled by speed functions. Namely, let $p$ parallel processors be used to execute the application, and let $s_i(x)$ be the speed of execution of the workload of size $x$ by processor $i$. Here the speed can be measured in floating point operations per second or any other fix-sized computation units per unit time. The size of workload can be characterized by the problem size (say, the number of cells in the computational domain or the matrix size) or just by the number of equal-sized computational units. Anyway, the speed $s_i(x)$ is calculated as $\frac{x}{t_i(x)}$, where $t_i(x)$ is the execution time of the workload of size $x$ on processor $i$. Using these definitions, we can formulate the following theorem.
*Theorem 1*: \[theo:speed\] Let $s_i(x)>0$ ($x>0$) be the speed of processor ${i \in \{1,\ldots,p\} }$, and $\forall{\Delta x>0} \colon$ $\frac{s_i(x)} {x}$ $\geq$ $\frac{s_i (x+\Delta x)} {x+\Delta x}$. Let $x_1+\ldots+x_p=n>0$ and $\frac{s_1(x_1)} {x_1}$ $=\ldots=$ $\frac{s_p(x_p)} {x_p}$. Then, $\forall{y_1,\ldots,y_p}$ $>$ $0$ such that $(y_1,\ldots,y_p)$ $\neq$ $(x_1,\ldots,x_p)$ and $y_1+\ldots+y_p=n \colon$ $\max_{i}\frac{y_i}{s_i(y_i)}$ $\geq$ $\frac{x_1}{s_1(x_1)}$.
*Proof*: As $(y_1,\ldots,y_p)$ $\neq$ $(x_1,\ldots,x_p)$ and $y_1+\ldots+y_p=x_1+\ldots+x_p$, then there exists ${k \in \{1,\ldots,p\} }$ such that $y_k > x_k$. Therefore, $\max_{i}\frac{y_i}{s_i(y_i)}$ $\geq$ $\frac{y_k}{s_k(y_k)}$ $=$ $\frac{x_k+(y_k-x_k)}{s_k(x_k+(y_k-x_k))}$ $\geq$ $\frac{x_k}{s_k(x_k)}$ $=$ $\frac{x_1}{s_1(x_1)}$.\
Theorem 1 states that in order to guarantee that the balanced configuration of the application will execute the workload of size $n$ faster than any unbalanced configuration, the speed functions $s_i(x)$, characterizing the performance profiles of the processors, should satisfy the condition: $$\forall{\Delta x>0} \colon \frac{s_i(x)} {x} \geq \frac{s_i (x+\Delta x)} {x+\Delta x}
\label{eq:cond}$$
Geometrically, it can be illustrated as follows. If we plot a speed function as shown in Figure \[fig:graph0\], then the angle $\alpha (x)$ between the straight line, connecting the point $(0,0)$ and the point $(x, s(x))$ on the speed curve, and the $x$-axis will be inversely proportional to the execution time of the workload of size $x$ by the processor. Indeed, the cotangent of this angle is directly proportional to the ratio $\frac{x}{s(x)}$ representing the execution time of the workload $x$. Therefore, larger angles correspond to shorter execution times. The condition \[eq:cond\] means that the increase of the workload, $x$, will never result in the decrease of the execution time, or equivalently in the increase of the angle $\alpha (x)$.
![Example of speed function suitable for minimization of computation time through load balancing. Angle $\alpha(x)$ represents the computation time: the greater the angle, the shorter the computation time.[]{data-label="fig:graph0"}](./graph0){width="50.00000%"}
The main body of the load balancing algorithms designed for performance optimization explicitly or implicitly assume that the speed of processor does not depend on the size of workload [@Fatica2009; @Yang2010; @Luk2009; @cierniak; @kalinov2001; @plaza2011]. In other words, the speed functions $s_i(x)$ are assumed to be positive constants, in which case the condition \[eq:cond\] is trivially satisfied. More advanced algorithms are based on functional performance models (FPMs), which represent the speed of processor by a continuous function of the problem size [@twamley2005; @ipdps2004]. However, the shape of the function is not arbitrary but has to satisfy the following assumption [@ijhpca2007]: Along each of the problem size variables, either the function is monotonically decreasing, or there exists point $x$ such that
- On the interval $[0, x]$, the function is
- monotonically increasing,
- concave, and
- any straight line coming through the origin of the coordinate system intersects the graph of the function in no more than one point.
- On the interval $[x, \infty)$, the function is monotonically decreasing.
These restrictions on the shape of speed functions guarantee that the efficient load balancing algorithms, proposed in [@Ilic2011; @Colaco; @lastovetsky2007data; @clarke2011dynamic; @clarke2012column; @alonazi2015], will always return a unique solution, minimizing the computation time. At the same time, it is easy to show that the restrictions imposed on FPMs will make them comfortably satisfy the condition \[eq:cond\].
Thus, the state-of-the-art load balancing algorithms designed for optimization of the computation performance of parallel applications assume that their performance profiles satisfy the condition \[eq:cond\]. Therefore, correct application of such algorithms requires that the experimental speed points be approximated by a function satisfying this condition. This approximation step may significantly distort the actual performance profile and lead to a substantially non-optimal solution.
Optimization of parallel applications through load imbalancing {#sec:opt}
==============================================================
In this section, we demonstrate that the performance profile of real-life scientific applications on modern parallel platforms may significantly deviate from the conditions, which guarantee that load balancing will always optimize their computation performance. Based on this observation, we propose an optimization method that uses the performance profile for optimization of the application through its imbalancing.
In this work, we build the performance profile of MPDATA on Intel Xeon Phi. MPDATA is a core component of EULAG (Eulerian/semi-Lagrangian fluid solver), which is an established computational model developed for simulating thermo-fluid flows across a wide range of scales and physical scenarios. Its carefully optimized data-parallel implementation on a 60-core Intel Xeon Phi [@SZU15] partitions the 3D rectilinear $n\times n\times l$ domain into four equal $\frac{n}{2}\times \frac{n}{2} \times l$ sub-domains, each allocated to a team of 15 cores. This configuration of the application is the best load-balanced configuration identified in [@SZU15]. The experimentally constructed speed functions of these four teams, each processing (in parallel) a $120\times m\times 128$ sub-domain, are shown in Figure \[fig:graph1\].
![ Speed functions of Intel MIC built for four 15-core teams, each processing in parallel a $120\times m\times 128$ sub-domain. The speed is measured in cells per second, while the problem size is represented by $m$. []{data-label="fig:graph1"}](./graph1){width="50.00000%"}
This graph clearly shows that for many $m$ and $\Delta m$ the speed of processing of the $120\times m\times 128$ sub-domain will be significantly lower than the speed of processing of the $120\times (m+\Delta m)\times 128$ sub-domain. Moreover, we can also see that $\alpha(m+\Delta m) >\alpha(m)$ for some such $m$ and $\Delta m$, which means that the time of processing of the $120\times m\times 128$ sub-domain will be longer than the time of processing of the $120\times (m+\Delta m)\times 128$ sub-domain. The latter observation can be used to speed up the execution of the application. Namely, if we re-partition the equally partitioned domain so that two teams get $120\times (m+\Delta m)\times 128$ sub-domains and two other teams get $120\times (m-\Delta m)\times 128$ sub-domains, and $\min\{\alpha(m+\Delta m), \alpha(m-\Delta m)\}>\alpha(m)$, than this unequal and unbalanced partitioning will result in faster execution.
In general, if the performance profile of an application violates the condition \[eq:cond\], that is, $$\exists {i \in \{1,\ldots,p\}} \exists{x>0} \exists{\Delta x>0}\colon \frac{s_i(x)} {x} < \frac{s_i (x+\Delta x)} {x+\Delta x}
\label{eq:not-cond}$$ and the balanced configuration of the application allocates the workload of size $x$ to processor $i$, then the application can be accelerated if we reduce the accumulated workload of all processors but processor $i$ by $\Delta x$ so that none of these processors would increase its execution time, and allocate this additional workload to processor $i$. This method can be applied to optimization of parallel applications on both heterogeneous and homogeneous platforms.
Model-based partitioning algorithm for optimal load imbalancing {#sec:model}
===============================================================
In this section, we develop the proposed approach for a relatively simple case and introduce a partitioning algorithm that finds the optimal distribution of computations of an application between homogeneous processors using the functional performance model of the application.
Consider the following problem. Let $p$ identical parallel processors be used to execute the workload of size $n$, and let $s(x)$ be the speed of execution of the workload of size $x$ by a processor. Let $\Delta x$ be the minimal granularity of workload so that each processor can be only allocated a multiple of $\Delta x$. The problem is to find the distribution of the workload of size $n$ between the $p$ processors, which minimizes the computation time of its parallel execution.
For simplicity, we assume that $\frac{n}{p}$ be a multiple of $\Delta x$ and $p$ be an even number. Then the following Algorithm \[Alg:Homo\] will solve this problem.
$x_{ropt}=x_{lopt}=x_r=x_l=\frac{n}{p}$ $t_{min}=\frac{\frac{n}{p}}{s(\frac{n}{p})}$ $x_r+=\Delta x$ $x_l-=\Delta x$ $t=\max(\frac{x_r}{s(x_r)},\frac{x_l}{s(x_l)})$ $t_{min}=t$ $x_{ropt}=x_r$ $x_{lopt}=x_l$
\[Alg:Homo\]
This algorithm returns the optimal distribution of the workload of size $n$ between $p$ processors so that each odd processor receives the workload of size $x_{lopt}$, and each even processor receives the workload of size $x_{ropt}$. It can be proved that the solution returned by this algorithm will always minimize the execution time of the given workload $n$. Note that the traditional load-balanced approach would assign the workload of size $\frac{n}{p}$ to all processors.
It is obvious that if we replace the speed function $s(x)$ by any function $a\times s(x)$, where $a=const$, then this algorithm will return the same solution. We will use this property when applying Algorithm \[Alg:Homo\] in Section \[subsec:model\].
Algorithm \[Alg:Homo\] can be easily generalized for an arbitrary (not only even) number of processors.
Application {#sec:app}
===========
In this section, we apply the partitioning algorithm proposed in Section \[sec:model\] to optimization of MPDATA on Intel Xeon Phi.
Intel MIC overview
------------------
The Intel MIC architecture is a relatively new system for high performance computing [@MICA]. Intel MIC combines many integrated Intel CPU cores into a single chip. This architecture is built to provide a general-purpose programming environment similar to that provided for Intel CPUs. It is capable of running applications written in industry-standard programming languages such as Fortran, C, and C++. The Intel Xeon Phi (codenamed Knights Corner) is the first product based on Intel MIC architecture. This coprocessor is delivered in form factor of a PCI express device, and can not be used as a stand-alone processor. However, it allows users to directly run individual applications in the native mode without the support of CPU.
In this study, we use the top-of-the-line Intel Xeon Phi 7120P coprocessor. It contains 61 cores clocked at 1.238 GHz, and 16 GB of on-board memory. As the Intel MIC architecture supports four-way hyper-threading, it totally gives 244 logical cores (threads) for a single chip. This coprocessor provides 352 GB/s of memory bandwidth. An important component of each Intel Xeon Phi processing core is its vector processing unit (VPU) [@SZU15], that significantly increases the computing power. Each VPU supports a new 512-bit SIMD instruction set called Intel Initial ManyCore Instructions. The theoretical peak performance of Intel Xeon Phi 7120P is 1208 GFlop/s for double precision numbers.
Introduction to MPDATA
----------------------
The MPDATA algorithm is a general approach for integrating the conservation laws of geophysical fluids on micro-to-planetary scales [@SMO01]. It belongs to the class of methods for the numerical simulation of fluid flows which are based on the sign-preserving properties of upstream differencing. The MPDATA scheme allows for solving advection problems, and offers several options to model a wide range of complex geophysical flows.
MPDATA corresponds to the group of nonoscillatory forward-in-time algorithms. The number of required time steps depends on a type of simulated physical phenomenon, and can exceed few millions especially when considering the MPDATA algorithm as a part of the EULAG model. For detailed description of the MPDATA mathematical scheme, the reader is referred to [@SMO06; @SMO90; @SMO01].
Each MPDATA time step is determined by a set of 17 computational stages, where each stage is responsible for calculating elements of a certain matrix. These stages represent stencil codes which update grid elements according to different patterns. Listing \[lst:MPDATAsrc\] shows a part of the 3D MPDATA stencil-based implementation for the 8-th stage.
/*...*/
//stage 8
for( ... ) // i - dimension
for( ... ) // j - dimension
for( ... ) // k - dimension
mx[i,j,k]=max(x[i][j][k],
x[i-1][j][k], x[i+1][j][k],
x[i][j-1][k], x[i][j+1][k],
x[i][j][k-1], x[i][j][k+1]);
/*...*/
The stages are dependent on each other: outcomes of prior stages are usually input data for the subsequent computations. Every stage reads a required set of matrices from the main memory, and writes results to the main memory after computation. In consequence, a significant traffic to the main memory is generated, which mostly limits the performance of novel architectures. A single MPDATA time step requires 5 input matrices, and returns one output matrix that is necessary for the next step.
Adaptation of 3D MPDATA to Intel Xeon Phi coprocessor {#subsec:adaptation}
-----------------------------------------------------
In our previous work [@SZU13; @SZU15], we proposed the adaptation of 3D MPDATA to Intel Xeon Phi coprocessors. The proposed decomposition contributes to ease the memory and communication bounds, and to better exploit computation resources of Intel Xeon Phi. The resulting adaptation is based on the following methodology:
- (3+1)D decomposition of MPDATA heterogeneous stencil computations;
- partitioning of threads into independent work teams;
- parallelization of MPDATA computations;
- scheduling for multicore and manycore systems.
To workaround the memory-bound nature of MPDATA, we proposed the (3+1)D decomposition of MPDATA stencil computation [@SZU15]. The main aim of this decomposition is to take advantage of cache memory reuse by transferring the data traffic associated with all intermediate computation from the main memory to the cache hierarchy. This aim is achieved by using a combination of two loop optimization techniques: loop tiling and loop fusion. Such an approach allows us to reduce the main memory traffic at the cost of additional computations associated with extra areas (*halo*) of all intermediate matrices. Another advantage of this approach is the possibility of reducing the main memory consumption because all intermediate results are stored only in the cache memory.
The proposed decomposition contributes to the data traffic from the main memory to the cache hierarchy. In consequence, a lot of inter- and intra-cache communications are generated between more than 200 Intel MIC’s processing cores. To improve the efficiency of the (3+1)D decomposition on Intel Xeon Phi, we provided partitioning of available cores (threads) into independent work teams. As a result, the MPDATA computing domain is partitioned into $P$ sub-domains of different sizes, each of which is processed by a single work team of threads, according to the proposed (3+1)D decomposition. The number of cores (threads) assigned to different teams can also be different. Figure \[fig:MPDATAteam\] shows an example of partitioning of 60 Intel MIC’s processing cores into $4$ teams, and partitioning of the MPDATA grid into $4$ pieces as well.
![Partitioning of Intel MIC’s processing cores into 4 work teams[]{data-label="fig:MPDATAteam"}](./Teams){width="50.00000%"}
Within every time step, the work teams execute computations in parallel and independently of each other. After each time step, the work teams are synchronized. Each sub-domain is further partitioned into a number of MPDATA blocks, where subsequent blocks are processed one by one, and each block is processed in parallel by the corresponding work team. Every block is further decomposed into sub-blocks, where each sub-block is processed by a certain thread of the work team. A sequence of all the MPDATA stages is executed within every sub-block, taking into account the data dependencies. This is achieved at the cost of some extra computations performed for halo regions by all teams.
Our best performance results on a single Intel Xeon Phi have been achieved so far by partitioning the 3D $n\times m\times l$ MPDATA domain in two dimensions $n$ and $m$ into four equal sub-domains, so that there is one-to-one mapping between these sub-domains and the teams of cores arranged in a $2\times2$ grid as illustrated in Figure \[fig:MPDATAteam\]. This partitioning allowed us to balance the load of the core teams and minimize the execution time of the application in comparison with all other partitioning shapes.
Applying model-based partitioning algorithm to MPDATA decomposition {#subsec:model}
-------------------------------------------------------------------
Thus, the best load-balanced configuration of the MPDATA application on a Intel Xeon Phi arranges its cores in four $15$-core teams as shown in Figure \[fig:MPDATAteam\] and evenly partitions the $n\times m\times l$ computation domain between these teams, allocating a $\frac{n}{2}\times\frac{m}{2}\times l$ sub-domain to each of the teams. In this subsection, we use the data-partitioning algorithm, presented in Section \[sec:model\], to find a better partitioning of the computation domain between these teams of cores.
As a first step, we build speed functions of the teams so that the speed of each team be represented by a function of problem size. In the case of MPDATA, the problem size is characterized by the size of the domain processed by the team and therefore represented by three parameters $n$, $m$ and $l$. In real-life NWP simulations $l$ is fixed [@SZU15]. Therefore, we build speeds of teams as functions of two parameters $n$ and $m$, setting $l$ to $128$, the value typically used in simulations.
In general, the speed should be measured in equal-sized computation units performed per one time unit [@ijhpca2007], for example, in flops. In the case of MPDATA, it is difficult to estimate the amount of arithmetic operations that will be executed during the processing of a $n\times m \times l$ computation domain. We know however that with a very high level of accuracy this amount is directly proportional to the number of cells in this domain. Therefore, we measure the speed in cells per second.
The speed functions are built empirically by benchmarking the work teams for a range of problem sizes. For each problem size $(n,m)$, the speed is calculated as $\frac{n\times m\times 128}{t}$, where $t$ is the measured execution time.
It has been shown [@ziming2011; @zimingTC2014] that in modern multicore, manycore and hybrid platforms, where processing elements are coupled and share resources, the speed of one group of elements may depend on the load of others due to resource contention. Therefore, the groups cannot be considered as independent processing units and their speed cannot be measured separately and independently. In this work, we use the performance measurement method proposed in [@zimingTC2014] . According to this method, the performance of the four teams of cores is measured simultaneously rather than separately, thereby taking into account resource contention. To ensure the reliability of measurements, we repeat measurements multiple times. We only measure the computation time of every team without the overheads of inter-team synchronization required after each time step. If the measurements were conducted separately, the measured performance of these teams would not reflect their actual performance during the execution of the application, and therefore performance optimization decisions based on the corresponding performance models would be inaccurate. Figure \[fig:graph3\_0\] demonstrates the difference between the speed of team $T_0$ measured separately and simultaneously with other teams.
![Comparison of speed functions of team $T_0$, measured separately($S_{T_0}^*(x)$) and simultaneously with other three teams ($S_{T_0}(x)$) executing the same workload[]{data-label="fig:graph3_0"}](./graph3_0){width="50.00000%"}
Figure \[fig:graph2\] illustrates the speed of team $T_0$ as a function of parameters $n$ and $m$ (remember that $l=128$). The experimental points for the speed function were obtained with steps $\Delta n=\Delta m=4$ for both $n$ and $m$ parameters.
![Experimentally built speed of execution of the MPDATA workload by team $T_0$ as function of two parameters $n$ and $m$ ($l=128$)[]{data-label="fig:graph2"}](./graph2){width="50.00000%"}
We can see that for a fixed value of $m$ the speed varies very slowly and very little with variation of $n$, staying nearly constant. More detailed analysis of the speed functions confirms that the speed of team strongly depends on $m$ and very little depends on $n$. This observation allows us to assume that with a high level of accuracy the optimal (or at least a near optimal) partitioning of a $N\times M \times 128$ domain between the four teams can be obtained from the optimal even load-balanced partitioning, which allocates a sub-domain of size $\frac{N}{2} \times \frac{M}{2} \times 128$ to each team, by fixing the first parameter $n$ to $\frac{N}{2}$ and varying $m$.
Mathematically, it means that we only have to deal with speed functions of just one parameter, $m$. These functions are obtained from the previously built speed functions of two parameters, $n$ and $m$, by fixing the parameter $n$. Geometrically, this can be illustrated as follows. The two-parameter speed functions are represented by surfaces. By fixing parameter $n$ to $\frac{N}{2}$, we cut the surfaces by a vertical plane $n=\frac{N}{2}$ as shown in Figure \[fig:graph2\], obtaining on this plane curves, representing the one-parameter speed functions as shown in Figure \[fig:graph3\] for $n=120$.
![Speeds of four teams built simultaneously as functions of parameter $m$ ( and )[]{data-label="fig:graph3"}](./graph3){width="50.00000%"}
Finally, as all four teams have very close speed functions (as can be seen in Figure \[fig:graph3\]), we calculate their average (shown in Figure \[fig:graph4\]) and use it as input to Algorithm \[Alg:Homo\] to find the optimal value of $m$ for each team.
![Averaged speed of teams built as a function of parameter $m$ ( and )[]{data-label="fig:graph4"}](./graph4){width="50.00000%"}
More specifically, let the MPDATA domain be of size $240 \times 240 \times 128$. Then, we consider our four teams as four identical abstract processors, $p=4$, the speed of each of which is given by the speed function shown in Figure \[fig:graph4\]. Note that in this function, the amount of workload is given in frames of cells of size $120 \times 128$, while the speed is given in cells per second. As pointed in Section \[sec:model\], despite the unit of workload used to measure the speed (axis $y$) is $120 \times 128$ times greater than the unit of workload used to measure the size of workload (axis $x$), we can safely use this function as input to Algorthm \[Alg:Homo\].
The solution returned by Algorthm \[Alg:Homo\] allocates $m=112$ frames to even abstract processors and $m=128$ frames to odd processors. This corresponds to partitioning of the $240 \times 240 \times 128$ domain into two sub-domains of size $120 \times 112 \times 128$, allocated to teams $T_0$ and $T_2$, and two sub-domains of size $120 \times 128 \times 128$, allocated to teams $T_1$ and $T_3$. The traditional load-balanced approach partitions the domain in four equal sub-domains of size $120 \times 120 \times 128$. This is illustrated in Figure \[fig:TeamsNew\].
![Optimal partitioning of MPDATA of size $240 \times 240 \times 128$ between 4 teams[]{data-label="fig:TeamsNew"}](./TeamsNew){width="50.00000%"}
The theoretical execution time of the even partitioning is 1.486 seconds, while the theoretical execution time of the uneven partitioning returned by Algorithm \[Alg:Homo\] is 1.386: $$\begin{split}
t=\max(\frac{x_r}{s(x_r)},\frac{x_l}{s(x_l)})
\end{split}$$ $$t= \max(\frac{120 \cdot 128 \cdot 128 }{1418579},\frac{120 \cdot 112 \cdot 128 }{ 1436742}) = 1.386[s].$$
Experimental results {#sec:result}
====================
In this section, we experimentally evaluate the optimization technique presented in Section \[sec:app\].
The performance results presented in this section are obtained for double precision MPDATA computations corresponding to 40 time steps. All the benchmarks are compiled as native executables using the Intel compiler (v.15.0.2), and run on the Intel Xeon Phi 7120P coprocessor. To ensure the reliability of the results, measurements are repeated multiple times, and average execution times are used. We find the confidence interval and stop the measurements if the sample mean lies in the interval with the confidence level 95%. We use Student’s *t*-test, assuming that the individual observations are independent and their population follows the normal distribution.
Table \[tab:240\] includes both theoretical and experimental execution times of MPDATA for the domain of size $240 \times 240 \times 128$. These results are obtained for different configurations of partitioning, including the traditional “load-balanced” partitioning ($\Delta m = 0$) and a range of “unbalanced” partitioning for different $\Delta m >0$. The theoretically optimal $\Delta m$ returned by Algorithm \[Alg:Homo\] is equal to $8$, which corresponds to the configuration where each odd or even team processes the sub-domain of size $120\times128\times128$ or $120\times112\times128$ respectively. In this case, the estimated execution time of 1.386 seconds is very close to the real computation time which is 1.364 seconds. According to experiments, the shortest execution time is achieved for $\Delta m=9$, when computations take 1.348 seconds.
------------ ------------- -------------- ---------
Offset Theoretical Experimental
$\Delta m$ time \[s\] time \[s\] Speedup
0 1.486 1.548 1.000
4 1.470 1.470 1.053
6 1.401 1.374 1.127
7 1.422 1.361 1.137
8 1.386 1.364 1.135
9 1.398 1.348 1.148
10 1.397 1.352 1.145
11 1.429 1.372 1.129
12 1.402 1.368 1.131
------------ ------------- -------------- ---------
: Theoretical and experimental execution times for MPDATA domain of size $240 \times 240 \times 128$ with different configurations of partitioning. The odd work teams process the sub-domain of size $120 \times (120+\Delta m) \times 128$, while the even teams –$120 \times (120-\Delta m) \times 128$.[]{data-label="tab:240"}
Comparing the experimental and theoretical times, we can see that the accuracy of theoretical prediction is very good, with prediction errors being as small as $2-4\%$. In general, we can identify two main factors contributing into the prediction error:
- While the experimentally built speed functions of teams $T_0$, $T_1$, $T_2$ and $T_3$ are not identical, suggesting some degree of their heterogeneity in execution of the MPDATA workload, our theoretical model considers them homogeneous and represents their speed by the average of the real speed functions, which is then used as input to Algorithm \[Alg:Homo\].
- During the construction of the speed functions, the speed of a team for problem size $n\times m\times l$ is measured when other teams process in parallel sub-domains of the same size, $n\times m\times l$. However, during the execution of the application in our experiments different teams process sub-domains of slightly different sizes when $\Delta m\neq 0$.
Table \[tab:240\] also demonstrates the performance gain from applying the proposed load-imbalancing optimization. For the imbalanced configurations presented in this table, we notice a better performance than for the load-balanced configuration of the MPDATA decomposition. The largest performance gain is achieved for $\Delta m=9$, giving the speedup of 1.148x.
------------ -------- -------- -------- -------- -------
Offset
$\Delta m$ Team 0 Team 1 Team 2 Team 3 Total
0 1.515 1.498 1.518 1.503 1.548
4 1.456 1.247 1.455 1.249 1.470
6 1.364 1.161 1.359 1.162 1.374
7 1.355 1.161 1.341 1.168 1.361
8 1.355 1.166 1.349 1.172 1.364
9 1.340 1.155 1.335 1.161 1.348
10 1.345 1.141 1.337 1.152 1.352
11 1.363 1.156 1.357 1.154 1.372
12 1.360 1.163 1.353 1.165 1.368
------------ -------- -------- -------- -------- -------
: Experimental time for all work teams with different partitionings: the odd work teams process the sub-domain of size $120 \times (120+\Delta m) \times 128$, while the even teams –$120 \times (120-\Delta m) \times 128$.[]{data-label="tab:teams"}
Table \[tab:teams\] complements the results in Table \[tab:240\] giving experimental execution times of the individual teams. We can clearly see a significant difference between the execution times measured for the odd and even teams when $\Delta m \ne 0$. Obviously, this difference is caused by the unbalanced workloads for the odd and even teams. However, the total execution time is shorter than in the case of balanced workloads ($\Delta m=0$).
Table \[tab:teams\] also shows that the total execution time is always slightly longer than the maximum time among all teams. It is mainly due to the fact that the computation time of every team is measured without the overheads of inter-team synchronization required after each time step. In addition, the results in Table \[tab:teams\] are presented in in a graphical form.
![Experimental execution times measured for individual work teams and the total execution time measured for the whole MPDATA workload[]{data-label="fig:result"}](./result){width="50.00000%"}
Finally, we evaluate the proposed model-based partitioning algorithm for the MPDATA domain of size $480 \times 480 \times 128$. As in the previous case, the application is executed for different configurations of partitioning, for a range of $\Delta m$. In this case, however, the theoretically optimal configuration returned by Algorithm \[Alg:Homo\] is exactly the same as the experimentally optimal one, both achieved when $\Delta m=20$. The prediction errors are also smaller in this case, not exceeding $3\%$. The experimental execution time for $\Delta m=20$ is 5.338 seconds, in comparison with 6.140 seconds for the even partitioning.This allows us to accelerate the MPDATA computations by 1.15 times. Moreover, the performance gain is also observed for other unbalanced configurations, but it is smaller than 1.15x. The results of these experiments are included in Table \[tab:480\].
------------ ------------- -------------- ---------
Offset Theoretical Experimental
$\Delta m$ time \[s\] time \[s\] Speedup
0 6.136 6.140 1.000
4 5.731 5.681 1.081
8 5.809 5.806 1.058
12 5.543 5.453 1.126
16 5.509 5.418 1.133
20 5.499 5.338 1.150
24 5.624 5.477 1.121
------------ ------------- -------------- ---------
: Theoretical and experimental execution times for MPDATA domain of size $480 \times 480 \times 128$ with different configurations of partitioning. The odd work teams process the sub-domain of size $240 \times (240+\Delta m) \times 128$, while the even teams – $240 \times (240-\Delta m) \times 128$.[]{data-label="tab:480"}
Conclusion {#sec:con}
==========
Modern compute nodes are characterized by both the increasing number of (possibly, heterogeneous) processing elements and a high level of complexity of their integration. Various resources such as caches and data links are shared in an hierarchical and non-uniform way. This makes the development of efficient applications for such platforms a very difficult and challenging task. It would be naive to expect that the performance profile of real-life scientific applications on these platforms will always be comfortably nice and smooth to suit traditional load-balancing techniques used for minimization of their computation time. Therefore, new optimization approaches that do not rely on such increasingly unrealistic assumptions are needed. This work has presented one such approach and demonstrated its applicability to optimization of a real-life application on a modern HPC platform.
**Acknowledgments**\
This research was conducted with the financial support of NCN under grant no. UMO-2011/03/B/ST6/03500. We gratefully acknowledge the help and support provided by Jamie Wilcox from Intel EMEA Technical Marketing HPC Lab. This work is partially supported by EU under the COST Program Action IC1305: Network for Sustainable Ultrascale Computing (NESUS).
|
---
abstract: |
We show that for any regular bounded domain $\Omega{\subseteq}{\mathbb{R}}^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on ${\partial}\Omega$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.
0.5
[**Résumé**]{} 0.5Nous montrons que pour tout domaine borné régulier $ \Omega {\subseteq}{\mathbb{R}}^n $, $n=2,3$, il existe une infinité de difféomorphismes globaux solutions de l’équation iconale, égaux à l’identité sur $ {\partial}\Omega $. Nous donnons également des exemples explicites de telles cartes dans des domaines annulaires. Ceci implique que le systéme du type $ \infty $-Laplacien apparaissant dans le Calcul des Variations vectoriel dans $ L^\infty $ ne suffit pas à caractériser les limites pour $p\to \infty$ des cartes $ p $-harmoniques, ni les minimiseurs absolus au sens d’Aronsson. [**Contre-exemples dans le Calcul des Variations dans $ L^\infty $ par l’équation iconale vectorielle**]{}
address: 'Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, Berkshire, England, UNITED KINGDOM'
author:
- Nikos Katzourakis
- Giles Shaw
title: 'Counterexamples in Calculus of Variations in $L^\infty$ through the vectorial Eikonal equation'
---
,
Introduction {#section1}
============
Calculus of Variations in $L^\infty$ is concerned with the variational study of supremal functionals, as well as with the necessary conditions governing their extrema. The archetypal model of interest is the functional $$\label{eq:minproblem}
{\mathrm{E}}_\infty(u,{\mathcal{O}})\,:=\, {{\mathrm{ess}}\sup}_{\mathcal{O}}|{\mathrm{D}}u|,\ \ \text{ for } u\in\mathrm{W}^{1,\infty}(\Omega;{\mathbb{R}}^N), \ {\mathcal{O}}{\subseteq}{\Omega}\text{ measurable},$$ where $n,N\in{\mathbb{N}}$, ${\Omega}{\subseteq}{\mathbb{R}}^n$ is a fixed open set and ${\mathrm{D}}u(x)=({\mathrm{D}}_i u_{\alpha}(x))_{i=1...n}^{{\alpha}=1...N} \in {\mathbb{R}}^{N\times n}$ is the gradient matrix. We note that our general notation is either self-explanatory or standard. In and throughout the remainder of this note, all norms appearing will be the Euclidean ones. In particular, on ${\mathbb{R}}^{N\times n}$ we use the norm induced by the inner product $A:B:=\smash{\sum_{{\alpha}, i}A_{{\alpha}i}B_{{\alpha}i}}$. Aronsson was the first to consider such problems in the 1960s [@Aronss67EFL; @A4], in the scalar-valued case $N=1$. In the general case of , the PDE system which arises from as the analogue of the Euler-Lagrange equations is the $\infty$-Laplace system $$\label{eq:infinitylaplace}
\Delta_\infty u \, :=\, \Big({\mathrm{D}}u {\otimes}{\mathrm{D}}u \,+\, |{\mathrm{D}}u|^2[\![{\mathrm{D}}u ]\!]^\bot \! {\otimes}\mathrm I\Big) :{\mathrm{D}}^2u \, =\, 0,$$ and has its origins in the paper [@Katzou12LVP]. Here, for any linear map $A :{\mathbb{R}}^n {\longrightarrow}{\mathbb{R}}^N$, $[\![A]\!]^\bot$ denotes the orthogonal projection on the orthogonal complement of the range $(\mathrm R(A))^\bot {\subseteq}{\mathbb{R}}^N$. In index form, reads $$\sum_{1\leq {\beta}\leq N}\sum_{1\leq i,j \leq n} \Big({\mathrm{D}}_i u_{\alpha}\, {\mathrm{D}}_j u_{\beta}\,+\, |{\mathrm{D}}u|^2[\![{\mathrm{D}}u ]\!]_{{\alpha}{\beta}}^\bot {\delta}_{ij}\Big) {\mathrm{D}}_{ij}^2u_{\beta}\, =\, 0,\ \ \ \ 1\leq {\alpha}\leq N.$$ A fundamental difficulty in the variational study of is that the usual global minimisers in the space $g+\mathrm{W}_0^{1,\infty}(\Omega;{\mathbb{R}}^N)$ are not truly optimal and may not solve any PDE. To this end, the notion of absolute minimisers has been introduced. Indeed, in the scalar case it is known that absolute minimisers of correspond uniquely to (viscosity) solutions of the *scalar* version of , which reduces to $\sum_{i,j}{\mathrm{D}}_i u {\mathrm{D}}_j u {\mathrm{D}}^2_{ij}u=0$ (see e.g. [@BhDiMa89LPRE; @Jensen93ULEM; @ACJ; @C; @Kbook1]). The “localised" concept of absolute minimisers is what forces to define on subsets of ${\Omega}$. In the vectorial case, the situation is more delicate and not fully understood yet, particularly when ${\mathrm{D}}u$ has rank greater than two [@BJW1; @AyK; @AK; @K9]. A by now standard mechanism to study - is through approximation by the respective $L^p$ variational notions as $p\to \infty$, namely by using $$\label{LpObjects}
{\mathrm{E}}_p(u):= \|{\mathrm{D}}u\|_{L^p({\Omega})},\ \text{ for } u \in\mathrm{W}^{1,p}(\Omega;{\mathbb{R}}^N) \ \ \text{ and } \ \ \ {\Delta}_pu:= {\mathrm{div}}\big(|{\mathrm{D}}u|^{p-2}{\mathrm{D}}u \big)=0,$$ which are known as the $p$-Dirichlet functional and the $p$-Laplacian. Hence, the identification of necessary and sufficient conditions for a mapping $u\in\mathrm{W}^{1,\infty}(\Omega;{\mathbb{R}}^N)$ to occur as a (weak) limit $u$ of $p$-harmonic maps $u_p$ is of interest (see [@BJW1; @BhDiMa89LPRE; @Jensen93ULEM; @K9]). Intuitively, we expect such limits to be “optimal” solutions, possibly absolute minimisers of . In the case $N=1$, a complete picture is known: the family $(u_p)_{p\geq 1}$ converges to a unique limit which is an absolute minimiser. Additionally, it follows from the form of the PDE that differentiable Eikonal functions solving $|{\mathrm{D}}u|=\mathrm{const}$, also satisfy and therefore is a $p$-harmonic limit.
On the other hand, in the case $N\geq2$, one can show the existence of infinitely-many (appropriately defined) [*generalised $\mathrm{W}^{1,\infty}$ solutions*]{} to which are [*not minimising for* ]{}, let alone absolutely minimising, see [@K7; @CKP]. A natural question is whether this phenomenon is a defect of the notion of solution used. The principal results of this note are Theorem \[thm:mainthm\] and Corollary \[cor:mainthm\], which answer this to the negative. Accordingly, we show for $n=N\in\{2,3\}$ the existence of infinitely many [*arbitrarily regular*]{} orientation preserving Eikonal diffeomorphisms $u : \Omega{\longrightarrow}{\mathbb{R}}^n$ with given affine boundary conditions. These maps are [*a fortiori $\infty$-Harmonic*]{}, since $[\![{\mathrm{D}}u]\!]^\perp\! = 0$ when $\det({\mathrm{D}}u)\neq 0$ and can be recast as the two independent systems $${\mathrm{D}}u \,{\mathrm{D}}\big(|{\mathrm{D}}u|^2\big) = 0\ \ \text{ and } \ \ |{\mathrm{D}}u|^2[\![{\mathrm{D}}u]\!]^\perp\Delta u = 0.$$
\[thm:mainthm\] Let $\Omega{\subseteq}{\mathbb{R}}^n$ be a bounded connected domain such that $n\in\{2,3\}$ and ${\partial}\Omega$ is ${\mathrm{C}}^{m+4 }$ for $m \geq2$. Then, there exist infinitely many maps $u\in {\mathrm{C}}^m\big(\overline{\Omega};{\mathbb{R}}^n\big)$ satisfying $$\text{$|{\mathrm{D}}u |\equiv \mathrm{const}$ in $\overline{{\Omega}}$, \ $\det({\mathrm{D}}u)>0$ in $\overline{{\Omega}}$ \ and \ $u=\mathrm{id}$ on ${\partial}\Omega$.}$$ Any such $u$ is an Eikonal orientation preserving diffeomorphism, equal to the identity on the boundary.
\[cor:mainthm\] Let $n,m,{\Omega}$ be as in Theorem \[thm:mainthm\]. Then, the Dirichlet problem for the $\infty$-Laplacian $$\text{${\Delta}_\infty u =0 $ in ${\Omega}$ \ \ and \ \ $u=\mathrm{id}$ on ${\partial}\Omega$,}$$ possesses infinitely-many classical solutions $u\in {\mathrm{C}}^m\big(\overline{\Omega};{\mathbb{R}}^n\big) {\setminus}\{\mathrm{id}\}$. In addition, none of these solutions minimises ${\mathrm{E}}_\infty(\cdot,{\Omega})$ among all maps in $W^{1,\infty}({\Omega},{\mathbb{R}}^n)$ with $u=\mathrm{id}$ on ${\partial}{\Omega}$.
We note that the results above improve and supersede one of the main results in [@K6] which required ${\Omega}$ to be a punctured ball. Since the unique solution to the Dirichlet problem for $\Delta_p u = 0$ in ${\Omega}$ with $u=\mathrm{id}$ on ${\partial}{\Omega}$ is $u(x)\equiv x$ when $p<\infty$, it follows that none of our diffeomorphisms is a limit of $p$-harmonic maps as $p\to \infty$. Thus, we confirm that by itself cannot suffice to identify limits of $p$-harmonic maps and that [*additional selection criteria are needed*]{} to have a situation analogous to the scalar case.
The proof of Theorem \[thm:mainthm\] is based on the next result of independent interest.
\[prop:functionalroots\] Let $n,m,{\Omega}$ be as in Theorem \[thm:mainthm\]. Then, the nonlinear problem $$|{\mathrm{D}}u|^2 +\, 2\, {\mathrm{div}}\, u \,\equiv \,C \, \text{ in }{\Omega}\ \ \ \text{ and } \ \ \ u=0 \, \text{ on }{\partial}{\Omega},$$ has infinitely many non-trivial solutions $(u,C) \in ({\mathrm{C}}^m\cap{\mathrm{C}}^0_0)\big(\overline{\Omega};{\mathbb{R}}^n\big) {\times}(0,\infty)$. Additionally, the set of all solutions has the trivial solution $(0,0)$ as an accumulation point with respect to the topology of $\smash{{\mathrm{C}}^m\big(\overline{\Omega};{\mathbb{R}}^n\big)}$.
Since the proofs of the above results are non constructive, we include in Section \[ex:counterexample\] explicit examples of smooth $\infty$-Harmonic maps defined on annular domains which coincide with affine maps on the boundary.
Proofs
======
We begin with the proof of Proposition \[prop:functionalroots\], which is an immediate consequence of the next lemma and of the Morrey estimate, in the form of inclusion of spaces ${\mathrm{H}}^{m+2}(\Omega;{\mathbb{R}}^n){\subseteq}{\mathrm{C}}^m\big(\overline{\Omega};{\mathbb{R}}^n\big)$ (since $n\in\{2,3\}$).
\[lem:functionalroots\] Let $n,m,{\Omega},$ be as in Theorem \[thm:mainthm\] and let us define the nonlinear mapping $${\mathcal{M}}\ : \ \ ({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n){\longrightarrow}{\mathrm{H}}^{m+1}_\sharp(\Omega)\,:= \,\left\{w\in{\mathrm{H}}^{m+1}(\Omega) :\ \int_\Omega w(x)\;{\mathrm{d}}x = 0\right\}$$ by setting (here the slashed integral denotes the average) $${\mathcal{M}}[u] \, := \, \frac{1}{2}|{\mathrm{D}}u|^2 \, +\, {\mathrm{div}}\, u \, -\, \frac{1}{2} \, \dashint_\Omega |{\mathrm{D}}u(x) |^2\;{\mathrm{d}}x.$$ Then, the inverse image ${\mathcal{M}}^{-1}[\{0\}]$ contains infinitely-many elements accumulating at zero. In addition, for any $\varepsilon>0$, there exists $\varphi_{\varepsilon}\in({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)\setminus\{0\}$ such that ${\mathcal{M}}[\varphi_{\varepsilon}]=0$ and $\|\varphi_{\varepsilon}\|_{{\mathrm{H}}^{m+2}({\Omega})}<\varepsilon$.
[**Proof of Lemma** ]{}\[lem:functionalroots\]. First note that ${\mathcal{M}}$ is well defined, namely its image lies in the subspace $\smash{{\mathrm{H}}^{m+1}_\sharp(\Omega)}$ of zero average. Indeed, for any $u\in ({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)$, the divergence theorem gives $$\int_{\Omega}{\mathcal{M}}[u](x)\,{\mathrm{d}}x\, =\, \int_\Omega {\mathrm{div}}\, u (x)\,{\mathrm{d}}x \, =\, \int_{{\partial}\Omega} u(x)\cdot n(x)\,{\mathrm{d}}\mathcal{H}^{n-1}(x) \,=\, 0,$$ where $n : {\partial}\Omega{\longrightarrow}{\mathbb{R}}^n$ denotes the outward pointing normal vector to ${\partial}\Omega$ and $\mathcal{H}^{n-1}$ is the $(n-1)$-Hausdorff measure. Additionally, we need to confirm that $|{\mathrm{D}}^p(|{\mathrm{D}}u|^2)| \in \mathrm L^2({\Omega})$ for all $p\in\{0,...,m+1\}$. Indeed, by the Leibniz formula we have $|{\mathrm{D}}^p(|{\mathrm{D}}u|^2)|\leq \sum_{i=0}^{p}C_{i,p}|{\mathrm{D}}^{p+1-i}u| | {\mathrm{D}}^{1+i}u|$, where $C_{i,p}$ is the binomial coefficient. Since $\min\{p+1-i,1+i\}\leq m$ for all $p$, Hölder’s inequality gives $|{\mathrm{D}}^{p+1-i}u| | {\mathrm{D}}^{1+i}u| \in \mathrm L^2({\Omega})$ for any $i$ and $p$, because by the Sobolev inequality we have $u,|{\mathrm{D}}u|,...,|{\mathrm{D}}^m u| \in \mathrm L^\infty({\Omega})$. Next, note that ${\mathcal{M}}$ is Fréchet differentiable at each $u\in\smash{({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)}$ with $$\ \ \ \ \ {\mathcal{M}}'[u]\varphi \, =\, {\mathrm{D}}u :{\mathrm{D}}\varphi \, +\, {\mathrm{div}}\, \varphi \, -\, \dashint_\Omega ({\mathrm{D}}u :{\mathrm{D}}\varphi )(x)\;{\mathrm{d}}x,\qquad\text{ for all }\varphi\in ({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n).$$ In particular, ${\mathcal{M}}'[0]={\mathrm{div}}$ and also ${\mathcal{M}}'[0]$ is a bounded linear surjection from $({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)$ into $\smash{{\mathrm{H}}^{m+1}_\sharp(\Omega)}$; the surjectivity of ${\mathcal{M}}'[0]$ is a consequence of Lemma \[Lemma2.2\] that follows. Next, since $\ker({\mathcal{M}}'[0])=\big\{v\in({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n) : {\mathrm{div}}\, v\equiv 0 \big\}$ is a closed subspace of the Hilbert space $({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)$, it possesses an orthogonal complement $V{\subseteq}({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)$: $$\ker({\mathcal{M}}'[0]) \oplus V \, =\, ({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n).$$ By noting that ${\mathcal{M}}'[0]|_{ V} : V{\longrightarrow}\smash{{\mathrm{H}}^{m+1}_\sharp(\Omega)}$ is a linear isomorphism, the canonical isomorphism between $\ker({\mathcal{M}}'[0])\oplus V$ and $\ker({\mathcal{M}}'[0])\times V$ allows us to view ${\mathcal{M}}$ as a map on $\ker({\mathcal{M}}'[0])\times V$ by setting ${\mathcal{M}}[(u,v)]:={\mathcal{M}}[u+v]$. Then, the implicit function theorem (see e.g. [@Zeidle95AFA Th. 4.E]) implies that, for ${\varepsilon}>0$ small enough, there exists a continuous map $\gamma : \ker({\mathcal{M}}'[0]) \cap \{v: \|v \|_{{\mathrm{H}}^{m+2}({\Omega})}<{\varepsilon}\} {\longrightarrow}V$ with $\gamma(0)=0$ and $${\mathcal{M}}[\varphi+\gamma(\varphi)] = 0 \quad\text{ for all }\varphi\in\ker({\mathcal{M}}'[0])\text{ with }\|\varphi\|_{{\mathrm{H}}^{m+2}({\Omega})}<{\varepsilon}.$$ Consequently, since $\ker({\mathcal{M}}'[0]) \neq \{0\}$ (as for instance $\mathrm{curl}^*\psi \in \ker({\mathcal{M}}'[0])$ for any $\psi\in{\mathrm{C}}^\infty_c(\Omega;{\mathbb{R}}^{n\times n}_{\mathrm{skew}})$) and $\gamma$ is continuous with $\gamma(0)=0$ we deduce that, for every ${\varepsilon}>0$, there exists $\varphi_{\varepsilon}\in({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)$ such that ${\mathcal{M}}[\varphi_{\varepsilon}]=0$ and $\|\varphi_{\varepsilon}\|_{{\mathrm{H}}^{m+2}({\Omega})}<{\varepsilon}$.
The next result completes the proof of Lemma \[lem:functionalroots\].
\[Lemma2.2\] For any $f\in {\mathrm{H}}^{m+1}_\sharp(\Omega)$, the next Dirichlet problem admits a solution in $({\mathrm{H}}^{m+2}\cap{\mathrm{H}}^1_0)(\Omega;{\mathbb{R}}^n)$: $${\mathrm{div}}\, u = f \ \text{ in }\Omega \ \ \text{ and } \ \ u =0 \ \text{ on }{\partial}\Omega.$$
[**Proof of Lemma** ]{}\[Lemma2.2\]. The claim follows from the Sobolev version of arguments presented in [@CsDaKn12TPB Ch. 9] and standard regularity results for the Neumann problem, which we sketch briefly for completeness. Since ${\partial}\Omega$ is assumed to be ${\mathrm{C}}^{m+4 }$, regularity theory for Poisson’s equation implies that we can always find $w\in {\mathrm{H}}^{m+3}(\Omega)$ such that $\Delta w=f$ in $\Omega$ and ${\mathrm{D}}_n w = 0$ on ${\partial}\Omega$. It remains to show that we can find $b\in{\mathrm{H}}^{m+2}(\Omega;{\mathbb{R}}^{n})$ such that ${\mathrm{div}}\, b= 0$ in $\Omega$ and $b=-{\mathrm{D}}w$ on ${\partial}\Omega$ (since we can then take $u := {\mathrm{D}}w -b$ as our desired solution). To this end, let us fix $(i,j)\in\{1,\ldots,n\}^2$ and set $c_{ij}:=(n_j\, {\mathrm{D}}_i w -n_i\, {\mathrm{D}}_j w)n$. Consider then the Biharmonic function $d_{ij}\in {\mathrm{H}}^{m+3}(\Omega)$ solving the Dirichlet problem (see [@GGS Th. 2.2]) $$\Delta^2 d_{ij} =0 \ \text{in }\Omega \ \ \text{ and }\ \ d_{ij}=0\ \text{on }{\partial}\Omega \ \ \text{ and }\ \ {\mathrm{D}}d_{ij} = c_{ij}\ \text{on }{\partial}\Omega.$$ Defining $b:=\mathrm{curl}^*d \in{\mathrm{H}}^{m+2}(\Omega;{\mathbb{R}}^n)$, where $\big(\mathrm{curl}^*d\big)_i := \sum_{j<i}{\mathrm{D}}_j d_{ji}-\sum_{j>i}{\mathrm{D}}_j d_{ij}$, we see that ${\mathrm{div}}\, b =0$ in $\Omega$ and by using that ${\mathrm{D}}_n w=0$ on ${\partial}\Omega$, we can easily confirm that $b=-{\mathrm{D}}w$ on ${\partial}\Omega$.
Now we may establish our main result.
[**Proof of Theorem** ]{}\[thm:mainthm\]. By continuity of the determinant, there exists $\varepsilon>0$ such that $|A|<\varepsilon$ implies $\det(\mathrm I +A)>\frac{1}{2}$. Using Proposition \[prop:functionalroots\], we can find $\varphi\in{\mathrm{C}}^m\big(\overline{\Omega};{\mathbb{R}}^n\big){\setminus}\{0\}$ with $\phi|_{ {\partial}\Omega}=0$ and $\|{\mathrm{D}}\varphi\|_{C^0({\Omega})}<{\varepsilon}$, satisfying $$|{\mathrm{D}}\varphi |^2 +2{\mathrm{div}}\varphi \, \equiv \, C\ \ \text{ and } \ \
\det(\mathrm{I} +{\mathrm{D}}\varphi ) > {1}/{2}\ \text{ in }\Omega,$$ for some $C>0$. Defining $u\in{\mathrm{C}}^m\big(\overline{\Omega};{\mathbb{R}}^n\big)$ by $u:=\mathrm{id} + \varphi$, we have $u=\mathrm{id}$ on ${\partial}\Omega$. Additionally, $$\begin{aligned}
|{\mathrm{D}}u |^2=|\mathrm I + {\mathrm{D}}\varphi |^2 \, &=\, |\mathrm I|^2\,+\,2\,\mathrm I\!:\!{\mathrm{D}}\varphi \, +\, |{\mathrm{D}}\varphi |^2\, =\, n^2 \, +\, \big(2\,{\mathrm{div}}\,\varphi \, +\, |{\mathrm{D}}\varphi |^2\big)\, =\, n^2 +C\end{aligned}$$ and also $\det({\mathrm{D}}u)\geq 1/2$ on ${\Omega}$, as required. Evidently, $u$ is a local diffeomorphism from ${\Omega}$ into ${\mathbb{R}}^n$. The fact that $u$ is a global diffeomorphism follows from standard degree theory results (see e.g. [@CsDaKn12TPB Th. 19.12]).
[**Proof of Corollary** ]{}\[cor:mainthm\]. Evidently, for any $u$ as above we have ${\Delta}_\infty u=0$ in ${\Omega}$ and $u=\mathrm{id}$ on ${\partial}{\Omega}$. Further, since $|{\mathrm{D}}u |^2\equiv n^2 +C= |\mathrm I|^2+C>|\mathrm I|^2$ and also ${\mathrm{D}}(\mathrm{id})=\mathrm I$, we obtain ${\mathrm{E}}_\infty(u,{\Omega})>{\mathrm{E}}_\infty(\mathrm{id},{\Omega})$.
Explicit constructions {#ex:counterexample}
======================
\[explicit\]
Let $n\in2{\mathbb{N}}$, ${\Omega}:=\{x\in{\mathbb{R}}^{n}: 1<|x|< e^{2\pi}\}$ and $S\in{\mathbb{R}}^{n\times n}$ an orthogonal, skew-symmetric matrix whose spectrum satisfies $\sigma(S){\subseteq}\{\pm i, 0\}$ so that $e^{2\pi S} =\mathrm I$. Let $u : \overline{{\Omega}}{\longrightarrow}{\mathbb{R}}^{n}$ be given by $u(x):=\mathrm{e}^{\log(|x|)S}x$. Then, $u \in {\mathrm{C}}^\infty\big(\overline{{\Omega}};{\mathbb{R}}^{n}\big){\setminus}\{\mathrm{id}\}$, $u=\mathrm{id}$ on ${\partial}{\Omega}$, $|{\mathrm{D}}u|^2\equiv n^2+1$ in $\overline{{\Omega}}$ and $\det({\mathrm{D}}u)\equiv 1$ in $\overline{{\Omega}}$. In particular, $u$ is a global $\infty$-Harmonic orientation preserving diffeomorphism.
[**Proof of Lemma** ]{}\[explicit\]. It is clear that $u \in {\mathrm{C}}^\infty\big(\overline{{\Omega}};{\mathbb{R}}^{n}\big){\setminus}\{\mathrm{id}\}$, $u=\mathrm{id}$ on ${\partial}{\Omega}$. By using standard properties of the matrix exponential (in particular that $e^{f(t)S}S=Se^{f(t)S}$ and ${\mathrm{D}}_t(e^{f(t)X})=f'(t)Se^{f(t)S}$ for any $f\in {\mathrm{C}}^1({\mathbb{R}};{\mathbb{R}})$) and setting for convenience ${\mathrm{sgn}}(x):=x/|x|$, when $x\in{\mathbb{R}}^n {\setminus}\{0\}$, we easily compute that $${\mathrm{D}}u(x) = \mathrm{e}^{\log(|x|)S}\big(\mathrm I \, +\, (S \, {\mathrm{sgn}}(x)) \otimes {\mathrm{sgn}}(x)\big), \ \ \text{ for }x\in{\Omega}.$$ Since $\mathrm{e}^{\log(|x|)S}$ is orthogonal and $|OA|=|A|$ for any $A,O\in{\mathbb{R}}^{n\times n}$ with $O$ being orthogonal, we have $$\begin{aligned}
|{\mathrm{D}}u(x) |^2 \, = \, \big|\mathrm I + (S \, {\mathrm{sgn}}(x))\otimes {\mathrm{sgn}}(x) \big|^2\, =\, n^2 \,+ \, 2\left(S\, {\mathrm{sgn}}(x)\right) \cdot {\mathrm{sgn}}(x) \,+\,\left|S \,{\mathrm{sgn}}(x)\right|^2\left|{\mathrm{sgn}}(x)\right|^2.\end{aligned}$$ Because $S$ is both skew-symmetric and orthogonal, we have $(Se)\cdot e=0$ and also $|Se|=1$ when $|e|=1$. We therefore have $|{\mathrm{D}}u|^2 \equiv n^2 + 1$ on ${\Omega}$. By using once again that $\mathrm{e}^{\log(|x|)S}$ is orthogonal, we have $$\det({\mathrm{D}}u(x)) \, =\, \det\big(\mathrm{e}^{\log(|x|)S}\big)\det \!\big(\mathrm{I} + (S\, {\mathrm{sgn}}(x))\otimes {\mathrm{sgn}}(x)\big) \, =\, \det\!\big(\mathrm{I} + (S\, {\mathrm{sgn}}(x))\otimes {\mathrm{sgn}}(x)\big).$$ By the Matrix Determinant Lemma, $\det(\mathrm{I}+a\otimes b)=1+a\cdot b$ for any $a,b\in{\mathbb{R}}^n$ and so we can use again the skew-symmetry of $S$ to deduce $\det({\mathrm{D}}u(x)) =1+ (S{\mathrm{sgn}}(x))\cdot {\mathrm{sgn}}(x) = 1$ for any $x\in{\Omega}$.
\[lemma\] For any $n,N \geq 2$, there exists an explicit smooth $\infty$-Harmonic map defined on a cylindrical subdomain of ${\mathbb{R}}^n$ with values in ${\mathbb{R}}^N$ which coincides with an affine map on the boundary of the domain.
[**Proof of Lemma** ]{}\[lemma\]. Let $u : {\mathbb{R}}^2 \supseteq \overline{{\Omega}} {\longrightarrow}{\mathbb{R}}^2$ be the mapping constructed in Lemma \[explicit\]. Then, by setting $v(x):=(u(x),0)^\top$, we obtain a map $v: {\Omega}{\longrightarrow}{\mathbb{R}}^{2+k}$ for any $k\in{\mathbb{N}}$ with the desired properties. Indeed, we have $|{\mathrm{D}}v|^2=|{\mathrm{D}}u|^2$, which gives ${\mathrm{D}}v {\otimes}{\mathrm{D}}v :{\mathrm{D}}^2 v \equiv 0$. Further, for any $x\in{\Omega}$ we have $\mathrm R({\mathrm{D}}v(x))={\mathbb{R}}^2 {\times}\{0\}$ and ${\Delta}v(x) \in {\mathbb{R}}^2 {\times}\{0\}$, which gives $[\![{\mathrm{D}}v]\!]^\bot{\Delta}v\equiv 0$. Hence, ${\Delta}_\infty v \equiv 0$ in ${\Omega}$, whilst $v=(\mathrm{id},0)$ on ${\partial}{\Omega}$.
Further, by setting $w(x,y):=v(x)$, we obtain a map $w: {\mathbb{R}}^{2+l} \supseteq {\Omega}{\times}{\mathbb{R}}^l {\longrightarrow}{\mathbb{R}}^{2+k}$ for any $k,l\in{\mathbb{N}}$ defined on a cylindrical annulus with $|{\mathrm{D}}w|^2=|{\mathrm{D}}v|^2$, which gives ${\mathrm{D}}w {\otimes}{\mathrm{D}}w :{\mathrm{D}}^2 w \equiv 0$. Also, for any $(x,y)\in{\Omega}{\times}{\mathbb{R}}^l$ we have $\mathrm R({\mathrm{D}}w(x,y))=\mathrm R({\mathrm{D}}v(x))$ and ${\Delta}w(x,y)={\Delta}v(x)$, giving $[\![{\mathrm{D}}w]\!]^\bot{\Delta}w\equiv 0$ and thus ${\Delta}_\infty w \equiv 0$ in ${\Omega}{\times}{\mathbb{R}}^l$. Finally, note that $w=(\mathrm{Proj}_{{\mathbb{R}}^2},0)$ on ${\partial}({\Omega}{\times}{\mathbb{R}}^l)$.
**Acknowledgements.** N.K. would like to thank Roger Moser for numerous inspiring scientific discussion on Calculus of Variations in $L^\infty$. Additionally, both authors have benefited whilst preparing this note by the ideas emerging in an unpublished existential counterexample relevant to our results herein, which was kindly and selflessly shared with the authors. We are grateful to Roger Moser for this deep insight. We would also like to thank the referee for their constructive comments and suggestion which improved the content and the presentation. In particular, we are grateful for a correction in the proof of Lemma 2.1.
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|
---
abstract: 'We consider a lattice regularization for an ill-posed diffusion equation with trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of single-interface initial data that the lattice solutions satisfy asymptotically a free boundary problem with hysteretic Stefan condition. The key challenge in the proof is to control the microscopic fluctuations that are inevitably produced by the backward diffusion when a particle passes the spinodal region.'
author:
- 'Michael Helmers[^1]'
- 'Michael Herrmann[^2]'
title: |
Hysteresis and phase transitions in a lattice regularization\
of an ill-posed forward-backward diffusion equation
---
Keywords:
*multi-scale analysis for gradient flows, regularization of ill-posed diffusion equations\
hysteresis and phase transitions, interface propagation in discrete media*
MSC (2010):
34A33, 35R25, 37L60, 74N20, 74N30
Introduction {#sect:intro}
============
Forward-backward diffusion problems arise in many branches of physics and materials science [@Elliott85; @BaBeDaUg93], mathematical biology [@Padron04; @HoPaOt04], and technology [@PeMa90] and lead to complex and intriguing mathematical problems. The simplest dynamical model for a one-dimensional continuous medium would be the nonlinear parabolic PDE $$\begin{aligned}
\label{Eqn:PDEIllPosed}
\partial_\tau U =\partial_\xi^2 P,
\qquad
P:=\Phi^\prime{{\left({U}\right)}}\end{aligned}$$ with time $\tau\geq0$, space $\xi\in{{\mathbb{R}}}$, and non-monotone $\Phi^\prime$, but the corresponding Cauchy problem is ill-posed. To overcome this difficulty, a well-known approach is to consider microscopic regularizations with length parameter $0<{{\varepsilon}}\ll1$ that take into account small-scale effects and complement by additional terms and dynamical laws. The latter depend on the particular choice of $\Phi'$ and in what follows we focus on a typical setting in materials science, where $\Phi^\prime$ is the bistable derivative of a double-well potential $\Phi$. We also assume that $\Phi^\prime$ and $\Phi$ are odd and even, respectively, and mention that a bistable function is sometimes called cubic-type as its graph consists of two increasing branches which are separated by an decreasing one.
In the literature, a lot of attention has been paid to the Cahn-Hilliard equation $$\begin{aligned}
\label{Eqn:PDECahnHilliard}
\partial_\tau U
=
\partial_\xi^2 P- {{\varepsilon}}^2\partial_\xi^4 U\end{aligned}$$ and the so-called viscous approximation $$\begin{aligned}
\label{Eqn:ViscousApp}
\partial_\tau U
=
\partial_\xi^2 P + {{\varepsilon}}^2\partial_\tau\partial_\xi^2 U,\end{aligned}$$ but in this paper we study the spatially discrete regularization $$\label{eq:master-eq}
\dot u_j(t)
=
\operatorname{\Delta}p_j(t),
\qquad
p_j
=
\Phi'{{\big(u_j(t)\big)}}$$ with microscopic time $t\geq0$, particle index $j\in{{\mathbb{Z}}}$, and standard Laplacian $\Delta$ on ${{\mathbb{Z}}}$, that is $$\begin{aligned}
\label{Def:Laplacian}
\Delta v_j = v_{j+1}+v_{j-1}-2v_j.\end{aligned}$$ This lattice ODE is linked to the PDE by the parabolic scaling $$\begin{aligned}
\label{Eqn:Scaling}
\tau := {{\varepsilon}}^2 t,
\qquad
\xi := {{\varepsilon}}j\end{aligned}$$ and the formal identification $$\begin{aligned}
\label{Eqn:Identification}
u_j{{\left({t}\right)}}\cong U{{\left({{{\varepsilon}}^2 t},\,{{{\varepsilon}}j}\right)}},
\qquad
p_j{{\left({t}\right)}}\cong P{{\left({{{\varepsilon}}^2 t},\,{{{\varepsilon}}j}\right)}},\end{aligned}$$ whereby we can regard as a spatial semi-discretization of or, conversely, the PDE as the naive continuum limit of the lattice .
Of particular interest in the analysis of any regularization is the sharp-interface limit ${{\varepsilon}}\to0$ since it gives rise to phase interfaces, that is, curves $\xi=\Xi{{\left({\tau}\right)}}$ which separate space-time regions in which $U$ is confined to either one of the convex components of $\Phi$ (usually called *phases*). The dynamics of such interface curves have to be determined by a free boundary problem that couples the – now locally well-posed – bulk diffusion for $U$ on either side of the interface with certain conditions for $\Xi$. The *Stefan condition* $$\begin{aligned}
\label{Eqn:StefanCondition}
\tfrac{{\mathrm{d}\Xi}}{{\mathrm{d}\tau}}\,{{|\![U]\!|}}+
{{|\![\partial_\xi P]\!|}}=0,\qquad
{{|\![P]\!|}}=0,\end{aligned}$$ where ${{|\![\cdot]\!|}}$ denotes the jump across the interface, guarantees for all models that holds in a distributional sense across the interface but the evolution of ${{\left({U},\,{\Xi}\right)}}$ depends on another interface condition which encodes the details of the microscopic regularization. For the Cahn-Hilliard equation , the additional law reads $$\begin{aligned}
\label{Eqn:MaxRule}
P = 0\end{aligned}$$ and fixes the value of $P$ according to Maxwell’s local equilibrium criterion. The validity of the free boundary problem , and has been proven rigorously in [@BeBeMaNo12].
Heuristic arguments indicate that the sharp-interface limit of the viscous approximation is more involved since the interface value of $P$ is no longer known as ${{\varepsilon}}\to0$ but depends in a hysteretic manner on both the state of the system and the propagation direction of the interface. More precisely, numerical experiments and formal asymptotic analysis as carried out in [@Plotnikov94; @EvPo04] predict that the viscous approximation supports both standing and moving interfaces according to the flow rule $$\begin{aligned}
\label{Eqn:FlowRule}
P=-p_*
\text{ for }
\tfrac{{\mathrm{d}\Xi}}{{\mathrm{d}\tau}}{{|\![U]\!|}}>0,
\quad
P=+p_*
\text{ for }
\tfrac{{\mathrm{d}\Xi}}{{\mathrm{d}\tau}}{{|\![U]\!|}}<0,
\quad
\tfrac{{\mathrm{d}\Xi}}{{\mathrm{d}\tau}}=0
\text{ for }
P\in[-p_*,+p_*],\end{aligned}$$ where $\pm p_*$ are the two local extrema of the odd function $\Phi^\prime$. The key argument in this derivation is that any reasonable limit for ${{\varepsilon}}\to0$ satisfies the entropy inequality $$\begin{aligned}
\label{Eqn:EntropyLaw}
\partial_\tau \eta{{\left({U}\right)}}-\partial_\xi{{\big(\mu{{\left({P}\right)}}\partial_\xi {P}\big)}}
\leq
0,\end{aligned}$$ where the entropy flux $\eta$ and the entropy density $\mu$ can be chosen arbitrarily as long as they comply with $$\begin{aligned}
\label{Eqn:EntropyPair}
\eta^\prime
=
\mu\circ \Phi^\prime,\qquad \mu^\prime\geq0.\end{aligned}$$ The main tasks for a rigorous justification of the hysteretic flow rule or, equivalently, of is to show the existence of a smooth interface curve $\Xi$ and to derive ${{\varepsilon}}$-uniform a priori estimates that guarantee the strong convergence of the fields as well as the regularity of the limit $P$. Although there is an extensive literature on the viscous approximation, see the discussion below, we are not aware of any rigorous result that links the hysteretic free boundary problem to the sharp interface limit of .
![*Left Panel*. Graph of the piecewise linear function $\Phi^\prime$ as defined in . The gray boxes represent the intervals $I_*$ and $I_{**}$ from and the corresponding double well potential is given in . *Right panel*. Cartoons of the hysteresis loop for macroscopic phase interfaces. Notice that the interface moves from the phase $\Theta_+$ into the phase $\Theta_-$ if and only the particles at the interface transit the other way round from $\Theta_-$ to $\Theta_+$ and that ${{|\![P]\!|}}=0$ implies ${{|\![U]\!|}}=\pm2$.[]{data-label="fig:potential"}](potential_1){width=".91\textwidth"}
For the lattice ODE , which can also be written as $\dot{w}_j=\nabla_-\Phi^\prime{{\left({\nabla_+w_j}\right)}}$ with $u_j=\nabla_+ w_j=w_{j+1}-w_j$, one can easily adapt the asymptotic arguments from [@Plotnikov94; @EvPo04] to show heuristically that the limit dynamics are governed by the same hysteretic free boundary problem as for the viscous approximation. Moreover, this micro-to-macro transition has been made rigorous in two cases: $(i)$ in [@GeNo11; @BeGeNo13] for generic bistable $\Phi'$ and initial data that give rise to standing interfaces only, and ${{\left({ii}\right)}}$ by the authors in [@HeHe13] for bilinear $\Phi'$ and a suitable class of well-prepared initial data. The latter is to our knowledge the only available rigorous microscopic justification for macroscopic phase interfaces that are driven by hysteric jump conditions. We also refer to [@EsSl08; @EsGr09] for coarsening in discrete forward-backward diffusion lattices with monostable $\Phi^\prime$ and to [@GuShTi13; @GuTi16] for other systems with spatially distributed hysteresis.
In the current paper, we extend the rigorous analysis from [@HeHe13] to the case of trilinear $\Phi'$. At first glance, the step from bilinear to trilinear seems to be a minor improvement only but the mathematical analysis of the trilinear case is significantly more involved because the spinodal region is no longer degenerate. In particular, microscopic phase transitions are no longer instantaneous processes related to temporal jumps but take a certain time as the particles have to move through the spinodal region. The novel challenge is that the backward diffusion during each spinodal visit produces strong microscopic fluctuations which have to be controlled on the macroscopic scale. The main achievement of the present paper consists, roughly speaking, in the derivation of asymptotic formulas and estimates for the creation and subsequent amplitude decay of the fluctuations which finally ensure that the lattice data converge as ${{\varepsilon}}\to0$ to regular macroscopic fields. Moreover, some of the arguments derived below can be generalized to genuinely nonlinear bistable functions $\Phi^\prime$.
In what follows we always suppose – see Figure \[fig:potential\] for an illustration – that the lattice ODE is complemented by $$\label{eq:phi_prime}
\Phi'(u)
:=
\begin{cases}
u+1 &\text{if } u \leq - u_*, \\
u-1 &\text{if } u \geq +u_*, \\
- \kappa u &\text{if } -u_* < u < +u_*,
\end{cases}$$ where ${{\kappa}}\in(0,\infty)$ is a free slope-parameter and $$\begin{aligned}
\label{Eqn:Parameters}
\pm p_*=\Phi^\prime{{\left({\mp u_*}\right)}}=\Phi^\prime{{\left({\pm u_{**}}\right)}},
\qquad
u_*:=\frac{1}{1+{{\kappa}}},
\qquad
p_*:=\frac{{{\kappa}}}{1+{{\kappa}}},
\qquad
u_{**}:=\frac{1+2{{\kappa}}}{1+{{\kappa}}}\end{aligned}$$ In particular, the bilinear case $\Phi^\prime{{\left({u}\right)}}=u-\operatorname{sgn}{{\left({u}\right)}}$ corresponds to ${{\kappa}}=\infty$ while for ${{\kappa}}\to0$ there is no backward diffusion anymore and the PDE becomes degenerate-parabolic.
Before we discuss the dynamical properties of the lattice ODE , we give a brief and non-exhaustive overview of the literature concerning the viscous approximation , which can also be formulated as $\partial_\tau W = \partial_\xi\Phi^\prime{{\left({\partial_\xi W}\right)}}+{{\varepsilon}}\partial_\tau\partial_\xi^2W$, where $U=\partial_\xi W$. Moreover, some authors refer to interfaces as phase boundaries, and a standing interface is often called steady.
The initial value problem for has been studied in [@Padron04; @NoCoPe91], and [@BoCoTo17] provides existence and uniqueness results for a broader class of regularizing PDEs. Numerical schemes are proposed and analyzed in [@EvPo04; @Pierre10; @LaMa12] – see also the discussion at the end of §\[sect:intro\] – and [@NoCoPe91] investigates the multitude of steady states and their dynamical stability with respect to . Moreover, [@Plotnikov94; @EvPo04] characterize the limit ${{\varepsilon}}\to0$ in the framework of Young measures and entropy inequalities but we already mentioned that the rigorous justification of the limit model has not yet been achieved.
The existence and uniqueness of two-phase entropy solutions to the limiting problem , , and have been proven in [@MaTeTe09] for a trilinear nonlinearity as in , and [@Visintin06] studies the existence and uniqueness problem for an equivalent formulation in terms of a parabolic PDE that comprises a spatial family of temporal hysteresis operators. [@GiTe10; @LaMa12] discuss the special case of Riemann initial data and provide explicit formulas for the corresponding self-similar solutions with moving or standing interface. Notice also that the ill-posed forward-backward equation admits in general – i.e., without entropy conditions and two-phase assumption – a plethora of solutions, see [@Hoe83; @Zh06] as well as [@Terracina14; @Terracina15] for recent results and a discussion of the literature concerning solutions that penetrate the spinodal region. Measure-valued solutions to have also been studied, see [@Plotnikov94; @YiWa03; @EvPo04; @SmTe10; @SmTe11; @BeSmTe16] and the references therein.
Overview of the key effects
---------------------------
![Second numerical example with depinning of the macroscopic interface at $\tau\approx0.05$. On the moving interface, $P$ attains the value $+p_*$ and $\partial_\xi P$ exhibits a jump, but when the interface rests, $P$ is smooth across the interface with non-fixed value in $J_*$. This dichotomy gives rise to the hysteresis diagram in the right panel of Figure \[fig:potential\] and complies with both the Stefan condition and the flow rule .[]{data-label="Fig:Front_1"}](front_2){width=".9\textwidth"}
![Second numerical example with depinning of the macroscopic interface at $\tau\approx0.05$. On the moving interface, $P$ attains the value $+p_*$ and $\partial_\xi P$ exhibits a jump, but when the interface rests, $P$ is smooth across the interface with non-fixed value in $J_*$. This dichotomy gives rise to the hysteresis diagram in the right panel of Figure \[fig:potential\] and complies with both the Stefan condition and the flow rule .[]{data-label="Fig:Front_1"}](front_1){width=".9\textwidth"}
The nonlinear lattice , exhibits a complex dynamical behavior since the non-monotonicity of $\Phi^\prime$ implies that each particle $u_j$ can either diffuse forwards with regular coefficient $\Phi^{\prime\prime}{{\left({u_j{{\left({t}\right)}}}\right)}} > 0$ or backwards with $\Phi^{\prime\prime}{{\left({u_j{{\left({t}\right)}}}\right)}}<0$. In order to illustrate the different phenomena we next discuss some numerical simulations of finite lattices $j=1,\ldots, N$ with natural scaling parameter ${{\varepsilon}}:=1/N$ and homogeneous Neumann conditions, see §\[sect:numerics\] for more details. In particular, we regard the lattice data for large $N$ as discrete sampling of macroscopic fields by scaling time and space but not amplitude according to , and rely on the following conventions and abbreviations for the interpretation of the numerical results.
We refer to the different connected components of the set $\{u : \Phi^{\prime\prime}{{\left({u}\right)}}>0\}$ as *phases* and write $$\begin{aligned}
\Theta_-
:=
(-\infty,-u_*) \quad \text{for the $-$-phase}\,,
\qquad\qquad
\Theta_+
:=
(+u_*,\infty) \quad\text{for the $+$-phase}\,,\end{aligned}$$ while $\Theta_0:=(-u_*,+u_*) = \{u : \Phi^{\prime\prime}{{\left({u}\right)}}<0\}$ is called the *spinodal region*. For the analysis of the macroscopic dynamics it is also convenient to introduce the intervals $$\begin{aligned}
\label{Eqn.Intervals}
I_* := [-u_*,+u_*],
\qquad
I_{**}=[-u_{**},+u_{**}],
\qquad
J_*:=[-p_*,+p_*],\end{aligned}$$ where $I_*$ and $J_*$ are the closures of $\Theta_0$ and $\Phi^\prime{{\left({\Theta_0}\right)}}$, respectively, and $I_{**}$ denotes the inverse image of $J_*$ under $\Phi^\prime$.
Numerical simulation as depicted in Figures \[Fig:Front\_2\] and \[Fig:Front\_1\] provide – for well prepared single-interface initial data as defined in Assumption \[ass:macro\] – evidence for the existence and dynamical stability of a macroscopic phase interface that separates two space-time regions in which the lattice data are confined to either one of the phases $\Theta_-$ and $\Theta_+$. The key observations concerning the corresponding large scale dynamics can be summarized as follows.
\[Obs:Macro\] The macroscopic phase interface located at the curve $\xi=\Xi{{\left({\tau}\right)}}$ can either propagate or be at rest according to the following rules:
1. : At any time $\tau$ with $\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tau}\right)}}=0$ we have $P{{\left({\tau},\,{\Xi{{\left({\tau}\right)}}}\right)}}\in J_*$ and $P$ is smooth across the interface.
2. : $\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tau}\right)}}\neq0$ implies $P{{\left({\tau},\,{\Xi{{\left({\tau}\right)}}}\right)}}=+p_*$ or $P{{\left({\tau},\,{\Xi{{\left({\tau}\right)}}}\right)}}=-p_*$ depending on whether the interface propagates into the phase $\Theta_-$ or $\Theta_+$, respectively. The field $P$ is still continuous across the interface but $\partial_\xi P$ admits a jump that drives the interface.
Moreover, continuity of $P$ implies discontinuity for $U$ and the type of each interface can change in time by *pinning* or *depinning*.
A closer look to the evolution of single particles – see Figures \[Fig:Snapshots\] and \[Fig:Trajectories\] – reveals the following features of the small scale dynamics.
\[Obs:Micro\] The microscopic dynamics of the phase interface are driven by particles $u_j$ changing their phase as follows:
1. : A particle $u_j$ can enter the spinodal interval $I_*$ only when its two neighbors belong to different phases and when one of these neighbors takes value outside of $I_{**}$. The microscopic phase interface therefore propagates on the lattice because the particles undergo a phase transition sequentially, that is, they pass through the spinodal interval $I_*$ one after another.
2. : Not any spinodal visit is related to a proper phase transitions since it may happen that a particle enters and leaves the spinodal interval $I_*$ on the same side.
3. Each spinodal visit (passage or excursion) evokes strong microscopic fluctuations that are initially very localized but in turn diffusively spread over the lattice.
Observations \[Obs:Macro\] and \[Obs:Micro\] match perfectly in that they relate the macroscopic speed of propagation to the number of particles that undergo a phase transition during a given period of time. In Proposition \[pro:existence\] we prove the crucial one-after-another-property in a simplified single-interface setting, and we obtain macroscopic Lipschitz estimates for the interface after bounding the asymptotic waiting time between adjacent phase transitions from below in Proposition \[lem:waiting\] and Corollary \[cor:time-and-number-bounds\].
The regularity observations that the macroscopic field $P$ is continuous while the lattice data vary rapidly on the microscopic scale seem to contradict each other at first glance. The bridging idea is that macroscopic regularity can be observed in, loosely speaking, most of the macroscopic points ${{\left({\tau},\,{\xi}\right)}}$ while the rapid microscopic fluctuations with large amplitude dominate the dynamical behavior in a small subset of the macroscopic space-time only. These arguments are made rigorous in §\[sect:fluctuations\] and §\[sec:justification\] where we prove that the superposition of all microscopic fluctuations converges as ${{\varepsilon}}\to0$ pointwise almost everywhere to a continuous macroscopic field that drives the phase interface.
We also emphasize that Observation \[Obs:Micro\] combined with the trilinearity of $\Phi^\prime$ allows us to decompose the nonlinear lattice into linear subproblems as follows. As long as no particle is inside the spinodal region, the microscopic dynamics reduce – thanks to $\dot{u}_j=\dot{p}_j $ – to the discrete heat equation for $p$, and if some $u_j$ is inside the spinodal region we can derive a linear equation for $p$ where $p_j$ diffuses backwards; see §\[sec:prot-phase-trans\] for the details. Of course, the entire problem is still nonlinear since we have no a priori information about the spinodal entrance or exit times and hence do not know when to switch between the different linear evolutions. The linear decomposition is nonetheless very useful as it allows us to derive nearly explicit representation formulas for the lattice data in §\[sect:fluctuations\].
![*Left panel*. Temporal trajectories of $u_2$ (gray, dashed) and $u_4$ (black, solid) for the numerical data from Figure \[Fig:Snapshots\]. The $k$-th vertical boxes represents the spinodal passage of $u_{k-1}$ during which fluctuations are created. *Right panel*. The evolution of the lattice dissipation ${\mathcal{D}}$ from with localized peak for each phase transition. In this numerical example we have $N=200$ and relatively large initial dissipation ${\mathcal{D}}{{\left({0}\right)}}\approx 140$, so the amplitude separation between peaks and bulk is rather small though clearly visible.[]{data-label="Fig:Trajectories"}](snapshots){width=".95\textwidth"}
![*Left panel*. Temporal trajectories of $u_2$ (gray, dashed) and $u_4$ (black, solid) for the numerical data from Figure \[Fig:Snapshots\]. The $k$-th vertical boxes represents the spinodal passage of $u_{k-1}$ during which fluctuations are created. *Right panel*. The evolution of the lattice dissipation ${\mathcal{D}}$ from with localized peak for each phase transition. In this numerical example we have $N=200$ and relatively large initial dissipation ${\mathcal{D}}{{\left({0}\right)}}\approx 140$, so the amplitude separation between peaks and bulk is rather small though clearly visible.[]{data-label="Fig:Trajectories"}](trajectories){width=".96\textwidth"}
Multiple scales and fluctuations
--------------------------------
The dynamics of the fluctuations are governed by a subtle interplay between the backward diffusion inside the spinodal region and the regularizing effects of the forward diffusion inside each phase. We can think of the fluctuations produced by the spinodal visit of some particle as a localized ‘package’ of fluctuations, which after its creation interacts by forward diffusion with the entire lattice and hence also with all packages evoked by former or later phase transitions. In particular, the $\ell^\infty$-norm of each package (amplitude) decays algebraically in time while the $\ell^1$-norm (mass) remains conserved since the fluctuations are not damped out but merely spread over the lattice. The microscopic lattice dynamics is therefore related to the informal concepts
---- ---------------- --------------------------------------------------------------------
1. *passage time* (time to pass the spinodal interval $I_*$),
2. *decay time* (time needed to spread and regularize the localized fluctuations),
3. *waiting time* (time between the phase transitions of adjacent particles),
---- ---------------- --------------------------------------------------------------------
and any mathematical analysis of the macroscopic limit ${{\varepsilon}}\to0$ requires to understand the scaling relations of these times at least on a heuristic level.
We already mentioned that our asymptotic approach involves a precise lower bound for the waiting time as established for well-prepared initial data in Corollary \[cor:time-and-number-bounds\]. Moreover, in we identify a universal *impact profile*, which provides the asymptotic shape of each package in the limit ${{\varepsilon}}\to0$ and enables us in the proof of Lemma \[Lem:Hoelder\] to compute a microscopic time period of order ${{\varepsilon}}^{-1}$ after which each package has been sufficiently regularized by the forward diffusion. This result can be regarded as an upper bound for the decay time although we state it differently and focus on the implied Hölder estimates for the regular fluctuations.
The heuristic concept of the passage time is a bit more involved. By splitting the microscopic dynamics during a spinodal passage into their slow and fast parts, we show in §\[sec:prot-phase-trans\] that the *typical* passage time is of order $\ln {{\varepsilon}}$ due to the exponential growth of the fast variable. On the other hand, one can construct special initial data such that the first passage time is as large as the observation time. Even in this case, however, we can pass to the macroscopic limit since the interface does not move and because our results in §\[sec:glob-fluct-estim\] imply, roughly speaking, that the fluctuations remain localized for all times and hence small with respect to macroscopic norms. By similar arguments we also control the cumulative impact of the spinodal excursion in Corollary \[cor:bounds-neg-fluct\] and do not attempt to estimate their number or duration.
The fluctuations as well as the different times scales can also be related to energetic concepts by regarding the lattice ODE as gradient flow with respect to the spatially discrete analog to the ${{{\mathsf{H}}}}^{-1}$-metric structure. In particular, for finite systems with either periodic or homogeneous Neumann boundary conditions we readily verify the energy law $$\begin{aligned}
\tfrac{{\mathrm{d}}}{{\mathrm{d}t}}{\mathcal{E}}{{\left({t}\right)}}
=
-{{\varepsilon}}^2{\mathcal{D}}{{\left({t}\right)}},\end{aligned}$$ where $$\begin{aligned}
\label{Eqn.EnergDiss}
{\mathcal{E}}{{\left({t}\right)}}:=N^{-1} \sum_{j=1}^N\Phi{{\left({u_j}\right)}},
\qquad
{\mathcal{D}}{{\left({t}\right)}}:=N\sum_{j=1}^N{{\left({p_{j+1}-p_j}\right)}}^2\end{aligned}$$ denote the *averaged energy* and the *dissipation*, respectively and both have been scaled such that the formal identification complies with the macroscopic formulas $$\begin{aligned}
{\mathcal{E}}{{\left({t}\right)}}
\cong
\int\limits_{0}^1 \Phi{{\big(U{{\left({{{\varepsilon}}^2t},\,{\xi}\right)}}\big)}}{\mathrm{d}\xi},
\qquad
{\mathcal{D}}{{\left({t}\right)}}
\cong
\int\limits_{0}^1 {{\big(\partial_\xi P{{\left({{{\varepsilon}}^2t},\,{\xi}\right)}}\big)}}^2{\mathrm{d}\xi}.\end{aligned}$$ Notice that the single-particle energy follows from up to an additive constant and reads $$\begin{aligned}
\label{Eqn:DoubleWell}
\Phi{{\left({u}\right)}}
=
\frac12
\begin{cases}
{{\left({u+1}\right)}}^2 &\text{if } u \leq - u_*, \\
{{\left({u-1}\right)}}^2 &\text{if } u \geq +u_*, \\
p_*- \kappa u^2 &\text{if } -u_* < u < +u_*.
\end{cases}\end{aligned}$$ From we infer for small ${{\varepsilon}}>0$ the heuristic equivalence $$\begin{aligned}
{\mathcal{D}}{{\left({t}\right)}}\sim 1 \quad \quad \text{if and only if} \quad \quad
P {{\left({{{\varepsilon}}^2t},\,{\cdot}\right)}}
\text{ is regular with weak derivative }
\partial_\xi P{{\left({{{\varepsilon}}^2t},\,{\cdot}\right)}},\end{aligned}$$ and conclude that the localized lattice fluctuations give rise to a significant increase in the dissipation. In other words, the interface dissipation stemming from microscopic phase transitions exceeds the regular dissipation coming from the macroscopic bulk diffusion. See the right panel in Figure \[Fig:Trajectories\] for typical numerical data and note that our asymptotic formulas ensure that ${\mathcal{D}}{{\left({t}\right)}}\sim N={{\varepsilon}}^{-1}$ at the end of each microscopic phase transition.
The energy equality for gradient flows $$\begin{aligned}
\int\limits_0^\infty {\mathcal{D}}{{\left({{{\varepsilon}}^{-2}\tau}\right)}}{\mathrm{d}\tau}
=
{\mathcal{E}}{{\left({0}\right)}}-{\mathcal{E}}{{\left({\infty}\right)}},\end{aligned}$$ reveals that the initial energy bounds the total number of microscopic phase transitions and hence also the maximal propagation distance of the macroscopic interface as well as the averaged impact of all fluctuations. It seems therefore tempting to tackle the macroscopic limit ${{\varepsilon}}\to0$ by variational methods and to show that the gradient flow of the lattice $\Gamma$-converges to the hysteretic free boundary problem , and whose variational structure is described in [@Visintin06]. Such approaches have been exploited in other micro-to-macro transitions, see for instance [@OtRe07; @Serfaty11; @BeBeMaNo12; @MiTu12; @PeSaVe13; @Braides14] for different frameworks, and are usually quite robust. It is, however, not clear to the authors whether variational methods are capable of resolving the complicate dynamical behavior of with non-monotone dissipation and temporally varying regularity of the microscopic data.
We finally recall that the above heuristic discussion of the lattice dynamic is restricted to well-prepared macroscopic single-interface data. All arguments can be adapted to the case of finitely many phase interfaces but other classes of initial data are more crucial. For instance, numerical simulations with oscillatory single-interface indicate the existence of an initial transient regime during which the systems dissipates a huge amount of energy before it reaches a state with macroscopic regularity for the first time. It seems, however, that there is no simple way to estimate the duration of the transient regime because a large number of phase transitions might push the phase interface over a long distance and produce many additional fluctuations. The dynamics of multi-phase initial data with oscillatory phase fraction or data with many particles inside the spinodal region are even more complicated since we expect to find measure-valued solutions on the macroscopic scale as well as phase interfaces that connect a pure-phase region with a mixed-phase one. First results in this direction have been obtained in [@Holle16] for a bilinear nonlinearity and a periodic pattern for the microscopic phase field, but in the general case with an irregular distribution of phases it is not even clear what the analog to the hysteretic flow rule is. Moreover, for arbitrary initial data there is an extra transient regime related to the spinodal decomposition of particles but it seems hard to show that the latter happens in a sufficiently short period of time.
Main result and plan of paper {#sect:mainresult}
-----------------------------
In this paper we derive the hysteretic free boundary problem , and in the trilinear case and for well-prepared single-interface initial data on the ${{\mathbb{Z}}}$. The prototypical example of the latter stems – as in Figures \[Fig:Front\_2\] and \[Fig:Front\_1\] – from a macroscopic initial datum with single interface located at $\xi=\Xi_{{\rm ini}}$ and phases $\Theta_+$ and $\Theta_-$ corresponding to $\xi<\Xi_{{\rm ini}}$ and $\xi>\Xi_{{\rm ini}}$, respectively. More precisely, after choosing a bounded, continuous, and piecewise smooth function $P_{{\rm ini}}$ on ${{\mathbb{R}}}$ such that $$\begin{aligned}
P_{{\rm ini}}{{\left({\xi}\right)}}>-p_*
\quad \text{for}\quad \xi<\Xi_{{\rm ini}},
\qquad
P_{{\rm ini}}{{\left({\xi}\right)}}\in J_*
\quad \text{for}\quad \xi>\Xi_{{\rm ini}},\end{aligned}$$ we consistently set $$\begin{aligned}
{}U_{{\rm ini}}{{\left({\xi}\right)}}:=P_{{\rm ini}}{{\left({\xi}\right)}}+1\in\Theta_+
\quad \text{for}\quad
\xi<\Xi_{{\rm ini}},
\qquad
U_{{\rm ini}}{{\left({\xi}\right)}}:=P_{{\rm ini}}{{\left({\xi}\right)}}-1\in\Theta_-\cap I_{**}
\quad \text{for}\quad \xi>\Xi_{{\rm ini}}{}\end{aligned}$$ and initialize the lattice data by a discrete sampling via . Due to the upper bound $P_{{\rm ini}}{{\left({\xi}\right)}}\leq +p_*$ for $\xi>\Xi_{{\rm ini}}$, the phase interface can propagate only to the right but it can switch between standing and moving by (several) pinning or depinning events.
For such initial data, the macroscopic model predicts a unique interface curve $\Xi$ with phase field $$\begin{aligned}
M{{\left({\tau},\,{\xi}\right)}}
=
\operatorname{sgn}U{{\left({\tau},\,{\xi}\right)}}
=
\operatorname{sgn}{{\big(\Xi{{\left({\tau}\right)}}-\xi\big)}}=U{{\left({\tau},\,{\xi}\right)}} -P{{\left({\tau},\,{\xi}\right)}}\end{aligned}$$ as well as ${{|\![U]\!|}}={{|\![P+M]\!|}}={{|\![M]\!|}}=-2$ at $\xi=\Xi{{\left({\tau}\right)}}$ and for all times $\tau\geq0$. We can therefore eliminate both $U$ and $M$ in the limit problem and summarize our main findings as follows.
\[res:Main\] For macroscopic single-interface initial data as described above, the scaled lattice data converge as ${{\varepsilon}}\to0$ to a solution of the hysteretic free boundary problem. In particular, the limit consists of a macroscopic field $P$ along with a nondecreasing interface curve ${{\Gamma}}=\{{{\left({\tau},\,{\xi}\right)}}\;:\;\xi=\Xi{{\left({\tau}\right)}}\}$ such that the following equations are satisfied: $$\begin{aligned}
\text{linear bulk diffusion outside ${{\Gamma}}$}:
\quad
&\partial_\tau P = \partial_\xi^2 P \label{Eqn:LimitBulkDiff}
\\
\text{Stefan condition across ${{\Gamma}}$}:
\quad
&2\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi = {{|\![\partial_\xi P]\!|}}
\quad\text{and}\quad {{{|\![P]\!|}}}=0& \label{Eqn:LimitStefan}
\\
\text{hysteretic flow rule on ${{\Gamma}}$}:
\quad
&P=+p_* \text{ if } \tfrac{{\mathrm{d}\Xi}}{{\mathrm{d}\tau}}>0
\quad\text{and}\quad
\tfrac{{\mathrm{d}\Xi}}{{\mathrm{d}\tau}}=0 \text{ if } P\in {}{(-p_*,\,+p_*)}{}\label{Eqn:LimitFlowRule}
\end{aligned}$$ Moreover, $\Xi$ and $P$ are Lipschitz and locally Hölder continuous, respectively, and uniquely determined by $\Xi_{{\rm ini}}$ and $P_{{\rm ini}}$.
The conditions on the initial data are made precise in Assumption \[ass:macro\], and the limit is established in several steps in §\[sec:justification\]. Proposition \[pro:compactness\] first provides macroscopic compactness of the scaled lattice data and in Theorem \[Thm:Limit\] we verify the limit dynamics along convergent subsequences. Both the convergence and the uniqueness statement then follow because the Cauchy problem for , and is well-posed, see [@Visintin06] and [@MaTeTe09] for approaches via hysteresis operators and entropy inequalities, respectively.
The paper is organized as follows. In §\[sect:lattice\] we prove well-posedness for microscopic single-interface solutions, derive a lower bound on the waiting time, and establish the entropy balances on the discrete level. §\[sect:fluctuations\] is the main analytical part of this paper and concerns the macroscopic impact of the microscopic fluctuations. First, studying a linear model problem for a spinodal visit in §\[sec:prot-phase-trans\], we characterize the backward-diffusion inside the spinodal region as the interaction of a scalar unstable mode with infinitely many slowly varying variables (slow-fast splitting). Afterwards we identify in §\[sect:defFluct\] and §\[sec:local-fluct-estim\] the microscopic fluctuations produced by a single particle and separate their essential part from the negligible one, where the former is given by the universal impact profile and the latter can be estimated with the help of the slow variables from the model problem. In §\[sec:glob-fluct-estim\] and §\[sec:regul-fluct\] we deal with the superposition of all fluctuations and prove Hölder estimates for the regular part of the essential fluctuation as well as vanishing bounds for their residual part and for the negligible fluctuations. In §\[sec:justification\] we finally pass to the limit ${{\varepsilon}}\to0$ and derive the Main Result \[res:Main\]. Since the spinodal effects are well-controlled by the fluctuation estimates from §\[sect:fluctuations\], the corresponding arguments are similar to those from [@HeHe13] for the bilinear limiting case ${{\kappa}}=\infty$.
We emphasize that the results of §\[sect:lattice\] can be generalized to more general bistable nonlinearities while our analysis in §\[sect:fluctuations\] is intimately connected to the trilinearity of $\Phi'$ as it relies on linear substitute problems and the superposition principle. Moreover, for general nonlinearities it is not clear what the analog to the aforementioned slow-fast splitting is.
On the numerical simulations {#sect:numerics}
----------------------------
To conclude this introduction we describe the numerical scheme that was used for the computation of the examples in Figures \[Fig:Front\_2\] and \[Fig:Front\_1\]. Fixing a finite particle number $N$, we impose homogeneous Neumann boundary conditions $$\begin{aligned}
u_0\equiv u_1,\qquad
u_{N+1}\equiv u_N\end{aligned}$$ and prescribe the initial data by $$\begin{aligned}
u_j{{\left({0}\right)}}=c_\pm + d_\pm \arctan{{\left({{{\varepsilon}}j +e_\pm}\right)}} \quad \text{for}\quad j\gtrless j_*\end{aligned}$$ with ${{\varepsilon}}=1/N$. Here, $j_*$ denotes the initial position of the single interface and the constants $c_\pm$, $d_\pm$, and $e_\pm$ have been chosen carefully for any example to produce illustrative results, see the snaphots for $\tau=0$.
We solve the ODE analog to the lattice by the explicit Euler scheme, which is easy to implement. Of course, the numerical time step size $\delta{t}$ must be chosen sufficiently small and in accordance with the macrosocpic CFL condition $$\begin{aligned}
\frac{\delta{\tau}}{\delta\xi^2}=\frac{{{\varepsilon}}^2\delta t}{{{\left({{{\varepsilon}}\delta j}\right)}}^2}=\delta t
< {{\lambda}}_{\max}\,,\end{aligned}$$ where the largest eigenvalue ${{\lambda}}_{\max}$ of the discrete Laplacian $-\Delta$ is basically independent of the system size $N$ and can be computed by discrete Fourier transform.
The numerical properties of the Euler scheme have already been investigated in [@LaMa12], and the authors there regard the onset of strong oscillations as a drawback of the discretization. They also propose a semi-implicit scheme for the time integration of , which is unconditionally stable but requires to monitor the spinodal entrance and exit times, as well as a numerical algorithm for the computation of two-phase solutions to the free boundary problem –. The latter scheme provides approximate solutions without spatial and temporal fluctuations as it imposes microscopic transmission conditions at the interface which are derived from the macroscopic entropy inequalities .
The oscillations in the Euler scheme are caused by the spinodal visits of particles and correspond precisely to the fluctuations described above on the level of the lattice equation with continuous time variable. Moreover, in view of the macroscopic free boundary problem one might in fact regard the microscopic oscillations as incorrect or spurious, but our analysis suggests a complementary interpretation. The fluctuations are the inevitable echo of the microscopic phase transitions, which drive the interface on large scales according to the hysteric flow rule and explain why the thermodynamic fields comply with the entropy conditions at all. In this context we emphasize that the solutions to the viscous approximation also exhibit strong oscillations and one might argue that the rigorous passage to the limit ${{\varepsilon}}\to0$ is still open because the fine structure of these oscillations has not yet been investigated carefully.
Properties of the lattice dynamics {#sect:lattice}
==================================
In this section we investigate the dynamical properties of the diffusive lattice with trilinear $\Phi^\prime$ as in . All arguments, however, can be generalized to other bistable nonlinearities at the cost of more technical and notational efforts.
Existence of single-interface solutions
---------------------------------------
We first introduce the notion of single-interface solutions and establish their existence and uniqueness. Furthermore, we derive some basic properties concerning the dynamics of $p = \Phi'(u)$.
\[def:single-iface-solution\] A differentiable function $u \colon [0,\infty) \to \ell^\infty({\mathbb{Z}})$ is a *single-interface solution* to if $u$ satisfies the differential equation and if there exists a non-decreasing sequence $(t_k^*)_{k \geq k_1} \subset (0,\infty]$, $k_1 \in {\mathbb{Z}}$ such that the following conditions are satisfied for all $k \geq k_1$ and with $t_{k_1-1}^* := 0$:
1. We have either $t_k^* = \infty$ or $t_{k+1}^* > t_k^*$.
2. If $t_{k-1}^*<\infty$, then $u$ takes values in the state space $$\begin{aligned}
X_k
=
\Big\{u \in \ell^\infty({\mathbb{Z}}) \colon
&u_* < \inf_{j<k} u_j \leq \sup_{j<k} u_j < \infty, \quad
-u_{**} < \inf_{j>k} u_j \leq \sup_{j>k} u_j < -u_*, \\
&-u_{**} < u_k < u_* \Big\}
\end{aligned}$$ on the time interval $(t_{k-1}^*, t_{k}^*)$.
![Two examples of single-interface states from $X_k$ as in Definition \[def:single-iface-solution\], where $u_{k-1}$ and $u_k$ are highlighted. At the phase transition time $t^*_k$ we have $u_k{{\left({t^*_k}\right)}}=u_*$ as well as $\dot{u}_k{{\left({t^*_k}\right)}}>0$ according to Proposition \[pro:existence\] and the system moves into $X_{k+1}$.[]{data-label="fig:single-interface"}](single_interface){width=".85\textwidth"}
If $u$ is a single-interface solution with $u(t) \in X_k$ for some $k \in {\mathbb{Z}}$ and $t>0$ then $u_j(t)$ belongs to the *positive phase* $\Theta_+$ for $j<k$ and to the *negative phase* $\Theta_-$ for $j>k$, respectively; see Figure \[fig:single-interface\]. At the *microscopic interface* $j=k$, however, $u_k(t)$ may be either in the negative phase or in the spinodal interval $\Theta_0$. Moreover, $u_k$ may enter and leave the spinodal region via $u_k = -u_*$ several times during the dynamics of in $X_k$, and we refer to the time intervals where $u_k \in \Theta_0$ as *spinodal visits* of $u_k$. On the other hand, the evolution continues in $X_{k+1}$ once $u_k$ passes through $u_k = +u_*$ at some *phase transition time* $t_k^*$.
The following proposition adapts [@HeHe13 Theorem 3.2] to the present potential and provides the existence and uniqueness of single-interface solutions, where we assume from now on that $k_1=1$. The crucial argument is to show that the particles pass the spinodal region one after another. We derive this property in the framework of comparison principles but mention that a similar observation has been reported in [@LaMa12].
\[pro:existence\] For given initial data $u{{\left({0}\right)}} \in X_1$ there exists a unique single-interface solution $u$ to , and this solution satisfies $$\label{eq:bounds-on-u}
-u_{**}
\leq
u_j(t)
\leq {}\max \Big( u_{**}, \sup_{j \in {\mathbb{Z}}} u_j(0) \Big){}$$ for all $t \geq 0$ and $j \in {\mathbb{Z}}$. Moreover, the entrance condition $$u_k(t_k^*) = u_*,
\qquad
\dot u_k(t_k^*) > 0
\qquad\text{and}\qquad
t_{k+1}^* - t_k^*
\geq
C$$ holds for any $k \geq 1$ with $t_k^*<\infty$, where $C>0$ depends only on $\Phi$ and the initial data, and the exit condition $$\label{eq:spinodal-entrance-condition}
u_{k-1}(t) > u_{**},\qquad$$ holds at any time $t>0$ with $u_k(t) = -u_*$ and $\dot u_k(t) \geq 0$.
**: The right hand side $\operatorname{\Delta}\Phi'(\cdot)$ of is Lipschitz continuous with respect to the $\ell^\infty$-norm of $u$, so Picard’s theorem yields the local existence and uniqueness of a continuously differentiable solution with values in $\ell^\infty({\mathbb{Z}})$. Moreover, denoting the upper bound in by $D$ and introducing the state set $$\begin{aligned}
Y&:=\big\{ u\in\ell^\infty\;:\; -u_{**} \leq u_j \leq D\quad \text{for all $j\in{{\mathbb{Z}}}$}\big\}
\end{aligned}$$ we infer from the properties of $\Phi^\prime$ the implication $$\begin{aligned}
{{\big(u_j{{\left({t}\right)}}\big)}}_{j\in{{\mathbb{Z}}}}=Y\qquad \implies \qquad 2\Phi^\prime{{\left({-u_{**}}\right)}}\leq \dot{u}_j{{\left({t}\right)}}+ 2\Phi^\prime{{\big(u_j{{\left({t}\right)}}\big)}}\leq 2\Phi^\prime{{\left({D}\right)}}\;\;\text{for all}\;\;j\in{{\mathbb{Z}}}\,.\end{aligned}$$ The comparison principle for scalar ODEs reveals that $Y$ is a forwardly invariant region for , and this ensures the global existence of solutions with .
For $u(t) \in X_1$ the dynamics of $p_j(t) = \Phi'(u_j(t))$ are governed by $$\begin{aligned}
\dot p_j(t)
=
\dot u_j(t)
=
\operatorname{\Delta}p_j(t)
\qquad \text{for} \qquad
j\neq1,
\end{aligned}$$ and together with we obtain $$\begin{aligned}
{3}
- 2 p_* &\leq \dot p_j(t) + 2 p_j(t) & &\leq 2 \Phi'(D)
&\qquad\text{for } j<1,
\\
- 2 p_* &\leq \dot p_j(t) + 2 p_j(t) & &\leq 2p_*
&\qquad\text{for } j>1.
\end{aligned}$$ The comparison principle yields $$\begin{aligned}
{2}
p_j(t) &\geq - p_* \left( 1- e^{-2t} \right) + p_j(0) {\mathrm{e}}^{-2t}
& &\qquad\text{for } j \not= 1,
\\
p_j(t) &\leq + p_* \left( 1- e^{-2t} \right) + p_j(0) {\mathrm{e}}^{-2t}
& &\qquad\text{for } j > 1
\end{aligned}$$ and from the continuity of $u$ we infer that $u(t) \in X_1$ holds unless $u_1$ reaches either $-u_{**}$ or $u_*$. In addition, if $u_1(t)$ is not inside the spinodal region, that is if $u_1(t) < -u_*$, then we have $\dot p_1(t) \geq -2 p_* -2 p_1(t)$ and this implies that $u_1(t)$ cannot reach $-u_{**}$. Hence, $u(t)$ either remains inside $X_1$ forever, which means $t_1^*:=\infty$, or $u(t)$ reaches $\partial X_1 \cap \partial X_2$ at some time $t_1^* \in (0,\infty)$ with $u_1(t_1^*) = u_*$.
For $t^*_1<\infty$ we have $$\dot u_1(t_1^*)
=
\operatorname{\Delta}p_1(t_1^*)
=
p_0(t_1^*) + p_2(t_1^*) - 2 (-p_*)
>
0$$ since $p_j(t_1^*)> -p_*$ for $j \not= 1$, and we conclude that at the exit time $t_1^*$ the solution $u$ runs into $X_2$ with positive speed. Now suppose that $t \in (0,t_1^*)$ is an entrance time such that $u_1(t) = -u_*$ and $\dot u_1(t) \geq 0$. Then we compute $$0
\leq
\dot u_1(t)
=
p_0(t) + p_2(t) - 2 p_*
<
p_0(t) - p_*$$ and obtain .
Repeating the two preceding steps in the case of $t_1^*<t^*_2<\infty$, we see that $u(t) \in X_2$ for $t \in (t_1^*,t_2^*)$, and that $p_1{{\left({t}\right)}} > p_*$ holds at any entrance time with $u_2{{\left({t}\right)}}=-u_*$ and $\dot u_2{{\left({t}\right)}} \geq 0$. Moreover, for $t_2^*<\infty$ there exists a time $t_2^{\#} \in (t_1^*, t_2^*)$ such that $u_2(t_2^\#)=-u_*$ for the first time, and this implies $$\dot p_1(t)
=
\operatorname{\Delta}p_1(t)
\leq
\Phi'(D) + p_* - 2 p_1(t)
\quad \text{for}\quad
t \in (t_1^*,t_2^\#),
\qquad
p_1 {{\left({t_1^*}\right)}} = -p_*,
\qquad
p_1 {(t_2^{\#})} \geq +p_{*}.$$ The comparison principle for ODEs yields $$\begin{aligned}
p_*
\leq
p_1{(t_2^\#)}
\leq
\tfrac12
{{\big(\Phi^\prime{{\left({D}\right)}}+p_*\big)}}
{{\big(1-{\mathrm{e}}^{-2{(t_2^\#-t_1^*)}}\big)}}
- p_*{\mathrm{e}}^{-2{(t_2^\#-t_1^*)}}
\end{aligned}$$ and after rearranging terms we obtain via $$\begin{aligned}
\label{eq:bounds-on-u.PEqn1}
{\mathrm{e}}^{2{(t_2^*-t_1^*)}}
\geq
{\mathrm{e}}^{2{(t_2^\#-t_1^*)}}
\geq
\frac{\Phi^\prime{{\left({D}\right)}}+3p_*}{\Phi^\prime{{\left({D}\right)}}-p_*}
\end{aligned}$$ a lower bound for $t_2^*-t_1^*$, where the above choice of $D$ implies $\Phi^\prime{{\left({D}\right)}}\geq p_*$.
The proof can now be completed by iteration.
As an immediate consequence of Proposition \[pro:existence\] we obtain the following characterization of the dynamics of $p=\Phi'(u)$ which will be the starting point for our analysis of the spinodal fluctuations in §\[sect:fluctuations\].
\[cor:structure-of-p\] Let $u$ be a single-interface solution and denote by $$\label{eq:def-chi}
\chi_j(t) = 1 \quad\text{if }\; u_j(t) \in (-u_*,u_*),
\qquad
\chi_j(t) = 0 \quad\text{otherwise}$$ the indicator of spinodal visits of $u_j$. Then $p_j=\Phi^\prime(u_j)$ satisfies $$\label{eq:structure-of-p}
\dot p_j(t)
=
\big( 1- \chi_j(t) \big) \operatorname{\Delta}p_j(t) - \chi_j(t) \kappa \operatorname{\Delta}p_j(t)$$ for all $j\in{{\mathbb{Z}}}$ and almost all $t>0$.
Equation is true for times $t$ where $u_j(t) \not\in \{ \pm u_* \}$, because $p_j$ is continuously differentiable in a neighborhood of such $t$ and we have $\dot
p_j(t) = \Phi''(u_j(t)) \operatorname{\Delta}p_j(t)$ with either $\Phi''(u_j(t))=1$ or $\Phi''(u_j(t))=-{{\kappa}}$. Moreover, the set of times $\{ t \colon u_j(t)=+ u_* \text{ for some } j \in
{\mathbb{Z}}\}$ is by Proposition \[pro:existence\] contained in the countable set $\{ t_k^* \colon k\in{{\mathbb{N}}}\}$ and thus not relevant for our discussion. The same is true for each set $T_j:=\{t\colon u_j{{\left({t}\right)}}=-u_*,\, \dot{u}_j{{\left({t}\right)}}\neq0 \}$, which consists of isolated points and is hence also countable (it can be covered by disjoint open intervals, each of which containing a different rational number). It remains to consider ${\mathcal{T}}_j = \{ t
\colon u_j(t)=-u_*, \,\dot{u}_j{{\left({t}\right)}}=0 \}$ with fixed $j\in{{\mathbb{Z}}}$. For any given $t\in{\mathcal{T}}_j$ and all sufficiently small ${\left|{h}\right|}>0$ we observe that $u_j(t+h) = u_j(t) + \dot u_j(t)h + o(h) = -u_* + o(h)$ and find $$\left| p_j(t+h) - p_j(t) \right|
=
\left| \Phi'(-u_* + o(h)) - \Phi'(-u_*) \right|
\leq
\max(1,\kappa) o(h).$$ This estimate implies $\dot{p}_j(t)=0$, and combining this with $\operatorname{\Delta}p_j(t)=\dot u_j(t)=0$ we conclude that is satisfied for all times in ${\mathcal{T}}_j$.
Lower bound for the waiting time
--------------------------------
Proposition \[pro:existence\] reveals the following dynamical properties for single-interface data:
1. at any time $t$ there is at most one particle inside the spinodal region, and
2. the particles undergo their phase transition one after the other in the sense that $u_{k+1}$ can enter the spinodal region only when $u_k$ has completed its phase transition.
Our next goal is to show that the spinodal visits of neighboring particles are suitably separated. To this end we introduce the following times and refer to Figure \[fig:times\] for an illustration.
![Schematic representation of the times from Notation \[not:spinodal-entrance-times\]. We control neither the number nor the duration of spinodal excursions but estimate their cumulative impact in Corollary \[cor:bounds-neg-fluct\].[]{data-label="fig:times"}](times_1){width=".85\textwidth"}
\[not:spinodal-entrance-times\] Let $u$ be a single-interface solution as in Proposition \[pro:existence\]. For $k\geq 1$ we denote by $$\label{Eqn:DefSpinodalTimes}
t_k^{\#}
:=
\inf {\ensuremath{ \left\{ \ifx\emptyt > t_{k-1}^* \relax \else t > t_{k-1}^*: \fi u_k(t)>-u_* \right\} }}
\qquad\text{and}\qquad
t_k^{\flat}
:=
\inf {\ensuremath{ \left\{ \ifx\emptyt \geq t_{k}^\# \relax \else t \geq t_{k}^\#: \fi u_k(s)>-u_* \text{ for all } s>t \right\} }}$$ the *first* and the *final spinodal entrance time of $u_k$*, respectively. Moreover, we refer to spinodal visits of $u_k$ that occur in $(t_k^\#,t_k^\flat)$ as *spinodal excursions* and to the spinodal visit in $(t_k^\flat,t_k^*)$ as *spinodal passage*.
The quantity $t_{k+1}^\#-t_k^*$ is a lower bound for the difference $t_{k+1}^*-t_k^*$ between consecutive phase transition times and implies an upper bound for the microscopic interface speed. In the proof of Proposition \[pro:existence\], see , we have shown that $t_{k+1}^\#-t_k^*\geq
C$ for some constant $C$, but this bound is not sufficient for passing to the macroscopic limit as it scales like $1/{{\varepsilon}}^2$ under the parabolic scaling . In the next lemma, we therefore derive an improved estimate for the difference $t_{k+1}^\#-t_k^*$ by means of problem-tailored comparison principles as sketched in Figure \[fig:waiting\_lemma\]. To this end, we note that Proposition \[pro:existence\] combined with implies for any $k\geq1$ the estimates $$\begin{aligned}
\label{Eqn:Identities1}
-p_*
\leq
p_k{{\left({t}\right)}}\leq p_*
\quad
\text{for } 0\leq t \leq t^*_k,
\qquad
-p_*\leq p_k{{\left({t}\right)}}<\infty
\quad
\text{for } t\geq{t^*_k}\end{aligned}$$ as well as $$\begin{aligned}
\label{Eqn:Identities2}
p_k{{\left({t^*_k}\right)}} = -p_*,
\qquad
p_{k}{(t^\#_{k+1})} {}>+ p_* = p_{k+1}{(t^\#_{k+1})}.{}\end{aligned}$$ Moreover, we denote by $g$ the discrete heat kernel, which solves $$\begin{aligned}
\label{Eqn:HeatKernel}
\dot{g}_j = \Delta g_j,
\qquad g_j{{\left({0}\right)}}=\delta_j^0\end{aligned}$$ with Kronecker delta $\delta^{0}_j$ and discrete Laplacian $\Delta$ as in . Notice that $g$ can be computed explicitly by discrete Fourier transform, see for instance [@HeHe13 Appendix].
![Illustration of Lemma \[lem:waiting\] which provides a majorant for $p$ and bounds the waiting time. *Left panel*. Cartoon of $p{{\left({t}\right)}}$ (black) and the stationary, kink-type supersolution ${{\gamma}}$ (gray) for $k=1$ and times $t\in[t^*_0,t^*_1]$. At the phase transition time $t^*_1$, both the interface (vertical line) and ${{\gamma}}$ are shifted to the right by one lattice position. *Right panel*. Cartoon of $p{{\left({t^*_1}\right)}}$ and $\bar{{{\gamma}}}{{\left({t^*_1}\right)}}$ for $k=1$, where the time-dependent supersolution $\bar{{\gamma}}$ is used to estimate $t^\#_2-t^*_1$ from below. Notice that the phase interface has already been shifted to $j=2$ and that Proposition \[pro:existence\] yields the two key conditions $p_1{(t^*_1)}=-p_*$ and $p_1{(t^\#_2)}>+p_*$.[]{data-label="fig:waiting_lemma"}](waiting_lemma){width=".85\textwidth"}
\[lem:waiting\] Suppose there exists $b>0$ such that the single-interface initial data $u{{\left({0}\right)}}\in X_1$ satisfy $$\begin{aligned}
p_j(0) \leq \gamma_j := p_*+b \max\big\{1-j,0\}
\qquad
\text{for all } j \in {\mathbb{Z}}.
\end{aligned}$$ Then the solution $u$ from Proposition \[pro:existence\] satisfies $$\label{lem:waiting.Eqn1}
p_j(t) \leq \gamma_{j-k+1}
\qquad
\text{for } j \in {\mathbb{Z}}\text{ and } t \in [t_{k-1}^*,t_{k}^*)$$ as well as $$\label{lem:waiting.Eqn2}
t_{k+1}^* - t_k^*
\geq
t_{k+1}^{\#} - t_k^*
\geq
\frac{c_* p_*}{b}$$ for all $k \geq 1$. Here, the universal constant $c_*$ is determined by the discrete heat kernel, and makes sense for $t^*_k<\infty$ only.
We start with $k=1$ and suppose for contradiction that there exists a finite time $\tilde{t}_1\in(t^*_0,t^*_1]$ such that $$\begin{aligned}
0
<
\tilde{C}
:=
\sup_{t\in[t^*_0,\tilde{t}_1]} \sup_{j\in{{\mathbb{Z}}}}c_j{{\left({t}\right)}},
\qquad
c_j{{\left({t}\right)}}
:=
p_j {{\left({t}\right)}}-{{\gamma}}_j,
\end{aligned}$$ where $\tilde{C}\in{{\mathbb{R}}}$ is well-defined due to and $t^*_0=0$ holds by definition. By we have $$\begin{aligned}
\label{lem:waiting.PEqn1}
c_1{{\left({t}\right)}}\leq{}p_*-{{\gamma}}_1={}0
\qquad \text{for }
t^*_0 \leq t \leq \tilde{t}_1,
\end{aligned}$$ while for $j\neq 1$ our definitions imply $$\begin{aligned}
\dot{c}_j = \dot{p}_j = \Delta p_j = \Delta c_j =c_{j+1}+c_{j-1}-2c_j\leq
2{(\tilde{C}-c_j)}\end{aligned}$$ thanks to Corollary \[cor:structure-of-p\]. Therefore, and due to the initial condition $c_j{{\left({t^*_0}\right)}}\leq 0$, the comparison principle for ODEs guarantees that $$\begin{aligned}
\label{lem:waiting.PEqn2}
{c}_j{{\left({t}\right)}}\leq \tilde{C}{{\big(1-{\mathrm{e}}^{-2t}\big)}}
\qquad
\text{for } j\neq1 \text{ and } t^*_0\leq t\leq \tilde{t}_1.
\end{aligned}$$ The combination of and finally yields $0<\tilde{C}\leq \tilde{C}{{\big(1-{\mathrm{e}}^{-2\tilde{t}_1}\big)}}<\tilde{C}$ and hence the desired contradiction. In particular, we established the claim for $k=1$, and since this implies $p_j{{\left({t^*_1}\right)}}\leq {{\gamma}}_j\leq {{\gamma}}_{j-1}$ we can proceed iteratively.
Due to the shift invariance it suffices again to study the case $k=1$. As illustrated in Figure \[fig:waiting\_lemma\], we introduce $\bar \gamma$ as the solution to the initial value problem $$\begin{aligned}
\dot{\bar \gamma}_j{{\left({t}\right)}}
=
\operatorname{\Delta}\bar \gamma_j{{\left({t}\right)}},\qquad \bar{\gamma}_j(t^*_1) = \gamma_j - 2 p_*
\delta^{1}_j\qquad \text{for } j\in{{\mathbb{Z}}}\text{ and } t\geq t^*_1,
\end{aligned}$$ and using the discrete heat kernel $g$ from we write its explicit solution as $$\bar \gamma_j(t)
=
\sum_{n \in {\mathbb{Z}}} g_{j-n}(t-t^*_1) \bar \gamma_n(t^*_1)
=
-
2 p_* g_{j-1}(t-t^*_1)+\sum_{n \in {\mathbb{Z}}} g_{n}(t-t^*_1) \gamma_{j-n}.$$ By differentiation of $\bar{{{\gamma}}}_1$ and recalling that $$\begin{aligned}
\sum_{n}\dot{g}_{n}{{\left({s}\right)}}{{\gamma}}_{1-n}
=
\sum_{n}\Delta{g}_{n}{{\left({s}\right)}}{{\gamma}}_{1-n}
=
\sum_{n}{g}_{n}{{\left({s}\right)}}\Delta {{\gamma}}_{1-n}
{}= b\sum_{n}{g}_{n}{{\left({s}\right)}}\delta^0_{n}={}b g_{0}{{\left({s}\right)}},
\end{aligned}$$ we find $\dot{\bar \gamma}_1{{\left({t}\right)}} = - 2 p_*\dot{g}_0{{\left({t-t^*_1}\right)}}+b
g_0{{\left({t-t^*_1}\right)}}$, which yields $$\begin{aligned}
\bar{{{\gamma}}}_1{{\left({t}\right)}}
=
p_*-2p_* g_0{{\left({t-t^*_1}\right)}}+b\int\limits_0^{t-t^*_1}g_0{{\left({s}\right)}} \,{\mathrm{d}s}
\end{aligned}$$ by integration and due to the initial conditions $\bar{{{\gamma}}}_1{{\left({t^*_1}\right)}}=-p_*$, $g_0{{\left({0}\right)}}=1$. Since $g_0$ is positive and decreasing we conclude the existence of a unique time $\bar{t}_1>t^*_1$ such that $$\begin{aligned}
\label{lem:waiting.PEqn4}
\bar{{\gamma}}_1{{\left({\bar{t}_1}\right)}}=p_*
\qquad \text{and}\qquad
\bar{{\gamma}}_1{{\left({t}\right)}}< p_*
\qquad
\text{for all } t\in[t^*_1,\bar{t}_1],
\end{aligned}$$ and exploiting $g_0{{\left({s}\right)}}\sim {{\left({1+s}\right)}}^{-1/2}$ we justify that $$\begin{aligned}
\label{lem:waiting.PEqn5}
\bar{t}_1-t^*_1
\geq
\frac{c_*p_*}{b}
\end{aligned}$$ holds for some universal constant $c_*>0$. Moreover, $p$ solves the discrete heat equation for $t\in[t^*_1,t^\#_2]$, where we have $$\begin{aligned}
p_j{{\left({t^*_1}\right)}} \leq {{\overline{{{\gamma}}}}}_j{{\left({t^*_1}\right)}}
\qquad
\text{for all } j\in{{\mathbb{Z}}}\end{aligned}$$ according to and since $p_1{{\left({t^*_1}\right)}}=-p_*$ holds by . A standard comparison principle therefore yields $$p_j(t) \leq \bar \gamma_j(t)
\quad \text{for all } j\in{{\mathbb{Z}}}\text{ and } t\in[t^*_1,t^\#_2],$$ and in combination with we obtain $t^\#_2>\bar{t}_1$ since also guarantees that $p_1{(t^\#_2)}\geq p^*$. The desired estimate now follows from .
Family of entropy inequalities
------------------------------
We finally establish the discrete analog to the weak formulation of the entropy relation as well as the local variant of the energy-dissipation relation.
\[Prop:Entropy\] Let $\psi\in\ell^1{{\left({{{\mathbb{Z}}}}\right)}}$ be an arbitrary but nonnegative test function, $t\geq0$ a given time, and $u$ be a solution to . Then we have $$\begin{aligned}
\label{Prop:Entropy.Eqn1}
{\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{t}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{t}} \fi }
}\sum_{j\in{{\mathbb{Z}}}} \eta{{\big(u_j{{\left({t}\right)}}\big)}} \psi_j\leq -\sum_{j \in
{\mathbb{Z}}} \mu{{\big(p_j{{\left({t}\right)}}\big)}}{{\big(\nabla_+ \psi_j\big)}}{{\big( \nabla_+ p_j{{\left({t}\right)}}\big)}}
\end{aligned}$$ for any smooth entropy pair $(\eta,\mu)$ satisfying as well as $$\begin{aligned}
\label{Prop:Entropy.Eqn2}
\sum_{j\in{{\mathbb{Z}}}} \int\limits_0^{t}{{\big(\nabla_+p_j{{\left({s}\right)}}\big)}}^2 \psi_j \,{\mathrm{d}s}
\leq
\sum_{j\in{{\mathbb{Z}}}}\Phi{{\big(u_j{{\left({0}\right)}}\big)}}
-
\sum_{j\in{{\mathbb{Z}}}}\int\limits_0^{t} p_j{{\left({s}\right)}} {{\big(\nabla_+\psi_j\big)}}
{{\big(\nabla_+p_j{{\left({s}\right)}}\big)}} \,{\mathrm{d}s}\end{aligned}$$ with energy $\Phi$ as in .
Since ensures ${\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{t}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{t}} \fi }
} \eta{{\left({u_j}\right)}} =
\eta^\prime{{\left({u_j}\right)}}\dot{u}_j= \mu{{\left({p_j}\right)}}\operatorname{\Delta}{p}_j$, we compute $$\begin{aligned}
\label{Prop:Entropy.PEqn1}
\begin{split}
{\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{t}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{t}} \fi }
} \sum_{j \in {\mathbb{Z}}} \eta{{\left({u_j}\right)}} \psi_j
&=
\sum_{j \in {\mathbb{Z}}} \psi_j \mu{{\left({p_j}\right)}}\nabla_-\nabla_+{p}_j
= - \sum_{j \in {\mathbb{Z}}} \nabla_+{{\big(\psi_j \mu{{\left({p_j}\right)}}\big)}} \nabla_+ p_j
\\
&= -\sum_{j \in {\mathbb{Z}}} \mu{{\left({p_j}\right)}}\nabla_+ \psi_j \nabla_+ p_j -
\sum_{j \in {\mathbb{Z}}}\psi_{j+1} \nabla_+ \mu{{\left({p_j}\right)}} \nabla_+ p_j\,,
\end{split}
\end{aligned}$$ where we used discrete integration by parts as well as the product rule ${{\left({a_{j+1}b_{j+1}-a_{j}b_{j}}\right)}} =
b_j{{\left({a_{j+1}-a_{j}}\right)}}+a_{j+1}{{\left({b_{j+1}-b_{j}}\right)}}$. The monotonicity of $\mu$ implies $$\begin{aligned}
\nabla_+ \mu{{\left({p_j}\right)}} \nabla_+ p_j
=
{{\big(\mu{{\left({p_{j+1}}\right)}}-\mu{{\left({p_j}\right)}}\big)}}{{\left({p_{j+1}-p_{j}}\right)}}
\geq
0\,,
\end{aligned}$$ so follows immediately thanks to the nonnegativity of $\psi$. Moreover, choosing $(\eta,\mu) = (\Phi,\mathrm{id})$ and integrating in time we obtain after rearranging terms and due to $\Phi{{\left({u_j{{\left({t}\right)}}}\right)}}\geq0$.
Analysis of the spinodal fluctuations {#sect:fluctuations}
=====================================
As already discussed in §\[sect:intro\], the analysis of the fluctuations is the very core of the convergence problem and so far we are only able to deal with trilinear nonlinearities $\Phi^\prime$ because for those we can decompose the nonlinear dynamics into linear subproblems and combine all partial results by the superposition principle. We also recall that the case ${{\kappa}}\in(0,\infty)$ is more involved than the bilinear limit ${{\kappa}}=\infty$ without spinodal excursions and with degenerate spinodal passages.
The asymptotic arguments below strongly rely on the regularity of the microscopic initial data. To keep the presentation as simple as possible we make from now on the following standing assumption, which guarantees that the initial data are well-prepared.
\[ass:macro\] The initial data $u(0)$ belong to $X_1$ and there exist constants ${{\alpha}}$, $\beta > 0$ such that $p{{\left({0}\right)}}=\Phi'(u(0))=u{{\left({0}\right)}}-\operatorname{sgn}{u{{\left({0}\right)}}}$ satisfies $$\sup_{j \in {\mathbb{Z}}} | p_j(0) | \leq \alpha,
\qquad
\sup_{j \in {\mathbb{Z}}} | \nabla_+ p_j(0) | \leq \alpha {{\varepsilon}},
\qquad
\sup_{j \in {\mathbb{Z}}\setminus\{1\}} | \operatorname{\Delta}p_j(0) | \leq \alpha {{\varepsilon}}^2$$ as well as $${\left|{\operatorname{\Delta}p_1(0)}\right|} \leq \beta {{\varepsilon}},
\qquad \qquad
p_j{{\left({0}\right)}} \leq p_*+{{\varepsilon}}\beta \max\big\{0,1-j\big\}
\quad \text{for all } j\in{{\mathbb{Z}}}$$ for ${{\varepsilon}}>0$. Moreover, for convenience we assume that $u_1(0) \not\in (-u_*,+u_*)$.
![Typical initial data (black dots) as in Assumption \[ass:macro\] which sample macroscopic functions $U_{{\rm ini}}$ and $P_{{\rm ini}}=U_{{\rm ini}}-\operatorname{sgn}U_{{\rm ini}}$ (gray curves) that are compatible with the limit model from Main Result \[res:Main\]. The gray dots represent the kink-type majorant for $p{{\left({0}\right)}}$ which enables us to bound all microscopic waiting times from below and hence the macroscopic interface speed from above, see Lemma \[lem:waiting\] and Corollary \[cor:time-and-number-bounds\].[]{data-label="Fig:InitialData"}](initial_data){width="90.00000%"}
Assumption \[ass:macro\] is motivated by the limit dynamics, see Figure \[Fig:InitialData\] for an illustration, and the prototypical example from §\[sect:mainresult\] corresponds to $\Xi_{{\rm ini}}=0$ and $$\begin{aligned}
\alpha
=
\sup_{\xi\in{{\mathbb{R}}}}{{\left({{\left|{P_{{\rm ini}}{{\left({\xi}\right)}}}\right|}
+
{\big|{P_{{\rm ini}}^\prime{{\left({\xi}\right)}}}\big|}+{\big|{P_{{\rm ini}}^{\prime\prime}{{\left({\xi}\right)}}}\big|}}\right)}},
\qquad
\beta
=
\max\Big\{
{\big|{ {{|\![\partial_\xi P_{{\rm ini}}{{\big(\Xi_{{\rm ini}}\big)}}]\!|}} }\big|},
\sup_{\xi<\Xi_{{\rm ini}}}{{\big({{\left({P_{{\rm ini}}{{\left({\xi}\right)}}-p_*}\right)}}/\xi\big)}}\Big\}.\end{aligned}$$ An important consequence of Assumption \[ass:macro\] and Lemma \[lem:waiting\] are the following bounds for the microscopic waiting time and the number of microscopic phase transitions.
\[cor:time-and-number-bounds\] The microscopic single-interface solution from Proposition \[pro:existence\] satisfies $$t_{k+1}^* - t_k^*
\geq
t_{k+1}^\# - t_k^*
\geq
\frac{2 d_*}{{{\varepsilon}}}$$ for all $k\geq1$ with $t_k^*<\infty$ and some constant $d_*>0$, which depends only on the potential parameter $\kappa$ and on the initial data via the parameters ${{\alpha}}$, ${{\beta}}$. In particular, for any macroscopic final time $\tau_{{\rm fin}}>0$ we have $$K_{{\varepsilon}}:=
\max \{k \geq 1: t_k^* \leq \tau_{{\rm fin}}/{{\varepsilon}}^2 \}
\leq
\frac{\tau_{{\rm fin}}}{2d_*{{\varepsilon}}},$$ where $K_{{\varepsilon}}$ abbreviates the number of phase transitions in the corresponding microscopic time interval $[0,t_{{\rm fin}}]$ with $t_{{\rm fin}}:= \tau_{{\rm fin}}/{{\varepsilon}}^2$.
In the following, we always suppose that $0<\tau_{{\rm fin}}<\infty$ is fixed and denote by $C$ a generic constant that depends on ${{\kappa}}$, ${{\alpha}}$, $\beta$, and $\tau_{{\rm fin}}$ but not on ${{\varepsilon}}>0$.
Prototypical spinodal problem {#sec:prot-phase-trans}
-----------------------------
Equation reveals that during a spinodal visit of some $u_k$ the corresponding $p_k=\Phi^\prime{{\left({u_k}\right)}}$ satisfies $\dot p_k = - \kappa \operatorname{\Delta}p_k$ while all other $p_j$ adhere to forward diffusion $\dot p_j = \operatorname{\Delta}p_j$. For this reason, we first consider a prototypical spinodal problem $$\label{Eqn:ToyProblem}
\dot z_j(t)
=
\begin{cases}
-\kappa \operatorname{\Delta}z_0(t) + (1+\kappa) f(t)
&\text{if } j=0,
\\
+ \operatorname{\Delta}z_j(t)
&\text{if } j \not=0,
\end{cases}
\qquad\text{for } j \in {\mathbb{Z}}, t \geq 0,$$ where $z$ represents some part of $p$ and where $f$ is a perturbation whose purpose will become clear later. Given bounded initial data at time $t=0$, the ODE admits a unique solution, and our goal in this section is to understand how the backward diffusing $z_0$ interacts with the forward diffusing background and the source term $(1+\kappa)f$. A typical numerical simulation is shown in Figure \[Fig:SlowFast\].
![Solution to the spinodal problem with ${{\kappa}}=1$, vanishing initial data, and source term $f{{\left({t}\right)}}\equiv 0.02$. Due to the backward diffusion of $z_0$, *all* lattice data $z_j$ change rapidly in time and explode exponentially but the slow variables from Lemma \[lem:prototypical\] behave much nicer.[]{data-label="Fig:SlowFast"}](slow_fast){width="95.00000%"}
Splitting the solution $z$ into its even and odd parts according to $$\begin{aligned}
z_{{{\rm \,even}},j}{{\left({t}\right)}}:=\tfrac12{{\big(z_{+j}{{\left({t}\right)}}+z_{-j}{{\left({t}\right)}}\big)}}\qquad\text{and}\qquad
z_{{{\rm \,odd}},j}{{\left({t}\right)}}:=\tfrac12{{\big(z_{+j}{{\left({t}\right)}}-z_{-j}{{\left({t}\right)}}\big)}},\end{aligned}$$ respectively, we first observe that $z_{{{\rm \,even}}}$ also satisfies , whereas $z_{{{\rm \,odd}}}$ solves the discrete heat equation. Next, introducing the variables $$\label{Eqn:SlowVariables}
\zeta_n(t)
=
\tfrac{1+2\kappa}{2\kappa} z_{{{\rm \,even}},n}(t)
- \tfrac{1}{2\kappa} z_{{{\rm \,even}},n-1}(t),
\qquad
n\geq1$$ we verify by direct computation the identities $$\label{Eqn:SlowDynamics}
\dot z_0(t)
=
\tfrac{(2\kappa)^2}{1+2\kappa} \big( z_0(t) - \zeta_1(t) \big)
+ (1+\kappa) f(t)$$ and $$\dot \zeta_n(t)
=
\begin{cases}
\zeta_2(t) - \zeta_1(t) - \tfrac{1+\kappa}{2\kappa} f(t)
&\text{if } n=1,
\\
\operatorname{\Delta}\zeta_n
&\text{if } n>1.
\end{cases}$$ The key observation is that $\zeta$ solves the discrete heat equation on the semi-infinite domain $n \geq 1$ with inhomogeneous Neumann boundary condition at $n=1$. Therefore, if the initial data $z(0)$ and the source term $f(t)$ are uniformly small in $j$ and $t$, respectively, then all components of $\zeta$ evolve *slowly*, and the same is true for $z_{{{\rm \,odd}}}$ as well. On the other hand, the *fast* variable $z_0$ exhibits a strong tendency to grow exponentially and changes generically by an order $1$ in times of order $1$. In this sense, the change of variables $$z \in \ell^1({\mathbb{Z}})
\qquad\rightsquigarrow\qquad
(z_0,z_{{{\rm \,odd}}},\zeta) \in {\mathbb{R}}\times \ell^1({\mathbb{N}}) \times \ell^1({\mathbb{N}})$$ separates the slow and fast dynamics of and allows us to isolate a single ‘unstable mode’ as follows.
Any solution to \[lem:prototypical\] can be written as $$z_j(t) = z_{{{\mathrm{fast}}},j}(t) + z_{{{\mathrm{slow}}},j}(t)
\qquad{}\text{with}{}\qquad
z_{{{\mathrm{fast}}},j}(t) := \frac{z_0(t)}{(1+2\kappa)^{|j|}},\qquad
z_{{{\mathrm{slow}}}} := z-z_{{{\mathrm{fast}}}},$$ and we have $$\sum_{j \in {\mathbb{Z}}} |z_{{{\mathrm{slow}}},j}(t)|
\leq
C
\Bigg(
\sum_{j \in {\mathbb{Z}}} |z_j(0)| + \int\limits_0^t |f(s)| \,{\mathrm{d}s}
\Bigg)$$ for some constant $C$ which depends only on the parameter $\kappa$.
In view of the even-odd parity of the prototypical phase-transition model it suffices to consider solutions that are either even or odd. For odd initial data, we always have $z_j{{\left({t}\right)}}=-z_{- j}{{\left({t}\right)}}$ and the assertions follow with $$\begin{aligned}
z_{{{\mathrm{fast}}},j}(t) = 0, \qquad z_{{{\mathrm{slow}}},j}(t) = z_j(t) = z_{{{\rm \,odd}},j}(t)\end{aligned}$$ since $z$ satisfies the discrete heat equation.
Using $z_j(t) = z_{{{\rm \,even}},j}(t)$ as well as the definition of $\zeta$ in we verify the representation formula $$z_{-j}(t)
=
z_{j}(t)
=
\frac{z_0(t)}{(1+2\kappa)^j}
+
\frac{2\kappa}{(1+2\kappa)^{j+1}} \sum_{n=1}^j (1+2\kappa)^n \zeta_n(t)
\qquad
\text{for all } j \geq 1,$$ where the first and the second term on the right hand side represent $z_{{\mathrm{fast}}}$ and $z_{{\mathrm{slow}}}$, respectively. In particular, we estimate $$\begin{aligned}
\sum_{j \in {\mathbb{Z}}} |z_{{{\mathrm{slow}}},j}(t)|
&\leq
\sum_{j=1}^\infty \frac{4\kappa}{(1+2\kappa)^{j+1}}
\sum_{n=1}^j (1+2\kappa)^n |\zeta_n(t)|
\\&=
\sum_{n=1}^\infty |\zeta_n(t)|
\sum_{j=n}^\infty \frac{4\kappa}{(1+2\kappa)^{j-n+1}}
=
2 \sum_{n=1}^\infty |\zeta_n(t)|
\end{aligned}$$ for all $t \geq 0$. Next, an off-site reflection with respect to $j=1/2$, that is, $$\widetilde \zeta_j(t) =
\begin{cases}
\zeta_j(t) &\text{if } j\geq1, \\
\zeta_{1-j}(t) &\text{if } j\leq0,
\end{cases}$$ transforms the boundary value problem for $\zeta$ into the discrete diffusion system $${{\textstyle{\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{t}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{t}} \fi }
}}} \widetilde \zeta_j(t)
=
\operatorname{\Delta}\widetilde\zeta_j(t) {}- {}\left( \delta_j^0 + \delta_j^1 \right)
\tfrac{1+\kappa}{2\kappa} f(t)
\qquad\text{for all}\quad j \in {\mathbb{Z}}\quad\text{and}\quad t \geq0$$ with source term at $j=0$ and $j=1$. Duhamel’s Principle gives $$\zeta_j(t)
=
\widetilde \zeta_j(t)
=
\sum_{n \in {\mathbb{Z}}} g_{j-n}(t) \widetilde \zeta_n(0)
{}- {}\int\limits_0^t \big( g_j(t-s) + g_{j-1}(t-s) \big) \tfrac{1+\kappa}{2\kappa}
f(s) \,{\mathrm{d}s}$$ for all $j \geq 1$, and the claim follows from $$\sum_{j \in {\mathbb{Z}}} |\widetilde \zeta_j(0)|
\leq
C \sum_{j \in {\mathbb{Z}}} |z_j(0)|$$ and the mass conservation property of the discrete heat kernel.
The proof of Lemma \[lem:prototypical\] is intimately related to the linearity of the spinodal problem as it allows us to construct the slow variables explicitly. For a general bistable nonlinearity, it remains a challenging task to identify the analog to and . We also mention that the existence of a single unstable mode has been shown in [@LaMa12] for a finite dimensional analog to using spectral analysis of tridiagonal matrices. It has also been argued that spinodal passages are typically fast with respect to the disffusive time scale. Lemma \[lem:prototypical\] extends these results to unbounded domains and quantifies the asymptotic slowness of the stable modes in a robust and reliable way.
Spinodal fluctuations {#sect:defFluct}
---------------------
As indicated in the previous section, we think of spinodal fluctuations as unstable modes in an otherwise diffusive evolution, which are evoked by spinodal visits of the $u_j$’s or, equivalently, by the linear backward diffusion of the corresponding $p_j$’s. To study this systematically, we define the *$k$-th spinodal fluctuation* $r^{(k)} := (r^{(k)}_j)_{j \in {\mathbb{Z}}}$ to be $$\label{Eqn:FA.DefFluctuations}
r^{(k)}_j(t)
:=
\begin{dcases}
0 &\text{for } 0 \leq t \leq t_k^\#,
\\
-p_j(t) + q^{(k)}_j(t) &\text{for } t_k^\# \leq t \leq t_k^*,
\\
\sum_{n \in {\mathbb{Z}}} g_{j-n}(t-t_k^*) r_n^{(k)}(t_k^*) &\text{for }
t_k^* < t,
\end{dcases}$$ where $g$ is the discrete heat kernel from and $$\label{Eqn:FA.DefDiffPart}
q^{(k)}_j(t)
:=
\begin{dcases}
0
&\text{for } t < t_k^\#,
\\
\sum_{n \in {\mathbb{Z}}} g_{j-n}(t-t_k^\#) p_n(t_k^\#)
&\text{for } t > t_k^\#
\end{dcases}$$ solves the discrete heat equation for $t>t_k^\#$ with initial data $p(t_k^\#)$.
Formula is at the heart of our asumptotic analysis and enables us to characterize both the local and the global behavior of the fluctuations. On the local side, we infer from and Corollary \[cor:structure-of-p\] that the evolution of each $r^{(k)}$ is determined by the initial condition $$\label{eq:FA.idata}
r_j^{(k)}(t_k^\#) = 0
\qquad\text{for all } j \in {\mathbb{Z}}$$ as well as the equations $$\label{Eqn:FA.DynLaw1}
\dot r_j^{(k)}(t)
=
\begin{cases}
\big( 1- \chi_k(t) \big) \operatorname{\Delta}r_k^{(k)}(t)
+
\chi_k(t) \Big( -\kappa \operatorname{\Delta}r_k^{(k)}(t)
+ (1+\kappa) \dot q_k^{(k)}(t) \Big)
&
\text{if } j=k,
\\
\operatorname{\Delta}r_j^{(k)}(t)
&
\text{if } j\not=k
\end{cases}$$ for almost every $t \in (t_k^\#,t_k^*)$ and $$\label{Eqn:FA.DynLaw2}
\dot r_j^{(k)}(t) = \operatorname{\Delta}r_j^{(k)}(t),
\qquad
j \in {\mathbb{Z}}$$ for $t>t_k^*$, where the indicator function $\chi_k$ has been introduced in . In particular, $r^{(k)}(t)$ satisfies – at any time $t$ with $\chi_k{{\left({t}\right)}}=1$ and hence on the entire interval ${(t^\flat_k,\,t^*_k)}$ – a shifted and delayed variant of the prototypical phase transition problem with forcing term $\dot q_k^{(k)}(t)$, and this gives rise to the local fluctuation estimates in §\[sec:local-fluct-estim\]. On the other hand, arguing recursively we derive from and the representation formula $$\label{eq:p-from-idata-and-fluctuation}
p_j(t)
=
\sum_{n \in {\mathbb{Z}}} g_{j-n}(t) p_j(0)
-
\sum_{k \geq 1} r_j^{(k)}(t)
\qquad
\text{for all } j\in{{\mathbb{Z}}}\text{ and } t \geq 0,$$ where the first and the second sum on the right hand side account for the initial data and the cumulative impact of all phase transitions, respectively. This identity allows us in §\[sec:glob-fluct-estim\] to sheave the local fluctuation estimates into global ones and to quantify how much $p$ deviates from the diffusive reference data due to the spinodal visits of all particles. Finally, since $p$ and $q^{(k)}$ are uniformly bounded due to and , the maximum principle for the discrete heat equation guarantees $$\label{eq:unif-bound-fluct}
\sup_{k \geq 1} \,
\sup_{j \in {\mathbb{Z}}} \,
\sup_{t \geq 0} |r^{(k)}_j(t)|
\leq C,$$ where the constant $C$ depends only on the potential $\Phi$ and the initial data $p(0)$.
The remainder of §\[sect:fluctuations\] deals with the analysis of the spinodal fluctuations. As indicated in Figure \[fig:all\_fluctuations\], it turns out that spinodal excursions and the spinodal passage of a $u_k$ lead to two distinguishable parts of $r^{(k)}$, namely the *negligible fluctuations* $r_{{{\mathrm{neg}}}}^{(k)}$ and the *essential fluctuations* $r_{{{\mathrm{ess}}}}^{(k)}$, respectively. We will show that the negligible fluctuations are not relevant for the macroscopic dynamics, whereas the essential fluctuations contribute significantly to them. More precisely, $r_{{{\mathrm{ess}}}}^{(k)}$ can be split further into a *regular* part, which leads to a sufficiently regular limit contribution, and a *residual* part which vanishes in suitable function spaces, see the proof of Proposition \[pro:compactness\] below.
We finally emphasize that phase transitions in the bilinear case ${{\kappa}}=\infty$ are instantaneous processes since the spinodal region has shrunk to a point. In particular, at the phase transition time $t^*_k=t^\#_k$, the value of $u_k$ is continuous but changes its sign from negative to positive while $p_k$ is discontinuous as it jumps down from $+p^*$ to $-p_*$. We therefore have $$\begin{aligned}
r^{(k)}_j{{\left({t^*_k+0}\right)}} = r^{(k)}_{{{\mathrm{ess}}},j}{{\left({t^*_k+0}\right)}}=2p_*\delta_j^k
\qquad \text{for}\qquad
\kappa = \infty\end{aligned}$$ and no negligible fluctuations at all.
![Life span of the total fluctuations and their parts defined in , and . Both the negligible and the residual fluctuations vanish in the macroscopic limit, see Corollary \[cor:bounds-neg-fluct\] and Lemma \[lem:bounds-res-fluct\], while the sum of all regular fluctuations drives the interface in the free boundary problem as shown in §\[sect:limit\].[]{data-label="fig:all_fluctuations"}](times_2 "fig:"){width=".75\textwidth"}\
![Life span of the total fluctuations and their parts defined in , and . Both the negligible and the residual fluctuations vanish in the macroscopic limit, see Corollary \[cor:bounds-neg-fluct\] and Lemma \[lem:bounds-res-fluct\], while the sum of all regular fluctuations drives the interface in the free boundary problem as shown in §\[sect:limit\].[]{data-label="fig:all_fluctuations"}](all_fluctuations "fig:"){width=".5\textwidth"}
Local fluctuation estimates {#sec:local-fluct-estim}
---------------------------
In the next two lemmas, we study the fluctuations $r^{(k)}$ for a fixed $k \geq 1$, and a key quantity for the analysis is $$\label{Eqn.FA.DefDk}
D_k
:=
\int\limits_{t_k^\#}^{t_k^*}
|\dot q^{(k)}_k(s)|\,{\mathrm{d}s},$$ which allows us to bound the source term in . Specifically, employing a slow-fast splitting as in §\[sec:prot-phase-trans\] we characterize the fluctuations induced by $u_k$ at the end of its phase transition and show that these are – up to small error terms – given by a shifted variant of the universal *impact profile* $\varrho$ with $$\begin{aligned}
\label{Eqn:ImpactProfile}
\varrho_j := \frac{2p_*}{(1+2\kappa)^{|j|}},\end{aligned}$$ which depends only on $\kappa$ and is illustrated in Figure \[Fig:EssFluktuations\]. Notice that the definition of $p_*$ in ensures $\sum_{j\in{{\mathbb{Z}}}}\varrho_j=2$ for all $\kappa \in (0,\infty)$ as well as $\varrho_j=2\delta_j^0$ for $\kappa=\infty$ and $\varrho_j\to0$ pointwise as ${{\kappa}}\to0$.
\[lem:loc-excursions\] For any $k \geq 1$ we have $$\label{eq:loc-excursions1}
\sup_{t \in [t_k^\#,t_k^\flat]} \sum_{j \in {\mathbb{Z}}} |r^{(k)}_j(t)|
\leq
C (1+D_k)$$ as well as $$\label{eq:loc-excursions2}
\sum_{j \in {\mathbb{Z}}} |r^{(k)}_j(t_k^\flat)|
\leq
C D_k$$ for some constant $C>0$ and spinodal entrance times $t_k^\#$, $t_k^\flat$ as in .
Throughout the proof we drop the upper index $k$ to ease the notation. Equation can be written as $$\dot r_j(t)
=
\operatorname{\Delta}r_j(t) + \delta_j^k \chi_k(t) \frac{1+\kappa}{\kappa}
{{\big( {}\dot r_k{{\left({t}\right)}} - \dot q_k{{\left({t}\right)}}{}\big)}}$$ for $t \in (t_k^\#,t_k^\flat)$, and using discrete integration by parts we find $$\begin{aligned}
\label{eq:lem-loc-excursions-eq2}
\begin{split}
{\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{t}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{t}} \fi }
} \sum_{j \in {\mathbb{Z}}} |r_j(t)|
&=
\sum_{j \in {\mathbb{Z}}} \operatorname{sgn}r_j(t) \operatorname{\Delta}r_j(t)
+ \operatorname{sgn}r_k(t) \, \chi_k(t) \frac{1+\kappa}{\kappa}{{\big({}\dot r_k{{\left({t}\right)}} - \dot q_k{{\left({t}\right)}}{}\big)}}
\\
&=
- \sum_{j \in {\mathbb{Z}}} \nabla_+ \operatorname{sgn}r_j(t) \nabla_+ r_j(t)
+ \operatorname{sgn}r_k(t) \, \chi_k(t) \frac{1+\kappa}{\kappa}{{\big({}\dot r_k{{\left({t}\right)}} - \dot q_k{{\left({t}\right)}}{}\big)}}
\\
&\leq
C \left( {{\textstyle{\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{t}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{t}} \fi }
}}} |r_k(t)| + {}|\dot q_k{{\left({t}\right)}}|{}\right),
\end{split}
\end{aligned}$$ where we used the monotonicity of the sign function. Thanks to , the fluctuations $r$ vanish at time $t_k^\#$, so an integration yields $$\label{eq:lem-loc-excursions-eq1}
\sum_{j \in {\mathbb{Z}}} |r_j(t)|
\leq
C |r_k(t)| + C \int\limits_{t_k^\#}^{t} |\dot q_k(s)| \,{\mathrm{d}s}$$ for all $t \in [t_k^\#,t_k^\flat]$, and this proves due to the bound . Moreover, by $$\begin{aligned}
q_k(t_k^\#) = p_k(t_k^\#) = p_k(t_k^\flat) = p_*
\end{aligned}$$ we have $$|r_k(t_k^\flat)|
=
|q_k(t_k^\flat) - p_*|
\leq
\int\limits_{t_k^\#}^{t_k^\flat} |\dot q_k(s)| \,{\mathrm{d}s}
+ |q_k(t_k^\#) - p_*|
\leq
D_k + 0$$ and obtain as a further consequence of .
![ The impact profile $\varrho$ from as function of $j$ for several values of the spinodal parameter ${{\kappa}}$. The essential fluctuations produced by each microscopic phase transition are given by a shifted and delayed variant of $g\ast \varrho$, see and , and contribute to the driving force of the macroscopic phase interface.[]{data-label="Fig:EssFluktuations"}](ess_fluctuations){width="85.00000%"}
\[lem:loc-passage\] For any $k \geq 1$ we have $$\label{eq:loc-passage1}
\sup_{t \in [t_k^\flat,t_k^*]} \sum_{j \in {\mathbb{Z}}} |r^{(k)}_j(t)|
\leq
C (1+D_k)$$ as well as $$\label{eq:loc-passage2}
\sum_{j \in {\mathbb{Z}}} \left|
r^{(k)}_j(t_k^*) - \varrho_{j-k} \right|
\leq
C D_k$$ for some constant $C$.
The proof of is identical to the one of in the previous lemma because is also true for $t\in[t^\#_k,t^*_k]$. To derive let us consider times $t \in (t_k^\flat, t_k^*)$, so that $u_k$ is located inside the spinodal region and can be written as $$\dot r_j(t) =
\begin{cases}
-\kappa \operatorname{\Delta}r_k(t) + (1+\kappa) \dot q_k(t) &\text{if } j=k, \\
+ \operatorname{\Delta}r_j(t) &\text{if } j\not=k,
\end{cases}$$ where we dropped the upper index $k$ for simplicity of notation. After shifting time and space by $t_k^\flat$ and $k$, respectively, this is the prototypical phase transition problem with $z=r$ and $f=\dot q$, and from Lemma \[lem:prototypical\] we obtain $$\sum_{j \in {\mathbb{Z}}}
\Big| r_j(t) - \frac{r_k(t)}{(1+2\kappa)^{|j-k|}} \Big|
\leq
C \Bigg( \sum_{j \in {\mathbb{Z}}} |r_j(t_k^\flat)|
+ \int\limits_{t_k^\flat}^t |\dot q_k(s)| \,{\mathrm{d}s} \Bigg)\leq C D_k,$$ where the second inequality is due to and . The claim now follows because $p_k(t_k^*) = -p_*$ and $q_k(t_k^\#) = p_*$ provide $${\big|{r_k(t_k^*) - 2p_*}\big|}
=
{\big|{q_k(t_k^*) - p_*}\big|}
\leq
\int\limits_{t_k^\#}^{t_k^*} |\dot q(s)| \,{\mathrm{d}s}
+
|q_k(t_k^\#) - p_*|
\leq
D_k+0$$ and since $\sum_{j \in {\mathbb{Z}}} \varrho_j$ is finite.
For small $D_k$ we infer from that at the end of the spinodal passage of $u_k$ the induced fluctuations $r^{(k)}(t_k^*)$ are in fact close to the shifted impact profile from . This observation together with the definition of $r^{(k)}(t)$ for $t>t_k^*$ – see , , and – motivates the splitting of $r^{(k)}$ into an *essential* part $$\label{Eqn:DefEssFluct}
r^{(k)}_{{{\mathrm{ess}}},j}(t)
:=
\chi_{\{t \geq t_k^*\}}\sum_{n \in {\mathbb{Z}}} g_{j-n}(t-t_k^*) \varrho_{n-k}$$ and the remainder $$\label{Eqn:DefNegFluct}
r^{(k)}_{{{\mathrm{neg}}},j}
:=
r^{(k)}_j(t) - r^{(k)}_{{{\mathrm{ess}}},j}(t),$$ which we call the *negligible* fluctuations. We prove in §\[sec:glob-fluct-estim\] below that these names are justified since Assumption \[ass:macro\] implies that $r^{(k)}_{{{\mathrm{ess}}}}$ is relevant for the limit dynamics, whereas $r^{(k)}_{{{\mathrm{neg}}}}$ is not.
Notice also that Lemma \[lem:loc-excursions\] and Lemma \[lem:loc-passage\] are again intimately related to the trilinearity of $\Phi^\prime$. It remains open to identify more robust proof strategies that cover general bistable nonlinearities as well and provide the analog to the impact profile and the splitting – for a broader class of nonlinear lattices .
Global fluctuation estimates {#sec:glob-fluct-estim}
----------------------------
In view of §\[sec:local-fluct-estim\], the main technical task for collectively controlling the fluctuations for all $k \geq 1$ is to estimate the sum of the quantities $D_k$ from . Our starting point is the representation formula $$\label{eq:q-k-formula}
q^{(k)}_j(t)
=
\sum_{n \in {\mathbb{Z}}} g_{j-n}(t) p_n(0)
-
\sum_{l=1}^{k-1} \sum_{n \in {\mathbb{Z}}} g_{j-n}(t-t_l^*) r^{(l)}_n(t_l^*)
\qquad
\text{for all}
\quad j \in {\mathbb{Z}}\quad\text{and}\quad t \geq t_k^\#,$$ which follows from and by induction over $k$ and splits $q^{(k)}$ into one part stemming from the initial data and another one from the previous phase transitions.
\[lem:D\_k-upper-bound\] There exists a constant $C$ such that $$\sum_{k=1}^{K_{{\varepsilon}}} D_k
\leq
\frac{C}{\sqrt{{{\varepsilon}}}}$$ for all sufficiently small ${{\varepsilon}}>0$.
By we have $$\label{eq:dot-q-k}
{}\dot q^{(k)}_k{{\left({t}\right)}}{}=
\sum_{n \in {\mathbb{N}}} \dot g_{k-n}(t) p_n(0)
-
\sum_{l=1}^{k-1} \sum_{n \in {\mathbb{Z}}} \dot g_{k-n}(t-t_l^*) r^{(l)}_n(t_l^*)$$ for all $t \in (t_k^\#,t_k^*)$, and due to Assumption \[ass:macro\] we can estimate the contribution from the initial data by $$\left| \sum_{n \in {\mathbb{Z}}} \dot g_{k-n}{{\left({t}\right)}} p_n(0) \right|
=
\left| \sum_{n \in {\mathbb{Z}}} g_{k-n} {{\left({t}\right)}}\operatorname{\Delta}p_n(0) \right|
\leq
C{{\left({ \alpha {{\varepsilon}}^2 + \frac{\beta {{\varepsilon}}}{(1+t)^{1/2}}}\right)}}$$ because the discrete heat kernel $g$ from is nonnegative and satisfies $\sum_{j\in{{\mathbb{Z}}}}g_j{{\left({t}\right)}}=1$ as well as $\sup_{j\in{{\mathbb{Z}}}} g_j{{\left({t}\right)}}\leq C{{\left({1+t}\right)}}^{-1/2}$. Moreover, the contributions from the previous phase transitions $l=1,\ldots,k-1$ satisfy $$\begin{aligned}
\left| \sum_{n \in {\mathbb{Z}}} \dot g_{k-n}(t-t_l^*) r^{(l)}_n(t_l^*) \right|
\leq
\| \dot g(t-t_l^*) \|_{\ell^\infty}
\sum_{n \in {\mathbb{Z}}} | r^{(l)}_n(t_l^*) |
\leq
C \frac{1+D_l}{(1+t-t_l^*)^{3/2}}
\end{aligned}$$ thanks to Lemma \[lem:loc-passage\] and $\|\dot g_j(s)\|_{\ell^\infty} \leq - \dot g_0(s) \leq C/(1+s)^{3/2}$. Combining these estimates with and integrating we thus find $$\label{eq:D-k-est}
D_k
\leq
\int\limits_{t_k^\#}^{t_k^*} \alpha {{\varepsilon}}^2 + \frac{\beta {{\varepsilon}}}{(1+t)^{1/2}}
\,{\mathrm{d}t}
+
C \sum_{l=1}^{k-1} \int\limits_{t_k^\#}^{t_k^*} \frac{1+D_l}{(1+t-t_l^*)^{3/2}}
\,{\mathrm{d}t}.$$ Summing over all phase transitions in $[0,t_{{\rm fin}}]$, we estimate the first integral in by $$\begin{aligned}
\label{eq:D-k-1}
\sum_{k=1}^{K_{{\varepsilon}}}
\int\limits_{t_k^\#}^{t_k^*} \alpha {{\varepsilon}}^2 + \frac{\beta {{\varepsilon}}}{(1+t)^{1/2}}
\,{\mathrm{d}t}
&\leq
\int\limits_{0}^{t_{{{\rm fin}}}} \alpha {{\varepsilon}}^2 + \frac{\beta {{\varepsilon}}}{(1+t)^{1/2}}
\,{\mathrm{d}t}
\leq
\alpha \tau_{{{\rm fin}}} + 2 \beta \sqrt{{{\varepsilon}}^2+\tau_{{{\rm fin}}}}
\leq C
\end{aligned}$$ and the second one by $$\begin{aligned}
\label{eq:D-k-2}
\begin{split}
\sum_{k=1}^{K_{{\varepsilon}}}
\sum_{l=1}^{k-1} \int\limits_{t_k^\#}^{t_k^*} \frac{1+D_l}{(1+t-t_l^*)^{3/2}}
\,{\mathrm{d}t}
&=
\sum_{l=1}^{K_{{\varepsilon}}} (1+D_l) \sum_{k=l+1}^{K_{{\varepsilon}}}
\int\limits_{t_k^\#}^{t_k^*} \frac{{\mathrm{d}t}}{(1+t-t_l^*)^{3/2}}
\\
&\leq
\sum_{l=1}^{K_{{\varepsilon}}} (1+D_l)
\int\limits_{t_{l+1}^\#}^{\infty} \frac{{\mathrm{d}t}}{(1+t-t_l^*)^{3/2}}
\\
&\leq
2 \sum_{l=1}^{K_{{\varepsilon}}} \frac{1+D_l}{(1+t_{l+1}^\#-t_l^*)^{1/2}}.
\end{split}
\end{aligned}$$ Moreover, Corollary \[cor:time-and-number-bounds\] provides $(1+t_{l+1}^\#-t_l^*)^{-1/2} \leq C \sqrt{{{\varepsilon}}}$. Adding the partial estimates and we thus arrive at $$\sum_{k=1}^{K_{{\varepsilon}}} D_k
\leq
C \bigg( 1 + \sqrt{{{\varepsilon}}} K_{{\varepsilon}}+ \sqrt{{{\varepsilon}}} \sum_{k=1}^{K_{{\varepsilon}}} D_k
\bigg),$$ and the thesis follows by rearranging terms since Corollary \[cor:time-and-number-bounds\] ensures that $K_{{\varepsilon}}\leq C/{{\varepsilon}}$.
As a consequence of Lemma \[lem:D\_k-upper-bound\], we obtain an upper bound for the sum of all negligible fluctuations.
\[cor:bounds-neg-fluct\] We have $$\label{eq:bounds-neg-fluct.Eqn1}
\sup_{0\leq t \leq t_{{\rm fin}}}\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{K_{{\varepsilon}}}
| r^{(k)}_{{{\mathrm{neg}}},j}(t) |
\leq
\frac{C}{\sqrt{{{\varepsilon}}}}$$ for some constant $C$ and all sufficiently small ${{\varepsilon}}>0$.
Fix $t \in [0,t_{{\rm fin}}]$ and note that if $t \leq t_1^\#$ then there are no fluctuations at all and the claim is trivially true at $t$. Otherwise the single-interface property from Proposition \[pro:existence\] provides exactly one $l \in \{1,\ldots,K_{{\varepsilon}}\}$ such that $$\text{either}\quad
t \in [t_l^\#,t_l^*)
\qquad\text{or}\qquad
t \in [t_l^*,t_{l+1}^\#),$$ where $t_{K_{{\varepsilon}}+1}^\#$ may be larger than $t_{{{\rm fin}}}$ or even infinite. In the first case we have $$\begin{aligned}
r^{(l)}_{{\mathrm{neg}}}{{\left({t}\right)}}=r^{(l)}{{\left({t}\right)}},\qquad\qquad r^{(k)}_{{\mathrm{neg}}}{{\left({t}\right)}}=0\quad \text{for $k>l$}\end{aligned}$$ according to the definitions in and , and using the local fluctuation estimates from Lemmas \[lem:loc-excursions\] and \[lem:loc-passage\] we find $$\begin{aligned}
\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{K_{{\varepsilon}}} | r^{(k)}_{{{\mathrm{neg}}},j}(t) |
&\leq
\sum_{j \in {\mathbb{Z}}} | r^{(l)}_{{{\mathrm{neg}}},j}(t) |
+
\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{l-1} | r^{(k)}_{{{\mathrm{neg}}},j}(t) |
=
\sum_{j \in {\mathbb{Z}}} | r^{(l)}_{j}(t) |
+
\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{l-1} | r^{(k)}_j(t) - r^{(k)}_{{{\mathrm{ess}}},j}(t) |
\\
&\leq
C (1+D_l)
+
\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{l-1} \sum_{n \in {\mathbb{Z}}}
g_{j-n}(t-t_k^*) \left| r_n^{(k)}(t_k^*) -
\varrho_{n-k} \right|
\\
&\leq
C (1+D_l)
+
C \sum_{k=1}^{l-1} D_k.
\end{aligned}$$ The discussion of the second case $t \in [t_l^*,t_{l+1}^\#)$ is even simpler since the contributions for $k=l$ and $k<l$ can be bounded in the same way. In particular, arguing as above we find $$\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{K_{{\varepsilon}}} | r^{(k)}_{{{\mathrm{neg}}},j}(t) |
\leq
\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{l} | r^{(k)}_j(t)-r^{(k)}_{{{\mathrm{ess}}},j}(t) |
\leq
C \sum_{k=1}^l D_k,$$ and the claim follows in both cases from Lemma \[lem:D\_k-upper-bound\].
Notice that the superposition of all essential fluctuations satisfies $$\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{K_{{\varepsilon}}} r^{(k)}_{{{\mathrm{ess}}},j}(t)
=
2\max {\ensuremath{ \left\{ \ifx\emptyk \relax \else k: \fi t_k^* \leq t \right\} }}$$ since we have $\sum_{j\in{{\mathbb{Z}}}}\varrho_j=2$ and because the convolution with the discrete heat kernel $g$ preserves mass as well as positivity. Consequently, the sum of all essential fluctuations is of order $1/{{\varepsilon}}$ and hence larger than the right hand side in , provided that the interface propagates on the macroscopic scale. In other words, the negligible fluctuations are in fact smaller than the essential ones.
We further emphasize that we are not able to estimate the number of spinodal excursions or their duration. Corollary \[cor:bounds-neg-fluct\], however, controls the impact of the corresponding fluctuations even in the worst-case-scenario that a single particle is either always inside the spinodal region or enters and leaves it repeatedly over a very long period of time. More precisely, combining the estimate with the scaling we show in §\[sect:compactness\] that the sum of all negligible fluctuations is small in the macroscopic ${{{\mathsf{L}}}}^1$-norm and confirm in this way that spinodal excursions are not related to proper phase transitions and do not drive the interface in the macroscopic free boundary problem –.
Regularity of fluctuations {#sec:regul-fluct}
--------------------------
![Cartoon of the essential fluctuations $r^{(k)}_{{{\mathrm{ess}}},j}$ and the corresponding regular and residual ones, see , depicted as functions of $t$ for $j=k$ (black, dashed) and $j=k+1$ (gray, solid). The shaded boxes indicate the different life spans.[]{data-label="fig:splitting"}](splitting){width=".95\textwidth"}
A fundamental ingredient for passing to the macroscopic limit in §\[sec:justification\] is to ensure that the superposition of all fluctuations converges to a continuous function. The essential fluctuations $r^{(k)}_{{{\mathrm{ess}}},k}$, however, are discontinuous in time as they jump at every $t_k^*$, see Figure \[fig:splitting\]. To overcome this problem we observe that the lower bound for the waiting time guarantees that the diffusion effectively regularizes $r^{(k)}_{{{\mathrm{ess}}}}$ in the time between $t^*_k$ and $t^\#_{k+1}$. We therefore split the latter into two parts and denote by $$\begin{aligned}
\label{Eqn:DefRegAndResFluct}
r^{(k)}_{{{\mathrm{reg}}},j}(t)
:=
r^{(k)}_{{{\mathrm{ess}}},j}(t)\chi_{[t^*_k+d_*/{{\varepsilon}},t_{{\rm fin}})}{{\left({t}\right)}}
\qquad\text{and}\qquad
r^{(k)}_{{{\mathrm{res}}},j}(t)
:=
r^{(k)}_{{{\mathrm{ess}}},j}(t)\chi_{[t^*_k,t^*_k+d_*/{{\varepsilon}})}{{\left({t}\right)}}\end{aligned}$$ the $k$-th *regular* and *residual* fluctuations, respectively, where $d_*$ is the constant from Corollary \[cor:time-and-number-bounds\]. The regular fluctuations are still discontinuous in time but it turns out that the jumps are small and disappear as ${{\varepsilon}}\to 0$. On the other hand, the sum of all residual fluctuations is very irregular but the Lebesgue measure of its domain of definition becomes small under the scaling .
\[lem:bounds-res-fluct\] We have $$\sup\limits_{0\leq t\leq t_{{\rm fin}}}
\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{K_{{\varepsilon}}} |r^{(k)}_{{{\mathrm{res}}},j}(t)|
\leq
C$$ for some constant $C$ and all sufficiently small ${{\varepsilon}}>0$.
By Corollary \[cor:time-and-number-bounds\] the intervals $[t_k^*,t_k^*+d_*/{{\varepsilon}}]$ are mutually disjoint and we conclude that for any $t$ only one $k$ contributes to the double sum. Combining this with , and the mass conservation of the discrete heat equation we find $$\sum_{j \in {\mathbb{Z}}} \sum_{k=1}^{K_{{\varepsilon}}} |r^{(k)}_{{{\mathrm{res}}},j}(t)|
=
\sup_{1\leq k\leq K_{{\varepsilon}}}\sum_{j \in {\mathbb{Z}}} |r^{(k)}_{{{\mathrm{res}}},j}(t)|
\leq
C$$ with $C := \sum_{j\in{{\mathbb{Z}}}}\varrho_j$.
The key result of this section is the following lemma, which shows that the regular fluctuations are Hölder continuous up to a small error that vanishes in the limit ${{\varepsilon}}\to 0$.
\[Lem:Hoelder\] There exists a constant $C$, which depends on $\kappa$ and $d_*$ such that $$\begin{aligned}
\label{Lem:Hoelder.Eqn1}
\left|
\sum_{k=1}^{K_{{\varepsilon}}} r_{{{\mathrm{reg}}},j_2}^{(k)}(t_2)
- \sum_{k=1}^{K_{{\varepsilon}}} r_{{{\mathrm{reg}}},j_1}^{(k)}(t_1)
\right|
\leq
C{{\varepsilon}}^{1/2}
{{\left({|t_2-t_1|^{1/4}+|j_2-j_1|^{1/2}}\right)}} + C {{\varepsilon}}^{1/2}
\end{aligned}$$ holds for any $j_1, j_2 \in {{\mathbb{Z}}}$ and all $0 \leq t_1, t_2 \leq
t_{{\rm fin}}$.
Elementary arguments reveal that the discrete heat kernel satisfies $$\label{Lem:Hoelder.PEqn1}
{\big|{g_{j_2}{{\left({t_2}\right)}}-g_{j_1}{{\left({t_1}\right)}}}\big|}
\leq
\frac{C}{{{\left({1+\min\{t_1,t_2\}}\right)}}^{3/4}}
{{\big({\left|{t_2-t_1}\right|}^{1/4}+{\left|{j_2-j_1}\right|}^{1/2}\big)}},$$ see for instance [@HeHe13 Appendix] for the details. In what follows we denote the argument of the modulus on left hand side of by $D(t_1,t_2,j_1,j_2)$ and study the cases $j_1=j_2$ and $t_1=t_2$ separately. The general result is then a consequence of the triangle inequality.
: For $t_1=t_2=t$, inequality along with and implies $$\begin{aligned}
{\big|{D(t,t,j_1,j_2)}\big|}
&\leq
\sum_{k:\, t_k^* \leq t-d_*/{{\varepsilon}}}
\sum_{n\in{{\mathbb{Z}}}}
\varrho_{n-k}
{\big|{g_{j_2-n}(t-t_k^*)-g_{j_1-n}(t-t_k^*)}\big|}
\\
&\leq
C |j_2-j_1|^{1/2}
\sum_{k:\, t_k^* \leq t-d_*/{{\varepsilon}}}
\frac{1}{(1+t-t_k^*)^{3/4}}
\end{aligned}$$ with $\varrho$ as in . Moreover, the lower bound for the waiting time in Corollary \[cor:time-and-number-bounds\] guarantees that all phase transition times are sufficiently separated from each other, and hence also that $$\begin{aligned}
\#\big\{k: t^*_k<t-d_*/{{\varepsilon}}\big\}
\leq
\left\lfloor\frac{{{\varepsilon}}t}{2d_* }\right\rfloor,
\end{aligned}$$ where $\lfloor\cdot\rfloor$ denotes the floor function. As illustrated in Figure \[fig:PastTimes\], we can therefore estimate $$\begin{aligned}
\label{Lem:Hoelder.PEqn2}
\sum_{k:\, t_k^* \leq t-d_*/{{\varepsilon}}}
\frac{1}{(1+t-t_k^*)^{3/4}}
\leq
{{}C{}}\sum_{{{}l {}}=1}^{\lfloor{{\varepsilon}}t/{{\left({ 2d_* }\right)}}\rfloor}
\left( \frac{{{\varepsilon}}}{{{}l {}} d_*} \right)^{3/4}
=
C {{\varepsilon}}t^{1/4}
\leq
C \tau_{{\rm fin}}^{1/4}{{\varepsilon}}^{1/2}{}\leq C{{\varepsilon}}^{1/2},
\end{aligned}$$ where we interpreted the sum as a discretized Riemann integral, and obtain via $$|D(t,t,j_1,j_2)|
\leq
C{{\varepsilon}}^{1/2} |j_2-j_1|^{1/2},$$ the claim in the first case.
: Supposing $j_1=j_2=j$ and $t_1<t_2$, we write $$D(t_1,t_2,j,j)
=
D_1(t_1,t_2,j) + D_2(t_1,t_2,j),$$ where $$D_1(t_1,t_2,j)
=
\sum_{k:\, t_k^*+d_*/{{\varepsilon}}<t_1}
\sum_{n\in{{\mathbb{Z}}}}
\varrho_{n-k}
\Big( g_{j-n}(t_2-t_k^*)-g_{j-n}(t_1-t_k^*) \Big)$$ and $$D_2(t_1,t_2,j)
=
\sum_{k:\, t_1 \leq t_k^* + d_*/{{\varepsilon}}< t_2}
\sum_{n\in{{\mathbb{Z}}}}
\varrho_{n-k}
g_{j-n}(t_2-t_k^*)$$ account for the phase transitions that occur in the intervals $[0,t_1]$ and $[t_1,t_2]$, respectively. To estimate the first term, we employ and Corollary \[cor:time-and-number-bounds\] as in the above discussion and infer that $$|D_1(t_1,t_2,j)|
\leq
C |t_2-t_1|^{1/4}
\sum_{k:\, t_k^*+d_*/{{\varepsilon}}<t_1}
\frac{1}{(1+t_1-t_k^*)^{3/4}}
\leq
C {{\varepsilon}}^{1/2} |t_2-t_1|^{1/4}.$$ Moreover, the decay $g_j(t) \leq C/(1+t)^{1/2}$ for all $j \in {\mathbb{Z}}$ and $t \geq 0$ yields $$|D_2(t_1,t_2,j)|
\leq
\sum_{k:\, t_1 \leq t_k^*+d_*/{{\varepsilon}}< t_2}
\frac{C}{(1+t_2-t_k^*)^{1/2}},$$ and Corollary \[cor:time-and-number-bounds\] combined with ${\left|{t_2-t_1}\right|}\leq t_{{\rm fin}}=\tau_{{\rm fin}}/{{\varepsilon}}^2$ allows us to estimate $$\begin{aligned}
|D_2(t_1,t_2,j)|
&\leq
\sum_{{{}l {}}=1}^{\lceil {{\varepsilon}}(t_2-t_1)/{{\left({2d_* }\right)}} \rceil}
\frac{C{{{\varepsilon}}}^{1/2}}{{\displaystyle}({{}l {}} d_*)^{1/2}}
\\
&\leq
C {{\varepsilon}}|t_2-t_1|^{1/2} + C {{\varepsilon}}^{1/2}
\leq C {{\varepsilon}}^{1/2} |t_2-t_1|^{1/4} + C {{\varepsilon}}^{1/2},
\end{aligned}$$ where $\lceil\cdot\rceil$ denotes the ceiling function.
![To control the regularity of the fluctuations in the proof of Lemma \[Lem:Hoelder\], we look from a given time $t$ backward and label the past phase transitions in reversed order by the index $l$.[]{data-label="fig:PastTimes"}](past_times){width=".55\textwidth"}
As a consequence of Lemma \[Lem:Hoelder\] we obtain the following bound for the regular fluctuations.
\[cor:infty-bounds-all-fluct\] There exists a constant $C$ such that $$\sup_{t\in[0,t_{{\rm fin}}]}
\sup_{j\in{{\mathbb{Z}}}}
{{\left({
{\big|{\sum_{k=1}^{K_{{\varepsilon}}}r^{(k)}_{{{\mathrm{reg}}},j}{{\left({t}\right)}}}\big|}
+{\big|{\sum_{k=1}^{K_{{\varepsilon}}}r^{(k)}_{{{\mathrm{res}}},j}{{\left({t}\right)}}}\big|}
+{\big|{\sum_{k=1}^{K_{{\varepsilon}}}r^{(k)}_{{{\mathrm{neg}}},j}{{\left({t}\right)}}}\big|}}\right)}}
\leq C$$ for all sufficiently small ${{\varepsilon}}>0$.
The claimed estimate for the residual fluctuations is a direct consequence of Lemma \[lem:bounds-res-fluct\] while the bound for the regular fluctuations follows from Lemma \[Lem:Hoelder\] with $t_2=t$, $t_1=0$, $j_1=j_2=j$ and due to $t^{1/4}\leq t_{{\rm fin}}^{1/4}=\tau_{{\rm fin}}^{1/4}{{\varepsilon}}^{-1/2}$. Moreover, the representation formula implies $$\begin{aligned}
\sup_{t\in[0,t_{{\rm fin}}]}
\sup_{j\in{{\mathbb{Z}}}}{\big|{\sum_{k=1}^{K_{{\varepsilon}}}r^{(k)}_{j}{{\left({t}\right)}}}\big|}\leq C\end{aligned}$$ thanks to Proposition \[pro:existence\], Assumption \[ass:macro\], and the maximum principle for diffusion equations. The assertion for the negligible fluctuations thus follows from $r^{(k)}_{{{\mathrm{neg}}},j}{{\left({t}\right)}} = r^{(k)}{{\left({t}\right)}} - r^{(k)}_{{{\mathrm{reg}}},j}{{\left({t}\right)}}
-r^{(k)}_{{{\mathrm{res}}},j}{{\left({t}\right)}} $, which is provided by , , and .
We conclude this section with an estimate for the spatial gradient of the regular fluctuations. To begin with, setting $j_1=j$, $j_2=j+1$ and $t_1=t_2=t$ in provides $$\left|
\sum_{k=1}^{K_{{\varepsilon}}} \nabla_+ r_{{{\mathrm{reg}}},j}^{(k)}(t)
\right|
\leq
C {{\varepsilon}}^{1/2},$$ so the corresponding macroscopic gradient is bounded pointwise in space and time by ${{\varepsilon}}^{-1/2}$ but not by some quantity of order $1$. The following result, however, establishes an improved $\ell^2$-estimate which enables us to pass to the macroscopic limit pointwise in time.
\[lem:bound-grad-reg-fluct\] We have $$\sup_{0\leq t\leq t_{{\rm fin}}} \sum_{j \in {\mathbb{Z}}}
\bigg| \sum_{k=1}^{K_{{\varepsilon}}} \nabla_+ r^{(k)}_{{{\mathrm{reg}}},j}(t)
\bigg|^2
\leq
C {{\varepsilon}}$$ for some constant $C$ and all sufficiently small ${{\varepsilon}}>0$.
The gradient of the discrete heat kernel satisfies $$\sum_{j \in {\mathbb{Z}}}
|\nabla_+ g_j(t)|^2
\leq
C(1+t)^{-3/2},$$ for all $t \geq 0$, see for instance [@HeHe13 Appendix], and , ensure $$\begin{aligned}
\nabla_+r^{(k)}_{{{\mathrm{reg}}},j}(t)=
\sum_{n \in {\mathbb{Z}}}
\varrho_{n-k}\nabla_+ g_{j-n}(t-t_k^*)
\end{aligned}$$ for any $k$, all $j$, and every $t$ with $t_k^*+d_*/{{\varepsilon}}\leq t$. Young’s inequality for convolutions implies via ${\|{\varrho \ast \cdot}\|}_2\leq{\|{\varrho}\|}_1{\|{\cdot}\|}_2$ the estimate $$\begin{aligned}
\bigg(\sum_{j \in {\mathbb{Z}}} \big| \nabla_+ r^{(k)}_{{{\mathrm{reg}}},j}(t)\big|^2\bigg)^{1/2}
\leq
\frac{C}{(1+t-t_k^*)^{3/4}},
\end{aligned}$$ and as in the proof of Lemma \[Lem:Hoelder\] – see and Figure \[fig:PastTimes\] – we deduce $$\sum_{k=1}^{K_{{\varepsilon}}} \bigg( \sum_{j \in {\mathbb{Z}}}
\big|\nabla_+ r^{(k)}_{{{\mathrm{reg}}},j}(t)\big|^2 \bigg)^{1/2}
\leq
C \sum_{{{}l {}}=1}^{\lfloor{{\varepsilon}}t/{{\left({ 2d_* }\right)}}\rfloor}
\left( \frac{{{\varepsilon}}}{d_* {{}l {}}} \right)^{3/4}
\leq
C {{\varepsilon}}^{1/2}.$$ The assertion is now a direct consequence of the triangle inequality.
Justification of the hysteretic free boundary problem {#sec:justification}
=====================================================
In order to pass to the macroscopic limit, we choose a scaling parameter $0<{{\varepsilon}}\ll1$ and regard the lattice data as continuous functions in the macroscopic time $\tau$ that are piecewise constant with respect to the macroscopic space variable as they depend only on the integer part of $\xi/{{\varepsilon}}$. More precisely, in accordance with we write $$\begin{aligned}
\label{Eqn:DefIntegerPart}
\xi={{\varepsilon}}{{\left({j_\xi+\zeta_\xi}\right)}}\qquad\text{ with}\qquad j_\xi\in{{\mathbb{Z}}},\quad\zeta_\xi\in {(-1/2,\,+1/2]},\end{aligned}$$ and define $$P_{{\varepsilon}}(\tau,\,\xi)
:=
p_{j_\xi}(\tau/{{\varepsilon}}^2),
\qquad
R_{{{\mathrm{reg}}},\,{{\varepsilon}}}(\tau,\xi)
:=
\sum_{k \geq 1} r^{(k)}_{{{\mathrm{reg}}},\,j_\xi}(\tau/{{\varepsilon}}^2).$$ Furthermore, by similar formulas we construct functions $U_{{\varepsilon}}$, $R_{{{\mathrm{res}}},{{\varepsilon}}}$, and $R_{{{\mathrm{neg}}},{{\varepsilon}}}$ from their microscopic counterparts, and setting $$Q_{{\varepsilon}}(\tau,\xi)
:=
\sum_{n \in {\mathbb{N}}} g_{j_\xi-n}(\tau/{{\varepsilon}}^2)p_n(0)$$ we infer from the identity$$\label{eq:FormulaForP}
P_{{{\varepsilon}}}
=
Q_{{\varepsilon}}- \left( R_{{{\mathrm{reg}}},{{\varepsilon}}} + R_{{{\mathrm{neg}}},{{\varepsilon}}} + R_{{{\mathrm{res}}},{{\varepsilon}}} \right).$$ Finally, we introduce two discrete analogs to the macroscopic interface curve via $$\Xi^*_{{\varepsilon}}(\tau)
:=
{{\varepsilon}}\sum_{k \geq 1} k \chi_{[t_{k-1}^*,t_k^*)}(\tau/{{\varepsilon}}^2),
\qquad
\Xi^\#_{{\varepsilon}}(\tau)
:=
{{\varepsilon}}\sum_{k \geq 1} k \chi_{[t_{k-1}^\#,t_k^\#)}(\tau/{{\varepsilon}}^2)$$ and approximate the macroscopic phase field by $$\begin{aligned}
\label{Eqn:DefMEps}
M_{{\varepsilon}}{{\left({\tau},\,{\xi}\right)}}
:=
\begin{cases}
-1 & \text{if } \xi > \Xi^\#_{{\varepsilon}}{{\left({\tau}\right)}}, \\
+1 & \text{if } \xi < \Xi^*_{{\varepsilon}}{{\left({\tau}\right)}}, \\
0 & \text{otherwise};
\end{cases}\end{aligned}$$ see Figure \[fig:interface\] for an illustration.
![Cartoon of the macroscopic phase interface, both on the discrete level (piecewise constant graphs $\Xi_{{\varepsilon}}^\#$ and $\Xi_{{\varepsilon}}^*$ in dark and light gray, respectively) and in the continuum limit (black curve $\Xi$). All spinodal passages and excursions take place inside the shaded region, whose macroscopic area is bounded from above by ${{\varepsilon}}\tau_{{\rm fin}}$ and typically of order ${{\varepsilon}}^2{\left|{\ln{{\varepsilon}}}\right|}$.[]{data-label="fig:interface"}](interface){width=".45\textwidth"}
Compactness results {#sect:compactness}
-------------------
Our first result concerns the compactness of the scaled lattice data and extends the arguments for the bilinear case ${{\kappa}}=\infty$ from [@HeHe13].
\[pro:compactness\] Under Assumption \[ass:macro\] there exist (not relabeled) sequences such that the following statements are satisfed for ${{\varepsilon}}\to0$:
1. *(convergence of interfaces)* We have $$\begin{aligned}
\label{pro:compactness.Eqn1}
{\big|{\Gamma_{{\varepsilon}}}\big|}\to0
\quad \text{where} \quad
\Gamma_{{\varepsilon}}:=
\big\{(\tau,\xi) :
{}\Xi^*_{{\varepsilon}}{{\left({\tau}\right)}} \leq \xi\leq \Xi^\#_{{\varepsilon}}{{\left({\tau}\right)}},\;{}0\leq\tau\leq\tau_{{\rm fin}}\big\},
\end{aligned}$$ and both $\Xi^\#_{{\varepsilon}}$ and $\Xi_{{\varepsilon}}^*$ converge strongly in ${{{\mathsf{L}}}}^\infty([0,\tau_{{\rm fin}}])$ to the same Lipschitz function $\Xi$.
2. *(strong convergence of fields)* There exist bounded functions $U$, $P$, and $M$ such that $$\begin{aligned}
\label{pro:compactness.Eqn2}
U_{{\varepsilon}}\to U,
\qquad
P_{{\varepsilon}}\to P,
\qquad
M_{{\varepsilon}}\to M
\qquad \text{strongly in} \qquad
{{{\mathsf{L}}}}^s_{\mathrm{loc}}{{\big([0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}\big)}}
\end{aligned}$$ for any $1\leq s<\infty$. Moreover, $P$ is locally Hölder-continuous in space and time on $[0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}$ and we have $P_{{\varepsilon}}(\tau,\cdot) \to P(\tau,\cdot)$ strongly in ${{{\mathsf{L}}}}^s_{\mathrm{loc}}{{\left({{{\mathbb{R}}}}\right)}}$ for any $\tau\in[0,\tau_{{\rm fin}}]$.
3. *(weak convergence of spatial derivatives)* $P$ admits the weak derivative $\partial_\xi P$ for any $\tau \in [0,\tau_{{\rm fin}}]$ and we have $$\begin{aligned}
\label{pro:compactness.Eqn3}
\nabla_{+{{\varepsilon}}} P_{{\varepsilon}}\to \partial_\xi P
\qquad \text{weakly in} \qquad
{{{\mathsf{L}}}}^2_{\mathrm{loc}}{{\big([0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}\big)}},
\end{aligned}$$ where $\nabla_{+{{\varepsilon}}}$ denotes the right-sided difference approximation of $\partial_\xi$ on ${{\varepsilon}}{{\mathbb{Z}}}$.
Here, $0<\tau_{{\rm fin}}<\infty$ denotes a fixed macroscopic time that is independent of ${{\varepsilon}}$.
**: The Lebesgue measure of ${{\Gamma}}_{{\varepsilon}}$ can be estimated by $$\begin{aligned}
\label{pro:compactness.PEqn1}
{\big|{{{\Gamma}}_{{\varepsilon}}}\big|}\leq {{\varepsilon}}\tau_{{\rm fin}}\end{aligned}$$ because Proposition \[pro:existence\] ensures that for each time $\tau$ there is at most one particle inside the spinodal region. Moreover, the jumps of both $\Xi_{{\varepsilon}}^*$ and $\Xi_{{\varepsilon}}^\#$ are always of size ${{\varepsilon}}$ and the time between two jumps is bounded from below by $2 d_* {{\varepsilon}}$ due to Corollary \[cor:time-and-number-bounds\]; see Figure \[fig:interface\] for an illustration. By approximation with piecewise linear functions we thus deduce the strong compactness of both $\Xi^*_{{\varepsilon}}$ and $\Xi^\#_{{\varepsilon}}$ as well as the Lipschitz continuity of any accumulation point, see [@HeHe13 Lemma 3.9] for the details. Finally, implies that the accumulations points of $\Xi_{{\varepsilon}}^\#$ and $\Xi_{{\varepsilon}}^*$ coincide.
**: For given $\tau$, Corollary \[cor:bounds-neg-fluct\] and Lemma \[lem:bounds-res-fluct\] yield $$\| R_{{{\mathrm{neg}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}} \|_{{{{\mathsf{L}}}}^1({{\mathbb{R}}})} \leq C\sqrt{{{\varepsilon}}}
\qquad\text{and}\qquad
\| R_{{{\mathrm{res}}},{{\varepsilon}}}{{\left({\cdot},\,{\tau}\right)}} \|_{{{{\mathsf{L}}}}^1({{\mathbb{R}}})} \leq C{{\varepsilon}},$$ and Corollary \[cor:infty-bounds-all-fluct\] provides $$\| R_{{{\mathrm{neg}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}} \|_{{{{\mathsf{L}}}}^\infty({{\mathbb{R}}})}
+
\| R_{{{\mathrm{res}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}} \|_{{{{\mathsf{L}}}}^\infty({{\mathbb{R}}})}
\leq
C.$$ By Hölder’s inequality and interpolation we thus find $$\label{pro:compactness.PEqn2}
R_{{{\mathrm{neg}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}} \to 0
\qquad\text{and}\qquad
R_{{{\mathrm{res}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}} \to 0
\qquad\text{strongly in}\quad{{{\mathsf{L}}}}^s{{\left({{{\mathbb{R}}}}\right)}},$$ as well as a corresponding convergence result in ${{{\mathsf{L}}}}^s{{\left({[0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}}\right)}}$.
**: From Lemma \[Lem:Hoelder\] we infer the estimate $$| R_{{{\mathrm{reg}}},{{\varepsilon}}}(\tau_2,\xi_2) - R_{{{\mathrm{reg}}},{{\varepsilon}}}(\tau_1,\xi_1) |
\leq
C\left( |\tau_2-\tau_1|^{1/4} + |\xi_2-\xi_1|^{1/2} \right) + C{{\varepsilon}}^{1/2}$$ and conclude that the piecewise constant function $R_{{{\mathrm{reg}}},{{\varepsilon}}}$ is almost Hölder continuous with small spatial jumps of order ${{\varepsilon}}^{1/2}$. A variant of the Arzelá-Ascoli theorem – see [@HeHe13 Lemma 3.10] – provides a Hölder continuous function $R$ along with a subsequence of ${{\varepsilon}}\to0$ such that $$\begin{aligned}
R_{{{\mathrm{reg}}},{{\varepsilon}}}\to R
\qquad\text{strongly in}\quad
{{{\mathsf{L}}}}^\infty{{\big([0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}\big)}}
\end{aligned}$$ as well as $$\begin{aligned}
\label{pro:compactness.PEqn3}
R_{{{\mathrm{reg}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}}\to R{{\left({\tau},\,{\cdot}\right)}}
\qquad\text{strongly in}\quad
{{{\mathsf{L}}}}^\infty({{\mathbb{R}}})
\end{aligned}$$ for any given $\tau$.
**: The compactness of $Q_{{\varepsilon}}$, which represent the scaled solutions of the discrete heat equation with macroscopic initial data as in Assumption \[ass:macro\], as well as the regularity of any accumulation point can be proven in many ways; see for instance [@HeHe13 Lemma 3.11] for an approach via Hölder regularity. Extracting another subsequence we can therefore assume that $$\begin{aligned}
\label{pro:compactness.PEqn4a}
Q_{{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}}\to Q{{\left({\tau},\,{\cdot}\right)}}
\qquad\text{strongly in}\quad
{{{\mathsf{L}}}}^s_{\mathrm{loc}}({{\mathbb{R}}})
\end{aligned}$$ and $$\begin{aligned}
\notag Q_{{{\varepsilon}}}\to Q
\qquad\text{strongly in}\quad
{{{\mathsf{L}}}}^s_{\mathrm{loc}}{{\big([0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}\big)}}
\end{aligned}$$ hold for some continuous function $Q$, and together with , , and we obtain the claimed convergence properties of $P_{{\varepsilon}}$. Moreover, and imply the convergence of $M_{{\varepsilon}}$.
**: For fixed $\tau$, Lemma \[lem:bound-grad-reg-fluct\] ensures that $$\begin{aligned}
{\|{{}\nabla_{+{{\varepsilon}}}{}R_{{{\mathrm{reg}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}}}\|}_{{{{\mathsf{L}}}}^2{{\left({{{\mathbb{R}}}}\right)}}}
\leq
C,
\end{aligned}$$ while Assumption \[ass:macro\] combined with the properties of the discrete heat kernel guarantees $$\begin{aligned}
{\|{{}\nabla_{+{{\varepsilon}}}{}Q_{{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}}}\|}_{{{{\mathsf{L}}}}^2_{{}{\mathrm{loc}}{}}{{\left({{{\mathbb{R}}}}\right)}}}
\leq
\alpha.
\end{aligned}$$ In particular, $\nabla_{+{{\varepsilon}}}
{{\big(R_{{{\mathrm{reg}}},{{\varepsilon}}}{{\left({\tau},\,{\cdot}\right)}}+Q_{{\varepsilon}}{{\left({\tau},\,{\cdot}\right)}}\big)}}$ is weakly compact in ${{{\mathsf{L}}}}^2_{{}{\mathrm{loc}}{}}{{\left({{{\mathbb{R}}}}\right)}}$ and any accumulation point $Z{{\left({\tau},\,{\cdot}\right)}}$ satisfies $$\begin{aligned}
\int\limits_{{\mathbb{R}}}Z{{\left({\tau},\,{\xi}\right)}}\Psi{{\left({\xi}\right)}}{\mathrm{d}\xi}=-\lim\limits_{{{\varepsilon}}\to0}\int\limits_{{\mathbb{R}}}{{\big(R_{{{\mathrm{reg}}},{{\varepsilon}}}{{\left({\tau},\,{\xi}\right)}}+Q_{{\varepsilon}}{{\left({\tau},\,{\xi}\right)}}\big)}}\nabla_{-{{\varepsilon}}}\Psi{{\left({\xi}\right)}}{\mathrm{d}\xi}=-\int\limits_{{\mathbb{R}}}P{{\left({\tau},\,{\xi}\right)}}\partial_\xi \Psi{{\left({\xi}\right)}}{\mathrm{d}\xi}\end{aligned}$$ thanks to –, where $\Psi\in{{{\mathsf{C}}}}_c^\infty{{\big({\mathbb{R}}\big)}}$ is an arbitrary smooth test function and $\nabla_{-{{\varepsilon}}}$ abbreviates the left-sided difference operator on ${{\varepsilon}}{{\mathbb{Z}}}$. This implies the existence of the weak derivative $\partial_\xi P {{\left({\tau},\,{\cdot}\right)}}\in{{{\mathsf{L}}}}^2_{{}{\mathrm{loc}}{}}{{\left({{{\mathbb{R}}}}\right)}}$ for any $\tau$. Towards we fix $\lambda>0$, define a nonnegative and piecewise constant function $\Psi_{{\varepsilon}}\in{{{\mathsf{L}}}}^\infty{{\left({{{\mathbb{R}}}}\right)}}$ in consistency with by $$\begin{aligned}
\Psi_{{\varepsilon}}{{\left({\xi}\right)}}:=\exp{{\big(-\lambda{{\varepsilon}}{\left|{j_\xi}\right|}\big)}},
\end{aligned}$$ and verify by direct computations that $$\begin{aligned}
{\big|{{}\nabla_{+{{\varepsilon}}}{}\Psi_{{\varepsilon}}{{\left({\xi}\right)}}}\big|}
\leq
C\lambda{{\Big( \Psi_{{\varepsilon}}{{\left({\xi}\right)}}+\chi_{{[-{{\varepsilon}}/2,\,+{{\varepsilon}}/2]}}{{\left({\xi}\right)}}\Big)}}.
\end{aligned}$$ Evaluating Proposition \[Prop:Entropy\] with $\psi_j=\Psi{{\left({{{\varepsilon}}j}\right)}}$ and inserting the scaling we then find $$\begin{aligned}
\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}}\Psi_{{\varepsilon}}{{\big({}\nabla_{+{{\varepsilon}}}{}P_{{\varepsilon}}\big)}}^2{\mathrm{d}\xi}{\mathrm{d}\tau}
&\leq
\int\limits_{{\mathbb{R}}}\Psi_{{\varepsilon}}\, \Phi{{\big(U_{{\varepsilon}}\big)}} \,{\mathrm{d}\xi}
-
\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}}P_{{\varepsilon}}{{\big({}\nabla_{+{{\varepsilon}}}{}\Psi_{{\varepsilon}}\big)}}
{{\big({}\nabla_{+{{\varepsilon}}}{}P_{{\varepsilon}}\big)}} \,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
\\
&\leq
C
+
C\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}}\Psi_{{\varepsilon}}{\big|{{}\nabla_{+{{\varepsilon}}}{}P_{{\varepsilon}}}\big|} \,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
\\
&\leq
C
+
C\int\limits_0^{\tau_{{\rm fin}}}\int\limits_{{\mathbb{R}}}\Psi_{{\varepsilon}}\,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
+
\frac{1}{2}\int\limits_0^{\tau_{{\rm fin}}}\int\limits_{{\mathbb{R}}}\Psi_{{\varepsilon}}{{\big({}\nabla_{+{{\varepsilon}}}{}P_{{\varepsilon}}\big)}}^2 \,{\mathrm{d}\xi} \,{\mathrm{d}\tau}\,.
\end{aligned}$$ Here, $C$ depends on $\lambda$ but not on ${{\varepsilon}}$ and we omitted the arguments of the functions to ease the notation. Since $\lambda$ is arbitrary we conclude that $\nabla_{+{{\varepsilon}}} P_{{\varepsilon}}$ is weakly compact in ${{{\mathsf{L}}}}^2_{\mathrm{loc}}{{\big([0,\tau_{{\rm fin}}] \times {{\mathbb{R}}}\big)}}$. Moreover, any accumulation $Z$ point fulfills $$\begin{aligned}
\int\limits_0^{\tau_{{\rm fin}}}\int\limits_{{\mathbb{R}}}Z\,\Psi\,{\mathrm{d}\xi}{\mathrm{d}\tau}=
-\lim\limits_{{{\varepsilon}}\to0}\int\limits_0^{\tau_{{\rm fin}}}\int\limits_{{\mathbb{R}}}P_{{\varepsilon}}\nabla_{-{{\varepsilon}}}\Psi{\mathrm{d}\xi}{\mathrm{d}\tau}=
\int\limits_0^{\tau_{{\rm fin}}}\int\limits_{{\mathbb{R}}}\partial_\xi P\,\Psi\,{\mathrm{d}\xi}{\mathrm{d}\tau}\end{aligned}$$ for any test function $\Psi\in {{{\mathsf{C}}}}_c^\infty{{\big((0,\tau_{{\rm fin}}){\times}{\mathbb{R}}\big)}}$, so follows from the standard argument that compactness and uniqueness of accumulation points imply convergence, which holds also with respect to the weak topology in ${{{\mathsf{L}}}}^2_{\mathrm{loc}}$.
Passage to the macroscopic limit {#sect:limit}
--------------------------------
Next we derive the hysteretic free boundary problem from Main Result \[res:Main\] along converging sequences and justify the hysteretic flow rule. In the bilinear case ${{\kappa}}=\infty$, there exists a straightforward argument based on the Hölder continuity of $P$ and the precise information on the microscopic phase transitions; see [@HeHe13 proof of Theorem 3.6]. In the trilinear case, however, we have to argue in a more sophisticated way due to the lack of vanishing $\ell^\infty$-bounds for the negligible fluctuations. In what follows we therefore employ the notion of entropy solutions that has been introduced in [@Plotnikov94; @EvPo04] in the context of the viscous regularization .
![*Left panel.* Illustration of the entropy argument in the proof of Theorem \[Thm:Limit\], which reveals that $P{{({\tilde{\tau}},\,{\tilde{\xi}})}}<p_*$ implies $\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tilde{\tau}}\right)}}=0$. *Center and right panel.* Smooth approximations of the entropy pair ${{\left({\tilde{\eta}},\,{\tilde{\mu}}\right)}}$ from .[]{data-label="fig:entropy"}](flow_rule){width="\textwidth"}
![*Left panel.* Illustration of the entropy argument in the proof of Theorem \[Thm:Limit\], which reveals that $P{{({\tilde{\tau}},\,{\tilde{\xi}})}}<p_*$ implies $\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tilde{\tau}}\right)}}=0$. *Center and right panel.* Smooth approximations of the entropy pair ${{\left({\tilde{\eta}},\,{\tilde{\mu}}\right)}}$ from .[]{data-label="fig:entropy"}](entropy){width="\textwidth"}
\[Thm:Limit\] Any limit from Proposition \[pro:compactness\] has the following properties, where ${{\Omega}}:={[0,\,\tau_{{\rm fin}}]}\times{{\mathbb{R}}}$ and ${{\Gamma}}:=\{{{\left({\tau},\,{\xi}\right)}}\in{{\Omega}}\,:\, \xi=\Xi{{\left({\tau}\right)}}\}$:
1. *(free boundary problem with Stefan condition)* $(P,\Xi)$ is a distributional solution of $$\label{Thm:Limit.Eqn1}
\partial_\tau P = \partial^2_\xi P
\quad\text{in}\quad \Omega{\setminus}{}{{\Gamma}}{},
\qquad
{{|\![P]\!|}} = 0
\quad\text{and}\quad
2 {{\textstyle{\ensuremath{ \ifx\empty\empty \frac{\mathrm{d}}{\mathrm{d}{\tau}} \else \frac{\mathrm{d}{\empty}}{\mathrm{d}{\tau}} \fi }
}}} {}\Xi{}= {{|\![\partial_\xi P]\!|}}
\quad\text{on}\quad {}\Gamma{}$$ and attains the initial data $(P(0),0)$. Moreover, we have $$\begin{aligned}
\label{Thm:Limit.Eqn2}
M(\tau,\xi)
=
\operatorname{sgn}{{\left({U(\tau,\xi)}\right)}}
=
\operatorname{sgn}{{\big(\Xi(\tau)-\xi\big)}}\quad \text{for}\quad \xi\neq\Xi{{\left({\tau}\right)}}
\end{aligned}$$ as well as $$\begin{aligned}
\label{Thm:Limit.Eqn3}
P(\tau,\xi)
\geq
-p_*
\quad \text{for}\quad \xi\leq\Xi{{\left({\tau}\right)}},
\qquad
P(\tau,\xi) \in [-p_*,+p_*]
\quad \text{for}\quad \xi\geq\Xi{{\left({\tau}\right)}}
\end{aligned}$$ and $\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tau}\right)}}\geq0$ for almost all $\tau$.
2. *(hysteretic flow rule and entropy balances)* The implication $$\begin{aligned}
\label{Thm:Limit.Eqn5}
P \big( \tau, \Xi{{\left({\tau}\right)}}\big) < p_*
\quad\implies\quad \tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tau}\right)}}=0
\end{aligned}$$ holds for almost all $\tau$ and the entropy inequality $$\begin{aligned}
\label{Thm:Limit.Eqn4}
\partial_\tau \eta{{\left({U}\right)}}-\partial_\xi{{\big(\mu{{\left({P}\right)}}\partial_\xi P\big)}}\leq0
\end{aligned}$$ is satisfied in the sense of distributions for any smooth entropy pair $(\eta,\mu)$ as in .
: By construction we have $$\begin{aligned}
M_{{\varepsilon}}(\tau,\xi)
=
U_{{\varepsilon}}(\tau,\xi) - P_{{\varepsilon}}(\tau,\xi)
=
\operatorname{sgn}{U}_{{\varepsilon}}(\tau,\xi)
\quad \text{for} \quad (\tau,\xi) \notin \Gamma_{{\varepsilon}}\end{aligned}$$ with $\Gamma_{{\varepsilon}}$ as in , and taking the limit ${{\varepsilon}}\to0$ we obtain by and the pointwise convergence of both $\Xi_{{\varepsilon}}^*$ and $\Xi_{{\varepsilon}}^\#$ to $\Xi$. Moreover, the lattice ODE combined with the scaling gives rise to $$\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}} U_{{\varepsilon}}\partial_\tau \Psi
\,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
=
-\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}} P_{{\varepsilon}}\Delta_{{\varepsilon}}\Psi
\,{\mathrm{d}\xi} \,{\mathrm{d}\tau},$$ for any test function $\Psi \in {{{\mathsf{C}}}}_c^\infty{{\big((0,\tau_{{\rm fin}}){\times}{\mathbb{R}}\big)}}$, where $\Delta_{{\varepsilon}}=\nabla_{-{{\varepsilon}}}\nabla_{+{{\varepsilon}}}$ is the finite difference approximation of $\partial_\xi^2$ on ${{\varepsilon}}{{\mathbb{Z}}}$. Using and we pass again to the limit ${{\varepsilon}}\to0$ and find $$\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}}
{{\big(P+M\big)}} \partial_\tau \Psi \,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
=
-\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}}
P \partial^2_\xi \Psi \,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
{}={}\int\limits_0^{\tau_{{\rm fin}}} \int\limits_{{\mathbb{R}}}
\partial_\xi P \partial_\xi \Psi
\,{\mathrm{d}\xi} \,{\mathrm{d}\tau}.$$ This is the weak formulation of since the properties of $\Phi$ from along with and the continuity of $P$ ensure that $$\begin{aligned}
{{|\![U{{\left({\tau}\right)}}]\!|}}
=
U\big(\tau, \Xi{{\left({\tau}\right)}}+0\big) - U\big(\tau, \Xi{{\left({\tau}\right)}}-0 \big)
=
{}- {}2 u_*
\end{aligned}$$ holds for almost all $\tau$ and $\xi$. Moreover, and the monotonicity of $\Xi$ also follow from their discrete counterparts, see Definition \[def:single-iface-solution\] and Propositions \[pro:existence\] and \[pro:compactness\].
: Let $0\leq\tau_1<\tau_2\leq \tau_{{\rm fin}}$ be given and $\Psi\in{{{\mathsf{C}}}}_c^\infty{{\big([0,\tau_{{\rm fin}}]\times{\mathbb{R}}\big)}}$ be a nonnegative test function. Proposition \[Prop:Entropy\] gives rise to the entropy inequality $$\begin{aligned}
\left.\int\limits_{{{\mathbb{R}}}}
\eta{{\big(U_{{\varepsilon}}\big)}}\Psi_{{\varepsilon}}\,{\mathrm{d}\xi}\right|^{\tau=\tau_2}_{\tau=\tau_1}
\leq
\int\limits_{\tau_1}^{\tau_2}\int\limits_{{{\mathbb{R}}}}
{{\Big( \eta{{\big(U_{{\varepsilon}}\big)}}\partial_\tau\Psi_{{\varepsilon}}- \mu{{\big(P_{{\varepsilon}}\big)}}{{\big(\nabla_{+{{\varepsilon}}}\Psi_{{\varepsilon}}\big)}}{{\big({}\nabla_{+{{\varepsilon}}}{}P_{{\varepsilon}}\big)}}\Big)}} \,{\mathrm{d}\xi} \,{\mathrm{d}\tau}
\end{aligned}$$ where $\Psi_{{\varepsilon}}$ denotes the ${{\varepsilon}}$-approximation of $\Psi$, which is piecewise constant in space and defined by $$\begin{aligned}
\Psi_{{\varepsilon}}{{\left({\tau},\,{{{\varepsilon}}j+\zeta}\right)}}
=
\Psi{{\left({\tau},\,{{{\varepsilon}}j}\right)}}
\qquad \text{for all}\quad
j\in{{\mathbb{Z}}},\; \tau\in[0,\tau_{{\rm fin}}],\; \zeta \in [-{{\varepsilon}}/2,{{\varepsilon}}/2).
\end{aligned}$$ Thanks to the smoothness of $\Psi$, the compactness of $\mathrm{supp}\,\Psi$, the weak convergence of ${}\nabla_{+{{\varepsilon}}}{}P_{{\varepsilon}}$, and the strong convergence of $P_{{\varepsilon}}$ – see and – we can pass to the limit ${{\varepsilon}}\to0$ and obtain $$\begin{aligned}
\label{Thm:Limit.PEqn4}
\left.\int\limits_{{{\mathbb{R}}}}
\eta{{\big(U\big)}}\Psi\,{\mathrm{d}\xi}\right|_{\tau=\tau_1}^{\tau=\tau_2}
\leq
\int\limits_{\tau_1}^{\tau_2}\int\limits_{{{\mathbb{R}}}}
{{\Big( \eta{{\big(U\big)}}\partial_\tau\Psi
- \mu{{\big(P\big)}}\partial_\xi\Psi\partial_\xi P \Big)}}
\,{\mathrm{d}\xi}\,{\mathrm{d}\tau},
\end{aligned}$$ which in turn yields in the sense of distributions if we choose $\tau_1=0$, $\tau_2=\tau_{{\rm fin}}$ and a test function $\Psi$ that vanishes for $\tau=0$ and $\tau=\tau_{{\rm fin}}$.
: Let $\tilde{\tau} \in [0,\tau_{{\rm fin}}]$ be fixed with $$\begin{aligned}
\label{Thm:Limit.PEqn0}
-p_*\leq P\big(\tilde\tau, \tilde\xi \big) < +p_*,
\qquad
\tilde{\xi} := \Xi{{\left({\tilde{\tau}}\right)}}.
\end{aligned}$$ Thanks to the continuity of both $\Xi$ and $P$ we can choose positions $\xi_1<\xi_2$ and times $\tau_1<\tau_2$ along with a number $\tilde{p}$ such that $$\begin{aligned}
\xi_1\leq \tilde{\xi}\leq \xi_2,
\qquad\quad
\tau_1\leq \tilde{\tau}\leq \tau_2,
\qquad\quad
\xi_1<\Xi{{\left({\tau}\right)}}<\xi_2
\quad\text{for all}\quad \tau\in [\tau_1,\tau_2]
\end{aligned}$$ and $$\begin{aligned}
-p_*\leq P(\tau,\xi) < \tilde{p} < p_*
\quad \text{for all}\quad
(\tau,\xi) \in [\tau_1,\tau_2] \times [\xi_1,\xi_2].
\end{aligned}$$ This construction is illustrated in the left panel in Figure \[fig:entropy\]. Moreover, considering nonnegative test functions $\Psi\in{{{\mathsf{C}}}}_c{{\big((\xi_1,\xi_2)\big)}}$ in we obtain $$\begin{aligned}
\label{Thm:Limit.PEqn1}
\int\limits_{\xi_1}^{\xi_2}
\eta{{\big(U(\tau_2,\xi)\big)}}\Psi{{\left({\xi}\right)}} \,{\mathrm{d}\xi}
-
\int\limits_{\xi_1}^{\xi_2}
\eta{{\big(U(\tau_1,\xi)\big)}}\Psi{{\left({\xi}\right)}} \,{\mathrm{d}\xi}
\leq
-\int\limits_{\tau_1}^{\tau_2} \int\limits_{\xi_1}^{\xi_2}
\mu {{\big(P(\tau,\xi)\big)}} \partial_\xi\Psi{{\left({\xi}\right)}}
\partial_\xi P(\tau,\xi)
\,{\mathrm{d}\xi} \,{\mathrm{d}\tau},
\end{aligned}$$ and by approximation with smooth densities and fluxes we deduce that holds also for the non-smooth entropy pair $$\label{Eqn:SpecialEntropy}
\tilde{\mu}{{\left({p}\right)}}:=
\begin{cases}
0 & \text{for } p \leq \tilde{p},
\\
+1 & \text{for } p > \tilde{p},
\end{cases}
\qquad
\tilde{\eta}{{\left({u}\right)}}
=
\int\limits_{-\infty}^u
\tilde{\mu}{{\big(\Phi^\prime{{\left({\bar{u}}\right)}}\big)}} \,{\mathrm{d}\bar{u}}.$$ Direct computations reveal that reduces to $$\tilde{c}\int\limits_{\xi_1}^{\Xi{{\left({\tau_2}\right)}}} \Psi{{\left({\xi}\right)}}\,{\mathrm{d}\xi}-
\tilde{c}\int\limits_{\xi_1}^{\Xi{{\left({\tau_1}\right)}}} \Psi{{\left({\xi}\right)}}\,{\mathrm{d}\xi}\leq 0$$ for some constant $\tilde{c}>0$, and since $\Psi$ was arbitrary we get $$\begin{aligned}
\Xi{{\left({\tau_2}\right)}}\leq\Xi{{\left({\tau_1}\right)}}.
\end{aligned}$$ On the other hand, $\Xi$ is also non-decreasing by construction. We thus arrive at $$\begin{aligned}
\Xi{{\left({\tau}\right)}}
=
\Xi{{\left({\tilde{\tau}}\right)}}
\qquad \text{for all}\qquad \tau\in [\tau_1,\tau_2]
\end{aligned}$$ and conclude that implies $\tfrac{{\mathrm{d}}}{{\mathrm{d}\tau}}\Xi{{\left({\tau}\right)}}=0$ for almost all $\tau\in [\tau_1,\tau_2]$. In particular, the interface satisfies .
The final ingredient to the proof of the main result from §\[sect:intro\] is to extend the convergence along sequences to convergence of the whole family ${{\varepsilon}}\to0$. This follows from the fact that for given macroscopic initial data there exists precisely one solution to the limit model from §\[sect:intro\]. Since the arguments are the same for the bilinear and the trilinear case, we refer to [@HeHe13 Theorem 3.18] for the proof and to [@Hilpert89; @Visintin06] for the key estimates. A similar uniqueness result can be found in [@MaTeTe09].
Conflict of Interest {#conflict-of-interest .unnumbered}
====================
The authors declare that they have no conflict of interest.
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[^1]: Rheinische Friedrichs-Wilhelm-Universität Bonn, .
[^2]: Westfälische Wilhelms-Universität Münster, .
|
---
abstract: 'We generalize several schedule matching theorems of Baiou–Balinski (Math. Oper. Res., 27 (2002), 485) and Alkan–Gale (J. Econ. Th. 112 (2003), 289) by applying a fixed point method of Fleiner (Math. Oper. Res., 28 (2003), 103). Thanks to a more general construction of revealing choice maps we develop an algorithm to solve rather complex matching problems. The flexibility and efficiency of our approach is illustrated by various examples. We also revisit the mathematical structure of the matching theory by comparing various definitions of stable sets and various classes of choice maps. We demonstrate, by several examples, that the revealing property of the choice maps is the most suitable one to ensure the existence of stable matchings; both from the theoretical and the practical point of view.'
address:
- |
Département de mathématique\
Université de Strasbourg\
7 rue René Descartes\
67084 Strasbourg Cedex, France
- |
UFR de mathématique et d’informatique\
Université de Strasbourg\
7 rue René Descartes\
67084 Strasbourg Cedex, France
- |
University of Maryland Baltimore County\
Department of Economics\
1000 Hilltop Circle\
Baltimore, MD 21250, USA
author:
- Vilmos Komornik
- Zsolt Komornik
- 'Christelle K. Viauroux'
date: 'Version 2010-05-11-a'
title: Stable schedule matchings by a fixed point method
---
[^1]
Introduction {#s1}
============
Since the pioneering paper of Gale and Shapley [@GalSha1962] on *stable matchings,* many studies have been devoted to the adaptations and the generalizations of their algorithm. Stable matching algorithms have found use in diverse economic applications ranging from labor markets to college admissions or even kidney exchanges.
In these two-sided matching markets, two sets of agents have preferences over the opposite set: on one side of the market, there are individuals (students, interns or employees) and on the other side there are institutions (colleges, hospitals or firms). A “stable match” is realized when all agents have been matched with the opposite side such that neither could obtain a more mutually beneficial match on their own.
The original strict preference ordering assumptions proved to be too restrictive for many real world problems. Following an influential contribution of Kelso and Crawford [@CraKel1982], Roth [@Rot1984] made a systematic study of a more flexible approach based on *choice functions*. The monograph of Roth and Sotomayor [@RotSot1990] provides an overview of the state of the art up to 1990 and it still serves as an excellent introduction to the subject. Feder [@Fed1992], Subramanian [@Sub1994] and Adachi [@Ada2000] discovered a close relationship between stable matchings and fixed points of set-valued maps. Then Fleiner [@Fle2003] demonstrated that many classical results may be obtained by a straightforward application of an old theorem of Knaster [@Kna1928] and Tarski [@Tar1928], [@Tar1955]. See also Hatfield and Milgrom [HatMil2005]{} for an economically motivated presentation of the fixed point method.
More recently, Baiou and Balinski [@BaiBal2002] introduced the notion of *schedule matching* which made it possible to consider, as a part of the contract not only the hiring of a particular worker by a particular firm, but also the number of hours of employment of the worker in the firm. In their setting, “stability” means that no pair of opposite agents can increase their hours together either due to unused capacity or by giving up hours with less desirable partners. They assumed that all agents have strict preference orderings. Alkan and Gale [@AlkGal2003] extended their model by using incomplete revealed preference ordering via choice functions instead.
In this paper, we generalize the notion of schedule matching of Baiou and Balinski [@BaiBal2002] to allow for schedule and preference constraints on each side of the market. We define a revealing choice map for each agent on the acceptable opposite side agent(s), possible days and (combinations of) restrictions or “subsets” placed on the opposite side agent and/or days worked. In particular, our framework allows for possible quotas placed by workers on firms and days worked, allowing him to work part-time for different firms on the same day or on different days, excluding some firms on some given days or excluding some days of work. In the same manner, it allows firms to adjust their labor force on certain days depending on their anticipated activity, or on the requirements associated to different activities on different days (or the same day). We show that the allocation of days, firms and workers is stable in the sense that given their schedule constraints, their preference orderings and constraints, there is no better schedule for both parties; moreover the stable allocation is shown to be worker optimal or firm optimal. This is done by using a slightly simplified version of Fleiner’s theorem and by giving a general construction of choice maps having the revealed preference property. We illustrate the power of our theorems by several examples. We provide the algorithm that can be used to obtain the optimal allocation: we will solve a deliberately complex example to explain its technical execution. Furthermore, in order to discuss the optimality of our results, we clarify the relationships between various properties of choice maps and between different definitions of stable sets, often used in the literature.
The plan of the paper is the following. In Section \[s2\] we formulate a model problem which will motivate our research and which may have natural real-word applications. In Section \[s3\] we present the mathematical framework for our model. In Section \[s4\] we solve the problems of Section \[s2\] and we also explain how our results cover some of the theorems of Alkan and Gale [@AlkGal2003]. In Section \[s5\] we illustrate the power and flexibility of our method by solving a number of more complex problems. Section \[s6\] concludes. The proofs of the theoretical results are given in Section \[s7\].
Schedule matching problems {#s2}
==========================
In order to illustrate the novelty of the present work we begin by recalling the first example of Gale and Shapley [@GalSha1962]. They considered three women: $w_1$, $w_2$, $w_3$ and three men: $f_1$, $f_2$, $f_3$ with the following preference orders (we change the notations for consistence with our later examples):
- Preference order of $w_1$: $f_1\succ f_2\succ f_3$;
- Preference order of $w_2$: $f_2\succ f_3\succ f_1$;
- Preference order of $w_3$: $f_3\succ f_1\succ f_2$;
- Preference order of $f_1$: $w_2\succ w_3\succ w_1$;
- Preference order of $f_2$: $w_3\succ w_1\succ w_2$;
- Preference order of $f_3$: $w_1\succ w_2\succ w_3$.
They looked for the possibilities of marrying all six people in a stable way. Instability would occur if there were a woman and a man, not married to each other who would prefer each other to their actual mates. It turns out that there are three solutions:
- each woman gets her first choice: $(w_1,f_1)$, $(w_2,f_2)$, $(w_3,f_3)$;
- each man gets his first choice: $(w_1,f_3)$, $(w_2,f_1)$, $(w_3,f_2)$;
- everyone get her or his second choice: $(w_{1},f_{2})$, $(w_{2},f_{3})$, $(w_{3},f_{1})$.
Now let us modify the problem to a simple job market problem as follows. Consider three workers: $w_{1}$, $w_{2}$, $w_{3}$ and three firms: $f_{1}$, $f_{2}$, $f_{3}$ with the same preference orders for hiring as above. Furthermore, assume that hiring is for two different days: $d_{1}$, $d_{2}$, with the following additional preferences and requirements:
- for each worker–firm pair $(w_i,f_j)$, the worker prefers $d_1$ to $d_2$ and the firm prefers $d_2$ to $d_1$;
- each worker may be hired by at most one firm on each given day (maybe different firms on different days);
- each firm may hire at most two workers per day; if they hire two workers for one given day, then they cannot hire anybody for the other day;
- no firm may hire the same worker for both days.
We are looking for a stable set of contracts, i.e., for a set $S$ of triplets $(w_i,f_j,d_k)$ having the following properties:
- each contract $(w_i,f_j,d_k)\in S$ is acceptable for both $w_i$ and $f_j$;
- for any other contract $(w_{i},f_{j},d_{k})\notin S$, either $w_{i}$ and/or $f_{j}$ prefers her/his contracts in $S$ to this new one.
The following section presents the theory necessary to address this type of problems. The solution to this example is given in Section \[s4\].
Existence of stable schedule matchings {#s3}
======================================
In this section we develop the theoretical framework required to solve problems like that of the preceding section. The main results are Theorems \[t37\], \[t310\] and \[t313\]. Propositions \[p35\] and \[p36\] contain useful complements and will be used in the proof of Theorem \[t37\] but they are not necessary for the understanding and the applications of our theorems. For the reader’s convenience, proofs are postponed to Section [s6]{}, which contains various remarks and examples discussing the optimality of the results formulated here.
Given a set $X$, we denote by $2^{X}$ the set of all subsets of $X$. By a *choice map* in $X$ we mean a function $C:2^{X}\rightarrow 2^{X}$ satisfying $$C(A)\subset A\quad \text{for all}\quad A\subset X. \label{31}$$
In economic applications $X$ is the set of all possible contracts, and for a given set $A$ of proposed contracts, $C(A)$ denotes the set of accepted contracts by some given rules of the market.
Assume that there are two competing sides, for example *workers and firms* and correspondingly two choice functions $C_{W},C_{F}:2^{X}\rightarrow 2^{X}$.
\[d31\] A set $S$ of contracts is said to be *stable* if there exist two sets $S_W,S_F\subset X$ satisfying the following three conditions: $$\begin{aligned}
&S_W\cup S_F=X; \label{32} \\
&C_W(A)=S\quad\text{for every}\quad S\subset A\subset S_W; \label{33} \\
&C_F(A)=S\quad\text{for every}\quad S\subset A\subset S_F. \label{34}\end{aligned}$$
Stable contract sets represent acceptable compromises.
\[r32\] A stable set[^2] $S$ is *individually rational* if $$C_{W}(S)=S=C_{F}(S), \label{35}$$and it is *not blocked by any other contract*, i.e., for each $x\in X$ we have $$\text{either}\quad C_{W}(S\cup \{x\})=S\quad \text{or}\quad C_{F}(S\cup
\{x\})=S\quad \text{(or both).} \label{36}$$
In order to ensure the existence of stable sets of contracts we need one additional assumption on the choice maps.
\[d33\] We say that a choice map $C:2^X\to 2^X$ is *revealing* (or satisfies the *revealed preference condition*) if $$\label{37}
A,B\subset X\quad\text{and}\quad C(A)\subset B\Longrightarrow A\cap
C(B)\subset C(A).$$
This means that if a contract is rejected from some proposed set $A$ of contracts, then it will also be rejected from every other proposed set $B$ which contains the accepted contracts.
\[e34\]
\(a) For any fixed set $Y\subset X$ the formula $C(A):=A\cap Y$ defines a revealing choice map on $X$. This example illustrates a situation where some contracts are unacceptable to certain agents.
\(b) More generally, given a finite subset $Y\subset X$, a nonnegative integer $q$ (called *quota*) and a strict preference ordering $y_{1}\succ y_{2}\succ \cdots $ on $Y$, we define a map $C(A)$ for any given $A\subset X$ as follows. If $\operatorname{Card}(A\cap Y)\leq q$, then we set $C(A):=A\cap Y
$. If $\operatorname{Card}(A\cap Y)\succ q$, then let $C(A)$ be the set of the first $q$ elements of $A\cap Y$ according to the ordering of $Y$. Then $C:2^{X}\rightarrow 2^{X}$ is a revealing choice map on $X$.
Choice maps of this kind are frequently used in classical matching problems such as the marriage problem, the college admission problem and various many-to-many matching problems; see, e.g., [@AlkGal2003], [RotSot1990]{} and the references of the latter.
Before stating our main theorem, we further clarify the relationships between the revealed preference condition and other usual properties of choice maps (Proposition 3.5.). We also discuss alternative equivalent definitions of stable sets (Proposition 3.6.).
\[p35\]
\(a) A choice map $C:2^X\to 2^X$ is revealing if and only if it is *consistent*: $$\label{38}
C(A)\subset B\subset A\Longrightarrow C(B)= C(A)$$ and *persistent* (or satisfies the *substitute condition*): $$\label{39}
A\subset B\Longrightarrow A\cap C(B)\subset C(A).$$
\(b) A choice map is persistent if and only if the *rejection map* $R:2^X\to 2^X$ defined by $R(A):=A\setminus C(A)$ is *monotone*, i.e., $$\label{310}
A\subset B\Longrightarrow R(A)\subset R(B).$$
\(c) A choice map satisfying either or is *idempotent*: $$\label{311}
C(C(A))=C(A)\quad\text{for all}\quad A\subset X.$$
\[p36\] We consider two choice maps $C_W, C_F:2^X\to 2^X$ and three sets $S, S_W, S_F\subset X$ satisfying $$\label{312}
S_W\cup S_F=X\quad\text{and}\quad C_W(S_W)=S=C_F(S_F).$$
\(a) If the choice maps $C_W,C_F:2^X\to 2^X$ are idempotent, then $S$ is individually rational, i.e., it satisfies .
\(b) If at least one of the two choice maps $C_W,C_F:2^X\to 2^X$ is consistent, then we may modify $S_W$ or $S_F$ such that that $S_W\cap S_F=S$ and remains valid.
\(c) If, moreover, both choice maps are consistent, then is equivalent to the stability – of $S$.
\(d) If both choice maps $C_W,C_F:2^X\to 2^X$ are revealing, then a set $S$ is stable if and only if it is individually rational, and it is not blocked by any other contract, i.e., – are equivalent to – (for all $x\in X$.
Observe that property below follows from the definition of stable sets.
Our main theorem below shows that the revealed preference condition ensures the existence of stable sets of contracts:
\[t37\] If $C_W,C_F:2^X\to 2^X$ are two revealing choice maps, then there exists at least one stable set of contracts.
\[r38\]
\(a) The proof of the theorem, provided in Section \[s7\], will show that the stable sets form a complete lattice for a natural order relation. In particular, there exists a worker-optimal and a firm-optimal stable set.
\(b) In case $X$ is a finite set, the proof of the theorem provides an efficient algorithm to find a stable set. Starting with $X_0:=X$ we compute successively $Y_1, X_2, Y_3, X_4,\ldots $ by using the recursive formulae $$Y_{n+1}:=(X\setminus X_n)\cup C_W(X_n) \quad\text{and}\quad
X_{n+1}:=(X\setminus Y_n)\cup C_F(Y_n).$$ There exists a first index $n\ge 1$ such that $X_{n-1}=X_{n+1}$, and then $S:=C_W(X_{n-1})$ is the worker-optimal stable set.
Similarly, starting with $Y_0:=X$ we may compute successively $X_1, Y_2,
X_3, Y_4,\ldots $ by the same recursive formulae. There exists a first index $n\ge 1$ such that $Y_{n-1}=Y_{n+1}$, and then $S:=C_F(Y_{n-1})$ is the firm-optimal stable set. See Remark \[r62\] below for the details.
\(c) The definitions of revealing choice maps, stable sets, the theorem and the preceding remarks remain valid if we replace $2^X$ by a complete sublattice $L$ of $2^X$, i.e., a subfamily $L$ of $2^X$ such that the union and the intersection of any system of sets $A\in L$ still belongs to $L$. See, e.g., [@Fle2003] for more details on lattice properties.
\(d) Part (a) of Proposition \[p35\] shows that Theorem \[t37\] is mathematically equivalent to a theorem of Fleiner [@Fle2003].
\(e) We will show in Examples \[e61\] (a)–(b) and \[e65\] of Section [s6]{} that the revealing condition cannot be weakened in Theorem \[t37\].
In order to apply Theorem \[t37\] for the solution of the problem stated in Section \[s2\], we need a generalization of the construction of revealing choice maps recalled in Example \[e34\]. Such a construction is provided by Theorem \[t310\] below.
Let we are given a finite subset $Y\subset X$, a family $\{Y_n\}$ of subsets $Y_n\subset X$, and corresponding nonnegative integers (called *quotas*) $q$ and $q_n$. We assume that the sets $Y_n\cap Y$ are disjoint. Furthermore, let be given a strict preference ordering $y_1\succ y_2\succ
\cdots $ on $Y$. Given any set $A\subset X$, we define a nondecreasing sequence $C_0(A)\subset C_1(A)\subset\cdots$ of subsets of $A\cap Y$ by recursion as follows. First we set $C_0(A)=\varnothing$. If $C_{k-1}(A)$ has already been defined for some $k$, then we set $C_k(A):=C_{k-1}(A)\cup\{y_k\}
$ if $$\begin{aligned}
&y_k\in A, \\
&\operatorname{Card}C_{k-1}(A) < q, \\
&\operatorname{Card}\left( C_{k-1}(A)\cap Y_n\right) < q_n\text{ if }y_k\in Y_n;\end{aligned}$$ otherwise we set $C_k(A):=C_{k-1}(A)$. Finally, we define $C(A):=\cup C_k(A)$.
\[r39\] It follows from the construction that $$\begin{aligned}
&C(A)\subset A\cap Y; \label{313} \\
&\operatorname{Card}C(A)\le q; \label{314} \\
&\operatorname{Card}\left( C(A)\cap Y_n\right) \le q_n\text{ for all }n. \label{315}\end{aligned}$$
\[t310\] $C:2^{X}\rightarrow 2^{X}$ is a revealing choice map.
\[r311\]
- If $q_{n}\geq q$ or $q_{n}\geq \operatorname{Card}(Y)$ for some $n$, then we may eliminate $Y_{n}$ and $q_{n}$ without changing the construction.
- If there are no sets $Y_{n}$, then our construction reduces to Example \[e34\] (b).
- If, moreover, $q\geq \operatorname{Card}(Y)$, then our construction reduces to Example \[e34\] (a). (In this case the choice of the order relation is irrelevant.)
- Instead of a finite subset $Y\subset X$, we can also consider arbitrary subsets $Y\subset X$ with a well-ordered preference relation: the construction and the proof of the proposition remain valid.
\[e312\] The disjointness condition is necessary. To show this, consider the sets $X=Y=\{a,b,c\}$, $Y_{1}=\{a,b\}$, $Y_{2}=\{b,c\}$ with the quotas $q=2$, $q_{1}=q_{2}=1$ and the preference order $a\succ b\succ c$. Then for $A=\{b,c\}$ and $B=\{a,b,c\}$ we have $C(A)=\{b\}$ and $C(B)=\{a,c\}$, so that $A\subset B$ but $A\cap C(B)\not\subset C(A)$.
Theorem \[t310\] can be often used for the construction of *individual* revealing choice functions. The following result enables us to combine individual revealing choice functions into *global* revealing choice functions.
\[t313\] Given a set function $C:2^X\to 2^X$ and a partition $X=\cup X_i$ with disjoint sets $X_i$, we define the set functions $C_i:2^{X_i}\to 2^{X_i}
$ by the formula $C_i(A_i):=C(A_i)\cap X_i$. Then $C$ is a revealing choice map on $X$ if and only if each $C_i$ is a revealing choice map on $X_i$.
Solution to the simple job market problem {#s4}
=========================================
For the solution we set $$W:=\{w_1, w_2, w_3\},\quad F:=\{f_1, f_2, f_3\},\quad D:=\{d_1,d_2\}$$ and we proceed in several steps.
*Step 1.* For each fixed worker $w_i$ we define a revealing choice map $C_{w_i}$ on $\{w_i\}\times F\times D$ by applying Theorem \[t310\] with $Y$, $q$, $Y_n$ and $q_n$ given below. For brevity we write $(i,j,k)$ instead of $(w_i,f_j,d_k)$ in the preference relations.
- For worker $w_1$ we choose $$\begin{aligned}
&Y:=\{w_1\}\times \{f_1,f_2,f_3\}\times \{d_1,d_2\}, \\
&Y_1:=\{w_1\}\times \{f_1,f_2,f_3\}\times \{d_1\}, \\
&Y_2:=\{w_1\}\times \{f_1,f_2,f_3\}\times \{d_2\}\end{aligned}$$ with quotas $q=6$ (which is ineffective), $q_1=q_2=1$ and the following preference relation on $Y$: $$(1,1,1)\succ (1,1,2)\succ (1,2,1)\succ (1,2,2)\succ (1,3,1)\succ (1,3,2).$$
- For worker $w_2$ we choose $$\begin{aligned}
&Y:=\{w_2\}\times \{f_1,f_2,f_3\}\times \{d_1,d_2\}, \\
&Y_1:=\{w_2\}\times \{f_1,f_2,f_3\}\times \{d_1\}, \\
&Y_2:=\{w_2\}\times \{f_1,f_2,f_3\}\times \{d_2\}\end{aligned}$$ with quotas $q=6$, $q_1=q_2=1$ and the following preference relation on $Y$: $$(2,2,1)\succ (2,2,2)\succ (2,3,1)\succ (2,3,2)\succ (2,1,1)\succ (2,1,2).$$
- For worker $w_3$ we choose $$\begin{aligned}
&Y:=\{w_3\}\times \{f_1,f_2,f_3\}\times \{d_1,d_2\}, \\
&Y_1:=\{w_3\}\times \{f_1,f_2,f_3\}\times \{d_1\}, \\
&Y_2:=\{w_3\}\times \{f_1,f_2,f_3\}\times \{d_2\}\end{aligned}$$ with quotas $q=6$, $q_1=q_2=1$ and the following preference relation on $Y$: $$(3,3,1)\succ (3,3,2)\succ (3,1,1)\succ (3,1,2)\succ (3,2,1)\succ (3,2,2).$$
*Step 2.* Applying Theorem \[t313\] we combine the three choice maps of the preceding step into a global revealing choice map $C_{W}$ on $W\times
F\times D$ by setting $$C_{W}(A):=\bigcup_{i=1}^{3}C_{w_{i}}\left( A\cap (\{w_{i}\}\times F\times
D)\right)$$for every $A\subset W\times F\times D$.
*Step 3.* For each firm $f_{j}$ we define a revealing choice map $C_{f_{j}}$ on $W\times \{f_{j}\}\times D$ by applying Theorem \[t310\] with $Y$, $q$, $Y_{n}$ and $q_{n}$ given below and still writing $(i,j,k)$ instead of $(w_{i},f_{j},d_{k})$ for brevity.
- For firm $f_1$ we choose $$\begin{aligned}
&Y:=\{w_1, w_2, w_3\}\times \{f_1\}\times \{d_1,d_2\}, \\
&Y_1:=\{w_1\}\times \{f_1\}\times \{d_1,d_2\}, \\
&Y_2:=\{w_2\}\times \{f_1\}\times\{d_1,d_2\}, \\
&Y_3:=\{w_3\}\times \{f_1\}\times\{d_1,d_2\},\end{aligned}$$ with quotas $q=2$, $q_1=q_2=q_3=1$ and the following preference relation on $Y$: $$(2,1,2)\succ (2,1,1)\succ (3,2,2)\succ (3,2,1)\succ (1,3,2)\succ (1,3,1).$$
- For firm $f_2$ we choose $$\begin{aligned}
&Y:=\{w_1, w_2, w_3\}\times \{f_2\}\times \{d_1,d_2\}, \\
&Y_1:=\{w_1\}\times \{f_2\}\times \{d_1,d_2\}, \\
&Y_2:=\{w_2\}\times \{f_2\}\times\{d_1,d_2\}, \\
&Y_3:=\{w_3\}\times \{f_2\}\times\{d_1,d_2\},\end{aligned}$$ with quotas $q=2$, $q_1=q_2=q_3=1$ and the following preference relation on $Y$: $$(3,2,2)\succ (3,2,1)\succ (1,2,2)\succ (1,2,1)\succ (2,2,2)\succ (2,2,1).$$
- For firm $f_3$ we choose $$\begin{aligned}
&Y:=\{w_1, w_2, w_3\}\times \{f_3\}\times \{d_1,d_2\}, \\
&Y_1:=\{w_1\}\times \{f_3\}\times \{d_1,d_2\}, \\
&Y_2:=\{w_2\}\times \{f_3\}\times\{d_1,d_2\}, \\
&Y_3:=\{w_3\}\times \{f_3\}\times\{d_1,d_2\},\end{aligned}$$ with quotas $q=2$, $q_1=q_2=q_3=1$ and the following preference relation on $Y$: $$(1,3,2)\succ (1,3,1)\succ (2,3,2)\succ (2,3,1)\succ (3,3,2)\succ (3,3,1).$$
*Step 4.* Applying Theorem \[t313\] we combine the three choice maps of the preceding step into a global revealing choice map $C_F$ on $W\times
F\times D$ by setting $$C_F(A):= \bigcup_{j=1}^3C_{f_j}\left( A\cap (W\times \{f_j\}\times D)\right)
,\quad A\subset W\times F\times D.$$
*Step 5.* The choice maps $C_W$ and $C_F$ satisfy the hypotheses of Theorem \[t37\]. We apply the algorithm as described in Remark \[r38\] (b) by starting with $X_0:=X$ and computing $Y_1, X_2, Y_3,X_4$ by the formulae $$Y_{n+1}:=(X\setminus X_n)\cup C_W(X_n) \quad\text{and}\quad
X_{n+1}:=(X\setminus Y_n)\cup C_F(Y_n).$$ We obtain that $X_2=X_4$ and therefore $S=C_W(X_2)$. The results are summarized in the following table:
$w_i$ $f_j$ $d_k$ $X_0$ $Y_1$ $X_2$ $Y_3$ $X_4$ $S$
------- ------- ------- ------- ------- ------- ------- ------- -----
1 1 1 $x$ $x$ $x$
1 1 2 $x$ $x$ $x$ $x$ $x$ $x$
1 2 1 $x$ $x$ $x$ $x$ $x$
1 2 2 $x$ $x$ $x$
1 3 1 $x$ $x$ $x$
1 3 2 $x$ $x$ $x$
2 1 1 $x$ $x$ $x$
2 1 2 $x$ $x$ $x$
2 2 1 $x$ $x$ $x$
2 2 2 $x$ $x$ $x$ $x$ $x$ $x$
2 3 1 $x$ $x$ $x$ $x$ $x$
2 3 2 $x$ $x$ $x$
3 1 1 $x$ $x$ $x$ $x$ $x$
3 1 2 $x$ $x$ $x$
3 2 1 $x$ $x$ $x$
3 2 2 $x$ $x$ $x$
3 3 1 $x$ $x$ $x$
3 3 2 $x$ $x$ $x$ $x$ $x$ $x$
In this *worker-optimal* solution each worker is hired by the second most preferred firm for the first day and by the most preferred firm for the second day.
*Step 6.* Applying the algorithm of Remark \[r38\] (b) by starting with $Y_0:=X$ and computing $X_1, Y_2, X_3, Y_4$ by the above formulae we obtain that $Y_2=Y_4$ and therefore $S=C_F(Y_2)$. The results are summarized in the following table:
$w_i$ $f_j$ $d_k$ $Y_0$ $X_1$ $Y_2$ $X_3$ $Y_4$ $S$
------- ------- ------- ------- ------- ------- ------- ------- -----
1 1 1 $x$ $x$ $x$
1 1 2 $x$ $x$ $x$
1 2 1 $x$ $x$ $x$
1 2 2 $x$ $x$ $x$ $x$ $x$ $x$
1 3 1 $x$ $x$ $x$ $x$ $x$
1 3 2 $x$ $x$ $x$
2 1 1 $x$ $x$ $x$ $x$ $x$
2 1 2 $x$ $x$ $x$
2 2 1 $x$ $x$ $x$
2 2 2 $x$ $x$ $x$
2 3 1 $x$ $x$ $x$
2 3 2 $x$ $x$ $x$ $x$ $x$ $x$
3 1 1 $x$ $x$ $x$
3 1 2 $x$ $x$ $x$ $x$ $x$ $x$
3 2 1 $x$ $x$ $x$ $x$ $x$
3 2 2 $x$ $x$ $x$
3 3 1 $x$ $x$ $x$
3 3 2 $x$ $x$ $x$
In this *firm-optimal* solution each firm hires the most preferred worker for the first day and by the second most preferred worker for the second day.
\[r41\] The stable schedule matchings as studied by Baiou and Balinski [@BaiBal2002] and Alkan and Gale [@AlkGal2003] enter the present framework as a special case. For simplicity we consider the discrete case and we denote by $D=\{1,2,\ldots \}$ the possible number of working hours with $k$ meaning the $k$th working hour. For each worker $w_{i}$, if there is a preference ranking $f_{j_{1}}\succ f_{j_{2}}\succ \cdots $ among the firms, then we extend it to the preference ranking $$\begin{aligned}
& (i,j_{1},1)\succ (i,j_{1},2)\succ \cdots \succ (i,j_{1},q_{i,j_{1}}^{w}) \\
\succ & (i,j_{2},1)\succ (i,j_{2},2)\succ \cdots \succ
(i,j_{2},q_{i,j_{2}}^{w}) \\
\succ & \cdots \\
& \vdots \end{aligned}$$where $q_{i,j}^{w}$ denotes the maximum number of working hours accepted by worker $w_{i}$ in firm $f_{j}$. Similarly, for each firm $f_{j}$, if there is a preference ranking $w_{i_{1}}\succ w_{i_{2}}\succ \cdots $ among the workers, then we extend it to the preference ranking $$\begin{aligned}
& (i_{1},j,1)\succ (i_{1},j,2)\succ \cdots \succ (i_{1},j,q_{i_{1},j}^{f}) \\
\succ & (i_{2},j,1)\succ (i_{2},j,2)\succ \cdots \succ
(i_{2},j,q_{i_{2},j}^{f}) \\
\succ & \cdots \\
& \vdots \end{aligned}$$where $q_{i,j}^{f}$ denotes the maximum number of working hours accepted by firm $f_{j}$ for worker $w_{i}$. Once a stable set $S$ found, the number of working hours of worker $w_{i}$ in firm $f_{j}$ is the biggest integer $k$ such that $(i,j,k)\in S$.
More complex examples {#s5}
=====================
We illustrate in this section the strength and flexibility of our theorems and algorithms by solving some more complex problems.
We consider the following modeling issue. We are given a finite number of workers $w_{i}$, firms $f_{j}$ and days $d_{k}$ (days of a week or days of a month for instance). Each worker may work at one or several firms per day, maybe at different firms on different days. Similarly, each firm may hire a given number of workers per day, maybe differents numbers on different days.
A *contract* is by definition a triple $(w_{i},f_{j},d_{k})$ meaning that worker $w_{i}$ is hired by firm $f_{j}$ for day $d_{k}$, and we are looking for an acceptable set of contracts, subject to various requirements of both workers and firms. Thus, each worker $w_{i}$
- may exclude some firm–day pairs $(f_j,d_k)$ considered unacceptable;
- has a strict preference ordering among the remaining firm–day pairs;
- may put some other restrictions, such as
- to set a maximum quota $q^w_i$ of accepted firm–day pairs;
- not to work on day $d_k$ at more than a given number $q^w_{i,k}$ of firms;
- or not to work at firm $f_j$ more than a given number $\tilde q^w_{i,j}
$ of days.
Similarly, each firm $f_j$
- may exclude some worker–day pairs $(w_i,d_k)$ considered unacceptable;
- has a strict preference ordering among the remaining worker–day pairs;
- may put some other restrictions, such as
- to set a maximum quota $q^f_j$ of worker–day pairs for hiring;
- not to hire on day $d_k$ more than a given number $q^f_{j,k}$ of workers;
- or not to hire worker $w_i$ for more than a given number $\tilde
q^f_{i,j}$ of days.
\[r51\]
- Although we keep strict preference ordering on worker-day pairs or on firm-day pairs, these preference ordering are not sufficient to characterize the choice map of each agent: they also depend on the quota system.
- In most applications we may assume that a worker does not work at more than one firm per day, so that $q^w_{i,k}=1$ for every $k$; then $q_i$ means the maximum number of working days for the worker $w_i$.
First problem {#ss51}
-------------
Assume that we have four workers $w_1$, $w_2$, $w_3$, $w_4$ and three firms $f_1$, $f_2$, $f_3$. Each worker may work at most at one firm per day, maybe at different firms on different days of the week. The further requirements of the agents are listed below.
- Worker $w_1$ can work at most 4 days per week, with the following strict preference order of the firm–day pairs $(f_j,d_k)$ where we write $(j,k)$ instead of $(f_j,d_k)$ for brevity: $$\begin{aligned}
\label{51}
&(2,1)\succ (3,1)\succ (2,2)\succ (3,2)\succ (2,3)\succ (3,3)\succ (2,4) \\
\succ &(3,4)\succ (2,5)\succ (3,5)\succ (2,6)\succ (3,6)\succ (2,7)\succ
(3,7) . \notag\end{aligned}$$ This list shows for instance that worker $w_1$ prefers most to be hired by firm $f_2$ for Mondays ($d_1$), then by firm $f_3$ always for Mondays, next by firm $f_2$ for Tuesdays ($d_2$), and so on. The absence of firm $f_1$ in the list shows that worker $w_1$ refuses to be hired by that firm.
- Worker $w_2$ can work at most 3 days per week, with the following strict preference order: $$\begin{aligned}
\label{52}
&(1,1)\succ (1,2)\succ (1,3)\succ (1,4)\succ (1,5)\succ (1,6)\succ (1,7) \\
\succ &(2,1)\succ (2,2)\succ (2,3)\succ (2,4)\succ (2,5)\succ (2,6)\succ
(2,7) \notag \\
\succ &(3,1)\succ (3,2)\succ (3,3)\succ (3,4)\succ (3,5)\succ (3,6)\succ
(3,7). \notag\end{aligned}$$
- Worker $w_3$ can work at most 2 days per week, with the following strict preference order: $$\begin{aligned}
\label{53}
&(2,2)\succ (2,3)\succ (3,2)\succ (3,3)\succ (1,2)\succ (1,3) \\
\succ &(2,4)\succ (2,5)\succ (2,6)\succ (1,4)\succ (1,5)\succ (1,6) \notag
\\
\succ &(3,6)\succ (3,4)\succ (3,5). \notag\end{aligned}$$ The list shows in particular that he/she does not work on Mondays ($d_1$) and Sundays ($d_7$).
- Worker $w_4$ accepts to work on all days of the week, with the following strict preference order: $$\begin{aligned}
\label{54}
&(1,1)\succ (1,2)\succ (1,3)\succ (1,4)\succ (1,5)\succ (1,6)\succ (1,7) \\
\succ &(2,1)\succ (2,2)\succ (2,3)\succ (2,4)\succ (2,5)\succ (2,6)\succ
(2,7) \notag \\
\succ &(3,1)\succ (3,2)\succ (3,3)\succ (3,4)\succ (3,5)\succ (3,6)\succ
(3,7). \notag\end{aligned}$$
- Firm $f_1$ may hire up to 4 workers per day when it is open, with the following strict preference order of the worker–day pairs $(w_i,d_k)$ where we write $(i,k)$ instead of $(w_i,d_k)$ for brevity: $$\begin{aligned}
\label{55}
&(1,1)\succ (1,2)\succ (1,3)\succ (1,4)\succ (1,5)\succ (1,6) \\
\succ &(2,1)\succ (2,2)\succ (2,3)\succ (2,4)\succ (2,5)\succ (2,6) \notag
\\
\succ &(3,1)\succ (3,2)\succ (3,3)\succ (3,4)\succ (3,5)\succ (3,6). \notag\end{aligned}$$ The list shows in particular that the firm is closed on Sundays and that it doesn’t hire worker $w_4$. Otherwise, it prefers most to hire worker $w_1$ for Mondays, then worker $w_1$ for Tuesdays, and so on.
- Firm $f_2$ may also hire up to 4 workers per day, with the following strict preference order: $$\begin{aligned}
\label{56}
&(3,7)\succ (3,6)\succ (3,5)\succ (3,4)\succ (3,3)\succ (3,2)\succ (3,1) \\
\succ &(4,7)\succ (4,6)\succ (4,5)\succ (4,4)\succ (4,3)\succ (4,2)\succ
(4,1) \notag \\
\succ &(1,7)\succ (1,6)\succ (1,5)\succ (1,4)\succ (1,3)\succ (1,2)\succ
(1,1) \notag \\
\succ &(2,7)\succ (2,6)\succ (2,5)\succ (2,4)\succ (2,3)\succ (2,2)\succ
(2,1). \notag\end{aligned}$$
- Firm $f_3$ is closed on Saturdays and Sundays; for the other days it may hire up to 4 workers per day, with the following strict preference order: $$\begin{aligned}
\label{57}
&(4,1)\succ (4,2)\succ (4,3)\succ (4,4)\succ (4,5) \\
\succ &(3,1)\succ (3,2)\succ (3,3)\succ (3,4)\succ (3,5) \notag \\
\succ &(2,1)\succ (2,2)\succ (2,3)\succ (2,4)\succ (2,5) \notag \\
\succ &(1,1)\succ (1,2)\succ (1,3)\succ (1,4)\succ (1,5). \notag\end{aligned}$$
Our task is to find an acceptable firm-worker assignment and work schedule under these constraints. For the solution we set $$W:=\{w_1, w_2, w_3, w_4\},\quad F:=\{f_1, f_2, f_3\},\quad
D:=\{d_1,\ldots,d_7\}$$ and we proceed in several steps.
*Step 1.* For each fixed worker $w_i$ we define a revealing choice map $C_{w_i}$ on $\{w_i\}\times F\times D$ by applying Theorem \[t310\] with $Y:=Y^w_i$, $q:=q^w_i$, $Y_n:=Y^w_{i,n}$ and $q_n:=q^w_{i,n}$ given below. For brevity we write $(i,j,k)$ instead of $(w_i,f_j,d_k)$ in the preference relations.
- For worker $w_1$ we choose $$Y^w_1:=\{w_1\}\times \{f_2,f_3\}\times D$$ representing the set of acceptable firms and days of worker $w_1$, with quota $q^w_1=4$ and the following preference relation on $Y^w_1$ (see ): $$\begin{aligned}
&(1,2,1)\succ (1,3,1)\succ (1,2,2)\succ (1,3,2)\succ (1,2,3) \\
\succ &(1,3,3)\succ (1,2,4)\succ (1,3,4)\succ (1,2,5)\succ (1,3,5) \\
\succ &(1,2,6)\succ (1,3,6)\succ (1,2,7)\succ (1,3,7) .\end{aligned}$$ Furthermore, we set $$Y^w_{1,k}:=\{w_1\}\times F\times \{d_k\}\text{ and } q^w_{1,k}=1\text{ for }
k=1,\ldots, 7.$$
- For worker $w_2$ we choose $$Y^w_2:=\{w_2\}\times F\times D$$ with quota $q^w_2=3$ and the preference relation $$\begin{aligned}
&(2,1,1)\succ (2,1,2)\succ (2,1,3)\succ (2,1,4)\succ (2,1,5) \\
\succ &(2,1,6)\succ (2,1,7)\succ (2,2,1)\succ (2,2,2)\succ (2,2,3) \\
\succ &(2,2,4)\succ (2,2,5)\succ (2,2,6)\succ (2,2,7)\succ (2,3,1) \\
\succ &(2,3,2)\succ (2,3,3)\succ (2,3,4)\succ (2,3,5)\succ (2,3,6)\succ
(2,3,7).\end{aligned}$$ on $Y^w_2$ (see ). Furthermore, we set $$Y^w_{2,k}:=\{w_2\}\times F\times \{d_k\}\text{ and } q^w_{2,k}=1\text{ for }
k=1,\ldots, 7.$$
- For worker $w_3$ we choose $$Y^w_3:=\{w_3\}\times F\times \{d_2,\ldots,d_6\}$$ with quota $q^w_3=2$ and the preference relation $$\begin{aligned}
&(3,2,2)\succ (3,2,3)\succ (3,3,2)\succ (3,3,3)\succ (3,1,2) \\
\succ &(3,1,3)\succ (3,2,4)\succ (3,2,5)\succ (3,2,6)\succ (3,1,4) \\
\succ &(3,1,5)\succ (3,1,6)\succ (3,3,6)\succ (3,3,4)\succ (3,3,5).\end{aligned}$$ on $Y^w_3$ (see ). Furthermore, we set $$Y^w_{3,k}:=\{w_3\}\times F\times \{d_k\}\text{ and } q^w_{3,k}=1\text{ for }
k=1,\ldots, 7.$$
- For worker $w_4$ we choose $$Y^w_4:=\{w_4\}\times F\times D$$ with quota $q^w_4=7$ and the preference relation $$\begin{aligned}
&(4,1,1)\succ (4,1,2)\succ (4,1,3)\succ (4,1,4)\succ (4,1,5) \\
\succ &(4,1,6)\succ (4,1,7)\succ (4,2,1)\succ (4,2,2)\succ (4,2,3) \\
\succ &(4,2,4)\succ (4,2,5)\succ (4,2,6)\succ (4,2,7)\succ (4,3,1) \\
\succ &(4,3,2)\succ (4,3,3)\succ (4,3,4)\succ (4,3,5)\succ (4,3,6)\succ
(4,3,7).\end{aligned}$$ on $Y^w_4$ (see ). Furthermore, we set $$Y^w_{4,k}:=\{w_4\}\times F\times \{d_k\}\text{ and } q^w_{4,k}=1\text{ for }
k=1,\ldots, 7.$$
*Step 2.* Applying Theorem \[t313\] we combine the four choice maps of the preceding step into a global revealing choice map $C_{W}$ on $W\times
F\times D$ by setting $$C_{W}(A):=\bigcup_{i=1}^{4}C_{w_{i}}\left( A\cap (\{w_{i}\}\times F\times
D)\right)$$for every $A\subset W\times F\times D$.
*Step 3.* For each firm $f_{j}$ we define a revealing choice map $C_{f_{j}}$ on $W\times \{f_{j}\}\times D$ by applying Theorem \[t310\] again, this time with $Y:=Y_{j}^{f}$, $q:=q_{j}^{f}$, $Y_{n}:=Y_{j,n}^{f}$ and $q_{n}:=q_{j,n}^{f}$ given below and still writing $(i,j,k)$ instead of $(w_{i},f_{j},d_{k})$ for brevity.
- For firm $f_1$ we choose $$Y^f_1:=W\times \{f_1\}\times \{d_1,\ldots,d_6\}$$ with quota $q^f_1=24$ and the following preference relation on $Y^f_1$ (see ): $$\begin{aligned}
&(1,1,1)\succ (1,1,2)\succ (1,1,3)\succ (1,1,4)\succ (1,1,5) \\
\succ &(1,1,6)\succ (2,1,1)\succ (2,1,2)\succ (2,1,3)\succ (2,1,4) \\
\succ &(2,1,5)\succ (2,1,6)\succ (3,1,1)\succ (3,1,2)\succ (3,1,3) \\
\succ &(3,1,4)\succ (3,1,5)\succ (3,1,6).\end{aligned}$$
- For firm $f_2$ we choose $$Y^f_2:=W\times \{f_2\}\times D$$ with quota $q^f_2=28$ and the following preference relation on $Y^f_2$ (see ): $$\begin{aligned}
&(3,2,7)\succ (3,2,6)\succ (3,2,5)\succ (3,2,4)\succ (3,2,3) \\
\succ &(3,2,2)\succ (3,2,1)\succ (4,2,7)\succ (4,2,6)\succ (4,2,5) \\
\succ &(4,2,4)\succ (4,2,3)\succ (4,2,2)\succ (4,2,1)\succ (1,2,7) \\
\succ &(1,2,6)\succ (1,2,5)\succ (1,2,4)\succ (1,2,3)\succ (1,2,2) \\
\succ &(1,2,1)\succ (2,2,7)\succ (2,2,6)\succ (2,2,5)\succ (2,2,4) \\
\succ &(2,2,3)\succ (2,2,2)\succ (2,2,1).\end{aligned}$$
- For firm $f_3$ we choose $$Y^f_3:=W\times \{f_3\}\times \{d_1,\ldots,d_5\}$$ with quota $q^f_3=20$ and the following preference relation on $Y^f_3$ (see ): $$\begin{aligned}
&(4,3,1)\succ (4,3,2)\succ (4,3,3)\succ (4,3,4)\succ (4,3,5) \\
\succ &(3,3,1)\succ (3,3,2)\succ (3,3,3)\succ (3,3,4)\succ (3,3,5) \\
\succ &(2,3,1)\succ (2,3,2)\succ (2,3,3)\succ (2,3,4)\succ (2,3,5) \\
\succ &(1,3,1)\succ (1,3,2)\succ (1,3,3)\succ (1,3,4)\succ (1,3,5).\end{aligned}$$
*Step 4.* Applying Theorem \[t313\] we combine the three choice maps of the preceding step into a global revealing choice map $C_F$ on $W\times
F\times D$ by setting $$C_F(A):= \bigcup_{j=1}^3C_{f_j}\left( A\cap (W\times \{f_j\}\times D)\right)
,\quad A\subset W\times F\times D.$$
*Step 5.* The choice maps $C_{W}$ and $C_{F}$ satisfy the hypotheses of Theorem \[t37\]. Applying the algorithm as described in Remark \[r38\] (b) by starting with $X_{0}:=X$, we use a computer program to make the otherwise tedious computation. We obtain the following worker-optimal stable schedule: $$(w_{1},f_{2},1-4),\quad (w_{2},f_{1},1-3),\quad (w_{3},f_{2},2-3),\quad
(w_{4},f_{2},1-7).$$The notations means that
- $f_1$ hires worker $w_2$ for Mondays, Tuesdays and Wednesdays;
- $f_2$ hires worker $w_1$ for Mondays, Tuesdays, Wednesdays and Thursdays, worker $w_3$ for Tuesdays and Wednesdays, and worker $w_4$ for all seven days of the week;
- $f_3$ does not hire anybody.
*Step 6.* Applying the algorithm of Remark \[r38\] (b) by starting with $Y_{0}:=X$ we obtain the same solution. This means that the worker-optimal and firm-optimal solutions coincide, and that there is a unique stable schedule matching in this case.
The remaining of this section investigates the changes in the solutions if we modify our requirements in various ways.
Second problem {#ss52}
--------------
If worker $w_3$ accepts to work up to four days per week (so we change $q^w_3=2$ to $q^w_3=4$), then the worker-optimal and firm-optimal solutions still coincide: the stable schedule is given by the list $$(w_1,f_2,1-4),\quad (w_2,f_1,1-3),\quad (w_3,f_2,2-5),\quad (w_4,f_2,1-7).$$ The only change with respect to the preceding case is that $f_2$ now hires $w_3$ for Thursdays and Fridays, too.
Third problem {#ss53}
-------------
We modify the problem such that $f_2$ hires at most one worker per day, so that for the construction of the choice map $C_{f_2}$ we add the extra conditions $$Y^f_{1,k}:=W\times \{f_2\}\times \{d_k\}\text{ and } q^f_{1,k}=1\text{ for }
k=1,\ldots, 7.$$ The changes are more important. Both the worker-optimal solution and firm-optimal solutions are given by the list $$(w_1,f_3,1-4),\quad (w_2,f_1,1-3),\quad (w_3,f_2,2-3),\quad (w_4,f_2,1,
4-7),\quad (w_4,f_3,2-3).$$
Fourth problem {#ss54}
--------------
Now assume that
- firm $f_1$ does not hire any worker for more than two days;
- firm $f_2$ does not hire Worker $w_1$ for more than three days;
- firm $f_2$ does not hire Worker $w_4$ for more than three days either.
We proceed as in Subsection \[ss51\] but in constructing $C_{1}^{f}$ we add the extra conditions $$\tilde{Y}_{i,1}^{f}:=\{w_{i}\}\times \{f_{1}\}\times D\text{ and }\tilde{q}_{i,1}^{f}:=2\text{ for }i=1,2,3,4,$$and in constructing $C_{2}^{f}$ we add the extra conditions $$\begin{aligned}
& \tilde{Y}_{1,2}^{f}:=\{w_{1}\}\times \{f_{2}\}\times D\text{ and }\tilde{q}_{1,2}^{f}:=3, \\
& \tilde{Y}_{4,2}^{f}:=\{w_{4}\}\times \{f_{2}\}\times D\text{ and }\tilde{q}_{1,2}^{f}:=3.\end{aligned}$$Now the worker-optimal and firm-optimal stable schedules differ: they are given by $$\begin{aligned}
& (w_{1},f_{2},2-4),\quad (w_{1},f_{3},1),\quad (w_{2},f_{1},1-2),\quad
(w_{2},f_{2},3), \\
& (w_{3},f_{2},2-3),\quad (w_{4},f_{2},5-7),\quad (w_{4},f_{3},1-4)\end{aligned}$$and $$\begin{aligned}
& (w_{1},f_{2},5-7),\quad (w_{1},f_{3},1),\quad (w_{2},f_{1},1-2),\quad
(w_{2},f_{2},3), \\
& (w_{3},f_{2},2-3),\quad (w_{4},f_{2},5-7),\quad (w_{4},f_{3},1-4),\end{aligned}$$respectively.
Concluding remarks {#s6}
==================
The *schedule matching* problem extends the standard matching procedure to the allocation of real numbers (days, hours or quantities) between two separate sets of agents. The present paper generalizes the notion of schedule matching to allow for schedule and preference constraints on each side of the market. We demonstrate, by several example, that the revealing property of the choice maps is the most suitable one to ensure the existence of stable matchings. We also revisit the mathematical structure of the matching theory by comparing various definitions of stable sets and various classes of choice maps.
The generality of our analysis is not only theoretically interesting but is potentially useful in application as well.
In certain, highly-competitive, labor markets employers perceive a shortage of top-level candidates that lead to hiring strategies intended to hire those who are believed to be the best. Competition within these paradigms inevitably leads to ever-evolving, if not escalating, dynamic reactions on both sides in an effort to maximize overall gain. As intended, the strategy forces the candidates make ever quicker decisions, before they can know, and weigh, other offers that may be proffered in the near future. As a consequence, candidates end up having less opportunities and employers less potential candidates than were originally available in the market. This results in sub-optimal matches that spawn a myriad of both observable and hidden costs on both sides of the market. The eventual failure of this common strategy ultimately mandates sets of new rules and procedures or market re-design. The algoritm proposed here could be used as a “clearinghouse” in situations where quotas are placed by workers on firms and days worked, allowing him to work part-time for different firms on the same day or on different days, excluding some firms on some given days or excluding some days of work. In the same manner, the algorithm is applicable to situations where firms to need to adjust their labor force on certain days depending on their anticipated activity, or on the requirements associated to different activities on different days or the same day.
Proof of the theorems of Section \[s3\] and supplementary results {#s7}
=================================================================
First we prove Propositions \[p35\] and \[p36\]. Then we apply them to establish Theorem \[t37\]. In the second, independent part of the section we prove Theorems \[t310\] and \[t313\].
\(a) Assume that $C:2^X\to 2^X$ is revealing. Then it is persistent because $C(A)\subset A$ for every choice map.
In order to prove the consistence first we observe that in case $C(A)\subset
B$ we infer from our hypothesis $C(A)\subset B\subset A$ and from the choice map property $C(B)\subset B$ that $C(B)\subset A$. Therefore using we have $$\begin{aligned}
\intertext{and}
&C(B)\subset A \Longrightarrow B\cap C(A)\subset C(B) \Longleftrightarrow
C(A)\subset C(B),\end{aligned}$$ so that $C(A)= C(B)$.
Now assume that $C:2^X\to 2^X$ is consistent and persistent, and consider two sets satisfying $C(A)\subset B$. We have to prove that $A\cap
C(B)\subset C(A)$.
Since $A\subset A\cup B$, applying we obtain that $$\label{61}
A\cap C(A\cup B)\subset C(A).$$ The proof will be completed by showing that $C(A\cup B)=C(B)$.
Using the hypothesis $C(A)\subset B$ we deduce from that $C(A\cup
B)\subset B$. Therefore $C(A\cup B)\subset B\subset A\cup B$, and the equality $C(A\cup B)=C(B)$ follows by applying .
\(b) If the choice map is consistent, then its idempotence follows by applying with $B=C(A)$. If $C$ is persistent, then applying with $A=C(B)$ we get $C(B)\subset C(C(B))$. The converse inclusion also holds because $C$ is a choice map.
\(c) If $A\subset B$, then $$\begin{aligned}
R(A)\subset R(B)& \Longleftrightarrow A\setminus C(A)\subset B\setminus C(B)
\\
& \Longleftrightarrow A\setminus C(A)\subset A\setminus C(B) \\
& \Longleftrightarrow A\cap C(B)\subset C(A).\qedhere\end{aligned}$$
\[e61\]
\(a) Consider a two-point set $X=\{a,b\}$ and the four choice maps defined by the following formulae:
$A$ $\varnothing$ $\{a\}$ $\{b\}$ $\{a,b\}$
---------- --------------- --------------- --------------- ---------------
$C_1(A)$ $\varnothing$ $\{a\}$ $\{b\}$ $\{a\}$
$C_2(A)$ $\varnothing$ $\varnothing$ $\{b\}$ $\varnothing$
$C_3(A)$ $\varnothing$ $\varnothing$ $\varnothing$ $\{a,b\}$
$C_4(A)$ $\varnothing$ $\varnothing$ $\varnothing$ $\{a\}$
One may readily verify that
- $C_1$ is revealing,
- $C_2$ is persistent but not consistent,
- $C_3$ is consistent but not persistent,
- $C_4$ is not idempotent.
One may check that every idempotent choice map on $X$ is either consistent or persistent (or both).
\(b) Consider a three-point set $X=\{a,b,c\}$ and the choice map $C_{5}:2^{X}\rightarrow 2^{X}$ defined by
$\begin{array}{ccc}
C_{5}(\{a\})=\varnothing , & C_{5}(X)=\{b\}, & C_{5}(A)=A\quad \text{otherwise.}\end{array}$
Then $C_{5}$ is idempotent but neither consistent, nor persistent.
\(a) Using the idempotence of $C_{W}$ and $C_{F}$ we deduce from that $$S=C_{F}(S_{F})=C_{F}(C_{F}(S))=C_{F}(S).$$
\(b) Assume that $C_{F}$ is consistent (the other case is similar) set $S_{F}^{\prime }:=S\cup (X\setminus S_{W})$. Then $S_{W}\cup S_{F}^{\prime }=X
$, $S_{W}\cap S_{F}^{\prime }=S$ and we still have $C_{W}(S_{W})=S$. Furthermore, since $$C_{F}(S_{F})=S\subset S_{F}^{\prime }\subset S_{F},$$using the consistence of $C_{F}$ we conclude that $C_{F}(S_{F}^{\prime })=S$.
\(c) As we already observed – imply . Conversely, contains ; furthermore, by the consistency of $C_{W}$ and $C_{F},$ implies –.
\(d) If $S$ is a stable set, then – follow from –. Now assume and . Setting $$S_{W}:=\{x\in X\ :\ C_{W}(S\cup \{x\})=S\}\quad \text{and}\quad
S_{F}:=\{x\in X\ :\ C_{F}(S\cup \{x\})=S\}$$we have $S_{W}\cup S_{F}=X$ by . In view of the consistence it remains to show that $C_{W}(S_{W})=S=C_{F}(S_{F})$.
If $x\in S_{W}\setminus S$, then applying the revealed preference property and using we deduce from the inclusion $C_{W}(S\cup \{x\})\subset
S_{W}$ that $$(S\cup \{x\})\cap C_{W}(S_{W})\subset C_{W}(S\cup \{x\})=S$$and hence $x\notin C_{W}(S_{W})$. We have thus $C_{W}(S_{W})\subset S$. Applying again the revealed preference property we deduce from this last inclusion that $$S_{W}\cap C_{W}(S)\subset C_{W}(S_{W}).$$Since $C_{W}(S)=S$ by , it follows that $S\subset C_{W}(S_{W})$, so that finally $C_{W}(S_{W})=S$. The proof of $C_{F}(S_{F})=S$ is similar.
Let us introduce the map $f:2^{X}\times 2^{X}\rightarrow 2^{X}\times 2^{X}$ by the formula $$f(A,B):=(X\setminus R_{F}(B),X\setminus R_{W}(A))$$where $R_{F}$, $R_{W}$ denote the rejection maps corresponding to $C_{F}$ and $C_{W}$. We observe that $2^{X}\times 2^{X}$ is a non-empty complete lattice with respect to the order relation $$(A,B)\leq (A^{\prime },B^{\prime })\Longleftrightarrow A\subset A^{\prime
}\quad \text{and}\quad B\supset B^{\prime }.$$Furthermore, $f$ is monotone with respect to this order relation. Indeed, using the monotonicity of the rejection maps we have $$\begin{aligned}
(A,B)\leq (A^{\prime },B^{\prime })& \Longleftrightarrow A\subset A^{\prime
}\quad \text{and}\quad B\supset B^{\prime } \\
& \Longrightarrow R_{W}(A)\subset R_{W}(A^{\prime })\quad \text{and}\quad
R_{F}(B)\supset R_{F}(B^{\prime }) \\
& \Longrightarrow X\setminus R_{F}(B)\subset X\setminus R_{F}(B^{\prime })\text{ and }X\setminus R_{W}(A)\supset X\setminus R_{W}(A^{\prime }) \\
& \Longleftrightarrow f(A,B)\leq f(A^{\prime },B^{\prime }).\end{aligned}$$Applying a fixed point theorem of Knaster and Tarski [@Kna1928], [Tar1928]{}, [@Tar1955] we conclude that $f$ has at least one fixed point and that the fixed points of $f$ form a complete lattice. It remains to show that the fixed points of $f$ coincide with the stable sets. More precisely, in view of Proposition \[p36\] it is sufficient to prove that $$f(A,B)=(A,B)\Longleftrightarrow A\cup B=X\quad \text{and}\quad
C_{W}(A)=A\cap B=C_{F}(B).$$
If $f(A,B)=(A,B)$, then $A=X\setminus R_{F}(B)$ and $B=X\setminus R_{W}(A)$. Since $R_{F}(B)\subset B$, it follows from the first relation that $A\cup B=X
$. Furthermore, the first relation also implies that $A$ is the disjoint union of the sets $X\setminus B$ and $C_{F}(B)$ and hence that $A\cap
B\subset C_{F}(B)\subset A$. Since $C_{F}$ is a choice map, we also have $C_{F}(B)\subset B$ and therefore $C_{F}(B)=A\cap B$. The proof of the equality $C_{W}(A)=A\cap B$ is analogous.
Conversely, if $A\cup B=X$ and $C_{W}(A)=A\cap B=C_{F}(B)$, then $$\begin{gathered}
X\setminus R_{F}(B)=(X\setminus B)\cup C_{F}(B) \\
=((A\cup B)\setminus B)\cup (A\cap B)=(A\setminus B)\cup (A\cap B)=A\end{gathered}$$and $$\begin{gathered}
X\setminus R_{W}(A)=(X\setminus A)\cup C_{W}(A) \\
=((A\cup B)\setminus A)\cup (A\cap B)=(B\setminus A)\cup (A\cap B)=B,\end{gathered}$$so that $f(A,B)=(A,B)$.
\[r62\]
\(a) In case $X$ is a finite set, the proof of the theorem provides an efficient algorithm to find a stable set. Starting with $(X_{0},Y_{0}):=X\times \varnothing $ we define a sequence $(X_{1},Y_{1})$, $(X_{2},Y_{2})$,…by the recursive relations $$(X_{n+1},Y_{n+1}):=(X\setminus R_{F}(Y_{n}),X\setminus R_{W}(X_{n})),\quad
n=0,1,\ldots ,$$i.e., $$X_{n+1}:=(X\setminus Y_{n})\cup C_{F}(Y_{n})\quad \text{and}\quad
Y_{n+1}:=(X\setminus X_{n})\cup C_{W}(X_{n}),\quad n=0,1,\ldots .$$Since we have obviously $X_{1}\subset X=X_{0}$ and $Y_{1}\supset \varnothing
=Y_{0}$, by the monotonicity of $f$ we conclude that $$X_{0}\supset X_{1}\supset \cdots \quad \text{and}\quad Y_{0}\subset
Y_{1}\subset \cdots .$$Since $X$ has only finitely many subsets, there exists an index $n$ such that $$(X_{n+1},Y_{n+1})=(X_{n},Y_{n}),$$and then $S:=X_{n}\cap Y_{n}$ is a stable set. As a matter of fact, we obtain in this way the worker-optimal stable set. Similarly, we may construct the firm-optimal stable set by the same recurrence relations if we start from $(X_{0},Y_{0}):=\varnothing \times X$.
\(b) If we start with $(X_{0},Y_{0}):=X\times \varnothing $, then we obtain $X_{1}=X\setminus R_{F}(\varnothing )=X$ and therefore $$X_{0}=X_{1},\quad Y_{1}=Y_{2},\quad X_{2}=X_{3},\quad Y_{3}=Y_{4},\ldots .
\label{62}$$This implies that the above algorithm is equivalent to the more economical Gale–Shapley algorithm. There we start with $X_{0}:=X$ and we compute successively $$Y_{1},X_{2},Y_{3},X_{4},\ldots$$by using the recursive formulae $$Y_{n+1}:=(X\setminus X_{n})\cup C_{W}(X_{n})\quad \text{and}\quad
X_{n+1}:=(X\setminus Y_{n})\cup C_{F}(Y_{n}).$$We stop when we obtain $X_{n-1}=X_{n+1}$ for the first time, and we set $S=C_{W}(X_{n-1})$. Indeed, the equalities and $X_{n-1}=X_{n+1}$ imply that $$X_{n-1}=X_{n}=X_{n+1}=X_{n+2}=\cdots \quad \text{and}\quad
Y_{n}=Y_{n+1}=Y_{n+2}=Y_{n+3}\cdots .$$Therefore $(X_{n+1},Y_{n+1})=(X_{n},Y_{n})$, and $$S=X_{n}\cap Y_{n}=X_{n-1}\cap \left( (X\setminus X_{n-1})\cup
C_{W}(X_{n-1})\right) =C_{W}(X_{n-1}).$$In the last step we used that $C_{W}(X_{n-1})\subset X_{n-1}$ because $C_{W}$ is a choice map.
Analogously, we may construct the firm-optimal stable set by starting with $Y_{0}:=X$, computing successively $X_{1},Y_{2},X_{3},Y_{4},\ldots $ by the same formulae as above, and setting $S=C_{F}(Y_{n-1})$ for the first $n$ such that $Y_{n-1}=Y_{n+1}$.
\[e63\]
\(a) We cannot replace the revealed preference condition with the substitutes condition in Theorem \[t37\]. To show this consider the choice maps $C_{W}:=C_{1}$ and $C_{F}:=C_{2}$ of Example \[e61\] (a) on the set $X=\{a,b\}$. Then $C_{W}$ is revealing and $C_{F}$ is persistent. However, there is no stable set. Indeed, we have $C_{W}(S)=S=C_{F}(S)$ only if $S=\varnothing $ or $S=\{b\}$, so that only these two sets are individually rational (see Remark \[r32\]). However, $S=\varnothing $ is blocked by $\{b\}$ because $$C_{W}(S\cup \{b\})=C_{F}(S\cup \{b\})=\{b\}\neq S,$$and $S=\{b\}$ is blocked by $\{a\}$ because $$C_{W}(S\cup \{a\})=\{a\}\neq S\quad \text{and}\quad C_{F}(S\cup
\{a\})=\varnothing \neq S.$$Hence none of these sets is stable.
\(b) We cannot replace the revealed preference condition with the consistence in Theorem \[t37\] either. To show this consider the choice maps $C_{W}:=C_{1}$ and $C_{F}:=C_{3}$ of Example \[e61\] (a) on the set $X=\{a,b\}$. Then $C_{W}$ is revealing and $C_{F}$ is consistent. However, there is no stable matching. Indeed, we have $C_{W}(S)=S=C_{F}(S)$ only if $S=\varnothing $, so this is the only individually rational set. For $S=\varnothing $ the condition is satisfied only if $S_{W}=\varnothing $, and then $S_{F}=X$ by . However, then $C_{F}(S_{F})=X\neq S$, so that fails.
Now we turn to the proofs of Theorems \[t310\] and \[t313\]. They are independent of the preceding part of the present section.
The choice map $C$ remains the same if we change each $Y_n$ to $Y_n\cap Y$ in the construction. The choice map does not change either if we complete the family $\{Y_n\}$ with $Y^{\prime }:=Y\setminus\cup Y_n$ corresponding to the quota $q^{\prime }:=\operatorname{Card}Y^{\prime }$. Without loss of generality we assume henceforth that $\{Y_n\}$ is a *partition* of $Y$, i.e., $Y$ is the *disjoint union* of the sets $Y_n$.
Let $A,B\subset X$ be two sets satisfying $C(A)\subset B$; we have to show that if $y_k\in A\cap C(B)$ for some $k$, then $y_k\in C(A)$.
First we establish by induction on $j$ the following inequalities: $$\label{63}
\operatorname{Card}(C_j(A)\cap Y_n)\le \operatorname{Card}(C_j(B)\cap Y_n)\text{ for all }n,\quad
j=0,\ldots, k-1.$$
For $j=0$ our claim reduces to the trivial equality $0=0$. Assuming that the inequalities hold until some $j<k-1$, consider the (unique) index $m$ for which $y_{j+1}\in Y_m$. For each $n\ne m$ we have $$C_j(A)\cap Y_n=C_{j+1}(A)\cap Y_n\text{ and }C_j(B)\cap Y_n=C_{j+1}(B)\cap
Y_n$$ and therefore $$\operatorname{Card}(C_{j+1}(A)\cap Y_n)\le \operatorname{Card}(C_{j+1}(B)\cap Y_n)$$ by our induction hypothesis. For $n=m$ the only critical case is when $$y_{j+1}\in C_{j+1}(A)\setminus C_{j+1}(B).$$ Since $y_{j+1}\in C(A)$ implies $y_{j+1}\in B$ and since $$\operatorname{Card}C_j(B)\le \operatorname{Card}C_{k-1}(B)\le q-1$$ because $y_k\in C(B)$ and therefore $$\operatorname{Card}C_{k-1}(B)=\operatorname{Card}C_k(B)-1\le q-1,$$ by the construction this can only happen if $$\operatorname{Card}(C_j(A)\cap Y_m)\le q_m-1\text{ and }\operatorname{Card}(C_j(B)\cap Y_m)=q_m.$$ But then we have $$\begin{aligned}
\operatorname{Card}(C_{j+1}(A)\cap Y_m) &=\operatorname{Card}(C_j(A)\cap Y_m)+1 \\
&\le q_m \\
&=\operatorname{Card}(C_j(B)\cap Y_m) \\
&=\operatorname{Card}(C_{j+1}(B)\cap Y_m)\end{aligned}$$ as required.
Since $y_k\in A\cap C(B)$, we have $y_k\in A$. Furthermore, since $C(A)\subset Y$ and the sets $Y_n$ form a partition of $Y$, it follows from that $$\begin{aligned}
\operatorname{Card}C_{k-1}(A) &=\cup_{n}\operatorname{Card}(C_{k-1}(A)\cap Y_n) \\
&\le \cup_{n}\operatorname{Card}(C_{k-1}(B)\cap Y_n) \\
&= \operatorname{Card}C_{k-1}(B) \\
&=\operatorname{Card}C_k(B)-1 \\
&\le q-1\end{aligned}$$ because $C_k(B)\setminus C_{k-1}(B)=\{y_k\}$.
Furthermore, in case $y_k\in Y_n$ we have $$(C_k(B)\cap Y_n)\setminus (C_{k-1}(B)\cap Y_n)=\{y_k\}$$ and therefore $$\begin{aligned}
\operatorname{Card}\left( C_{k-1}(A)\cap Y_n\right) &\le\operatorname{Card}\left(C_{k-1}(B)\cap
Y_n\right) \\
&=\operatorname{Card}\left(C_k(B)\cap Y_n\right)-1 \\
&\le q_n-1.\end{aligned}$$
Summarizing, the conditions – are satisfied and we conclude that $y_{k}\in C(A)$ by construction. This completes the proof.
\[r64\]
\(a) The choice map constructed in Theorem \[t310\] is consistent even if the sets $Y_n\cap Y$ are not disjoint. Indeed, if $C(A)\subset B\subset A$, then comparing the construction of $$C_0(A)\subset C_1(A)\subset\cdots\text{ and }C_0(B)\subset
C_1(B)\subset\cdots,$$ we see that $C_k(A)=C_k(B)$ for every $k$ and therefore $C(A)=C(B)$. The equality $C_k(A)=C_k(B)$ is obvious for $k=0$ because both sides are equal to zero. If it is true for some $k-1\ge 0$, then we have $y_k\in C_k(A)$ if and only if $$\begin{aligned}
&y_k\in A, \\
&\operatorname{Card}\left( C_{k-1}(A)\cup\{y_k\}\right) \le q \\
&\operatorname{Card}\left( (C_{k-1}(A)\cup\{y_k\})\cap Y_n\right) \le q_n\text{ for all }n,\end{aligned}$$ and $y_k\in C_k(B)$ if and only if $$\begin{aligned}
&y_k\in B, \\
&\operatorname{Card}\left( C_{k-1}(B)\cup\{y_k\}\right) \le q \\
&\operatorname{Card}\left( (C_{k-1}(B)\cup\{y_k\})\cap Y_n\right) \le q_n\text{ for all }n.\end{aligned}$$ Since $C_{k-1}(A)=C_{k-1}(B)$ by the induction hypothesis, the equality $C_k(A)=C_k(B)$ will follow if we show that $y_k\in A\Longleftrightarrow
y_k\in B$ if the last two conditions are satisfied. Since $C(A)\subset
B\subset A$ and since $C(B)\subset B$ ($C$ is a choice map), we have $$\begin{aligned}
\intertext{and}
&y_k\in B\Longrightarrow y_k\in C(B)\Longrightarrow y_k\in B\Longrightarrow
y_k\in A.\end{aligned}$$
\(b) The range of an idempotent choice map coincides with the set of its fixed points: $$\{C(A)\ :\ A\subset X\}=\{A\subset X\ :\ C(A)=A\}.$$
\[e65\] In Example \[e63\] (b) the consistent choice map $C_F$ cannot be obtained by the construction of Theorem \[t310\] without the disjointness condition (see Remark \[r64\] (a)). A stronger counterexample is the following. We consider a three-point set $X=\{a,b,c\}$ and the following two choice maps:
$A$ $\varnothing$ $\{a\}$ $\{b\}$ $\{c\}$ $\{a,b\}$ $\{a,c\}$ $\{b,c\}$ $\{a,b,c\}$
---------- --------------- --------- --------- --------- ----------- ----------- ----------- -------------
$C_W(A)$ $\varnothing$ $\{a\}$ $\{b\}$ $\{c\}$ $\{a,b\}$ $\{c\}$ $\{b,c\}$ $\{b,c\}$
$C_F(A)$ $\varnothing$ $\{a\}$ $\{b\}$ $\{c\}$ $\{a\}$ $\{a,c\}$ $\{b\}$ $\{a,c\}$
Both choice maps are defined by the construction of Theorem \[t310\]. For $C_W$ we take $Y=X$ with the preference order $c\succ b\succ a$ and quota $q=2
$, and we set $Y_1=\{a,c\}$ with the quota $q_1=1$. This is a revealing choice map. The choice map $C_F$ is the one given in Example \[e312\] above: a consistent but not revealing choice map because the disjointness condition is not satisfied.
In order to find a stable set $S$ we have to cover $X=\{a,b,c\}$ by two sets $S_W$ and $S_F$ satisfying $C_W(S_W)=S=C_W(S)$ and $C_F(S_F)=S=C_F(S)$. The equalities $C_W(S)=S=C_F(S)$ are satisfied if and only if $S$ has at most one element, so that there are four candidates for the stable set $S$. We can see easily from the table that in order to have $C_W(S_W)=S=C_W(S)$,
- in case $S=\varnothing$ we must have $S_W=S_F=\varnothing$;
- in case $S=\{a\}$ we must have $S_W=\{a\}$ and $S_F\subset\{a,b\}$;
- in case $S=\{b\}$ we must have $S_W=\{b\}$ and $S_F\subset\{b,c\}$;
- in case $S=\{c\}$ we must have $S_W\subset\{a,c\}$ and $S_F=\{c\}$.
Since $S_W\cup S_F\ne X$ in all these cases, we conclude that there is no stable set.
If $C(A)\subset B$, then setting $A_i:=A\cap X_i$ and $B_i:=B\cap X_i$ we have $$\begin{aligned}
A\cap C(B)\subset C(A) &\Longleftrightarrow \left( A\cap C(B)\right) \cap
X_i\subset C(A)\cap X_i\text{ for all }i \\
&\Longleftrightarrow A_i\cap C_i(B_i)\subset C_i(A_i)\text{ for all }i. \qedhere\end{aligned}$$
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[^1]: Part of this research was realized during the stay of the first author at the Department of Mathematics of the University of Cincinnati as a Taft research fellow in March–June, 2007. He is grateful to the Charles Phelps Taft Research Center for their kind invitation and for the excellent working conditions.
[^2]: A more thorough investigation of stable sets is carried out in Proposition \[p36\] below.
|
harvmac
David S. Berman[^1][D.S.Berman@qmul.ac.uk]{}
*Department of Physics*
*Queen Mary College, University of London*
*Mile End Road, London E1 4NS, England*
[*and*]{}
Jeffrey A. Harvey[^2][harvey@theory.uchicago.edu]{}
*Enrico Fermi Institute and Department of Physics*
*University of Chicago*
*5640 S. Ellis Ave., Chicago, IL 60637, USA*
*Abstract*
We study the anomalies of a charge $Q_2$ self-dual string solution in the Coulomb branch of $Q_5$ M5-branes. Cancellation of these anomalies allows us to determine the anomaly of the zero-modes on the self-dual string and their scaling with $Q_2$ and $Q_5$. The dimensional reduction of the five-brane anomalous couplings then lead to certain anomalous couplings for D-branes.
There remain many puzzling aspects concerning coincident branes in M-theory. One of the central puzzles involves the lack of a microscopic derivation of the number of degrees of freedom on $Q_5$ coincident five-branes. Some information about the theory has been obtained through circuitous methods such as anomalies , low energy scattering and the AdS/CFT correspondence . All these methods show that in the large $Q_5$ limit the number of degrees of freedom scale as $Q_5^3$ for the five-brane. For the two-brane the number of degrees of freedom scales like $Q_2^{3/2}$.
In the five-brane itself there are self-dual strings . These appear as solitonic solutions to the nonlinear five-brane theory and are associated with the five-brane worldvolume description of a membrane ending on a five-brane. From the point of view of a spontaneously broken five-brane theory, these strings are similar to W-bosons, becoming tensionless when the five-brane separation of the branes vanishes. There is no adequate description of these self-dual strings. Apart from the problem of describing tensionless string dynamics, self-dual strings are never weakly coupled and so one cannot use standard perturbative techniques. Furthermore, the five-brane theory in which these strings live has no known description when there is more than one five-brane.
This paper will be concerned with the study of these strings, in particular, with $Q_2$ coincident self-dual strings. In it was found that the absorption cross section of $Q_2$ coincident self-dual strings (when the number of five-branes $Q_5=1$) is proportional to $Q_2$ indicating that number of degrees of freedom scales linearly with $Q_2$. A similar scattering calculation was done for a membrane probing a supergravity solution describing $Q_5$ coincident five-branes, indicating that the number of degrees of freedom scale as $Q_5$ (this is with $Q_2=1$). (Note, the same reasoning applied to D3 branes where the absorption cross section for $Q$ D3 branes was shown to scale as $Q^2$ indicates a $Q^2$ scaling in the number of light degrees of freedom as is expected for $Q$ coincident three-branes .)
Here we wish to consider a more general situation and use anomalies to determine the scaling of the number of degrees of freedom for $Q_2$ self-dual strings in a generic spontaneously broken five-brane theory associated with $Q_5$ five-branes in the Coulomb branch. We will be interested in the dependence of the anomaly on the charges $Q_2$ and $Q_5$ as well as the role of the unbroken gauge group. The power of anomalies is that they are topological in nature and can be studied in the low-energy theory, yet provide a probe into high energy physics. This will prove very powerful in this situation where a description of the fundamental degrees of freedom is lacking and we have only an effective low energy description.
We will consider a five-brane theory labelled by an ADE Lie algebra (how this symmetry is realised in terms of local fields is not known (see for a discussion of this problem). The simplest case is the U(N) theory, which is actually a $U(1) \times
SU(N)$ theory with the decoupled U(1) corresponding to the Nambu-Goldstone mode of translating the whole stack of branes. Spontaneous symmetry breaking of the theory occurs when one (or more) of the branes are separated off from the stack. There will then be a U(1) mode corresponding to fluctuations in the separation of the brane stacks. At low energies, i.e. at scales less than the inverse brane separation, that U(1) mode will be described by a U(1) (0,2) tensor multiplet.
Self-dual strings are solutions of the (0,2) abelian tensor multiplet. As such we can embed the known self-dual string solution into the U(1) tensor multiplet corresponding to this separation mode as opposed to the usual overall U(1) translation mode. This will allow us to investigate properties of the spontaneously broken five-brane theory. In particular, we will be able to see how anomalies in the normal bundle of the self-dual string may be cancelled by inflow from Wess-Zumino type terms in the five-brane world volume theory. Imposing this cancellation will then allow us to determine the scaling of the coefficient in terms of $Q_2$ and $Q_5$ and the dependence on the unbroken gauge group. The inflow mechanism is analogous to the sort used to cancel anomalies of intersecting D-branes or the M-theory five-brane itself, with an inflow from the supergravity bulk.
Now the [*[bulk]{}*]{} is the five-brane world volume and the defect is the self-dual string. Similar issues concerning the anomalies of self-dual strings have been discussed in ; here we expand their discussion to multiple branes and relate the couplings involved in anomaly cancellation to interactions discussed in the recent literature . While this paper was in preparation we became aware of which also considers anomaly cancellation for strings in a six dimensional theory but does not consider the anomalies normal to the five-brane nor allow for a scaling of the zero modes with the charge.
Apart from providing an insight into the self-dual string, the terms that are required by anomaly inflow on the five-brane imply certain anomalous couplings for D-branes via dimensional reduction. These terms are closely related to terms discussed elsewhere in the literature and will be discussed in the penultimate section of the paper.
The Bosonic field content of the 5+1 dimensional (0,2) tensor multiplet theory consists of a two form field $b$ whose three form field strength $h$ obeys a self-duality constraint and five scalar fields, $\phi^i, \,
i=1..5$. The self-dual string is a half BPS solution with the two form $b$ field excited along with a single scalar field, denoted here as $\phi$. The solution is given explicitly by: r is the radial coordinate of the space transverse to the string ie. for a string lying along $x^1$, $r^2=x_2^2 + x_3^2 +x_4^2 +x_5^2$ and $\epsilon_{mnpq}$ is the associated epsilon tensor of this transverse space. The charge of the string is, $Q_2=\mp Q$.
This solution can be viewed as the worldvolume description of an M2 or anti-M2 brane ending on the M5-brane. The field $\phi$ represents the value of one of the coordinates transverse to the M5. With the convention that $\phi$ increases from left to right, the solution with the upper choice of $\pm$ sign and $Q>0$ corresponds to an M2 coming in from the right and terminating on the $M5$ located at $\phi_0$. The lower choice of sign with $Q>0$ is then an anti-M2 coming in from the right while the upper choice of sign with $Q<0$ is an M2 coming in from the left and the lower choice of sign with $Q<0$ is an anti-M2 coming in from the left.
This self-dual string solution preserves eight supercharges (one half BPS with respect to the five-brane and one quarter BPS with respect to M-theory). As usual for such BPS objects, the fermion zero modes of the lowest charge solution are generated by the broken supersymmetries, that is by $\epsilon^{\alpha l}$ which satisfy: where $\alpha, \beta=1..4$ are (Weyl) spinor indices of Spin(1,5) and $i,j=1..4$ are spinor indices of USp(4) the Spin cover of the SO(5) R-symmetry group. The choice of plus or minus is associated with the choice of sign in the solution in . We now wish to decompose these Fermionic zero modes into representations of:
This decomposition is the Spin cover of the Lorentz group that is preserved by the self-dual string solution. (The original $SO(1,5)
\times SO(5)$ group which is preserved by the five-brane becomes broken by the self-dual string solution to $SO(1,1) \times SO(4)_T
\times SO(4)_N$). The subscripts T, N denote tangent and normal to the five-brane world volume respectively. An eigenspinor of $\gamma^{01}$ will be a 1+1 chiral spinor and an eigenspinor of $\gamma_{5'}$ will be a Weyl spinor of the $Spin(4)_N$. Importantly, the 6d Weyl spinors that are also eigenspinors of $\gamma^{01}$ are Weyl spinors of $Spin(4)_T$. Putting these facts together implies the BPS self-dual string has (4,4) supersymmetry in 1+1 dimensions with the Fermions lying in the following representations of : and the anti-BPS (negative sign in ) string has the Fermions lying in:
The superscript labels the $SO(1,1)$ helicity with the numbers in brackets labelling the $Spin(4)=SU(2) \times SU(2)$ representations.
For solutions with charge $|Q| \ge 1$ there will be $|Q|$ such multiplets. One can argue for this result in several ways. Since the self-dual string solution is BPS, solutions with $|Q|>1$ can be deformed into $|Q|$ separated string solutions, each with its own centre of mass Bosonic zero modes. Supersymmetry then requires that each of these Bosonic multiplets be accompanied by fermion zero modes. Alternatively, we can reduce this system to a D-brane configuration (as will be discussed in detail later) and then the $|Q|$ fermion zero modes arise from the usual Chan-Paton factors. In principle it should also be possible to show this by analysing an index theorem for the corresponding Dirac operator on the brane, but to our knowledge this has not been done in the literature. (It would be interesting to see this explicitly since the Dirac operator on a brane in the presence of a background field is somewhat different from the usual Dirac operator).
The above analysis has all been concerned with $Q_5=1$, that is membranes ending on a single M5-brane. For $Q_5>1$ the zero mode structure on strings corresponding to membranes ending on the stack of fivebranes is unknown. The main result of this paper will be to put constraints on the zero mode structure for this case.
We now turn to a computation of the anomaly of the fermion zero modes for $Q_5=1$. We are interested in anomalies in diffeomorphisms of the eleven-dimensional spacetime which preserved the self-dual string solution of the M5-brane, or equivalently which preserve the configuration of M2-branes ending on the M5-brane. These diffeomorphisms act as diffeomorphisms of the string world-sheet, or as gauge transformations of the $SO(4)_T \times SO(4)_N$ normal bundle to the string. Since there are equal numbers of left and right-movers, there is no anomaly in world-sheet diffeomorphisms. However, the left and right moving fermions are in different representations of the normal bundle (the R-symmetry group) which will give rise to a normal bundle anomaly. (In a field theory context these would be the ’t Hooft anomalies).
The anomaly can be computed by treating the each $SO(4)$ symmetry as a gauge symmetry (see e.g.the discussion in ). In two dimensions the anomaly is derived by descent from a four-form characteristic class. $SO(4)$ has two such classes, the first Pontryagin class and the Euler class and the anomaly in this case is proportional to the Euler class. For an $SO(4)$ field strength two-form $F^{ab}$, a,b,=1,2,3,4 the Euler class is which in terms of $SU(2)$ field strengths is, The descent procedure involves writing and the gauge variation as The normal bundle anomaly for each SO(4) is then proportional to with $\Sigma_2$ the self-dual string world-volume. (The factor of $\pi$ appears instead of the usual $2 \pi$ since the Fermions are Majorana and Weyl.) The total anomaly is with the sign correlated with the sign in . From now on we work with the upper sign in and .
In the next section we will consider the self-dual strings in a (0,2) multiplet arising from the low energy description of fivebranes in the Coulomb branch. We will show how the anomaly may cancelled by an inflow mechanism and in doing so we will also see how the anomaly must scale with the number of fivebranes.
There are various approaches and levels of analysis one can take in dealing with anomaly cancellation in string theory and M-theory, particularly in analysing anomalies for extended objects.
As usual in trying to cancel anomalies, one is free to add local counterterms to the Lagrangian. In theories with UV divergences these can be viewed as part of the definition of the theory. In theories such as string theory, and presumably M-theory, such counterterms should in principle be computable in the underlying microscopic theory. Thus, in the original analysis of Green and Schwarz , anomaly cancellation was understood both by a direct string calculation, and in the low-energy effective Lagrangian, and the local counterterms needed to cancel the anomaly in the low-energy effective theory could be verified directly. Similarly, the analysis of anomaly cancellation on D-branes in string theory requires certain anomalous, or Wess-Zumino, terms in the low-energy effective action on the D-brane, and it is possible to check that the required terms are present by a direct calculation .
The situation for NS 5-branes and M5-branes in IIA string theory and M-theory is less satisfactory. It was shown in that the anomalies cancel for NS 5-branes after addition of a local counterterm to the 5-brane low-energy effective action, but to our knowledge this counterterm has not been verified by a direct calculation in string theory. Similarly, anomalies cancel for the M5-brane after one gives a careful definition of the $C$ field and its action in the presence of an M5-brane . This definition involves adding local counterterms to the M-theory low-energy effective action. There is however no microscopic derivation of these terms. However,the counterterm required in does follow from the analysis of , so at least they are not independent problems .
In dealing with anomalies on extended objects there are two different points of view one can take. The NS 5-brane and M5-brane can be viewed as smooth soliton solutions of string theory or M-theory. Viewed this way, the zero modes localised on the 5-brane arise from a collective coordinate expansion of the bulk fields . However, it has proved difficult to analyse anomalies directly in this framework because this involves dealing with the bulk Rarita-Schwinger equation in the non-trivial fivebrane geometry. The point of view which is most commonly taken is to split the degrees of freedom into bulk degrees of freedom and degrees of freedom localised on the brane without taking into account the detailed form of the brane soliton solution or the relation between localised zero modes and bulk fields. This is the point of view adopted in .
In this paper we will take this last point of view. As we discussed earlier, the M2-brane ending on an M5-brane can be viewed as a soliton solution to the M5-brane equations of motion, but for analysing anomalies we will treat the M2-brane and M5-brane zero modes as independent of the bulk fields. We also allow ourselves the freedom to add local counterterms to the Lagrangian as long as they respect the symmetries of the system. As we discussed earlier, this means they should respect the $Spin(1,1) \times SO(4)_T \times SO(4)_N$ symmetry of the M2-M5 configuration.
The cancellation of the $SO(4)_T$ anomaly in is the most straightforward to understand and was already discussed in . We redo this analysis here using the formalism developed in .
We denote the M5-brane and M2-brane world-volumes by $\Sigma_6$ and $\Sigma_3$ respectively. The M2-brane boundary on the M5-brane is the worldvolume of the self-dual string with worldvolume $\Sigma_2 \equiv \partial \Sigma_3$. We introduce two bump forms $d \rho (r)$ and $d \rho'(r')$ where $r$ is the radial direction transverse to $\Sigma_2$ in $\Sigma_6$ and $r'$ is the radial direction transverse to $\Sigma_6$ in the 11-dimensional spacetime manifold $M_{11}$. M-theory in $M_{11}$ has, in the absence of M5-branes, a four-form field strength $G_4$ with $G_4= dC_3$ and a Bianchi identity $dG_4=0$. As we mentioned earlier, the M5-brane worldvolume theory has, in the absence of self-dual strings, a three-form field strength $h_3$ with a Bianchi identity $d h_3 = - G_4|_{\Sigma_6}$. In the presence of M5-branes and self-dual strings the resulting Bianchi identities become Note the factor of $\pi$ as opposed to $2 \pi$ on the right hand side for the Bianchi identity of $h_3$. This arises because the flux quantisation for Dyonic strings in six dimensions is given by $eg=\pi n$ . Physically, the quantities $\delta_5$ and $\delta_4$ in are often thought of as delta functions with integral one in the spaces transverse to $\Sigma_6$ and $\Sigma_2$ respectively. However, a more careful mathematical treatment has turned out to be necessary in dealing with anomalies in the M5 system in which we think if these as the Poincare duals to $\Sigma_6$ in $M_{11}$ and $\Sigma_2$ in $\Sigma_6$ respectively. Using the isomorphism between the Poincare dual and the Thom class of the normal bundle we can choose explicit representatives for $\delta_5$ and $\delta_4$ given by where $e_4$ is the global angular form with integral two over the $S^4$ fibres transverse to $\Sigma_6$ in $M_{11}$, and $e_3$ is the global angular form with integral two over the $S^3$ fibres transverse to $\Sigma_2$ in $\Sigma_6$.
Following we write with $d \Omega_3 = - \chi(F_T)$ and solve these Bianchi identities so that $G_4$ and $h_3$ are non-singular on $\Sigma_6$ and $\Sigma_2$ respectively: Since $h_3$ must be gauge invariant, and the variation of $\Omega_3$ under $SO(4)_T$ gauge variations is $\delta \Omega_3 = d \chi_2^{(1)}(A_T)$, we learn using $\rho(0)=-1$ that $b_2$ has a $SO(4)_T$ gauge variation so that the minimal coupling of the string to the two-form field on the M5-brane has a variation which cancels the $SO(4)_T$ anomaly in .
Where we differ from is in the treatment of the $SO(4)_N$ anomaly. In the above formalism $G_4|_{\Sigma_6} = d C_3|_{\Sigma_6}$ and there does not seem to be room for the additional $SO(4)_N$ variation of $b_2$ found in .
Fortunately, the $SO(4)_N$ anomaly can be cancelled by adding a local counterterm to the M5-brane action: It is easy to see using the second equation in that this term has an anomalous variation localised on $\Sigma_2$ that cancels the anomaly in . In the following section we will see that a coupling of the form is already know to exist in the theory of $Q_5>1$ M5-branes.
We have thus demonstrated anomaly cancellation for $Q_2$ M2-branes ending on a single M5-brane. For $Q_5>1$ M5-branes we do not know how to carry out such a general analysis, because neither the fermion zero mode structure nor the $Q_5$ dependence of the coupling is known. However, we can obtain some partial results by utilising a known generalisation of the coupling which appears on the Coulomb branch of the theory with $Q_5>1$.
The analysis in the previous section involved coupling to the $U(1)$ centre of mass multiplet of the $(2,0)$ theory. We now consider a general M5-brane theory with an ADE symmetry $G \times U(1)$ broken down to $(H \times U(1)) \times U(1)$.
As in the analogous D1-D3 system, it is expected that this theory will have self-dual strings which couple to the $U(1) \subset G$ factor. These are the analogs of the non-Abelian ’t Hooft-Polyakov monopoles in the D1-D3 system. Assuming that such solutions exist and act as sources for the $h$ field in the relative $U(1)$ multiplet, we can deduce some facts about the anomalies of the fermion zero modes on such strings.
The coupling in the five-brane theory which is anomalous in such a string background originates from a coupling derived in . Following the conventions in the coupling is given by where with $(D_i \phi)^a = \partial_i \phi^a -
A_i^{ab} \phi^b$ the covariant derivative of $\phi^a$ with $A_i^{ab}$ the SO(5) gauge field of the five-brane normal bundle and $\hat\phi^a=
{{\phi^a} \over{\| \phi \|}} $. The coupling constant $\alpha_e$ depends on the breaking of the five-brane theory gauge group. If $G$ is the ADE Lie algebra labelling the $(0,2)$ theory, then for a breaking given by $G \rightarrow H \times U(1)$, it was argued that For example if $G= SU(Q_5+1)$ and $H= SU(Q_5)$ then, $\alpha_e ={1
\over 2} Q_5$.
This coupling was derived by considering anomalies in the Coulomb branch of the five-brane theory, ie. when the scalars have non-zero vacuum expectation values. The point is that the U(1) multiplet at low energies must include an interaction term to compensate for what would otherwise be a difference in the anomaly at the origin of moduli space and the anomaly at a generic point in the Coulomb branch. Hence the decoupling of the U(1) multiplet never really happens; this term is always sensitive to the full theory and so even in the infrared there remains some information of the integrated out ultra-massive modes.
Here we will evaluate this interaction term in the presence of a self-dual string embedded in the naively decoupled U(1) tensor multiplet.
For the self-dual string solution we can take just one scalar, say $\phi^5$, to be non-zero and obeying , while also taking $A^{5a}=0$ to reduce the $SO(5)$ to the $SO(4)$ preserved by the string solution. Then it is easy to see that in the presence of the self-dual string $\Omega_3 $ reduces to $\chi^{(0)}(A)/2$. The effective coupling on the fivebrane in the presence of the self-dual string is thus In the presence of the self-dual string the coupling is not gauge invariant. Rather, its gauge variation is Now the self-dual string acts as a source of $h_3$ via the equation Using this, the variation becomes
Now, assuming the anomalous variation is, as before, cancelled by the fermion zero modes we deduce that the $SO(4)_N$ zero mode anomaly of the self-dual string in the Coulomb branch scales as: Using the above and the equation for $\alpha_e$, we may compute the anomaly for the string, that is $c$ in terms of the charges $Q_2,Q_5$.
Consider the case of pulling off a single five brane from a stack of $Q_5+1$ fivebranes and embedding the string in the relative U(1). The five-brane theory would have $G=SU(Q_5+1)$ and $H=SU(Q_5)$ giving, $\alpha_e ={1 \over 2} Q_5$. This would then give c =[1 4]{} Q\_2 Q\_5 .
We may then interpret $c$ as being related to the number of degrees of freedom of the string. The $Q_2 Q_5$ dependence of the anomaly is then consistent with the cross section scattering calculation of the self-dual string described in .
A more interesting situation arises if one considers a different breaking pattern for the five-branes. Take a stack and separate all the branes thus giving a maximal breaking with $G=SU(Q_5+1)$ and $H=U(1)^{Q_5}$. In this case simply applying the above formula yields, $ c= {1 \over 8} Q_2 (Q_5^2 +Q_5-1)$. In this case no cross section scattering calculation has been done to confirm the charge dependence. It would be interesting to find other ways to study this system which would confirm this behaviour.
Note that for this calculation we are really only using the supersymmetry/R-symmetry preserved by the string solution to determine the anomaly in conjunction with the Ganor, Motl, Intrilligator term for the cancellation and so the precise form of the solution should not matter. Thus even if one might be concerned about applying the solution of in the more exotic circumstances described above, provided the symmetries of the string are the same our results should remain valid.
We now consider the implications of these terms for IIA string theory by reducing M theory on an $S^1$. To be explicit we take the M5-worldvolume to lie in the $(0,1, \cdots 5)$ plane and the M2-worldvolume to lie in the $(016)$ plane. The self-dual string worldvolume then lies in the $(01)$ plane. Normal bundle gauge transformations that preserve this configuration act act on the $(2,3,4,5)$ coordinates ($SO(4)_T$) or the $(7,8,9,10)$ coordinates ($SO(4)_N$).
There are then two interesting reductions to IIA string theory. We can take one of the $(2,3,4,5)$ coordinates to be periodic. This turns the M2 into a D2-brane in IIA theory and the M5 which wraps this periodic coordinate into a D4-brane. The $SO(4)_T$ symmetry is broken to $SO(3)$ which has no Euler class, so there is no $SO(3)$ normal bundle anomaly. However, the $SO(4)_N$ symmetry is preserved and has an anomaly derived by descent from the Euler class.
Cancellation of this normal bundle anomaly for a D2 ending on a D4 requires a coupling on the D4-worldvolume of the form where $F_2$ is the $U(1)$ gauge field strength on the D4-brane. Note that this term is distinct from the usual anomalous couplings on D-branes . Because it is independent of the bulk Ramond-Ramond fields, it would arise at one-loop level rather than as a tree-level coupling, presumably explaining why it has not been seen in previous explicit calculations of anomalous couplings . In fact, closely related couplings (but for relative $U(1)$ factors) have been discussed in and .
The other inequivalent reduction takes one of the $(7,8,9,10)$ coordinates to be periodic. This turns the M2 into a D2 while converting the M5 into a NS5-brane. Now the $SO(4)_N$ symmetry is broken to $SO(3)$ with vanishing anomaly while the $SO(4)_T$ symmetry is preserved. Cancellation of this normal bundle anomaly then requires a coupling on the NS5-brane worldvolume analogous to .
We would like to thank C. Bachas, N. Copland, D. Kutasov, M. Green, P. Howe, G. Moore, G. Papadopolous, M. Perry and A. Ritz for relevant discussions. JH thanks D. Freed for discussions of his unpublished work with E. Diaconescu on related material . DSB is supported by EPSRC grant GR/R75373/02 and would like to thank DAMTP and Clare Hall college Cambridge for continued support and CERN for hospitality during the final stage of the project. JH is supported by NSF grant PHY-0204608 and acknowledges the stimulating atmosphere provided by the Aspen Center for Physics during the completion of this work.
[^1]: $^a$
[^2]: $^b$
|
---
author:
- Hirofumi Takesue
- Akira Ozawa
- So Morikawa
title: 'Evolution of favoritism and group fairness in a co-evolving three-person ultimatum game'
---
Introduction
============
The evolution of cooperation is one of the most actively investigated subjects in the physical and biological sciences [@Nowak2006; @Szabo2007]. It is known that *fair division* of the benefit of collaborative behavior is important for maintaining cooperative relationships [@Tomasello2009]. Thus, the fairness of resource division has been studied using ultimatum games [@Guth1982]. In ultimatum games, two players (a proposer and a responder) divide the resource. The proposer makes an offer regarding the resource division. If the responder accepts the offer, the resource is divided accordingly. If the responder rejects the offer, both players gain nothing.
The standard equilibrium notion in classical game theory predicts that the proposer almost monopolizes the resource. The responder gains nothing by rejection, so they should accept any positive offer. Expecting this reaction, a rational proposer should claim most of the resource. However, experimental evidence has repeatedly falsified this prediction [@Camerer2003]. An excessively low offer is often rejected, and the proposer offers nearly half of the resource to the responder. These observations are explained better by a model that incorporates the disutility due to inequity [@Fehr1999].
Many theoretical models have been proposed to explain the evolutionary origin of fairness. One approach stresses that the opportunity to choose the interaction partner is crucial for the evolution of fairness [@Andre2011a]. Other studies have shown that error [@Santos2015a] and weak selection [@Rand2013] can also explain the preference for fairness in ultimatum games. The roles of reputation [@Nowak2000] and empathy [@Page2002] have also been investigated.
Among the mechanisms proposed for the evolution of fairness, the effect of the network (spatial) structure is among the most intensively studied. A seminal study has shown that introducing lattice structure can facilitate the emergence of fairness [@Page2000]. This positive effect of the network structure was also confirmed using more complex network structures [@Kuperman2008]. The role of the network structure has also been studied in combination with other mechanisms, including empathy [@Sinatra2009; @Szolnoki2012], the fineness of the strategy [@Szolnoki2012a], role switching [@Wu2013], allocation mechanism of divided resource [@Wang2014a; @Chen2015a], migration [@Wang2015d] and simple strategy updating after the breakdown of bargaining [@Duan2010].
In addition, recent research considers the possibility that both the interaction structure and the strategy employed by players could co-evolve. In these studies, players can change interaction partners based on the neighbors’ strategy or the payoff from the game. The results obtained in previous studies have shown that fairness can evolve more easily if opportunities exist for partner switching (choice) [@Deng2011; @Gao2011; @Miyaji2013; @Yang2015]. The role of partner switching (choice) has also been investigated with respect to the evolution of cooperation. Previous studies have shown that partner switching (choice) enhances cooperation in the prisoner’s dilemma [@Zimmermann2004; @Pacheco2006a; @Santos2006; @Fu2008; @Szolnoki2008b; @VanSegbroeck2008; @Fu2009; @Szolnoki2009b; @Tanimoto2009; @Perc2010; @Cong2014; @Chen2016] and public goods game [@Wu2009a; @Wu2009c; @Zhang2011].
These previous studies on the ultimatum game have deepened our understanding of fairness in *dyadic* relationships. However, human fairness is not limited to two-person situations. Indeed, Boehm [@Boehm2012] noted that universal egalitarianism among group members is widely observed in hunter gatherers and tribal agriculturalists. Despite this empirical observation, there have been few theoretical analyses of group fairness. In contrast, the evolution of cooperation in public goods games on networks has been widely investigated [@Hauert2002; @Santos2008; @Szolnoki2009c; @Helbing2010; @Shi2010; @Szolnoki2010a; @Xu2010a; @Szolnoki2011b; @Chen2012; @Wu2014a; @Chen2015; @Chen2016a].
In this study, we investigated the evolution of *group* fairness in the three-person ultimatum game (TUG) based on numerical simulations. In the TUG, one proposer makes a proposal regarding resource division to two responders. The proposal is accepted, and resource division occurs if *at least* one responder accepts the offer. In the TUG, the proposer can secure the support of one responder while excluding the rest. As a result, the possibility of *favoritism* is introduced. The possibility of favoritism questions the effectiveness of the mechanism that is assumed to support the evolution of group fairness. For example, Boehm [@Boehm2012] suggested that punishment via the coalition of weaker individuals can support the emergence of group fairness. However, a favored agent has little incentive to conduct punishment, so a coalition may be unstable.
Some important previous studies [@Santos2015; @Santos2016] investigated the multiplayer ultimatum game in a well-mixed population. In the multiplayer ultimatum game, one proposer offers part of the resource to multiple responders and each responder accepts or rejects the offer. These studies have shown that the minimum number of individual acceptances needed for group level acceptance is important. Here, the proposer has to offer the *same* amount of the resource to all responders, and the possibility of favoritism is not considered. In fact, one study [@Santos2016] noted that the effect of allowing the proposer to target offers to specific responders is an open question.
In this study, we introduced a co-evolutionary mechanism where the strategies used in the game and individual partnerships can both evolve. Our simulations showed that the ratio of the frequency of partner switching events relative to strategy updating events has a profound effect on the distribution of the resource in the group.
Model
=====
First, we explained the TUG. In the TUG, there is one proposer and two responders. The proposer offers $p_A$ and $p_B$ to the two responders. Who will be offered $p_A \ (p_B)$ is determined by the agents’ *type*, $\theta$ (range: 0-1). The distance between two agents $i$ and $j$ is defined as $r_{ij} = \min(|\theta_i - \theta_j|, 1- |\theta_i - \theta_j|)$. Note that this definition means that the type is a circular variable where 0 and 1 are equivalent. This circular nature ensures that no agent is in advantageous position because of its type. The proposer offers $p_A \ (p_B)$ to the responder if their distance from the proposer is small (large). If $p_A > p_B (p_A < p_B)$, then the proposer favors (disfavors) the responder with a similar type. Without $\theta$, even if $p_A \neq p_B$, agents decide who will be favored randomly in each game and their behavior cannot be interpreted as favoritism in the long term. Each responder compares their minimum demand $q$ and $p_A \ (p_B)$ and accepts or rejects the offer. We introduced the group decision variable ($g$), which takes a value of 1 if the proposal is accepted at the group level and 0 otherwise.
The group makes a decision based on the proposal to offer $p_{iA} \ (p_{iB})$ to $j \ (k)$ in the following manner. $$g_{ijk} =
\begin{cases}
1, & \text{if}\ p_{iA} \geq q_j \lor p_{iB} \geq q_k \\
0, & \text{otherwise}
\end{cases}$$ Thus, the proposal is accepted if at least one responder accepts the offer. Using $g_{ijk}$, the payoff from the game for the proposer $i$ is calculated as $\pi_i = g_{ijk}[1 - (p_{iA} + p_{iB})]$. The payoff for the responders $j$ and $k$ are calculated as $\pi_j = g_{ijk}p_{iA}$ and $\pi_k = g_{ijk}p_{iB}$, respectively. Classical game theory predicts that the proposer will monopolize the resource in the same manner as in the two-person ultimatum game [@Santos2016]. Furthermore, the proposer obviously has a strong incentive to ignore one responder because their approval is not needed for group level acceptance.
Next, we explained the evolutionary process for the network structure and the strategy in the TUG. Let us assume that $N$ agents are located in the network, which is defined by the neighbors of each agent. The edges between agents represent social relationships. Initially, all agents have the same number of edges (${\ensuremath{\langlek\rangle} }$), which are randomly connected to other agents [@Santos2005a]. The size of the proposal has random values under the restriction that the sum of the proposals, $p_A + p_B$, does not exceed 1. The responder’s minimum acceptance threshold also has a random value. The type ($\theta$) also follows a standard uniform distribution, $U(0, 1)$. In each time step, a strategy updating event or a partner switching event occurs.
Strategy updating occurs with a probability of $1-w$. We used a link-based update rule to reduce the effects of large degree nodes in the neighborhood [@Fu2009]. This rule is similar to the “pairwise comparison process” in the well-mixed population model [@Traulsen2006]. In a strategy updating event, we first chose one edge ($E_{ij}$) randomly. Next, agent $i \ (j)$ plays TUG with their direct neighbors $k_i \ (k_j)$ times, where $k_i \ (k_j)$ is node $i$’s ($j$’s) number of neighbors. The number of games corresponds to the ordinary evolutionary game on the graph, where agents engage in two-person games with all their neighbors. In each game, two agents who play TUG with $i \ (j)$ are chosen randomly from the neighbors and the proposer is also chosen randomly from the three agents. Agents $i$ and $j$ gain a payoff from each game and accumulate total payoffs of $\Pi_i$ and $\Pi_j$, respectively. Next, $\Pi_i$ and $\Pi_j$ are compared and strategy updating occurs. Specifically, the strategy of agent $j$ replaces that of agent $i$ with a probability of $$P(s_i \leftarrow s_j) = [1 + \exp(-\beta(\Pi_j - \Pi_i))]^{-1}.$$ The value of $\beta$ is the intensity of selection ($\beta \to 0$ leads to random drift whereas $\beta \to \infty$ leads to imitation dynamics); otherwise, the strategy of agent $i$ replaces that of agent $j$. As a result, one of the two agents copies the other agent’s strategy. In addition, the agent’s type is copied at the same time. This corresponds to the assumption in previous studies regarding the evolution of favoritism where the strategies of agents as well as group membership evolve according to the payoff [@Fu2012].
We assume that a small error accompanies the copying of attributes, where the error follows a uniform distribution, $U(-\epsilon, \epsilon)$. Note that if the strategic variable ($p_A$, $p_B$ or $q$) takes value outside the defined condition (range: 0-1), then we set the value to the nearest boundary value. In addition, if the sum of $p_A$ and $p_B$ exceeds 1, then we compressed it to one while keeping the ratio between $p_A$ and $p_B$. We also set the value of the type ($\theta$) to $\theta-1 (\theta+1)$ if it exceeds 1 (goes below 0) due to its circular nature.
A partner switching event is chosen with a probability of $w$. In a partner switching event, one agent ($i$) is chosen randomly and plays TUG with their direct neighbors $k_i$ times in the same manner as the strategy updating event. The agent decides the continuation of the social relationship based on the payoff from the games. Specifically, the agent finds one game where they gained the smallest payoff and breaks the relationship with the two neighbors who participated in that game [@Deng2011; @Zhang2011]. If the agent earned the same smallest payoff in multiple games, one game is chosen randomly. Next, agent $i$ creates new links with two randomly chosen agents. We imposed the restriction that an agent with two edges does not lose links so that they can participate in TUG if they are a focal agent.
Intuitively, partner switching appears to engender a generous offer because responders will not be satisfied with a small offer and they will sever the link. In contrast, link adaptation appears to foster a “rational” acceptance threshold because rejection may cause the negotiation to break down (with the smallest payoff, 0). These opposite possible effects complicate the prediction.
Results
=======
To investigate the emerging resource distribution in the TUG, we conducted numerical simulations. The simulations continued for $2 \times 10^7$ time steps, and we computed the quantities of interest by averaging over the values from the last $10^4$ time steps. We conducted 50 independent simulations for each combination of parameters and calculated the mean values from these simulation runs.
First, we observed the impact of the frequency of partner switching on the TUG results. The parameter $w$ controls the frequency of partner switching, and a larger value implies that agents have more opportunities to adjust their social relationships. Figure \[fig\_papbq\_w\] shows the mean values of $p_A$, $p_B$, and $q$ as a function of $w$ for different values of $\beta$. The figure shows that $p_A$ was larger than $p_B$ with a smaller value of $w$, which means that proposers consistently discriminated in favor of one responder with a similar type. Proposers treated one responder indifferently because this responder was strategically irrelevant for securing support at the group level, and the type functioned as a *tag* when deciding the favored agent. The sum of the offer for two agents was larger when selection was weak (small $\beta$), and this result is similar to that obtained in previous studies [@Rand2013; @Santos2015]. However, the condition that induces fairness in the two-person ultimatum game led to *favoritism* rather than *group* fairness.
Figure \[fig\_papbq\_w\] also shows that as the value of $w$ increased, the value of $p_A$ started to decrease, whereas that of $p_B$ started to increase. Proposers stopped discriminating responders to prevent the relationship being severed by the indifferently treated responder. This increase in the offer raised the probability of $p_B$ being accepted, which deteriorated the importance of the favored agent in securing the $group$ level acceptance. Taking the result of $\beta = 0.1$ as an example, when $w = 0.05$, the number of cases where only $p_A$ was accepted was 13.7 times larger than the cases where only $p_B$ was accepted. Conversely, this ratio was only 1.8 when $w = 0.4$ and the importance of favored agents actually diminished. As a result, proposers could lower the degree of favoritism while keeping their own share ($p_A + p_B$).
In addition, the values of $p_A$ and $p_B$ both started to increase as $w$ increased further. In this situation, the proposers stopped monopolizing the resource and *group* fairness was achieved. We noted that with a very large value for $w$, the resource distribution was slightly unfavorable to the proposer, especially when selection was weak. This result is obtained when a deterministic rule is used for partner switching [@Miyaji2013] although the disadvantage was much weaker in our results.
The acceptance threshold of the responder decreased when $w$ was small, thereby supporting the intuitive prediction. Intuitively, larger opportunities for partner switching (higher $w$) were *disadvantageous* to agents with a larger $q$ because rejecting the offer could lead to a 0 payoff and losing links with other players. However, the value of $q$ increased rapidly as $w$ increased. Furthermore, $p_A \ (p_B)$ and $q$ started to increase with almost the same value of $w$. This result suggested that the proposer’s strategy is strongly related to the character of the network.
Note that if the acceptance by $both$ responders was required, the favoritism was not observed because there was no reason to distinguish two responders. Group fairness was similarly observed when $w$ was large enough. Our point is that co-evolutionary mechanism can lead to group fairness even if the game involves one strategically irrelevant player (see supplementary fig.S1 for this result).
We also checked the results by investigating the strategy that emerged as a function of $\beta$. Figure \[fig\_papbq\_beta\] shows that without the opportunity for partner switching ($w = 0$), all the strategic variables decreased as the intensity of selection became stronger. In fact, the offer to a similar agent ($p_A$) exceeded 0.3 when the intensity of selection was extremely weak, and this value was almost the same as the result in fig. \[fig\_papbq\_w\] when the frequency of link adaptation was high. However, the value of $p_B$ (shown in panel (b)) was consistently smaller than the value of $p_A$ (panel (a)). Thus, a tendency toward favoritism was observed regardless of the value of $\beta$. The equality of the offer to the two agents was observed only after the combination of $w$ and $\beta$ was sufficiently large. Partner switching is beneficial to fair agents, so they can fully exploit this advantage under strong selection. We also observed that the value of $q$ exhibited a similar pattern to $p_A$ and $p_B$, which was similar to that shown in fig. \[fig\_papbq\_w\].
The effect of a higher frequency of partner switching can be understood in the following way. Agents who offer a small resource will lose edges and gain a smaller payoff. This fact encourages agents to make a generous offer (higher $p_A$ ($p_B$)). In addition, the smallest payoff determines the severance of relationships, so discriminating two responders is disadvantageous for the maintenance of edges. We noted that the offer size does not increase without limit. If the average of the sum of $p_A$ and $p_B$ is above 2/3, the worst payoff will probably be achieved when the focal agent is a proposer. Thus, the opportunity to sever relationships by unsatisfied responders will be reduced and a generous offer will not help agents to acquire more edges. In this situation, the acceptance threshold ($q$) is almost neutral unless it exceeds the offer size. An acceptance threshold above the offer size will simply lead to the rejection of a generous offer and the loss of social relationships. In fact, the value of $q$ fluctuated over time below $p_A$ ($p_B$) after the offer size became larger, which suggests that the acceptance threshold was not related to the resulting network when $w$ was large.
To examine the relationship between the game strategy and the resulting network, fig. \[fig\_corpapbq\_w\] shows the Pearson’s correlation coefficients between the strategy variables and the agent’s neighborhood size ($k$) as a function of $w$ for different values of $\beta$. We computed the correlation in the last time period. The figure shows that the correlation between $p_A \ (p_B)$ and $k$ was positive and that it increased weakly as $w$ increased. This result suggests that the co-evolutionary mechanism penalizes greedy proposers by depriving them of opportunities for interaction. In addition, the correlation between $p_B$ and $k$ was stronger than the correlation between $p_A$ and $k$ when $w$ was small, which suggests that an offer to a disfavored agent was more important before a jump in the value of $p_A \ (p_B)$ was observed.
The figure also shows that the monotonic relationship did not hold with respect to $q$. With smaller values of $w$, the correlation between $q$ and $k$ was negative and it became stronger as the frequency of partner switching opportunities increased. The proposer kept their own share with modest values of $w$, and the higher demand threshold increased the possibility of bargaining breaking down (0 payoff) as well as risking the relationships with other players. However, with a larger value of $w$, the value of the correlation coefficient started to increase and it approached 0. Therefore, the relationship between the value of $q$ and the disadvantage due to the co-evolutionary mechanism disappeared. This pattern supports an interpretation where the co-evolutionary mechanism influences the behavior in the ultimatum game mainly through the proposer’s strategy.
Next, we examined the effect of the average degree on the emerging behavior. Figure \[fig\_papbq\_k\] shows the results of TUG as the function of ${\ensuremath{\langlek\rangle} }$ for different values of $w$. Basically, the results obtained in previous studies [@Page2000; @Kuperman2008; @Deng2011; @Gao2011] were replicated and a smaller average degree led to fairness (higher $p_A$, $p_B$ and $q$; but see [@Wang2015d]). Because the number of successful bargaining rather than the gain from one game had a stronger effect on the responders ’ payoff with larger [$\langlek\rangle$ ]{}, responders had a stronger temptation to lower $q$ as the neighborhood size increased and the proposal also decreased accordingly [@Wang2014a; @Wang2015d]. This logic also seemed to apply in our TUG. An exception to this pattern occurred when the value of $w$ became large and the variables approached the values observed in the fair state. For example, when $w = 0.6$, $p_A$ ($p_B$, $q$) exhibited a different pattern and a smaller ${\ensuremath{\langlek\rangle} }$ hindered the fair strategy in some cases. Generous proposers enjoyed the benefit of larger degree when $w$ was high (fig.\[fig\_corpapbq\_w\]). Because larger variance of degree was observed with larger [$\langlek\rangle$ ]{} (fig. \[fig\_density\], this result seems to be independent of the resulting strategy since the same pattern was observed when $w \leq 0.5$), smaller [$\langlek\rangle$ ]{} lowered the benefit of degree heterogeneity for generous proposers.
Finally, we examined the impact of limiting the maximum number of games. The evolution of the network structure generated a heterogeneous neighborhood size (opportunities for social interactions) among agents. It is natural to assume that some agents engaged in more social interactions, but agents might not have been able to fully exploit the benefits of the larger neighborhood size. One method for dealing with this possibility is restricting the maximum number of interactions per unit time [@Poncela2011]. Thus, we restricted the number of games in terms of both strategy updating and partner switching. Figure \[fig\_papbq\_mk\] shows the results of TUG as a function of the maximum number of interactions ($k_{max}$) for different values of $w$. As expected, limiting the number of social interactions had a disadvantageous effect on the emergence of group fairness. Less egalitarian results emerged using the combination of parameters where group fairness evolved in fig. \[fig\_papbq\_w\]. However, with a higher frequency of partner switching ($w = 0.8$), agents did not have to fully exploit the full capacity of potential interactions and the number of games needed for fairness to emerge was less than the mean degree (in the figure, ${\ensuremath{\langlek\rangle} } = 20$).
Discussion
==========
In this study, we investigated the co-evolutionary TUG. In the TUG, there is a possibility that a proposer might discriminate in favor of a specific responder. Few studies have examined *group* fairness despite its empirical significance. The impact of a co-evolutionary mechanism is ambiguous because it appears to foster fair proposals and a rational response at the same time, but our results showed that the opportunities for partner switching led to the emergence of group fairness. With a small opportunity for partner switching, the proposer favored one responder. By increasing the frequency of link adaptation, we then observed that equality among responders was achieved while proposers maintained their own large share. With a higher frequency of partner switching, group fairness was observed. This state emerged because the co-evolutionary mechanism worked mainly through the proposer’s behavior. In fact, generous proposers achieved a larger neighborhood size and the responder’s strategy had no impact on the resulting network. We also observed that a smaller average degree enhanced fair behavior although the opposite pattern was observed in some cases when the frequency of partner switching was high. Finally, a higher frequency of partner switching could support the evolution of fairness when the maximum number of games was restricted.
Our results have similarities with some observational studies. For example, anthropologists have reported that the social relationships of individuals who violated the norm were severed [@Guala2012]. Our simulation results suggest that severing relationships functions as punishment [@Fehr2003] and can actually support group fairness.
Our study helps understand the emergence of fairness beyond *dyadic* relationships, but future extensions would be beneficial. First, to simplify the problem, we investigated the TUG with one proposer. Obviously, we could consider more complex games. For example, an $N$-person ultimatum game with one proposer [@Santos2015] would increase the complexity of the proposer’s strategy. In addition, we could also consider a game with more than one proposer, which was investigated in an experimental study [@Roth1991]. Furthermore, evolutionary or imitation dynamics were assumed in the present study, but the robustness of the results should be examined using another learning rule [@Santos2016]. We consider that this line of research would further deepen our understanding of the evolution of fairness.
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amstex =1200 =0.7in ¶
**AN UPDATE**
**ON SEMISIMPLE QUANTUM COHOMOLOGY**
**AND $F$–MANIFOLDS**
**Claus Hertling, Yuri I. Manin, Constantin Teleman**
[**Abstract.**]{} In the first section of this note we show that the Theorem 1.8.1 of Bayer–Manin (\[BaMa\]) can be strengthened in the following way: [*if the even quantum cohomology of a projective algebraic manifold $V$ is generically semi–simple, then $V$ has no odd cohomology and is of Hodge–Tate type.*]{} In particular, this addressess a question in \[Ci\].
In the second section, we prove that [*an analytic (or formal) supermanifold $M$ with a given supercommutative associative $\Cal{O}_M$–bilinear multiplication on its tangent sheaf $\Cal{T}_M$ is an $F$–manifold in the sense of \[HeMa\], iff its spectral cover as an analytic subspace of the cotangent bundle $T^*_M$ is coisotropic of maximal dimension.*]{} This answers a question of V. Ginzburg.
Finally, we discuss these results in the context of mirror symmetry and Landau–Ginzburg models for Fano varieties.
**§0. Introduction**
[**0.1. Contents of the paper.**]{} Semisimple Frobenius manifolds have many nice properties: see e. g. \[Du\], \[Ma\], \[Te\], \[Go1\], \[Go2\], and references therein. It is important to understand as precisely as possible, which projective algebraic manifolds $V$ have (generically) semi–simple quantum cohomology. In this case the quantum cohomology is determined by a finite amount of numbers, and a mirror (Landau–Ginzburg model) can in many cases be described explicitly.
If $V$ has non–trivial odd cohomology, its full quantum cohomology cannot be semi–simple, but its even part is a closed Frobenius subspace, and in principle it can be semisimple. In \[BaMa\], Theorem 1.8.1, it was proved that if $H^{ev}_{quant}(V)$ is generically semisimple, then $h^{p,q}(V)=0$ for $p+ q \equiv 0\,\roman{mod}\,2$, $p\ne q.$ In the first section of this note we show that in this case $h^{p,q}(V)=0$ for $p+ q \equiv 1\,\roman{mod}\,2$ as well.
Thus, the Theorem 1.8.1 of Bayer–Manin (\[BaMa\]) can be strengthened in the following way: [*if the even quantum cohomology of a projective algebraic manifold $V$ is generically semi–simple, then $V$ has no odd cohomology and is of Hodge–Tate type.*]{} In particular, for the 47 families of Fano threefolds with $b_3(V)>0$, pure even quantum cohomology cannot be semi–simple, cf. \[Ci\], p. 826.
The second section is dedicated to a previously unpublished result of C. Hertling (letter dated March 09, 2005, where it was stated for the pure even case). It shows that an analytic (or formal) supermanifold $M$ with a given supercommutative associative $\Cal{O}_M$–bilinear multiplication on its tangent sheaf $\Cal{T}_M$ is an $F$–(super)manifold in the sense of \[HeMa\], iff its spectral cover as an analytic subspace of the cotangent bundle $T^*_M$ is coisotropic of maximal dimension.
This answered a question posed to Yu. Manin by V. Ginzburg.
[*Acknowledgement.*]{} We are grateful to Arend Bayer for illuminating comments on the Proposition 1.2 and for sharing with us his version of Dubrovin’s conjecture.
**§1. Semisimple quantum cohomology**
**and Dubrovin’s conjecture**
[**1.1. Notation.**]{} Let $V$ be a projective manifold over $\bold{C}$. We denote by $H^{ev}_{quant}(V)$ its even quantum cohomology ring. As in \[BaMa\] and \[Ba\], it is a topological commutative algebra. Multiplication in it (the classical cup product plus “quantum corections”) is denoted $\circ$. The space $H^{ev}(V)$ is embedded in it as a field of flat vector fields on the respective formal Frobenius manifold.
[**1.2. Proposition**]{}. [*If $H^{ev}_{quant}(V)$ is generically reduced, i.e. has no nilpotents at (the local ring of) the generic point, then $H^{odd}(V)=0$.*]{}
[**Proof.**]{} Assume that $H^{odd}(V)\ne 0$. Let $\Delta$ be a non–zero class of an odd dimension.
First, we have $\Delta\circ \Delta=0$. In fact, $\Delta\cup \Delta=0$, because the cup produt is supercommutative. The quantum corrections vanish, because the correlators $\langle...\rangle$ are also supercommutative in their arguments, so $\langle\Delta\Delta\Delta^{\prime}...\rangle
=0$. This follows from the fact that the quantum correlators come from the $S_n$–covariant maps $H^*(V)^{\otimes n}\to H^*(\overline{M}_{0,n})$, induced by algebraic correspondences (push–forwards of virtual fundamental classes). Covariance holds with respect to the action of $S_n$ on the tensor power permuting factors and introducing signs as usual in $Z_2$-graded setting. On the target it renumbers points and hence leaves the fundamental class invariant.
Now, find another (odd) class $\Delta^{\prime}$ such that $g(\Delta ,\Delta^{\prime})=1$, where $g$ is the Poincare form. Then we have $1=g(\Delta,\Delta^{\prime})=g(\Delta\circ \Delta^{\prime}, e)$ where $e$ is the identity in quantum cohomology. Hence $\Delta\circ \Delta^{\prime}\in
H^{ev}_{quant}(V)$ must be generically non-zero. But its square is zero because of the first remark. This contradicts the generic absence of nilpotents in $H^{ev}_{quant}(V)$.
[**1.3. Theorem.**]{} [*If the even quantum cohomology of a projective algebraic manifold $V$ is generically semi–simple, then $V$ has no odd cohomology and is of Hodge–Tate type.*]{}
[**Proof.**]{} From the generic semisimplicity and the Proposition 1.2 it follows, that $h^{p,q}(V)=0$ for $p+q\equiv 1\,\roman{mod}\,2.$
To prove that $h^{pq}(V)=0$ for $p+q \equiv 0\,\roman{mod}\,2$ and $p\ne q$, we reproduce a short reasoning from \[BaMa\]. It compares the Lie algebra of Euler vector fields in the semi–simple case and in the quantum cohomology case.
Firstly, in he semisimple case each Euler vector field must be of the form $E=d_0\sum_iu_ie_i+\sum_j c_je_j$, where $d_0$ is a constant (weight of $E$, cf. \[Ma1\], \[Ma2\]), and $(u_i)$ are (local) Dubrovin’s canonical coordinates, that is, $e_i:=\partial /\partial u_i$ form a complete system of pairwise orthogonal idempotents in $H^*_{quant}(V)$. Moreover, $(c_j)$ are arbitrary constants.
From this explicit description it follows directly, that if two Euler fields of non–zero weights commute, they are proportional.
On the other hand, if $h^{p,q}(V)\ne 0$ for some $p+q \equiv 0\,\roman{mod}\,2$ and $p\ne q$, then $H^*_{quant}(V)$ admits two commuting and non–proportional Euler vector fields $E_1,E_2$ of weight 1. Namely, in the bihomogeneous (with respect to the $(p,q)$–grading) basis of flat vector fields $\Delta_a\in H^{p_a,q_a}(V)$, we can take $$E_1:=\sum_a (1-p_a)x_a\Delta_a +\sum _{p_b=q_b=1} r_b\Delta_b,$$ $$E_2:=\sum_a (1-q_a)x_a\Delta_a +\sum _{p_b=q_b=1} r_b\Delta_b.$$ Here $(x_a)$ are dual flat coordinates, and $-K_V=c_1(\Cal{T}_V)=\sum_b r_b\Delta_b.$
This completes the proof.
[**1.4. Dubrovin’s conjecture and related insights.**]{} In \[Du\] (p. 321) the problem of characterization of varieties $V$ with semisimple quantum cohomology was formulated explicitly. It was also stated there that a necessary condition for such $V$ is to be Fano. This was disproved by A. Bayer \[Ba\], who established that blowing up points on such a variety does not destroy semisimplicity. In particular, not only del Pezzo surfaces have semisimple quantum cohomology, but arbitrary blowups of $\bold{P}^2$ as well.
A. Bayer has later conjectured that the maximal length of a semi–orthogonal decomposition of $D^b(V)$ must coincide with the generic number of idempotents in $H^*_{quant}(V)$.
Combining the results of \[Ba\], of this note, and the further part of Dubrovin’s conjecture stated on p. 322 of \[Du\] (cf. also \[Z\]), one can now guess that a necessary and sufficient condition for semisimplicity is that $V$ is of Hodge–Tate type, whose bounded derived coherent category admits a full exceptional collection $(E_i)$. Moreover, after adjusting some arbitrary choices, in this case one should be able to identify the Stokes matrix of its second structure connection with the matrix $(\chi (E_i,E_j))$.
This last statement is now checked, in particular, for three–dimensional Fano varieties with minimal cohomology in \[Go2\]. The reader can find there more details and explanations about the involvement of the vanishing cycles in the mirror Landau–Ginzburg model.
All these constructions reflect some facets of Kontsevich’s homological mirror symmetry program. However, one should keep in mind that in this note we are concerned almost exclusively with a multiplication on the tangent bundle, i.e. with the structure of an $F$–manifold (see below). In order to invoke mirror symmetry, we need also to take in consideration a compatible flat metric. In quantum cohomology, it comes “for free” at the start; it is multiplication that requires a special construction. In various contexts relevant for mirror symmetry, the metric can be described implicitly by at least five different kinds of data which we list here for reader’s convenience.
\(a) Values of the diagonal coefficients of the flat metric $\sum_i\eta_i (du_i)^2$ in canonical coordinates and values of their first derivatives $\eta_{ij}$ at a tame semi–simple point. This is initial data for the second structure connection (cf. \[Ma1\], II.3).
\(b) Monodromy data for the first structure connection and oscillating integrals for the deformed flat coordinates (cf. \[Gi\], \[Du\], \[Sa\] and the references therein).
\(c) Choice of one of K. Saito’s primitive forms.
\(d) Choice of a filtration on the cohomology space of the Milnor fiber (M. Saito, cf. \[He2\] and the references therein).
\(e) Use of the semi–infinite Hodge structure. This is a refinement of (c), described by S. Barannikov (\[Bar1\], \[Bar2\]).
**§2. $F$–geometry and symplectic geometry**
[**2.1. $F$–structure and Poisson structure.**]{} Manifolds $M$ considered in this section can be $C^{\infty}$, analytic, or formal, eventually with even and odd coordintes (supermanifolds). The ground field $K$ of characteristic zero is most often $\bold{C}$ or $\bold{R}$. Each such manifold, by definition, is endowed with the structure sheaf $\Cal{O}_M$ which is a sheaf of (super)commutative $K$–algebras, and the tangent sheaf $\Cal{T}_M$ which is a locally free $\Cal{O}_M$–module of (super)rank equal to the (super)dimension of $M$. $\Cal{T}_M$ acts on $\Cal{O}_M$ by derivations, and is a sheaf of Lie (super)algebras with an intrinsically defined Lie bracket $[\, ,]$.
There is a classical notion of [*Poisson structure*]{} on $M$ which endows $\Cal{O_M}$ as well with a Lie bracket $\{\, ,\}$ constrained by a well known identity.
Similarly, [*an $F$–structure*]{} on $M$ endows $\Cal{T}_M$ with an extra operation: (super)commutative and associative $\Cal{O}_M$–bilinear multiplication. We denote it always $\circ$ and assume that it is endowed with identity: an even vector field $e$. Then $\Cal{O}_M$ is embedded in $\Cal{T}_M$ as a subalgebra: $f\mapsto fe.$
Given such a multiplication on the tangent sheaf, we can define its [*spectral cover $\widetilde{M}$*]{} which is a closed ringed (super)subspace (generally not a submanifold) in the cotangent (super)manifold $T^*M$. In the Grothendieck language, it is simply the relative affine spectrum of the sheaf of algebras $(\Cal{T}_M,\circ )$ on $M$.
More precisely, consider $Symm_{\Cal{O}_M}(\Cal{T}_M)$ as the sheaf of algebras of those functions on the cotangent (super)space $T^*_M$ that are polynomial along the fibres of the projection $T^*_M\to M$. The multiplication in this sheaf will be denoted $\cdot$. For example, for two local vector fields $X,Y\in \Cal{T}_M (U)$, $X\cdot Y$ denotes their product as an element of $Symm^2_{\Cal{O}_M}(\Cal{T}_M)$.
Consider the canonical surjective morphism of sheaves of $\Cal{O}_M$–algebras $$Symm_{\Cal{O}_M}(\Cal{T}_M)\to (\Cal{T}_M,\circ )$$ sending, say, $X\cdot Y$ to $X\circ Y$. Denote its kernel by $J(M,\circ )$, and let $\widetilde{M}$ be defined by the sheaf of ideals $J(M,\circ )$.
The spectral cover $\widetilde{M}\to M$ is flat, because $\Cal{T}_M$ is locally free.
Now we will describe the structure identities imposed onto $\{\,,\}$ on $\Cal{O}_M$, resp. $\circ$ on $\Cal{T}_M$. To this end, recall the notion of the Poisson tensor. Let generally $A$ be a $K$–linear superspace (or a sheaf of superspaces) endowed with a $K$–bilinear multiplication and a $K$–bilinear Lie bracket $[\, ,]$. Then for any $a,b,c\in A$ put $$P_a(b,c):=[a,bc]-[a,b]c-(-1)^{ab}b[a,c] .
\eqno(2.1)$$ (From here on, $(-1)^{ab}$ and similar notation refers to the sign occuring in superalgebra when the two neighboring elements get permuted.)
This tensor will be written for $A=(\Cal{O}_M, \cdot, \{\,,\})$ in case of the Poisson structure, and for $A=(\Cal{T}_M, \circ , [\,,])$ in case of an $F$–structure.
We will now present parallel lists of basic properties of Poisson, resp. $F$–manifolds.
[**2.2. Poisson (super)manifolds.**]{} (i)${}_P$. [*Structure identity:*]{} for all local functions $f,g,h$ on $M$ $$P_f(g,h)\equiv 0 .
\eqno(2.2)$$
(ii)${}_P$. [*Each local function $f$ on $M$ becomes a local vector field $X_f$ (of the same parity as $f$) on $M$ via $X_f(g):=\{f,g\}$.*]{}
This is a reformulation of (2.2).
(iii)${}_P$. [*Maximally nondegenerate case: symplectic structure.*]{} There exist local canonical coordinates $(q_i,p_i)$ such that for any $f,g$ $$\{f,g\}=\sum_{i=1}^n (\partial_{q_i}f\partial_{p_i}g
- \partial_{q_i}g\partial_{p_i}f).$$ Thus, locally all symplectic manifolds of the same dimension are isomorphic. The local group of symplectomorphisms is, however, infinite dimensional.
[**2.3. $F$–manifolds.**]{} (i)${}_F$. [*Structure identity:*]{} for all local vector fields $X,Y,Z,U$ $$P_{X\circ Y}(Z,U)-X\circ P_Y(Z,U)- (-1)^{XY}Y\circ P_X(Z,U)=0.
\eqno(2.3)$$
(ii)${}_F$. [*Each local vector field on $M$ becomes a local function on the spectral cover $\widetilde{M}$ of $M$*]{}.
As we already mentioned, generally $\widetilde{M}$ is not a (super)manifold. In the pure even case this often happens because of nilpotents in $\Cal{O}_{\widetilde{M}}$ and/or singularities. In the presence of odd coordinates on $M$ nilpotents by themselves are always present, but typically they cannot form an exterior algebra over functions of even coordinates because ranks do not match.
A theorem due to Hertling describes certain important cases when $\widetilde{M}$ is a manifold.
(iii)${}_F$. [*Maximally nondegenerate case: semisimple $F$–manifolds.*]{} $\widetilde{M}$ will be a manifold and even an unramified covering of $M$ in the appropriate “maximally nondegenerate case”, namely, when $M$ is pure even, and locally $(\Cal{T}_M,\circ )$ is isomorphic to $(\Cal{O}_M^d)$ as algebra, $d=\roman{dim}\,M.$
In this case there exist local canonical coordinates $(u_a)$ (Dubrovin’s coordinates) such that the respective vector fields $\partial_a:=\partial/\partial_a$ are orthogonal idempotents: $$\partial_a\circ \partial_a =\delta_{ab}\partial_a .$$ Thus, locally all semisimple $F$–manifolds of the same dimension are isomorphic. Local automorphisms of an $F$–semisimple structure are generated by renumberings and shifts of canonical coordinates: $$u_a\mapsto u_{\sigma (a)}+c_a$$ so that this structure is more rigid than the symplectic one.
[**2.4. Spectral cover as a subspace in symplectic supermanifold.**]{} There is a structure of sheaf of Lie algebras on $Symm_{\Cal{O}_M}(\Cal{T}_M)$. It is given by the Poisson brackets $\{\,, \}$ with respect to the canonical (super)symplectic structure on $T^*_M$.
It is easy to check that the ideal $J=J (M,\circ )\subset Symm_{\Cal{O}_M}(\Cal{T}_M)$ defining $\widetilde{M}$ in this sheaf of supercommutative algebras is generated by all expressions: $$e-1,\quad X\circ Y-X\cdot Y, \quad X,Y \in \Cal{T}_M.
\eqno(2.4)$$
[**2.5. Theorem.**]{} [*The multiplication $\circ$ satisfies the structure identity of $F$–manifolds (2.3), iff the ideal $J(M,\circ )$ is stable with respect to the Poisson brackets.* ]{}
[**Proof.**]{} From (2.2), one easily infers that stability of an ideal in a Poisson algebra with respect to the brackets can be checked on any system of generators of this ideal. In our case we choose (2.4).
Clearly, $\{e-1,e-1\} =0$.
If $X,Y$ are local vector fields, then $\{X,Y\}=[X,Y]$
We will establish by a direct computation that for all $X,Y,Z,W$ as above, $$\{X\circ Y-X\cdot Y,\, Z\circ W-Z\cdot W\} \equiv$$ $$P_{X\circ Y}(Z,W)-X\circ P_Y(Z,W)-(-1)^{XY}Y\circ P_X(Z,W)\ \roman{mod}\,
J(M,\circ )
\eqno(2.5)$$ and $$\{e-1, X\circ Y-X\cdot Y\} = [e,X\circ Y]-X\cdot [e,Y] -[e,X]\cdot Y.
\eqno(2.6)$$
Assume that this is done. From (2.5) and (2.6) it follows that if (2.3) holds, then $J(M,\circ )$ is stable with respect to the Poisson brackets. For (2.6), one uses the identity $[e, X\circ Y]=X\circ [e,Y] + [e,X]\circ Y$ which follows from (2.3) by choosing $X=Y=e$ and renaming $Z,U$.
Conversely, if $J(M,\circ )$ is stable with respect to the brackets, then the right–hand side of (2.5) must belong to $J(M,\circ )$. But it lies in the degree 1 part of the symmetric algebra of $\Cal{T}_M$, which projects onto $\Cal{T}_M$. Hence it must vanish, and as a result, the right hand side of (2.6) must belong to $J(M,\circ )$ as well.
It remains to check (2.5) and (2.6). We will briefly indicate how to do it, restricting ourselves to the clumsier case (2.5).
First of all, the right hand side of (2.5) can be rewritten as follows: $$P_{X\circ Y}(Z,W)-X\circ P_Y(Z,W)-(-1)^{XY}Y\circ P_X(Z,W)=$$ $$[X\circ Y,Z\circ W]-[X\circ Y,Z]\circ W-
(-1)^{(X+Y)Z}Z\circ [X\circ Y,W]$$ $$-X\circ [Y,Z\circ W] -(-1)^{XY}Y\circ [X,Z\circ W]+ X\circ [Y,Z]\circ W
+(-1)^{YZ} X\circ Z\circ [Y,W]$$ $$+ (-1)^{XY}Y\circ [X,Z]\circ W
+(-1)^{X(Y+Z)}Y\circ Z\circ [X,W] .
\eqno(2.7)$$ It turns out that (2.7) is in fact a tensor, that is $\Cal{O}_M$–polylinear in $X,Y,Z,W$. See \[Me1\], \[Me2\] for a discussion and operadic generalizations of the condition of its vanishing.
In our context, this formula is convenient, because a straightforward decomposition of the left hand side of (2.5) into Poisson monomials (constructed using two operations) gives exactly the same list of monomials as in (2.7) modulo $J(M,\circ )$, with the same signs.
Here are samples of calculations.
The first term $\{X\circ Y, Z\circ W\}$ at the left hand side of (2.5) coincides with the first term in (2.7).
Using the Poisson identity (2.2), we find further: $$-\{X\circ Y, Z\cdot W\} =-\{X\circ Y, Z\}\cdot W-
(-1)^{(X+Y)Z}Z\cdot \{X\circ Y, W\}.$$ Modulo $J(M, \circ )$, this can be replaced by $$-[X\circ Y, Z]\circ W-
(-1)^{(X+Y)Z}Z\circ [X\circ Y, W]$$ which corresponds to the second and third terms of (2.7).
We leave the rest as an exercise to the reader.
[**2.5.1. Reduced spectral cover.**]{} Contrary to what might be expected, the condition $$\{J(M,\circ ),J(M,\circ )\}\subset J(M,\circ )$$ [*does not*]{} imply the respective condition for the radical of $J(M,\circ )$ even in the pure even case. This means that $\widetilde{M}_{red}$ need not be a Lagrange subvariety, even if it comes from an $F$–manifold.
This can be shown on the following explicit examples.
We will construct two families of everywhere indecomposable (see 2.6 below) $F$-manifolds in terms of the ideals $J$, defining (nonreduced) subspaces $\widetilde{M} \subset T^*M$. In order to give rise to $F$-manifolds with $\pi :\,\widetilde M\to M$ as their spectral cover, they have to satisfy the following conditions:
\(a) The projection $\widetilde{M}\to M$ is flat of degree $n=\,\roman{dim}\, M$ and the canonical map $\Cal{T}_M\to \pi_*(\Cal{O}_{\widetilde{M}})$ is an isomorphism.
To check this by direct calculations, we will choose (pure even) local coordinates $(t_1,\dots ,t_n)$ on $M$ in such a way that $e=\partial /\partial t_1.$ By $(y_1,\dots ,y_n)$ we will denote the conjugate coordinates along the fibres of $T^*(M)$.
\(b) $\{J,J\}\subset J$.
We will see that in these examples $$\{\sqrt{J},\sqrt{J}\}\not\subset \sqrt{J}.$$
[**2.5.2. The first family.**]{} Here we put $$J=(y_1-1,\; (y_i-\rho_i)(y_j-\rho_j)),$$ with $\rho_1=1$ and $\rho_i\in \Cal{O}_M$ for $i\geq 2$ such that $\partial_1 \rho_i=0$. Clearly, (a) and (b) are satisfied. The radical of $J$ is $$\sqrt{J} = (y_1-1; y_2-\rho_2, \dots ,y_n-\rho_n).$$ We have $\{\sqrt{J},\sqrt{J}\}\not\subset \sqrt{J}$, if $$\partial_i \rho_j \neq \partial_j\rho_i
\quad \hbox{ for some }i,j\geq 2 \hbox{ with }i\neq j.$$ The algebra $T_tM$ at any point $t\in M$ is isomorphic to $\hbox{\bf C}[x_1,...,x_{n-1}]/(x_ix_j)$.
[**2.5.3. The second family.**]{} Here we put for any $n\ge 3$ $$J=(y_1-1,\; (y_2-\rho_2)^2,\; (y_2-\rho_2)\cdot y_3, \;
y_3^{n-1},\; y_4-y_3^2,\; y_5-y_3^3,...,\; y_n-y_3^{n-2}),$$ with $$\rho_2(y,t)= t_3y_1+\sum_{k=3}^{n-1} (k-1)t_{k+1}\cdot y_k.$$ Now, (a) is rather obvious, but checking (b) requires a calculation which we omit. The radical of $J$ is $$\sqrt{J} = (y_1-1,\; y_2-t_3\cdot y_1,\; y_3,y_4,y_5,...,y_n).$$
The algebra $T_tM$ at any point $t\in M$ is isomorphic to $\hbox{\bf C}[x_2,x_3]/(x_2^2,x_2x_3,x_3^{n-1})$.
We will now explain in which context the considerations of this section can be related to the problems, arising in the study of semisimple quantum cohomology
[**2.6. Hertling’s local decomposition theorem.**]{} For any point $x$ of a pure even $F$–manifold $M$, the tangent space $T_xM$ is endowed with the structure of a $K$–algebra. This $K$–algebra can be represented as a direct sum of local $K$–algebras. The decomposition is unique in the following sense: the set of pairwise orthogonal idempotent tangent vectors determining it is well defined.
C. Hertling has shown that this decomposition extends to a neighborhood of $x$.
More precisely, define the sum of two $F$–manifolds: $$(M_1,\circ_1,e_1) \oplus (M_2,\circ_2,e_2) :=
(M_1\times M_2,\circ_1\boxplus\circ_2,e_1\boxplus e_2)$$ A manifold is called [*indecomposable*]{} if it cannot be represented as a sum in a nontrivial way.
[**2.6.1. Theorem.**]{}
*Every germ $(M,x)$ of a complex analytic $F$–manifold decomposes into a direct sum of indecomposable germs such that for each summand, the tangent algebra at $x$ is a local algebra.*
This decomposition is unique in the following sense: the set of pairwise orthogonal idempotent vector fields determining it is well defined.
For a proof, see \[He\], Theorem 2.11.
Furthermore, we have (\[He\], Theorems 5.3 and 5.6):
[**2.7. Theorem.**]{}
*(i) The spectral cover space $\widetilde{M}$ of the $F$–structure on the germ of the unfolding space of an isolated hypersurface singularity is smooth.*
\(ii) Conversely, let $M$ be an irreducible germ of a generically semisimple $F$–manifold with the smooth spectral cover $\widetilde{M}.$ Then it is (isomorphic to) the germ of the unfolding space of an isolated hypersurface singularity. Moreover, any isomorphism of germs of such unfolding spaces compatible with their $F$–structure comes from a stable right equivalence of the germs of the respective singularities.
Recall that the stable right equivalence is generated by adding sums of squares of coordinates and making invertible analytic coordinate changes.
In view of this result, it would be important to understand the following
[**2.8. Problem.**]{} [*Characterize those varieties $V$ for which the quantum cohomology Frobenius spaces $H^*_{quant}(V)$ have smooth spectral covers.*]{}
Theorem 2.7 produces for such manifolds a weak version of Landau–Ginzburg model, and thus gives a partial solution of the mirror problem for them.
**References**
\[AuKaOr\] D. Auroux, L. Katzarkov, D. Orlov. [*Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves.*]{} Invent. Math. 166 (2006), no. 3, 537–582. e-arxiv 0506166
\[Bar1\] S. Barannikov. [*Semi–infinite Hodge structures and mirror symmetry for projective spaces.*]{} e-arxiv 0010157
\[Bar2\] S. Barannikov. [*Semi–infinite variations of Hodge structure and integrable hierarchies of KdV type.*]{} Int. Math. Research Notices, 19 (2002), 973–990. e-arxiv math.AG/00148.
\[Ba\] A. Bayer. [*Semisimple quantum cohomology and blowups.*]{} Int. Math. Research Notices, 40 (2004), 2069–2083. e-arxiv math.AG/0403260.
\[BaMa\] A. Bayer, Yu. Manin. [*(Semi)simple exercises in quantum cohomology.*]{} Proc. of the Fano conference, Univ. Torino, Turin, 2004, 143–173. e-arxiv math.AG/0103164.
\[Ci\] G. Ciolli. [*On the quantum cohomology of some Fano threefolds and a conjecture of Dubrovin.*]{} Int. Journ. of Math., vol. 16, No. 8 (2005), 823–839. e-arxiv math/0403300.
\[Du\] B. Dubrovin. [*Geometry and analytic theory of Frobenius manifolds.*]{} Proc. ICM Berlin 1998, vol. 2, 315 – 326.
\[Gi\] A. Givental. [*A mirror theorem for toric complete intersections.*]{} Progr. Mat., 160 (1998), 141–175.
\[Go1\] V. Golyshev. [*Riemann–Roch variations.*]{} Izv. Ross. AN, ser. mat., 65:5 (2001), 3–32.
\[Go2\] V. Golyshev. [*A remark on minimal Fano threefolds.*]{} e-arxiv 0803.0031
\[He\] C. Hertling. [*Frobenius spaces and moduli spaces for singularities.*]{} Cambridge University Press, 2002.
\[HeMa\] C Hertling, Yu. Manin. [*Weak Frobenius manifolds.*]{} Int. Math. Res. Notices, 6 (1999), 277–286. e-arxiv math.QA/9810132.
\[Ma1\] Yu. Manin. [*Frobenius manifolds, quantum cohomology, and moduli spaces.*]{} AMS Colloquium Publications, vol. 47, 1999.
\[Ma2\] Yu. Manin. [*Manifolds with multiplication on the tangent sheaf.*]{} Rendiconti Mat. Appl., Serie VII, vol. 26 (2006), 69–85. e-arxiv math.AG/0502016
\[Me1\] S. Merkulov. [*Operads, deformation theory and $F$–manifolds.*]{} In: Frobenius manifolds, quantum cohomology, and singularities (eds. C. Hertling and M. Marcolli), Vieweg 2004, 213–251. e-arxiv math.AG/0210478.
\[Me2\] S. Merkulov. [*PROP profile of Poisson geometry.*]{} Comm. Math. Phys., 262:1 (2006), 117–135. e-arxiv math.AG/0401034.
\[Or\] D. Orlov. [*Derived category of coherent sheaves, and motives.*]{} (Russian) Uspekhi Mat. Nauk 60 (2005), no. 6(366), 231–232; translation in Russian Math. Surveys 60 (2005), no. 6, 1242–1244 14F05 e-arxiv math/0512620
\[Sa\] C. Sabbah. [*Isomonodromic deformations and Frobenius manifolds.*]{} EDP Sciences and Springer, 2007
\[Te\] C. Teleman. [*The structure of 2D semi-simple field theories.*]{} e-arxiv 0712.0160
\[Z\] E. Zaslow. [*Solitons and helices: the search for a math–physics bridge.*]{} Comm. Math. Phys. 175, No. 2 (1996), 337–375.
SUGGESTED COMPLEMENTS 14.03.08:
1\) Rename subsection 1.4: [**1.4. Dubrovin’s conjecture and related insights I.**]{}
2\) Delete two last paragraphs of sec. 1. Add: We will continue this discussion in sec. 2.9 below.
3\) Add subsection:
[**2.9. Dubrovin’s conjecture and related insights II.**]{} We can now sketch a tentative path leading from a manifold $V$ with semisimple quantum cohomology to its Stokes matrix/monodromy as predicted by Dubrovin, in the landscape of Kontsevich’s mirror symmetry.
A. If one is lucky, the spectral cover of quantum cohomology will be semisimple at a sufficiently big neighborhood of the point of classical limit. Assume this.
B. Define the Landau–Ginzburg model of $V$ via unfolding of the respective germ of isolated hypersurface singularity, as in Theorem 2.7.
C. Calculate monodromy/Stokes matrix of the structure connections of the unfolding Frobenius manifold via Lagrangian vanishing cycles.
D. Identify an exceptional collection in $D^b(V)$ with an exceptional collection of the relevant derived Fukaya–Seidel category of Lagrangian vanishing cycles, cf. \[Se\].
For some calculations in this vein, see \[Go2\], \[AuKaOr\] and other papers quoted therein.
4\) Add in the references;
\[AuKaOr\] Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves. e-arxiv 0506166
\[Se\] P. Seidel. [*Vanishing cycles and mutation.*]{} Proc. 3rd ECM, Barcelona 2000, vol.2, Progr. in Math. 202, Birkhäuser 2001, 65–85. e-arxiv 0007115
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abstract: 'Since only recently, cosmological observations are sensitive to hot dark matter (HDM) admixtures with sub-eV mass, $\mnuseff < \eV$, that are not fully-thermalised, $\D\Neff< 1$. We argue that their almost automatic interpretation as a sterile neutrino species is neither from theoretical nor practical parsimony principles preferred over HDM formed by decay products (daughters) of an out-of-equilibrium particle decay. While daughters mimic sterile neutrinos in $\Neff$ and $\mnuseff$, there are opportunities to assess this possibility in likelihood analyses. Connecting cosmological parameters and moments of momentum distribution functions, we show that –also in the case of mass-degenerate daughters with indistinguishable main physical effects– the mimicry breaks down when the next moment, the skewness, is considered. Predicted differences of order one in the root-mean-squares of absolute momenta are too small for current sensitivities.'
author:
- Jasper Hasenkamp
bibliography:
- 'mimicry.bib'
title: ' **Daughters mimic sterile neutrinos (almost!) perfectly**'
---
Introduction
============
We are in the era of precision cosmology. By observing the cosmic microwave background (CMB), the Planck satellite provided data that allows us to determine base quantities in the standard model of cosmology ($\L$CDM) like the energy density of cold dark matter (CDM) on the percent level [@Ade:2013zuv]. Especially, if combined with further cosmological observations, most violations of assumptions and deviations from predictions are constrained severely.
We focus on recent hints for a hot dark matter (HDM) *admixture* [@Wyman:2013lza; @Hamann:2013iba; @Battye:2013xqa] parametrised by an effective number of additional neutrino species $\D\Neff$ and effective HDM mass of [@Hamann:2013iba] $$\label{hdmsignal}
\D\Neff = 0.61\pm0.30 \text{,} \quad m_\text{hdm}^\text{eff} = (0.41 \pm 0.13) \eV \, .$$ Only recently, precisions became high enough to possibly find evidence for such a *sub-eV, not fully-thermalised* species. Consequently, it is of utmost importance to scrutinise and improve the accuracy of the considered observations, see for example [@Costanzi:2013bha; @Efstathiou:2013via; @Paranjape:2014lga; @Leistedt:2014sia]. For the same reason, the time to consider such HDM admixtures independent of the exact confidence intervals is now. The common particle physics interpretation is an additional, uncharged (= sterile) neutrino $\nus$ species that mixes with the active neutrinos and, consequently, is produced when these scatter in the early universe. The beauty of this interpretation is its parsimony: Firstly, there are hints from neutrino oscillation experiments for sterile neutrinos with masses and mixings that thermalise them. No further physics needs to be introduced, actually, from symmetry arguments one might expect even three such particles. Secondly, the cosmological model is amended by only one free parameter per particle, the sterile neutrino’s mass, since full-thermalisation fixes its temperature to the neutrino temperature and thus implies $\D\Neff=1$. However, the cosmological signal just does not fit that interpretation as illustrated in Fig. \[fig:posterior\].
![Our nomenclature for a cosmological particle decay and the corresponding degrees of freedom. The, in principle, arbitrarily large number is argued to reduce to three, which is further reduced to only two actually independent degrees of freedom by a physical parameter degeneracy.[]{data-label="fig:dof"}](dof.pdf){width="0.8\columnwidth"}
Accepting the possibility of $\D\Neff < 1$, the cosmological model is amended by a second free parameter. Attempts to reconcile sterile neutrinos with cosmology include the addition of light, interacting particles [@Steigman:2013yua] and new neutrino interactions [@Hannestad:2013ana; @Dasgupta:2013zpn] that, indeed, lead to HDM improving the cosmological fit if they are gauged and shared by CDM [@Bringmann:2013vra]. Along the way, these ideas also give up the first parsimony argument. At that state we find it to be indicated to reconsider the interpretation of being due to a thermally produced sterile neutrino (HDM), even if for the sake of identifying the limits of our understanding. In this work, we consider the possibility that hot dark matter is formed by the decay products of an out-of-equilibrium particle decay (dpHDM). So it originates from particle decay instead of scattering. Cosmological particle decays with in part dramatic consequences are naturally expected in many theories beyond the Standard Model of particle physics (SM). They do not only occur naturally, if gravity with the prominent gravitino and Polonyi problems is considered, see [@Hasenkamp:2011xh; @Hasenkamp:2011em; @Hasenkamp:2012ii; @Bae:2013qr; @Graf:2013xpe; @Higaki:2013vuv; @DiBari:2013dna; @Conlon:2013isa; @Park:2013bza; @Hooper:2013nia; @Kelso:2013nwa; @Jeong:2013oza] and references therein. It is remarkable that dpHDM does not require the introduction of any small mass scales as HDM. From a particle physical point of view a decay, as depicted schematically in Fig. \[fig:dof\], might be described by an arbitrarily large number of degrees of freedom just like a “dark sector” with new interactions. As in that case the HDM component is, nevertheless, described by only two actually independent degrees of freedom. So from an observational point of view the complexity of the cosmological models is the same.
We will explore, which scenarios of dpHDM have for what reason the potential to be distinguished from HDM and in which case dpHDM is indistinguishable from HDM in analyses like [@Hamann:2013iba] utilising CMB and large-scale structure observations. In any case, the cosmological impact of dpHDM cannot be identical, because dpHDM possesses a different momentum distribution function than HDM. We will show at which point and how *any mimicry must break down*.
This work is organised as follows: In the next section we list main physical effects, define parameters and remind the case of HDM, where we also assign briefly the current observational evidence. In Sec. \[sec:mimicry\] we show how decay products mimic sterile neutrinos, translate observations, point out in which cases there are testable differences in the main physical effects, and how any mimicry must break down in the third cosmological parameter. In Sec. \[sec:conclusions\] we summarise and conclude.
Preliminaries {#sec:preliminaries}
=============
[***Physical effects (neutrino case) –***]{} A population of free-streaming particles, which becomes non-relativistic after photon decoupling, affects the cosmological background and the evolution of perturbations [@Lesgourgues:2006nd]. Its main physical effects are related to:
1\) its contribution to the radiation energy density $\rhorad$ of the Universe before photon decoupling. Given in terms of an effective number of neutrino species $\Neff$ the radiation energy density reads $$\label{rhorad}
\r_\text{rad} = \left( 1 + \Neff \frac{7}{8} \left(\frac{T_\n}{T_\g}\right)^4 \right) \r_\g \, ,$$ such that it is split into a sum of the energy density in photons $\r_\g=(\pi^2/15) T^4$ and the relativistic energy density in anything else. The Standard Model of particle physics (SM) contains three active neutrinos with $\Neff^\text{sm}=3.046$ [@Mangano:2005cc] and temperature ratio $T_\n/T_\g = (4/11)^{1/3}$. The small correction in $\Neff^\text{sm}$ is due to incomplete neutrino decoupling at $e^+e^-$-annihilation. Any departure from the standard scenario, which increases the expansion rate of the Universe and could shift the time of matter-radiation equality, is then parametrised as a summand in $\Neff = \Neff^\text{sm} + \D\Neff$, such that the active neutrinos correspond to $\D\Neff=0$ by construction. 2) its non-relativistic energy density today $\r_\text{hdm}^0=n_\text{hdm}^0 m_\text{hdm}$, which is given by the mass $m_\text{hdm}$ and today’s number density $n_\text{hdm}^0$ of the population. Free-streaming particles do not cluster on scales below the free-streaming scale and thus damp fluctuations. The extent of the arising amplitude reduction in the matter power spectrum due to the free-streaming population is $\simeq 8 \,\O_\text{hdm}/\O_\text{m}$ [@Lesgourgues:2006nd], so that the HDM fraction $f=\O_\text{hdm}/\O_\text{m}$ provides a useful parametrisation. Note that such a small HDM *admixture* leads to a step-like feature and not, for example, to a cut-off. Assuming the active neutrinos are degenerate in mass, their today’s energy density normalised by the critical energy density $\rhoc$ reads $$\label{rhonu}
\O_\n^0 =
\frac{\Neff^\text{sm}}{11} \frac{n_\g^0}{\rhoc} \sumnu \Leftrightarrow \O_\n^0 h^2 \simeq 0.0108 \frac{\sumnu}{\text{eV}}$$ with today’s number density of CMB photons given by $n_\g^0 = (2 \z(3)/\pi^2) T_0^3$, if $T_0=2.7255$ K denotes the CMB temperature today and $\z$ the zeta function. By construction corresponds to the $\D\Neff = 0$ case. In a $\L$CDM model amended to include massive (degenerate) active neutrinos this sum of neutrino masses $\sumnu$ is usually “observed,” when cosmological parameters are determined. If $\sumnu$ is not freed in an analysis (“pure” $\L$CDM), a minimal-mass normal hierarchy can be assumed, which is accurately approximated for current cosmological data as a single massive eigenstate with $m_\nu = 0.06 \eV$ [@Ade:2013zuv]. 3) the minimum of the comoving free-streaming wavenumber $k^\text{nr}$. This is the scale at which the suppression of fluctuations in the matter power spectrum sets in. It is given by the time of the transition when the population becomes non-relativistic [@Lesgourgues:2006nd]. In Appendix \[appendix:A\] we make this point explicit arriving at $$\label{knr}
k^\text{nr} \simeq 4.08 \times 10^{-4} \, \O_\text{m}^\frac{1}{2} \fb{\Tnr}{T_0}{\frac{1}{2}} h \Mpc^{-1} ,$$ which depends on the properties of the free-streaming population via the temperature of the Universe at transition $\Tnr$ only. It is defined by $$\label{defTnr}
\langle p \rangle(\Tnr) = m \, ,$$ where the angle brackets indicate the average of the particles absolute momenta, $p=|\vec p|$. If we kept the directional information, the population is on average at rest due to the isotropy of the Universe, $\langle \vec p \rangle =0$. The root-mean-square momentum of the population equals the mean of its distribution of absolute momenta, $\langle \vec p \rangle_\text{rms} = \langle p \rangle$, which is sometimes just called ”mean of the distribution“. For degenerate neutrinos reads $$\label{Tnrnu1}
\langle p_\nu \rangle (\Tnr_\nu)
= \frac{7 \pi^4}{180 \z(3)} \left(\frac{4}{11}\right)^\frac{1}{3} \Tnr_\nu
= \frac{1}{3} \sumnu$$ giving $$\label{Tnrnu2}
\Tnr_\nu = \frac{1}{3} \fb{11}{4}{\frac{1}{3}} \frac{180 \z(3)}{7 \pi^4} \sumnu \simeq 0.148 \sumnu \, .$$ We will use $\Tnr$ later on when comparing different origins of HDM. Insertion in yields the expected result $$k^\text{nr}_\nu \simeq 0.0103 \, \O_\text{m}^\frac{1}{2} \fb{\sumnu}{\text{eV}}{\frac{1}{2}} h \Mpc^{-1} \, .$$ The corresponding free-streaming scale evaluated today reads $$\l^\text{fs}_\nu \simeq 24 \, \frac{\text{eV}}{\sumnu} h^{-1} \Mpc .$$ Altogether, we see that $\sumnu$ sets both $\O_\nu^0$ (thus $f_\nu$) and $\Tnr_\nu$ (thus $k^\text{nr}_\nu$). At the same time, the neutrino temperature is fixed by SM weak interactions. [**Sterile neutrino (and current evidence) –**]{}
![Joint 68%- and 95%-credible contours of the marginalised posterior found in [@Hamann:2013iba] for $\D\Neff$ and $\mnuseff$ of a thermal sterile neutrino model with their 1-d means marked by a dot. Also marked is a best-fit value and the corresponding 3-$\s$ range from oscillation anomalies [@Giunti:2013aea] in a model with one sterile neutrino. The 1+3+1 mass scheme with two sterile neutrinos, which actually might be preferred depending on the datasets considered, implies a large minimal mass like $3.2\eV$ [@Kopp:2013vaa]. []{data-label="fig:posterior"}](Tnr.pdf){width="\columnwidth"}
As prime example for hot thermal relics we consider a light sterile neutrino $\nu_s$ that is thermalised but with a different temperature than the active neutrinos (HDM).[^1] In this case the parameters under consideration are given by:
1\) If $T_{\nu_s}$ denotes the temperature of the sterile neutrino population, $$\label{DNeffnus}
\D\Neff = \left(\frac{T_{\nu_s}}{T_\nu}\right)^4 \, .$$ Inspecting we see that holds by construction for a fermion. Thus a fully-thermalised neutrino species with the energy density $\r_{1\n}=(7/8) (4/11)^{4/3} \r_\g$ corresponds to $\D\Neff=1$. Other particle natures can be considered by appropriate factors in , for example, $4/7$ for a Nambu-Goldstone boson or $8/7$ for a massless $U(1)$ gauge boson. If the sterile neutrino shared the bath temperature once, but decoupled earlier than the active neutrinos, there is a one-to-one correspondence between its decoupling temperature $T_{\nu_s}^\text{dpl}$ and its temperature at later times. It is colder as it missed heating compared to the bath, when particles annihilated away. So $\D\Neff = (\gast(T_{\nu}^\text{dpl})/\gast(T_{\nu_s}^\text{dpl}))^{4/3}$, where $\gast$ is the effective number of relativistic degrees of freedom and the decoupling temperature depends on the sterile neutrino’s couplings to the bath. The evidence in [@Hamann:2013iba] for $\D\Neff > 0$ is mainly driven by local measurements of $H_0$ and seem to be supported by lensing observations [@Hamann:2013iba; @Battye:2013xqa]. Independent evidence arises from the large tensor-to-scalar ratio reported by the BICEP collaboration [@Ade:2014xna]. In the simplest model of inflation a large ratio implies a large scalar spectral index that increases the power, in particular, at higher multipoles. Whereas an increased $\Neff$ suppresses power at higher multipoles [@Hou:2012xq], which is due to increased Silk damping caused by the increased expansion rate [@Hou:2011ec]. 2) The sterile neutrino’s non-relativistic energy density today, $$\label{Onus}
\O_{\nu_s}^0 = \frac{3}{11} \frac{n_\g^0}{\rhoc} \fb{T_{\nu_s}}{T_\nu}{3} \frac{\mnus}{\text{eV}}
\Leftrightarrow \O_{\nu_s}^0 h^2 \simeq 0.0106 \frac{\mnuseff}{\text{eV}} \, ,$$ can be understood by comparison with . The case of interest is one sterile neutrino, which is much heavier than the active neutrinos. Thus the sum of neutrino masses is to be replaced by the sterile neutrino mass $\mnus$. The active neutrinos are then taken into account separately as a single massive eigenstate with $m_\nu = 0.06 \eV$. Number densities decrease as $\propto T^3$, which explains the temperature ratio to the third power. We just defined the effective hot dark matter mass $$\mnuseff = \mnus (\Tnus/T_\n)^{1/3} .$$ Other particle natures can be considered by appropriate factors in , for example, $2/3$ for a Nambu-Goldstone boson or $4/3$ for a massive $U(1)$ gauge boson. Finally, if a hot relic decouples earlier than the active neutrinos, it does not receive any corrections from $e^+e^-$ annihilation. Since this is the case of interest, the pre-factor is smaller by $3/\Neff^\text{sm}$. There is no factor of $1/3$, because the sterile neutrino possesses three times the mass of one degenerate neutrino in . The current evidence for $\mnuseff > 0$ is mainly driven by galaxy cluster data [@Wyman:2013lza; @Hamann:2013iba] and supported by galaxy [@Hamann:2013iba; @Battye:2013xqa] as well as CMB lensing data [@Battye:2013xqa]. Even before the Planck cluster count became available, cluster data has pushed evidence for $\mnuseff > 0$ [@Hou:2012xq; @Burenin:2013wg]. Corresponding HDM fractions $f$ are large enough to suppress the amplitude of matter fluctuations measured in $\s_8$ (= root-mean-square fluctuation in total matter in $8 \, h^{-1} \Mpc$ spheres at $z=0$, computed in linear theory) down to values inferred from these “local” observations. At the same time they appear small enough not to spoil the CMB fit, see also the next paragraph. 3) The average momentum of a sterile neutrino population is lowered (or increased) compared to the active neutrinos by their temperature ratio $T_{\nu_s}/T_\nu$. Comparing with and , we find $$\label{Tnrnus}
\Tnr_{\nu_s} = \frac{180 \z(3)}{7 \pi^4} \fb{11}{4}{\frac{1}{3}} \frac{T_{\nu}}{T_{\nu_s}} \mnus
\simeq 0.445 \, \frac{\mnuseff}{\D\Neff}$$ Now, there is a factor of three for the reason given two paragraphs above. For completeness, we insert into obtaining $$k^\text{nr}_{\nus} \simeq 0.0178 \, \O_\text{m}^\frac{1}{2} \fb{\mnuseff}{\text{eV}}{\frac{1}{2}} \D\Neff^{-\frac{1}{2}} h \Mpc^{-1}$$ and $$\lfs_{\nus} \simeq 8.1 \, \O_\text{m}^\frac{1}{2} \D\Neff \frac{\text{eV}}{\mnuseff} h^{-1} \Mpc .$$
In Fig. \[fig:posterior\] we re-plotted the countours found in [@Hamann:2013iba]. They lie roughly around the line $\Tnr_{\nus} = T_\g^\text{dpl}$ with an upper bound on $\D\Neff$. If a population becomes non-relativistic before photon decoupling, $\Tnr > T_\g^\text{dpl}$, it affects the CMB directly, see e.g. [@Dodelson:1995es]. The non-observation of such an impact constrains the size of $\mnuseff$, in particular, for large $\D\Neff$. For smaller $\D\Neff$ the impact on the CMB decreases accordingly which allows for somewhat larger $\Tnr_{\nus}$. However, the observations appear to be matched for the obtained best-fits. Even though $\Tnr_\nu$ and $\Tnr_{\nus}$ differ, there is neither preference in the data for $\sumnu >0$ with an additional source for “dark radiation” implying $\D\Neff>0$ nor for HDM. As only difference to HDM, dark radiation is still relativistic today. For such cosmologies parameters are obtained from our work in the limit $m_\text{hdm}\text{, }m_1\text{, }m_2 \rightarrow 0$.
Perfect(?) Mimicry {#sec:mimicry}
==================
We consider the two-body decay of a non-relativistic particle (matter or dust in the cosmological sense). We distinguish between “early” cosmological decays occurring before the onset of the BBN era, so at a time $\t<0.05 \seconds$, and “late” decays occurring during or after the BBN era with $\t>0.05 \seconds$. A decay during the CMB era deserves a dedicated study investigating its impact on the CMB power spectrum. Therefore, we restrict ourselves to decay times $\t \lesssim 5.2 \times 10^{10} \seconds$. This is before the first observable modes of the CMB enter the horizon [@Fischler:2010xz].
In principle, a decay as drawn schematically in Fig. \[fig:dof\] might have to be described by an arbitrarily large number of parameters: mass $m$ and yield $Y$ of the decaying particle (mother) and the masses of the decay products (daughters) $m_{1}^i$, $m_{2}^i$ in each existing decay mode $i$ that sum up to the total width $\G= \sum_i \G_\text{vis}^i + \sum_i \G_\text{dark}^i= \t^{-1}$, where $\G_\text{vis}^i$ and $\G_\text{dark}^i$ denote partial decay widths into particles with and without electromagnetic interactions, respectively. Thus one might naively expect that there is an arbitrarily large number of possibilities for the daughters to show up in the outlined physical effects. In the following, we argue that, as far as the cosmological observables are concerned, the decay can be described by a decisively smaller number of parameters [**Early decay –**]{} The case of interest is the decay of a mother that dominates the energy density of the Universe at her decay (the opposite is covered in the next subsection) and produces HDM in its decay that does not thermalise with the SM bath. The majority of the energy stored in the mother must be transferred to visible particles that unavoidably thermalise. The information how the mother decayed into visible particles –contained in all $\G_\text{vis}^i$– is irrelevant. Instead, the following cosmology will depend on branching ratios $B_i$ into hot dark matter $B_\text{hdm}$ and SM particles $B_\text{vis}$ only as they fix how the mother’s energy is distributed. We just argued $B_\text{vis}\simeq 1 \gg B_\text{hdm}$. If one allows for (additional) decay modes to produce the observed CDM, the corresponding branching ratio were restricted to be much smaller than the one into HDM, because the CDM number density needs to be much smaller. So $B_\text{vis} = 1- B_\text{hdm}$ is either exact, if there is no additional dark decay mode, or holds to a sufficient approximation.
Concerning the kinematics, in the case of mass-degenerate daughters, $m_1 =m_2$, we define $x_2 \equiv m_2/m = m_1/m$ as measure of the mass hierarchy between mother and daughters. It might well be that the well-motivated case of daughters with similar masses, $m_1 \lesssim m_2$, is indistinguishable in cosmological observations from the case of mass-degenerate daughters. This is a subtle case that we leave for future work. If the daughters possess a large mass hierarchy, $m_1 \ll m_2$, the interesting case would be that there is a heavier daughter forming (observable) HDM and an effectively massless one not doing so and thus forming dark radiation. In that case the number of daughters forming HDM $g_\text{hdm}=1$, in contrast to the mass-degenerate case with $g_\text{hdm}=2$. We define $\d \equiv ( m - m_2)/m_2$ as measure of the mass hierarchy between mother and heavier daughter. Altogether, we can describe the kinematics approximately using only one hierarchy measure, $x_2$ or $\d$, depending on the case.
The mother’s energy density together with her lifetime determine the temperature of the thermal bath after her decay $\Trh$ as[^2] $$\label{rhomotherinearlydec}
\r|_\text{dec}= m n|_\text{dec}= \mu^{-1} \frac{\pi^2}{30} \gast^\text{rh} B_\text{vis}^{-1} \Trh^4 \, ,$$ where the correction factor $\m = \m_\text{dom}\simeq 0.877$ considers the exponential decay law in the expanding Universe. We determined it by solving corresponding Boltzmann equations numerically in the limit of strong dominance, $\rho \gg \r_\text{rad}$, cf. [@Scherrer:1987rr]. Superscripts at $\gast$ –or, if the entropy density $s$ is considered, $\gasts$– always indicated the temperature at which the functions are evaluated.
From an, in principle, arbitrarily large number of degrees of freedom we are down to four: $\Trh$, $B_\text{vis}$ (or $B_\text{hdm}$), $x_2$ or $\d$ and potentially $m$. For an early decay the cosmological parameters under consideration are given by:
1\) The relativistic HDM energy density $\r_\text{hdm}|_\text{rel} = n_\text{hdm} \langle p \rangle \simeq n_\text{hdm} (\m/2) m$, where it has been exploited that the daughters must be much lighter than the mother due to structure formation constraints [@Hasenkamp:2012ii]. With $n_\text{hdm} = g_\text{hdm} n$ it can be written as $\r_\text{hdm}|_\text{rel} = \m_\text{dom} \r|_\text{dec} B_\text{hdm} (g_\text{hdm}/2) (T/\Trh)^4 (\gasts/\gasts^\text{rh})^{4/3}$. Inserting we see that $$\label{DNeffearlydec}
\D\Neff = \frac{\r_\text{hdm}|_\text{rel}}{\r_{1\n}} \simeq 8.67 \, \frac{B_\text{hdm}}{B_\text{vis}} \fb{\gastst}{\gast^\text{rh}}{\frac{1}{3}} \, .$$
2\) Its today’s (non-relativistic) energy density $\r_\text{hdm}^0 = n^0_\text{hdm} m_\text{hdm}$ can be written as $\r^0_\text{hdm}= \r|_\text{dec} x_2 g_\text{hdm} B_\text{hdm} (T_0/\Trh)^3 (\gastst/\gasts^\text{rh})$. Inserting we see that it reads $$\label{Ohdmearlydec}
\O^0_\text{ed} h^2 = \frac{\r_\text{hdm}^0 h^2}{\rhoc} \simeq 4.15 \, \mu_\text{dom}^{-1} \frac{x_2}{10^{-8}} \frac{B_\text{hdm}}{B_\text{vis}} \frac{\Trh}{\text{GeV}}\, .$$
3\) The temperature of the Universe when the population of free-streaming daughters becomes non-relativistic is found as $$\begin{aligned}
\Tnr_\text{ed} &=& \Trh \frac{2}{\mu} \frac{\d+1}{(\d+1)^2 -1} \fb{\gasts^\text{rh}}{\gastst}{\frac{1}{3}} \nonumber \\
\text{ or } \, &=& \Trh \frac{2}{\mu} (x_2^{-2} -4)^{-\frac{1}{2}} \fb{\gasts^\text{rh}}{\gastst}{\frac{1}{3}} ,
\label{Tnrdecay}\end{aligned}$$ see (8) and (50) of [@Hasenkamp:2012ii], while $\m = \m_\text{dom}$, here. In all cases $\Tnr_\text{dec}< \Teq$ and the minimal comoving free-streaming wavenumber of the daughters is set by with $\Tnr$ given by .
By inspection of and we see that the mother’s mass does not enter independently. Furthermore, the observables depend on the ratio of the branching ratios $B_\text{hdm}/B_\text{vis}$ only. In other words, they depend on the amount of HDM relative to the amount of visible matter at the decay. This reduces the number of degrees of freedom in the description of the early decay to three: $\Trh$, $B_\text{hdm}/B_\text{vis}$ and a hierarchy measure, either $\d$ or $x$, depending on the case under investigation. A dominating mother produces significant entropy that dilutes relic densities during her decay. The corresponding dilution factor $\D$ can be given as [@Hasenkamp:2010if] $$\D = \frac{\langle \gast^\frac{1}{3} \rangle^\frac{3}{4}}{(\gast^\text{rh})^\frac{1}{4}} \frac{mY}{\Trh} \, ,$$ where the angle brackets indicate the appropriately-averaged value of $\gast$ over the decay interval. We see that $\D$ is independent of the HDM observables. The dilution is an *additional, independent* physical effect. After discovering the primordial gravitational wave background in the CMB, we *know* that this impact of the decay will be ultimately tested by a local detection of this background radiation [@Durrer:2011bi] that would rule out any such early period of matter domination .
[***Late decay –***]{} For $\t > 0.05 \seconds$ current constraints on $\D\Neff$ forbid the particle to dominate the energy density of the Universe prior its decay, see Sec. 2.1 of [@Hasenkamp:2012ii].[^3] For such late decays, if the mother is sufficiently heavier than the proton, the branching ratio into any visible particles is constrained from BBN and CMB observations to be much smaller than one, $\sum_i \G_\text{vis}^i \ll \sum_i \G_\text{dark}^i$, see Sec. 3 of [@Hasenkamp:2012ii]. This holds even if photons and electrons are emitted at the end of a decay chain only, cp. [@Cline:2013fm]. If one allows for (additional) decay modes to produce the observed dark matter, the corresponding branching ratio were restricted to be much smaller than the one into hot dark matter, $\sum_i \G_\text{dark}^i \simeq \G_\text{hdm}$, see Sec. 4.2 of [@Hasenkamp:2012ii]. In other words, the HDM branching ratio is to a very good approximation one and the lifetime of the mother is given by her HDM decay width. This implies a decisive reduction of parameters. From an, in principle, arbitrarily large number we are down to four: mass $m$, yield $Y$ and lifetime $\t$ of the mother and one of the two hierarchy measures, $\d$ or $x_2$. For the late decay following [@Hasenkamp:2012ii] the cosmological parameters under consideration are given by:
1\) If $\Tdec= T(\t)$ denotes the temperature of the Universe at decay[^4], $$\label{DNeffdecay}
\D\Neff = 10.25 \, \frac{m Y}{\Tdec} \fb{\gastst}{\gastsd}{\frac{1}{3}} \frac{(\d+1)^2-1}{(\d+1)^2} \, .$$ Again taking into account the huge mass hierarchy, $\d\gg 1$, required from structure formation constraints, the last fraction is also in this case to a good approximation one. This holds analogously, if the daughters are mass degenerate, see (20) of [@Hasenkamp:2012ii]. 2) The non-relativistic energy density of the daughters today is given by, cf. (45) and (55) of [@Hasenkamp:2012ii], $$\begin{aligned}
\label{Odecay}
\O_\text{ld}^0 &= g_\text{hdm} \frac{mY}{\d +1} \frac{\gastsd}{\gastst} \frac{s_0}{\rhoc} \nonumber \\
& \Leftrightarrow \O_\text{ld}^0 h^2 = \frac{2.76 \times 10^8}{\d+1} \frac{mY}{\text{GeV}} g_\text{hdm} \frac{\gastsd}{\gastst}\, .\end{aligned}$$ For a decay into mass-degenerate daughters we should replace $1/(\d +1) \rightarrow x_2 $. Note that the yield $Y$ is the one in . So it is to be evaluated at the mother’s decay. nto two identical particles we should replace $1/(\d +1) \rightarrow 2 x_2 = 2 m_2 /m$. Note that by construction the same decay leads to both, $\D\Neff>0$ and $\mnuseff>0$. We discuss the implications of a second decay mode into dark radiation or HDM below.
3\) The temperature of the Universe when the population of free-streaming daughters becomes non-relativistic is given by with $\m = \sqrt{\pi}/2$.
We inspect and to complete our counting of degrees of freedom in the description of a late cosmological particle decay, for the moment. We see that both observables depend on the product $mY$ only and not on $m$ or $Y$ independently. They depend on the energy density at decay and not on how the energy density is built from a large mass or number density, as for an early decay. Thus observations are not sensitive to this degeneracy in determining the energy density of the mother. So instead of an arbitrarily large number of degrees of freedom we identify three: the lifetime $\t$, the energy density of the decaying particle $\r(=mYs)$ and a hierarchy measure, either $\d$ or $x$, depending on the case under investigation. Even though the descriptions for the early and late decay are qualitatively different, we find in both cases three degrees of freedom that are possibly affecting the outlined three main physical effects of a free-streaming population. One might compare this to HDM with two parameters, i.e., $\Tnus$ and $\mnus$. In contrast to HDM, no (possibly unnatural) small mass scale needs to be introduced.
Translating observations
------------------------
![ Determination of $mY(\t)$ depending on $\Tnus$ exploiting . A higher $\Tnus$ is mimicked by a correspondingly larger energy density of the decaying particle at decay. The solid line corresponds to $\Neff = 0.61 \pm 0.60$ with its 2-$\s$ range as grey band. The dashed line marks Planck’s CMB only mean $\D\Neff=0.29$ [@Ade:2013zuv]. ](DNeffdecay.pdf "fig:"){width="\columnwidth"} \[fig:DNeffdecay\]
![Determination of $mY(\d)$ depending only on $\mnuseff$. A larger $\mnuseff$ is mimicked by a correspondingly smaller mass hierarchy. The solid line corresponds to $\mnuseff/\text{eV} = 0.41 \pm 0.26$ with its 2-$\s$ range as grey band. The dashed line marks Planck’s CMB-only upper bound $\mnuseff < 0.59 \eV$ (95%) [@Ade:2013zuv]. We have chosen $g_\text{hdm}=1$ for the figure. If the decay is into mass-degenerate daughters, the horizontal axis should carry $x_2^{-1}$ and all lines shift down by $1/2$ as $g_\text{hdm}=2$. ](Odecay.pdf "fig:"){width="\columnwidth"} \[fig:Odecay\]
![Determination of $\d(\t)$ depending only on $\mnuseff$ exploiting and with fixed $\Tnus$. For a fixed $\mnuseff$ the later the decay the smaller the mass hierarchy might be. The solid line corresponds to $\mnuseff/\text{eV} = 0.41 \pm 0.26$ with its 2-$\s$ range as grey band, while $\D\Neff=0.61$ is kept fix. The jump around $t\sim 10 \seconds$ corresponds to the time of $e^+e^-$ annihilation. Planck’s CMB only upper bound $\mnuseff < 0.59 \eV$ and $\D\Neff < 0.86$ (both 95%) [@Ade:2013zuv] is barely visible right on the solid line. We have chosen $g_\text{hdm}=1$ for the figure. If the decay is into identical particles, the vertical axis should carry $x_2^{-1}$ and all lines shift up by $1/2$ as $g_\text{hdm}=2$. ](Odecay2.pdf "fig:"){width="\columnwidth"} \[fig:Odecay2\]
In the following we show actually that dpHDM and HDM are in general indistinguishable in CMB and large-scale structure observations using $\Neff$ and $\mnuseff$ only. First, we provide the prescription how to obtain (for any “observed” sterile neutrino parameter) the corresponding parameters describing the cosmological particle decay with exactly the same signal. In other words, parameters determined assuming a sterile neutrino cosmology can be translated back and forth into parameters describing a cosmology with HDM originating from a cosmological particle decay.
[**Early decay –**]{} Comparing and we see trivially that a higher sterile neutrino temperature is mimicked by a larger $B_\text{hdm}/B_\text{vis}$ (for fix $\gast^\text{rh}$). Furthermore, equating and we find how the effective HDM mass is mimicked, $$\label{mnuseffearlydec}
\mnuseff = \mu_\text{dom}^{-1} \frac{11 \pi^4 \gastst}{90 \zeta(3)} \frac{B_\text{hdm}}{B_\text{vis}} x_2 \Trh \, .$$ So the prescription is: i) Fix $B_\text{hdm}/B_\text{vis}$ to obtain the desired $\D\Neff$. ii) Determine $x_2 \Trh$, while $\D\Neff$ is kept fix. At this point keep in mind that the description is valid for $\Trh > \MeV$ only. For illustration, we translate into $$\label{sigDNeffearlydec}
\frac{B_\text{hdm}}{B_\text{vis}} \left( \gast^\text{rh}\right)^{-\frac{1}{3}} = 0.045 \pm 0.022
\text{ } (\text{``1-}\s\text{''} \text{; } \t < 0.05 \seconds)$$ and $$\begin{aligned}
\label{sigmnuseffearlydec}
\frac{B_\text{hdm}}{B_\text{vis}} \frac{x_2}{10^{-9}} \frac{\Trh}{\text{GeV}} = (9.2 \pm 2.9) & \times 10^{-3} \nonumber \\
& (\text{``1-}\s \text{''; } \t < 0.05 \seconds) \end{aligned}$$ for a scenario with mass-degenerate daughters. If the daughters possess a large mass hierarchy, such that $g_\text{hdm}=1$, a) nothing changes for $\D\Neff$ and b) $x_2 \rightarrow 1/(\d+1)$ in and its right-hand-side is to be divided by 2. However, in this case there is a shift in $\Tnr$ as shown below and thus a difference in the main physical effects. We would like to point out that and (later also and ) are by way of illustration only. We simply “translated” parameters, while the confidence intervals are not determined by a likelihood ratio, but by the area below the posterior function that is not invariant under non-linear parameter transformations. In Sec. \[sec:breakdown\] we identify (two, independent) parameters that should be determined in a Markov Chain Monte Carlo likelihood analysis with approriate priors.
[**Late decay –**]{} How to mimic a sterile neutrino in $\Neff$ is shown in Fig. \[fig:DNeffdecay\]. The horizontal axis spans the full range of considered lifetimes. We take the temperature dependence of $\gasts$ fully into account. Around $\t\sim10 \seconds$, when $e^+e^-$ annihilate away, $\gasts$ decreases. The curves stay straight as for the same reason the temperature of the Universe increases, both effects cancelling each other up to the difference between $\gastt\simeq3.384$ and $\gastst\simeq3.938$. It is roughly $(\gastst/\gastt)^{1/3} \simeq 1.05$. The time when BBN ends $t_\text{bbn}^\text{end}$ is highlighted. Decays before this time increase the effective number of neutrino species inferred from BBN $\D\Neff|_\text{bbn}$ partially as quantified in [@Menestrina:2011mz]. Comparing and we see that a higher $\Tnus$ is mimicked by a larger energy density of the mother at decay. This has been shown in [@Hooper:2011aj] solving the corresponding Boltzmann equations numerically. The values chosen for $\D\Neff$, which are in one-to-one correspondence with $\Tnus$ via , are given in the figure caption. We can see how a narrow range of $\Tnus$ maps onto a narrow range of $mY(\t)$ forming a strip in the $mY$-$\t$ plane. After having fixed $mY/\Tdec$ we can use Fig. \[fig:Odecay\], which is obtained by setting equal to , to determine the correct $\d(mY)$ mimicking the effective HDM mass $$\begin{aligned}
&\mnuseff \simeq 26.0 \, g_\text{hdm} \frac{mY}{\d+1} \frac{\gastsd}{\gastst} \nonumber \\
&\text{ or } \simeq 52.0 \, x_2 mY \frac{\gastsd}{\gastst} \, . \end{aligned}$$ Alternatively, we can use to re-write , which equals , obtaining $$\label{deltaofobs}
\d \simeq 5 \times 10^3 \, \frac{\Tdec}{\text{keV}} \D\Neff \frac{\text{eV}}{\mnuseff}
\fb{\gastsd}{\gastst}{\frac{4}{3}} \frac{g_\text{hdm}}{2} -1$$ which allows to provide Fig. \[fig:Odecay2\]. For a given lifetime we can thus read off the corresponding $mY$ and $\d$ to mimic any “observed” sterile neutrino parameters.
Altogether, we found the following prescription: i) Determine $mY(\t)$ using such that the desired $\D\Neff$ is obtained. ii) Determine $\d(\t)$ using such that the desired $\mnuseff$ is obtained while $\D\Neff$ is kept fixed. This makes dpHDM indistinguishable from HDM in $\Neff$ and $\mnuseff$. For illustration, we translate into $$\begin{aligned}
\label{sigDNeff}
\frac{mY}{\Tdec}\fb{\gastst}{\gastsd}{\frac{1}{3}} &= 0.060 \pm 0.029 \nonumber \\
& (\text{``1-}\s \text{''; } 0.05 \seconds <\t <5.2 \times 10^{10} \seconds)\end{aligned}$$ and $$\begin{aligned}
\label{sigmnuseff}
x_2 \, mY \frac{\gastsd}{\gastst} &= (7.9 \pm 2.5)\times 10^{-3} \eV \nonumber \\
&(\text{``1-}\s \text{``; } 0.05 \seconds <\t <5.2 \times 10^{10} \seconds)\end{aligned}$$ for a scenario with mass-degenerate daughters. If the daughters possess a large mass hierarchy, such that $g_\text{hdm}=1$, proceed as discussed below .
It is important that –in contrast to the early decay case– the assumption of BBN consistency is wrong, because for a late cosmological decay $\Neff|_\text{bbn} < \Neff|_\text{cmb}$. The possibility of late cosmological decays motivates determinations of $\Neff|_\text{cmb}$, where the primordial abundance of light elements is constrained from direct observations only, so *really* independent of cosmology at earlier times, and fixed $\Neff|_\text{bbn}=\Neff^\text{sm}$. For earlier related work with a different approach in a different scenario see [@GonzalezGarcia:2012yq].
[**A degeneracy –**]{} For the early decay we can exploit , and to obtain the following *parameter degeneracy* relation $$\label{Tnrobs2}
\Tnr_\text{ed} = \frac{180 \zeta(3)}{7 \pi^4} \fb{11}{4}{\frac{1}{3}} \frac{\mnuseff}{\D\Neff} \frac{2}{g_\text{hdm}} \, .$$ Analogously, if we use and eliminating $mY$, we can single out $\Tdec (\d+1)/ ((\d+1)^2 -1)$. Inserting the resulting expression into we obtain $$\label{Tnrobs}
\Tnr_\text{ld} \simeq 0.445 \frac{\mnuseff}{\D\Neff} \frac{\gastst}{\gastsd} \frac{2}{g_\text{hdm}} \, .$$ We see that the main physical effects are described by only two independent parameters. In other words, there is a degeneracy among the three physical parameters. The same degeneracy has been identified for the thermal relic case in [@Acero:2008rh].[^5] Indeed, we find exactly the same degeneracy, i.e. the same numerical value, if we consider $g_\text{hdm}=2$ and -for the late decay- $\gastsd = \gastst$, which is fulfilled after $e^+e^-$ annihilation, so in a large part of the parameter space. Viewed as a constraint equation the degeneracy in physical parameters reduces the number of independent degrees of freedom describing dpHDM from three to two. Considering $\Neff$ and $\mnuseff$ only, which can describe the main physical effects, an early cosmological particle decay, as well as a decay after $e^+e^-$ annihilation, as origin of HDM is indistinguishable from HDM, if the daughters are mass degenerate. The mimicry is ”perfect“ in the sense that there is no new signature in the main physical effects as those contained in HDM parameters ($\Tnus$ or $\D\Neff$ and $\mnus$ or $\mnuseff$).
[***More complex scenarios***]{} If the daughters do not possess a large mass hierarchy, $m_1\lesssim m_2$, their impact on observables is similar to the case of two sterile neutrinos with equal temperature but different mass. Moreover, considering two sterile neutrino populations with sufficiently different temperatures, one neutrino might not account for $\O_\text{hdm}$ as being still relativistic today and thus forming dark radiation. This situation would correspond to two dark decay modes in the cosmological particle decay, one for dark radiation and one for HDM. All we would like to point out here is that these scenarios are more complex also in the case of a thermal relic. Consequently, they are not discussed in this work and left for future work.
The arguably most interesting case is found if the heavier daughter forms HDM and the lighter one dark radiation. Then the temperature of the universe, when the HDM becomes non-relativistic is a factor of two larger as $g_\text{hdm}=1$ in and , respectively. Translated to sterile neutrinos, this situation is represented by two sterile neutrino populations with the same temperature, while one is massive and the other one is massless. Interestingly, current (non-zero) mean values for a HDM contribution correspond to $\Tnr$s around $T_\g^\text{dpl}$, see Fig. \[fig:posterior\]. If $\Tnr$ becomes significantly larger than $T_\g^\text{dpl}$ due this factor of two, this effect could be used to distinguish this case, because the impact on the CMB becomes qualitatively different for $\Tnr >T_\g^\text{dpl}$ [@Dodelson:1995es]. As mentioned already, the non-observation of such an impact constrains the size of $\mnuseff$. While we do not attempt quantitative statements, here, it seems that the increase in $\mnuseff$ would be in stronger tension with the CMB data for $g_\text{hdm}=1$. This could disfavour this case compared to the case of mass-degenerate daughters. On the other hand, if a late decay occurs before $e^+e^-$ annihilation, the daughters become non-relativistic at a time $\gastsd/\gastst \simeq 2.73$ later which reduces the constraining power from the direct impact on the CMB.
For lower $\Tnr$ it has been argued in [@Hannestad:2005bt] that an optimal LSST-type wide field survey might provide the ability to distinguish between thermal, fermionic HDM (Majorana fermion) and thermal, bosonic HDM (scalar). For fixed $\O_\text{hdm}$ the scalar becomes non-relativistic at a temperature, which is a factor of $3/2$ larger than in the case of a fermion. This is simply the increase in mass compensating the factor considering the scalar particle nature in . We might conclude that the same observations are able to differ between HDM and dpHDM where the heavier daughter forms HDM and the lighter one dark radiation. However, on the observed non-linear scales N-body simulations appear to become necessary [@Bird:2011rb] and HDM infall might become an observable effect with the increased precision [@Hannestad:2005bt]. The exploration of this interesting opportunity to probe the origin of HDM is beyond the scope of this work. In future work, all discussed scenarios can be implemented in CAMB[^6] or CLASS[^7] easily, because they can be represented by thermal fermion populations as just described and summarised in Tab. \[tab:summary\].
Breakdown in the Third {#sec:breakdown}
----------------------
![Normalised, time-invariant probability distribution $P(x=p/p(\t,t))$ for finding a relativistic particle from two-body decay (decaying particle at rest) within the infinitesimal momentum interval $[p,p+\d p]$. In the early decay case the mother may dominate (black, solid) the Universe. For a late decay the Universe is radiation dominated (black, dashed). The Universe becomes matter-dominated (grey, dotted) at very late times only. Highlighted with corresponding lines are the maxima and means. []{data-label="fig:Px_overplot"}](P_x__overplot.pdf){width="\columnwidth"}
It is a simplification to reduce the physical impact of any population of massive free-streaming particles to the effects listed in Sec. \[sec:preliminaries\]. The free-streaming effect must depend on the details of the momentum distribution $f(p)$ of the population. For illustration imagine a distribution with a sharp peak close to $p=0$. Such particles would act as cold dark matter and thus should not be counted within the massive free-streaming component . Furthermore, the average considered in were between cold and hot particles, so that the correct physical effects were not captured. [**Observables and statistical moments –**]{} Distribution functions can be specified by their moments $Q^{(n)}$. This has been done for the Fermi-Dirac (active neutrino) distribution $f_\text{fd}(y)=1/(e^y +1)$ with comoving momentum $y=pa$ in [@Cuoco:2005qr]. As their result can be adapted to the sterile neutrino case straightforwardly, we sketch decisive steps in Appendix \[appendix:nudistr\]. For a thermal population of sterile neutrinos we define $$\label{Qnus}
Q^{(n)}_{\nus} = \frac{1}{\pi^2} \fb{4}{11}{\frac{3+n}{3}} \fb{\Tnus}{T_\nu}{3+n} T^{3+n}
\int y^{2+n} f_\text{fd}(y) dy \, .$$ The (first two) cosmological parameters can be expressed by these moments as $$\label{NeffbyQnus}
\Neff = 3.046 + \frac{120}{7 \pi^2} \fb{11}{4}{\frac{4}{3}} T^{-4} Q^{(1)}_{\nus}$$ and $$\begin{aligned}
\O_{\nus}^0 &= \frac{\mnus}{\rhoc} \fb{T_0}{T}{3} Q_{\nus}^{(0)} \Leftrightarrow \nonumber \\
\O_{\nus}^0 h^2 & \simeq 0.160 \, \frac{\mnuseff}{\text{eV}} \D\Neff^\frac{3}{4} T^{-3} Q_{\nus}^{(0)} \, .
% f_{\nus} = 0.5086 \frac{\mnus}{\text{eV}} \frac{0.22}{\Odm} \frac{11}{4} T^{-3} Q^{(0)}_{\nus} \, .\end{aligned}$$ We see that $\Neff$ and $\O_{\nus}^0$ probe the first two moments of the momentum distribution function.
If the mother does not dominate the Universe at her decay, the momentum distribution function of relativistically emitted daughters (cf. appendix of [@Hasenkamp:2012ii] and [@Scherrer:1987rr]) as function of comoving momentum $y$ reads $$\label{decdistr}
df(y)= \nini c p_\text{ini}^{-c} y^{c-1} e^{-(y/p_\text{ini})^c} dy \, ,$$ where $n_\text{ini}$ denotes the number density of daughters at decay and $p_\text{ini}= m (1-(\d+1)^{-2})/2$ or $=m (1 - 4 x_2^2)^{1/2}/2$ their initial momentum. For the considered lifetime range it is well approximated by $p_\text{ini} \simeq m/2$. In Fig. \[fig:Px\_overplot\] we depicted the corresponding probability distribution function.
For the late decay we use explicitly that the Universe is radiation dominated, so that $c\equiv 3(1+\o)/2 =2$ as the equation of state of the Universe, $p = \o \r$, is given by $\o=1/3$ in that case. Nevertheless, our treatment could be applied to any expansion law with $c>0$. By inspection of we can see that the distribution function decays as $e^{-y}$ for large comoving momenta as the neutrino distribution function considered in [@Cuoco:2005qr]. We define the moments as $$\label{Qdec}
Q_\text{dec}^{(n)} = \frac{\nini}{\Tdec^3} \fb{\pini}{\Tdec}{n} \fb{\gastst}{\gastsd}{\frac{n}{3}} T^{3+n} \int c y^{c+n-1} e^{-y^c} dy$$ Then the (first two) cosmological parameters expressed by these moments read $$\Neff = 3.046 + \frac{120}{7\pi^2}\fb{11}{4}{\frac{4}{3}} T^{-4} Q_\text{dec}^{(1)}
%\simeq 3.046 + 5.93069 \frac{\nini \pini}{\Tdec^4} \fb{\gastst]}{\gastsd}{\frac{1}{3}}$$ and $$\begin{aligned}
\label{OQdec}
\O_\text{dec}^0 &= \frac{g_\text{hdm}}{2} \frac{m_2}{\rhoc} \fb{T_0}{T}{3} Q_\text{dec}^{(0)} \Leftrightarrow \nonumber \\
\O_\text{dec}^0 h^2 &\simeq 0.160 \, \frac{g_\text{hdm}}{2} \frac{m_2}{\text{eV}} T^{-3} Q_\text{dec}^{(0)}
% f = m_2 Q^{(0)} \rhoc^{-1} \Odm^{-1} = 1.3986 \frac{\nini}{\Tdec^3} \frac{m_2}{\text{eV}} \frac{0.22}{\Odm} \, .\end{aligned}$$ Note that $n_\text{ini}= 2 n$. We identify two dimensionless, independent parameters $\nini/\Tdec^3$ and $\pini/\Tdec$ in accordance with the previous discussion. In future work, these could be determined with a Markov Chain Monte Carlo likelihood analysis. Our discussion from to is an alternative way to demonstrate why the mimicry in $\Neff$ and $\mnuseff$ is ”perfect“.
In the case of a dominating mother, the simple analytic form is invalid. Boltzmann equations need to be solved numerically, cf. [@Scherrer:1987rr]. In Fig. \[fig:Px\_overplot\] we depicted the resulting probability distribution in the limit of strong dominance, $\r/\r_\text{rad} \gg 1$, at decay. Concerning the moments , the factor $\gastst/\gastsd$ were absent and the integral could not be performed analytically.
[**Root-mean-square of absolute momenta –**]{} The momentum distribution function of a thermal relic differs from the momentum distribution function of relativistic decay products, cp. and . The two available degrees of freedom have been used to achieve perfect mimicry in the first two cosmological parameters, which have been expressed by the first two moments of the distribution functions. Therefore, the *mimicry must break down* if the next (third) cosmological parameter, which is to be expressed by the next higher moment of the distribution function, is taken into account. The number of parameters to be mimicked then exceeds the number of degrees of freedom.
We identify the third cosmological parameter as the root-mean-square of absolute momenta $\langle p \rangle_\text{rms}$ –or, equivalently, the root-mean-square of absolute velocities today[^8]–, which is given by the quadratic mean of the distribution of absolute momenta. This quadratic mean is equal to the skewness of the momentum distribution keeping the directional information. In general, the skewness is the next higher moment following the quadratic mean. The next higher moment to the skewness is the kurtosis. Giving preference to dimensionless parameters we define the *normalised root-mean-square of absolute momenta* $$\label{defnormrmsmomentum}
\g^\text{rms} \equiv \frac{\langle p \rangle_\text{rms}}{\langle p_\nu \rangle_\text{rms}}$$ where $\langle p_\nu \rangle_\text{rms}$ is the root-mean-square of the absolute momentum of the active neutrinos , so that $\g_\nu^\text{rms} = 1$ by definition. The root-mean-squares of absolute momenta can be expressed by moments of the corresponding momentum distribution functions. This can be understood by reminding that the number density $n = Q^{(0)}$ and the relativistic energy density $\r =\langle p \rangle n = Q^{(1)}$, so that the mean absolute momentum $\langle p \rangle = Q^{(1)}/Q^{(0)}$. Analogously, $\langle p \rangle_\text{rms} = (Q^{(2)}/Q^{(0)})^{1/2}$.
For a thermal sterile neutrino population we find $$\label{nusprms}
\langle p_{\nu_s} \rangle_\text{rms}
= \fb{15 \zeta(5)}{\zeta(3)}{\frac{1}{2}} \fb{4}{11}{\frac{1}{3}} \frac{\Tnus}{T_\n} T \simeq 3.6 \, \Tnus \,,$$ which implies with a normalised root-mean-square of absolute momenta $$\label{grmsnus}
\g_{\nu_s}^\text{rms} = \frac{\Tnus}{T_\n} = \D\Neff^{\frac{1}{4}}$$ that can be expressed by lower momenta . This is to be expected, since $\g_{\nu_s}^\text{rms}$ cannot carry additional information as both neutrino species possess a Fermi-Dirac distribution.
The momentum distribution function of the sterile neutrinos in the popular Dodelson-Widrow (DW) scenario [@Dodelson:1993je] (a.k.a. ”non-resonant production scenario“) reads $$f_\text{dw}(y) = \frac{\chi}{e^y+1} = \chi f_\text{fd} \,.$$ The sterile neutrinos share the same ”observable“ parameters ($\D\Neff$, $\O_\text{hdm}^0$, $T^\text{nr}$) as a thermal population with $\mnus^\text{th} = \chi^{1/4} \mnus^\text{dw}$ and $\Tnus^\text{th} =\chi^{1/4} T_\nu$. For these two models it has been shown by a change of variable in the background and linear perturbation equations that the two models are strictly equivalent for cosmological observables [@Colombi:1995ze]. We confirm this result with the expansion of the distribution functions in moments. Compensating the scaling factor $\chi$, the two models share the same (Fermi-Dirac) distribution function. Higher moments can be expressed by lower ones and the distribution functions are identical. Therefore, they are indistinguishable for cosmological observations even if higher moments are taken into account. In other words, their *mimicry is perfect to arbitrary order in moments*. This is *qualitatively different for the decay products* of a cosmological particle decay mimicking a thermal sterile neutrino population. For daughters emitted in a late decay we find $$\langle p_\text{dec} \rangle_\text{rms} =
\fb{Q_\text{dec}^{(2)}}{Q_\text{dec}^{(0)}}{\frac{1}{2}}
= \frac{\pini}{\Tdec} T \fb{\gastst}{\gastsd}{\frac{1}{3}} \, ,$$ where we used that $\int c y^{c+n-1} e^{-y^c} dy =1 $ for $c=2$ in both cases, $n=0$ and $n=2$. We obtain a very simple, exact expression. With this implies $$\begin{aligned}
\g_\text{ld}^\text{rms} &=&
\fb{\zeta (3)}{15 \zeta (5)}{\frac{1}{2}}
\fb{11}{4}{\frac{1}{3}}
\fb{\gastst}{\gastsd}{\frac{1}{3}}
\frac{\pini}{\Tdec} \nonumber \\
&\simeq& 0.389 \, \frac{\pini}{\Tdec} \fb{\gastst}{\gastsd}{\frac{1}{3}} ,\end{aligned}$$ which can be expressed as $$\label{grmsdec}
\g_\text{ld}^\text{rms} \simeq 0.0105 \, \frac{\D\Neff}{\O_\text{ld}^0 h^2} \frac{g_\text{hdm}}{2}$$ by lower moments. Again, this must be, since there are no further parameters. All higher momenta are fixed by the first two.
![Momentum distribution functions with perfect mimicry in the first two cosmological observables corresponding to their first two moments and their next higher moment. Depicted are distributions for the early decay (black, solid), the late decay (black, dashed), Pauli-Dirac (black, dash-dotted), Einstein-Bose (grey, dash-dotted) and decay in a matter-dominated universe (grey, dashed). Zeroth and first moment have been set to unity. Vertical lines mark the corresponding second moments (= skewness). Units are arbitrary. []{data-label="fig:fp_overplot"}](f_p__overplot.pdf){width="\columnwidth"}
The predicted deviation between HDM and dpHDM (with $g_\text{hdm}=2$) in the normalised root-mean-squares of absolute momenta is thus found by combining and as $$\g_\text{ld}^\text{rms}/\g_{\nus}^\text{rms} \simeq 0.0105 \, \frac{\D\Neff^\frac{3}{4}}{\O_\text{dec}^0 h^2} \simeq 0.989 \, \frac{\D\Neff^\frac{3}{4}}{\mnuseff/\text{eV}} \, .$$ Under the crude assumption of Gaussian error propagation this implies for a prediction, $$\g_\text{ld}^\text{rms}/\g_{\nus}^\text{rms} = 1.66 \pm 0.81 \; (\text{1-}\s)\, .$$ Since the momentum distribution functions differ between early and late decay as can be seen in Fig. \[fig:fp\_overplot\], these numbers differ between these two cases. We find $ \langle p_\text{ed} \rangle_\text{rms} \simeq 1.03 \times \langle p_\text{ld} \rangle_\text{rms}$ and thus $$\g_\text{ed}^\text{rms}/\g_{\nus}^\text{rms} = 1.72 \pm 0.84 \; (\text{1-}\s)\, .$$ So there is a 66% (72%) difference in the mean for the late (early) decay but the uncertainty of the prediction is decisively larger. Obviously, the difference to HDM is too small to be detected in an extended analysis that would have to try to determine $\g^\text{rms}$ as additional free parameter. Not to mention that the observations need to be sensitive to this effect. We do not expect that current observations or observations in the near future will be able to distinguish dpHDM from HDM in all cases. In this sense, the mimicry can be ”perfect for our limited observational capabilities.“[^9] It is tempting to look for a first hint if this negative conclusion could change in the next decade. With full Planck data available the galaxy survey EUCLID (to be launched 2020) will increase sensitivities dramatically by roughly an order of magnitude, cf. [@Hamann:2012fe]. Including galaxy surveys, eBOSS and DESI, and a new (Stage-IV) CMB polarimeter, expected sensitivities go down to $\s(\sumnu) = 16 \meV$ and $\s(\Neff)= 0.020$ [@Abazajian:2013oma]. These sensitivities would imply 1-$\s$ errors in the prediction of $\g_\text{dec}^\text{rms}/\g_{\nus}^\text{rms}$ that are nine times smaller than the predicted deviation for the current mean values . However, these means can and will change, but for means half the size the 1-$\s$ errors were still $3.5$ times smaller than the predicted deviation. If not from testing our standard interpretation alone, the opportunity to distinguish $\nus$HDM from dpHDM, motivates to study which future observations can how far probe the root-mean-square of absolute momenta of a free-streaming population.
Summary and Conclusions {#sec:conclusions}
=======================
opportunity? equivalent
------------------- -- ------------------------ -------------
$m_1 \ll m_2$ $2\times$larger $\Tnr$
$m_1=m_2$
$m_1\lesssim m_2$ as equivalent
: Overview of different cases depending on the daughter masses. In every case there is mimicry in the first two cosmological observables. Given are opportunities to distinguish these cases from HDM and how they are represented by thermal populations.[]{data-label="tab:summary"}
As summarised in Tab. \[tab:summary\], it depends on the daughter masses where and how the mimicry breaks down. Every case can be implemented easily as populations in existing numerical tools. If the daughters do posses a large mass hierarchy, the temperature when dpHDM becomes non-relativistic is larger by a factor of two compared to HDM, so that this case might be in stronger tension with CMB data. If the daughters are mass degenerate, they are indistinguishable from HDM in analyses like [@Hamann:2013iba].
Connecting cosmological ”observables“ with moments of the momentum distribution functions depicted in Fig. \[fig:fp\_overplot\], we find that for mass-degenerate daughters the mimicry breaks down only, if the next higher moment of the momentum distribution, the skewness, is considered. We define the *normalised root-mean-square of absolute momenta* and find sizeable differences in its predicted value between dpHDM and HDM. While these are certainly too small for current observations, this is a *qualitative difference* compared to the attempt to distinguish different HDM models, where the mimicry is perfect to arbitrary order of moments.
Other opportunities depend on the time of decay: For certain times during the BBN era, dpHDM becomes non-relativistic later than HDM. A decay after BBN increases $\Neff|_\text{cmb} > \Neff|_\text{bbn}= \Neff^\text{sm}$, which motivates analyses that drop the BBN consistency relation. To conclude, current cosmological observations are sensitive to sub-eV, not fully-thermalised HDM characterised by $\D\Neff < 1$ and $\mnuseff < \eV$. In that case, from a cosmological point of view HDM is not preferred over dpHDM, neither from theoretical nor practical simplicity. Our (current) blindness for the case of mass-degenerate daughters, should prevent us from premature conclusions when interpreting signals like . Fortunately, there are various cases that can be considered easily and gainfully in likelihood analyses utilising available data already. After our proof of principle that, in contrast to different, thermal HDM models, the mimicry of dpHDM breaks down at least in the root-mean-square of absolute momenta, it is an open question which observation due to which effect will be able to distinguish the two possibilities.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Jan Hamann for valuable discussions. I acknowledge support from the German Academy of Science through the Leopoldina Fellowship Programme grant LPDS 2012-14.
Free-streaming scale and transition time {#appendix:A}
========================================
Starting from (93) in [@Lesgourgues:2006nd], $$k^\text{fs}(T) =\left(\frac{3}{2} \frac{H^2(T) a^2(T)}{\langle v \rangle(T)}\right)^2 \text{, } \l^\text{fs}(T)=2\pi\frac{a(T)}{k^\text{fs}(T)},$$ we approximate the average absolute velocity of the population $\langle v \rangle = \langle p\rangle/m$ at the transition, $\langle p \rangle =m$, as the speed of light $c$. We assume that it then decreases as $a^{-1} \propto T$, so that $\langle v \rangle(T) = c \, T/\Tnr $, if $\Tnr$ denotes the temperature of the Universe at the transition.[^10] The scale factor can be expressed in temperatures as $a(T)/a_0=T_0/T$ and $H^2 = H_0^2 \left(\O_\text{m} (T/T_0)^3 +\O_\L\right)$. Inserting yields $$k^\text{fs}(T) \simeq 4.08 \times 10^{-4} \left(\O_\text{m} \fb{T}{T_0}{3} +\O_\L \right)^\frac{1}{2} \frac{T_0 \Tnr}{T^2} \frac{h}{\text{Mpc}}$$ and $$\lfs(T) \simeq 1.54 \times 10^4 \left(\O_\text{m} \fb{T}{T_0}{3} +\O_\L \right)^{-\frac{1}{2}} \frac{T}{\Tnr} \frac{\text{Mpc}}{h} .$$ For $k^\text{nr}=k^\text{fs}(T^\text{nr})$ we obtain . For reference we provide the often used free-streaming scale of the population today, $$\lfs \simeq 1.54 \times 10^4 \frac{T_0}{\Tnr} h^{-1} \Mpc.$$ Consistently we see from (93) of [@Lesgourgues:2006nd] $$\lfs(\tnr) =2 \pi (2/3)^{1/2} \langle v(\tnr) \rangle/H(\tnr)=\sqrt{6} \pi \tnr \, ,$$ where we used $\langle v(\tnr) \rangle =1$ and $H=2/(3 t)$ in matter domination. We calculate the time when a population becomes non-relativistic in a Universe filled with radiation and matter as $
\tnr = \frac{\teq}{2-\sqrt{2}} \left( \left(\frac{\Teq}{\Tnr} -2 \right) \left( \frac{\Teq}{\Tnr} +1 \right)^{1/2} +2 \right)
$, where $\Teq$ $(\teq)$ denotes the temperature (time) at matter-radiation equality.
Neutrino distribution specified by its moments {#appendix:nudistr}
==============================================
This appendix shall improve the accessibility of Sec. \[sec:breakdown\] for the reader. We repeat findings of Cuoco, Lesgourgues, Mangano and Pastor in [@Cuoco:2005qr] and refer the reader to the original work.
The solution to the collisionless kinetic equations in a Lemaitre-Friedman-Robertson-Walker universe is a Fermi-Dirac function $f(\vec{p}) = 1/(e^{(E-\m)/T}+1)$ with particle energy $E^2=|\vec{p}|^2 +m^2= p^2+m^2$. We are interested in the standard situation with vanishing chemical potential $\m$ and early times, when neutrinos are ultra-relativistic. Considering a radiation dominated universe, where $T\propto a^{-1}$, and defining the comoving momentum $y=pa$ one finds $$\label{nudistr}
df(p,T_\nu)= \frac{1}{\pi^2} T_\nu^3 \frac{y^2}{e^y+1} dy \, ,$$ where isotropy has been exploited to reduce the dimension of the differential and which is normalised such that the integral yields the number density as required. One can define a set of moments $$\label{numoments}
Q^{(n)}_\nu= \frac{1}{\pi^2} \fb{4}{11}{\frac{3+n}{3}} T^{3+n} \int y^{2+n} f(y) dy$$ in terms of the neutrino temperature $T_\nu= (4/11)^{1/3} T$. These moments can specify the neutrino distribution $df(y)$ regardless of the specific case at hand. In [@Cuoco:2005qr] this is used to explore observation prospects for non-thermal contributions to the standard neutrino spectrum. If the distribution decays at large comoving momentum as $e^{-y}$, it admits moments of all orders. In [@Cuoco:2005qr] the neutrino distribution is given in terms of its moments as $$df(y)= \frac{y^2}{e^y+1} \sum_{m=0}^\infty \sum_{k=0}^m
c_k^{(m)} Q^{(k)}_\nu T_\nu^{-k} P_m(y) dy$$ with $P_m(y)= \sum_{k=0}^m c_k^{(m)} y^k$, $m$ being the degree of $P_m(y)$ and $c_k^{(m)}$ being a coefficient, denoting the set of polynomials orthonormal with respect to the measure $y^2/(e^y+1)$, i.e., $\int_0^\infty dy \frac{y^2}{e^y+1} P_n(y)P_m(y) =\d_{nm}$. For a Fermi-Dirac distribution all moments can be expressed in terms of the lowest moment $Q^{(0)}= n_\nu$ or, equivalently, as functions of the temperature $T_\nu$ since it is the only independent parameter. For neutrinos the (first two) cosmological observables can be written as $$\begin{aligned}
\Neff = \frac{120}{7 \pi^2} T_\nu^{-4} \sum_\a Q_{\nu \a}^{(1)}\end{aligned}$$ and $$\begin{aligned}
\O_\nu^0 &= \frac{\sumnu}{\rhoc} Q_\nu^{(0)} \fb{T_0}{T}{3} \nonumber \\
&\Leftrightarrow \O_\nu^0 h^2 \simeq 0.162 \, \frac{\sumnu}{\text{eV}} T^{-3} Q_\nu^{(0)} ,\end{aligned}$$ where it is assumed that the three neutrinos share the same distribution today and the small correction from $e^+e^-$ annihilation, $\Neff^\text{sm}=3.046 \neq 3$, is incorporated in the numerical prefactor. In [@Cuoco:2005qr] $\Neff$ and $f_\nu$ are used to probe deviations from the standard neutrino spectrum in the first two moments. Also the following facts are used in our discussion: The average absolute neutrino momentum $\langle p_\nu \rangle$ expressed by moments of their distribution reads $$\label{averagenumomentum}
\langle p_\nu \rangle = \frac{Q_\nu^{(1)}}{Q_\nu^{(0)}}
= \frac{7 \pi^4}{180 \zeta(3)} \fb{4}{11}{\frac{1}{3}} T \simeq 3.15 \, T_\nu$$ and the root-mean-square of absolute momenta $\langle p_\nu \rangle_\text{rms} =(Q_\nu^{(2)}/Q_\nu^{(0)})^{1/2}$ can be calculated analogously as $$\label{rmsnumomentum}
\langle p_\nu \rangle_\text{rms}
= \fb{15 \zeta(5)}{\zeta(3)}{\frac{1}{2}} \fb{4}{11}{\frac{1}{3}} T \simeq 3.6 \, T_\nu \, .$$
[^1]: In other scenarios sterile neutrinos might be produced non-thermally. As we are going to compare the decay case with thermal production, we do not include such possibilities and consider the sterile neutrino to possess a Fermi-Dirac distribution, see also Sec. \[sec:breakdown\].
[^2]: We define this temperature as the temperature of the standard thermal bath at the time of decay $T(\t)|_\text{rad-dom}$ calculated in radiation domination. This temperature is known as “reheating” temperature, therefore, the subscript. However, the Universe is not re-heated as at the end of inflation. It just cools more slowly during the decay period. From this point of view, the notation used in the next subsection, $\Tdec$, might be seen as appropriate for this case, too. Anyway, to prevent confusion we distinguish the two cases explicitly also in the notation.
[^3]: Of course, the following considerations hold for any earlier decay of a non-relativistic particle that does not dominate at its decay. There is just no observational reason that would forbid its domination.
[^4]: The Universe is radiation dominated in the time window under consideration.
[^5]: It has been used to show that a Dodelson-Widrow model shares the same ”observable“ parameters as a thermal sterile neutrino model with adjusted mass and temperature.
[^6]: http://camb.info/
[^7]: http://class-code.net/
[^8]: If clustering can be neglected, today’s root-mean-square of absolute velocities $\langle v^0 \rangle_\text{rms} = \langle p^0 \rangle_\text{rms}/m$.
[^9]: This usage of the notion of mimicry is actually closest to mimicry in biology.
[^10]: As a side note, we could also calculate $\langle v \rangle =\langle E_\text{kin} \rangle /m = \rho_\text{kin}/ (mn)$ and insert appropriate expressions for the kinetic energy density $\r_\text{kin}$.
|
---
abstract: |
This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras $\Theta $ and algebras $H\in \Theta $ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras $H_{1}$ and $H_{2}$ are called geometrically equivalent if and only if the $H_{1}$-closed sets coincide with the $H_{2}$-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras $H_{1}$ and $H_{2}$ are called automorphicaly equivalent if and only if the $H_{1}$-closed sets coincide with the $H_{2}$-closed sets after some “changing of coordinates”.
We can detect the difference between geometric and automorphic equivalence of algebras of the variety $\Theta $ by researching of the automorphisms of the category $\Theta ^{0}$ of the finitely generated free algebras of the variety $\Theta $. By [@PlotkinSame] the automorphic equivalence of algebras provided by inner automorphism degenerated to the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety $\Theta $ if the factor group $\mathfrak{A/Y}$ is big. Hear $\mathfrak{A}$ is the group of all automorphisms of the category* *$\Theta ^{0}$, $\mathfrak{Y}$ is a normal subgroup of all inner automorphisms of the category* *$\Theta
^{0}$.
In [@PlotkinZhitAutCat] the variety of all Lie algebras and the variety of all associative algebras over the infinite field $k$ were studied. If the field $k$ has not nontrivial automorphisms then group $\mathfrak{A/Y}$ in the first case is trivial and in the second case has order $2$. We consider in this paper the variety of all linear algebras over the infinite field $k$. We prove that group $\mathfrak{A/Y}$ is isomorphic to the group $\left(
U\left( k\mathbf{S}_{\mathbf{2}}\right) \mathfrak{/}U\left( k\left\{
e\right\} \right) \right) \mathfrak{\leftthreetimes }\mathrm{Aut}k$, where $\mathbf{S}_{\mathbf{2}}$ is the symmetric group of the set which has $2$ elements, $U\left( k\mathbf{S}_{\mathbf{2}}\right) $ is the group of all invertible elements of the group algebra $k\mathbf{S}_{\mathbf{2}}$, $e\in
\mathbf{S}_{\mathbf{2}}$, $U\left( k\left\{ e\right\} \right) $ is a group of all invertible elements of the subalgebra $k\left\{ e\right\} $, $\mathrm{Aut}k$ is the group of all automorphisms of the field $k$.
So even the field $k$ has not nontrivial automorphisms the group $\mathfrak{A/Y}$ is infinite. This kind of result is obtained for the first time.
The example of two linear algebras which are automorphically equivalent but not** **geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.
author:
- |
[A.Tsurkov]{}\
Institute of Mathematics and Statistics.\
University São Paulo.\
Rua do Matão, 1010\
Cidade Universitária\
São Paulo - SP - Brasil - CEP 05508-090\
arkady.tsurkov@gmail.com
title: '[AUTOMORPHIC EQUIVALENCE OF LINEAR ALGEBRAS.]{}'
---
Introduction.\[intro\]
======================
In the first two sections we consider some variety $\Theta $ of one-sorted algebras of the signature $\Omega $. Denote by $X_{0}=\left\{
x_{1},x_{2},\ldots ,x_{n},\ldots \right\} $ a countable set of symbols, and by $\mathfrak{F}\left( X_{0}\right) $ the set of all finite subsets of $X_{0} $. We will consider the category $\Theta ^{0}$, whose objects are all free algebras $F\left( X\right) $ of the variety $\Theta $ generated by finite subsets $X\in \mathfrak{F}\left( X_{0}\right) $. Morphisms of the category $\Theta ^{0}$ are homomorphisms of free algebras.
We denote some time $F\left( X\right) =F\left( x_{1},x_{2},\ldots
,x_{n}\right) $ if $X=\left\{ x_{1},x_{2},\ldots ,x_{n}\right\} $ and even $F\left( X\right) =F\left( x\right) $ if $X$ has only one element.
We assume that our variety $\Theta $ possesses the IBN property: for free algebras $F\left( X\right) ,F\left( Y\right) \in \Theta $ we have $F\left(
X\right) \cong F\left( Y\right) $ if and only if $\left\vert X\right\vert
=\left\vert Y\right\vert $. In this case we have [PlotkinZhitAutCat]{} this decomposition$$\mathfrak{A=YS}. \label{decomp}$$of the group $\mathfrak{A}$ of all automorphisms of the category* *$\Theta ^{0}$. Hear $\mathfrak{Y}$ is a group of all inner automorphisms of the category* *$\Theta ^{0}$ and $\mathfrak{S}$ is a group of all strongly stable automorphisms of the category* *$\Theta ^{0}$.
\[inner\]An automorphism $\Upsilon $ of a category $\mathfrak{K}$ is **inner**, if it is isomorphic as a functor to the identity automorphism of the category $\mathfrak{K}$.
This means that for every $A\in \mathrm{Ob}\mathfrak{K}$ there exists an isomorphism $s_{A}^{\Upsilon }:A\rightarrow \Upsilon \left( A\right) $ such that for every $\alpha \in \mathrm{Mor}_{\mathfrak{K}}\left( A,B\right) $ the diagram$$\begin{array}{ccc}
A & \overrightarrow{s_{A}^{\Upsilon }} & \Upsilon \left( A\right) \\
\downarrow \alpha & & \Upsilon \left( \alpha \right) \downarrow \\
B & \underrightarrow{s_{B}^{\Upsilon }} & \Upsilon \left( B\right)\end{array}$$commutes.
\[str\_stab\_aut\]**.** *An automorphism $\Phi
$ of the category* $\Theta ^{0}$* is called **strongly stable** if it satisfies the conditions:*
1. $\Phi $* preserves all objects of* $\Theta ^{0}$*,*
2. *there exists a system of bijections* $\left\{ s_{F}^{\Phi
}:F\rightarrow F\mid F\in \mathrm{Ob}\Theta ^{0}\right\} $* such that* $\Phi $* acts on the morphisms $\alpha :D\rightarrow F$ of* $\Theta ^{0}$* by this way:* $$\Phi \left( \alpha \right) =s_{F}^{\Phi }\alpha \left( s_{D}^{\Phi }\right)
^{-1}, \label{biject_action}$$
3. $s_{F}^{\Phi }\mid _{X}=id_{X},$ * for every free algebra* $F=F\left( X\right) $.
The subgroup $\mathfrak{Y}$ is a normal in $\mathfrak{A}$. We will calculate the factor group $\mathfrak{A/Y\cong S/S\cap Y}$. This calculation is very important for universal algebraic geometry.
All definitions of the basic notions of the universal algebraic geometry can be found, for example, in [@PlotkinVarCat], [@PlotkinNotions] and [@PlotkinSame]. In universal algebraic geometry we consider a “set of equations” $T\subset F\times F$ in some finitely generated free algebra $F$ of the arbitrary variety of universal algebras $\Theta $ and we “resolve” these equations in $\mathrm{Hom}\left( F,H\right) $, where $H\in \Theta $. The set $\mathrm{Hom}\left( F,H\right) $ serves as an “affine space over the algebra $H$”. Denote by $T_{H}^{\prime }$ the set $\left\{ \mu \in \mathrm{Hom}\left( F,H\right) \mid T\subset \ker \mu \right\} $. This is the set of all solutions of the set of equations $T$. For every set of “points” $R$ of the affine space $\mathrm{Hom}\left( F,H\right) $ we consider a congruence of equations defined by this set: $R_{H}^{\prime }=\bigcap\limits_{\mu \in
R}\ker \mu $. For every set of equations $T$ we consider its algebraic closure $T_{H}^{\prime \prime }$ in respect to the algebra $H$. A set $T\subset F\times F$ is called $H$-closed if $T=T_{H}^{\prime \prime }$. An $H $-closed set is always a congruence.
Algebras $H_{1},H_{2}\in \Theta $ are **geometrically equivalent** if and only if for every $X\in \mathfrak{F}\left( X_{0}\right) $ and every $T\subset F\left( X\right) \times F\left( X\right) $ fulfills $T_{H_{1}}^{\prime \prime }=T_{H_{2}}^{\prime \prime }$.
Denote the family of all $H$-closed congruences in $F$ by $Cl_{H}(F)$. We can consider the category $C_{\Theta }\left( H\right) $ of the *coordinate algebras* connected with the algebra $H\in \Theta $. Objects of this category are quotient algebras $F\left( X\right) /T$, where $X\in
\mathfrak{F}\left( X_{0}\right) $, $T\in Cl_{H}(F\left( X\right) )$. Morphisms of this category are homomorphisms of algebras.
\[automorphic\_equivalence\]Let $Id\left( H,X\right)
=\bigcap\limits_{\varphi \in \mathrm{Hom}\left( F\left( X\right) ,H\right)
}\ker \varphi $ be the minimal $H$-closed congruence in $\ F\left( X\right) $. Algebras $H_{1},H_{2}\in \Theta $ are **automorphically equivalent** if and only if there exists a pair $\left( \Phi ,\Psi \right) ,$ where $\Phi
:\Theta ^{0}\rightarrow \Theta ^{0}$ is an automorphism,* *$\Psi
:C_{\Theta }\left( H_{1}\right) \rightarrow $* *$C_{\Theta }\left(
H_{2}\right) $ is an isomorphism subject to conditions:
1. $\Psi \left( F\left( X\right) /Id\left( H_{1},X\right) \right)
=F\left( Y\right) /Id\left( H_{2},Y\right) $*, where* $\Phi \left(
F\left( X\right) \right) =F\left( Y\right) $*,*
2. $\Psi \left( F\left( X\right) /T\right) =F\left( Y\right) /\widetilde{T}$*, where* $T\in Cl_{H_{1}}(F\left( X\right) )$*,* $\widetilde{T}\in Cl_{H_{2}}(F\left( Y\right) )$*,*
3. $\Psi $* takes the natural epimorphism* $\overline{\tau }:F\left( X\right) /Id\left( H_{1},X\right) \rightarrow F\left( X\right) /T$* to the natural epimorphism* $\Psi \left( \overline{\tau }\right)
:F\left( Y\right) /Id\left( H_{2},Y\right) \rightarrow F\left( Y\right) /\widetilde{T}$*.*
Note that if such a pair $\left( \Phi ,\Psi \right) $ exists, then $\Psi $ is uniquely defined by $\Phi $.
We can say, in certain sense, that automorphic equivalence of algebras is a coinciding of the structure of closed sets after some “changing of coordinates” provided by automorphism $\Phi $.
Algebras $H_{1}$ and $H_{2}$ are geometrically equivalent if and only if an inner automorphism $\Phi :\Theta ^{0}\rightarrow \Theta ^{0}$ provides the automorphic equivalence of algebras $H_{1}$ and $H_{2}$. So, only strongly stable automorphism $\Phi $ can provide us automorphic equivalence of algebras which not coincides with geometric equivalence of algebras. Therefore, in some sense, difference from the automorphic equivalence to the geometric equivalence is measured by the factor group $\mathfrak{A/Y\cong
S/S\cap Y}$.
Verbal operations and strongly stable automorphisms.[operations]{}
==================================================================
For every word $w=w\left( x_{1},\ldots ,x_{k}\right) \in F\left( X\right) $, where $F\left( X\right) \in \mathrm{Ob}\Theta ^{0}$, $X=\left\{ x_{1},\ldots
,x_{k}\right\} $ and for every algebra $H\in \Theta $ we can define a $k$-ary operation $w_{H}^{\ast }$ on $H$ by $$w_{H}^{\ast }\left( h_{1},\ldots ,h_{k}\right) =w\left( h_{1},\ldots
,h_{k}\right) =\gamma _{h}\left( w\left( x_{1},\ldots ,x_{k}\right) \right)
,$$where $\gamma _{h}$ is a homomorphism $F\left( X\right) \ni x_{i}\rightarrow
\gamma _{h}\left( x_{i}\right) =h_{i}\in H$, $1\leq i\leq k$. This operation we call the **verbal operation** induced on the algebra $H$ by the word $w\left( x_{1},\ldots ,x_{k}\right) \in F\left( X\right) $. [A system of words ]{}$W=\left\{ w_{i}\mid i\in I\right\} $ such that $w_{i}\in F\left(
X_{i}\right) $ [, ]{}$X_{i}=\left\{ x_{1},\ldots ,x_{k_{i}}\right\} ,$ [determines a system of ]{}$k_{i}$[-ary operations ]{}$\left( w_{i}\right)
_{H}^{\ast }$ on $H$. Denote the set $H$ with the system of these operation by $H_{W}^{\ast }$.
We have a correspondence between strongly stable automorphisms and [systems of words which define ]{}the verbal operation and fulfill some conditions. This correspondence explained in [@PlotkinZhitAutCat] and [TsurkovAutomEquiv]{}: We denote the signature of our variety $\Theta $ by $\Omega $, by $k_{\omega }$ we denote the arity of $\omega $ for every $\omega \in \Omega $. We suppose that we have the system of words $W=\left\{
w_{\omega }\mid \omega \in \Omega \right\} $ satisfies the conditions:
1. $w_{\omega }\left( x_{1},\ldots ,x_{k_{\omega }}\right) \in
F\left( X_{\omega }\right) $, where $X_{\omega }=\left\{ x_{1},\ldots
,x_{k_{\omega }}\right\} $;
2. for every $F=F\left( X\right) \in \mathrm{Ob}\Theta ^{0}$ there exists an isomorphism $\sigma _{F}:F\rightarrow F_{W}^{\ast }$ such that $\sigma _{F}\mid _{X}=id_{X}$.
$F_{W}^{\ast }\in \Theta $ so isomorphisms $\sigma _{F}$ are defined uniquely by the system of words $W$.
The set $S=\left\{ \sigma _{F}:F\rightarrow F\mid F\in \mathrm{Ob}\Theta
^{0}\right\} $ is a system of bijections which satisfies the conditions:
1. for every homomorphism $\alpha :A\rightarrow B\in \mathrm{Mor}\Theta ^{0}$ the mappings $\sigma _{B}\alpha \sigma _{A}^{-1}$ and $\sigma
_{B}^{-1}\alpha \sigma _{A}$ are homomorphisms;
2. $\sigma _{F}\mid _{X}=id_{X}$ for every free algebra $F\in
\mathrm{Ob}\Theta ^{0}$.
So we can define the strongly stable automorphism* *by this system of bijections. This automorphism preserves all objects of $\Theta ^{0}$ and acts on morphism of $\Theta ^{0}$ by formula (\[biject\_action\]), where $s_{F}^{\Phi }=$ $\sigma _{F}$.
Vice versa if we have a strongly stable automorphism $\Phi $ of the category $\Theta ^{0}$ then its system of bijections $S=\left\{ s_{F}^{\Phi
}:F\rightarrow F\mid F\in \mathrm{Ob}\Theta ^{0}\right\} $ defined uniquely. Really, if $F\in \mathrm{Ob}\Theta ^{0}$ and $f\in F$ then$$s_{F}^{\Phi }\left( f\right) =s_{F}^{\Phi }\alpha \left( x\right) =\left(
s_{F}^{\Phi }\alpha \left( s_{D}^{\Phi }\right) ^{-1}\right) \left( x\right)
=\left( \Phi \left( \alpha \right) \right) \left( x\right) ,
\label{autom_bijections}$$where $D=F\left( x\right) $ - $1$-generated free linear algebra - and $\alpha :D\rightarrow F$ homomorphism such that $\alpha \left( x\right) =f$. Obviously that this system of bijections $S=\left\{ s_{F}^{\Phi
}:F\rightarrow F\mid F\in \mathrm{Ob}\Theta ^{0}\right\} $ fulfills conditions B1 and B2 with $\sigma _{F}=s_{F}^{\Phi }$.
If we have a system of bijections $S=\left\{ \sigma _{F}:F\rightarrow F\mid
F\in \mathrm{Ob}\Theta ^{0}\right\} $ which fulfills conditions B1 and B2 than we can define the system of words $W=\left\{ w_{\omega }\mid \omega \in
\Omega \right\} $ satisfies the conditions Op1 and Op2 by formula$$w_{\omega }\left( x_{1},\ldots ,x_{k_{\omega }}\right) =\sigma _{F_{\omega
}}\left( \omega \left( \left( x_{1},\ldots ,x_{k_{\omega }}\right) \right)
\right) \in F_{\omega }, \label{der_veb_opr}$$where $F_{\omega }=F\left( X_{\omega }\right) $.
By formulas (\[autom\_bijections\]) and (\[der\_veb\_opr\]) we can check that there are
1. one to one and onto correspondence between strongly stable automorphisms of the category $\Theta ^{0}$ and systems of bijections satisfied the conditions B1 and B2
2. one to one and onto correspondence between systems of bijections satisfied the conditions B1 and B2 and systems of words satisfied the conditions Op1 and Op2.
So we can find a strongly stable automorphism $\Phi $ of the category $\Theta ^{0}$ by finding a system of words which fulfills conditions Op1 and Op2.
Verbal operations in linear algebras.[operations\_in\_linear\_alg]{}
====================================================================
From now on, we consider the variety $\Theta $ of all linear algebras over infinite field $k$. We consider linear algebras as one-sorted universal algebras, i. e., multiplication by scalar we consider as $1$-ary operation for every $\lambda \in k$: $H\ni h\rightarrow \lambda h\in H$ where $H\in
\Theta $. Hence the signature $\Omega $ of algebras of our variety contains these operations: $0$-ary operation $0$; $\left\vert k\right\vert $ $1$-ary operations of multiplications by scalars; $1$-ary operation $-:h\rightarrow
-h$, where $h\in H$, $H\in \Theta $; $2$-ary operation $\cdot $ and $2$-ary operation $+$. We will finding the system of words $W=\left\{ w_{\omega
}\mid \omega \in \Omega \right\} $ satisfies the conditions Op1 and Op2. We denote the words corresponding to these operations by $w_{0}$, $w_{\lambda }$ for all $\lambda \in k$, $w_{-}$, $w_{\cdot }$, $w_{+}$.
For arbitrary $F\left( X\right) \in \mathrm{Ob}\Theta ^{0}$ we denote $F\left( X\right) =\bigoplus\limits_{i=1}^{\infty }F_{i}$ the decomposition to the linear spaces of elements which are homogeneous according the sum of degrees of generators from the set $X$. We also denote the two-sides ideals $\bigoplus\limits_{i=j}^{\infty }F_{i}=F^{j}$. From now on, the word “ideal” means two sided ideal of linear algebra.
We denote the group of all automorphisms of the field $k$ by $\mathrm{Aut}k$.
Our variety $\Theta $ possesses the IBN property, because $\left\vert
X\right\vert =\dim F/F^{2}$ fulfills for all free algebras $F=F\left(
X\right) \in \Theta $. So we have the decomposition (\[decomp\]) for group of all automorphisms of the category $\Theta ^{0}$.
Now we need to prove one technical fact about $1$-generated free linear algebra $F\left( x\right) $.
\[monomials\]Let $\left\{ u_{1},\ldots ,u_{r}\right\} $ is the set of all monomials of degree $n$ in $F\left( x\right) $ (basis of $F_{n}$), $\left\{ v_{1},\ldots ,v_{t}\right\} $ is the set of all monomials of degree $m$ in $F\left( x\right) $ (basis of $F_{m}$), $\varphi $ is an arbitrary function from $\left\{ 1,\ldots ,n\right\} $ to $\left\{ 1,\ldots ,t\right\}
$. Denote by $\varphi \left( u_{l}\right) $ the monomial which is a results of substitution into monomial $u_{l}$ ($1\leq l\leq r$) instead $j$-th from left entry of $x$ the monomial $v_{\varphi \left( j\right) }$ ($1\leq j\leq
n $). All these monomials are distinct, i. e., $\varphi _{1}\left(
u_{l_{1}}\right) =\varphi _{2}\left( u_{l_{2}}\right) $ if and only if $\varphi _{1}=\varphi _{2}$ and $u_{l_{1}}=u_{l_{2}}$, where $\varphi
_{1},\varphi _{2}:\left\{ 1,\ldots ,n\right\} \rightarrow \left\{ 1,\ldots
,t\right\} $, $u_{l_{1}},u_{l_{2}}\in \left\{ u_{1},\ldots ,u_{r}\right\} $.
We will prove this lemma by induction by $n$ - degree of monomials from $\left\{ u_{1},\ldots ,u_{r}\right\} $. The claim of the lemma is trivial for $n=1$. We assume that the claim of the lemma is proved for monomials which have degree $<n$. We suppose that $\varphi _{1}\left( u_{l_{1}}\right)
=\varphi _{2}\left( u_{l_{2}}\right) $, where $\deg u_{l_{1}}=\deg
u_{l_{2}}=n>1$, $\varphi _{1},\varphi _{2}:\left\{ 1,\ldots ,n\right\}
\rightarrow \left\{ 1,\ldots ,t\right\} $. $u_{l_{i}}=u_{l_{i}}^{(1)}\cdot
u_{l_{i}}^{(2)}$, where $i=1,2$. We denote $\deg u_{l_{i}}^{(1)}=c_{i}$. $1\leq c_{i}<n$ for $i=1,2$. For $i=1,2$ we have $\varphi _{i}\left(
u_{l_{i}}\right) =\varphi _{i}^{(1)}\left( u_{l_{i}}^{(1)}\right) \cdot
\varphi _{i}^{(2)}\left( u_{l_{i}}^{(2)}\right) $, where $\varphi
_{i}^{(1)}:\left\{ 1,\ldots ,c_{i}\right\} \rightarrow \left\{ 1,\ldots
,t\right\} $, $\varphi _{i}^{(2)}:\left\{ 1,\ldots ,n-c_{i}\right\}
\rightarrow \left\{ 1,\ldots ,t\right\} $, $\varphi _{i}^{(1)}\left(
j\right) =\varphi _{i}\left( j\right) $ for $1\leq j\leq c_{i}$, $\varphi
_{i}^{(2)}\left( j\right) =\varphi _{i}\left( c_{i}+j\right) $ for $1\leq
j\leq n-c_{1}$. $\varphi _{1}\left( u_{l_{1}}\right) =\varphi _{2}\left(
u_{l_{2}}\right) $ if and only if $\varphi _{1}^{(1)}\left(
u_{l_{1}}^{(1)}\right) =\varphi _{2}^{(1)}\left( u_{l_{2}}^{(1)}\right) $ and $\varphi _{1}^{(2)}\left( u_{l_{1}}^{(2)}\right) =\varphi
_{2}^{(2)}\left( u_{l_{2}}^{(2)}\right) $. If $c_{1}\neq c_{2}$ then $\deg
\varphi _{1}^{(1)}\left( u_{l_{1}}^{(1)}\right) =c_{1}m\neq \deg \varphi
_{2}^{(1)}\left( u_{l_{2}}^{(1)}\right) =c_{2}m$, hence $\varphi
_{1}^{(1)}\left( u_{l_{1}}^{(1)}\right) \neq \varphi _{2}^{(1)}\left(
u_{l_{2}}^{(1)}\right) $ and $\varphi _{1}\left( u_{l_{1}}\right) \neq
\varphi _{2}\left( u_{l_{2}}\right) $. So $c_{1}=c_{2}$ and, by our assumption, $\varphi _{1}^{(1)}=\varphi _{2}^{(1)}$, $u_{l_{1}}^{(1)}=u_{l_{2}}^{(1)}$, $\varphi _{1}^{(2)}=\varphi _{2}^{(2)}$, $u_{l_{1}}^{(2)}=u_{l_{2}}^{(2)}$. Therefore $\varphi _{1}=\varphi _{2}$ and $u_{l_{1}}=u_{l_{2}}$.
\[substitute\]Let $f\left( x\right) ,g\left( x\right) \in F\left(
X\right) $. $f\left( g\left( x\right) \right) $ is a result of substitution of $g\left( x\right) $ in $f\left( x\right) $ instead $x$. $f\left( g\left(
x\right) \right) \in F_{1}$ if and only if $f\left( x\right) ,g\left(
x\right) \in F_{1}$.
We write $f\left( x\right) $ and $g\left( x\right) $ as sum of its homogeneous components: $f\left( x\right) =f_{1}\left( x\right) +f_{2}\left(
x\right) +\ldots +f_{n}\left( x\right) $, $g\left( x\right) =g_{1}\left(
x\right) +g_{2}\left( x\right) +\ldots +g_{m}\left( x\right) $, $f_{i}\left(
x\right) ,g_{i}\left( x\right) \in F_{i}$. We assume that $n>1$ or $m>1$, $f_{n}\left( x\right) \neq 0$ and $g_{m}\left( x\right) \neq 0$. $f\left(
g\left( x\right) \right) =f_{1}\left( g\left( x\right) \right) +f_{2}\left(
g\left( x\right) \right) +\ldots +f_{n}\left( g\left( x\right) \right) $. Addenda of the maximal possible degree of $x$, which can appear in $f\left(
g\left( x\right) \right) $, i. e., addenda of degree $nm$ can appear in $f_{n}\left( g\left( x\right) \right) $. They coincide with addenda of $f_{n}\left( g_{m}\left( x\right) \right) $. Denote $f_{n}\left( x\right)
=\lambda _{1}u_{1}+\ldots +\lambda _{r}u_{r}$, $g_{m}\left( x\right) =\mu
_{1}v_{1}+\ldots +\mu _{t}v_{t}$, where $\left\{ u_{1},\ldots ,u_{r}\right\}
$ is the set of all monomials of degree $n$ in $F\left( x\right) $, $\left\{
v_{1},\ldots ,v_{t}\right\} $ is the set of all monomials of degree $m$ in $F\left( x\right) $, $\lambda _{i},\mu _{j}\in k$. Not all $\left\{ \lambda
_{1},\ldots ,\lambda _{r}\right\} $ and not all $\left\{ \mu _{1},\ldots
,\mu _{t}\right\} $ are equal to $0$ by our assumption. $f_{n}\left(
g_{m}\left( x\right) \right) =\lambda _{1}u_{1}\left( g_{m}\left( x\right)
\right) +\ldots +\lambda _{r}u_{r}\left( g_{m}\left( x\right) \right) $. If we open the brackets in $u_{l}\left( g_{m}\left( x\right) \right)
=u_{l}\left( \mu _{1}v_{1}+\ldots +\mu _{t}v_{t}\right) $ ($1\leq l\leq r$), we obtain addenda, which are results of substitution into monomial $u_{l}$ instead all entry of $x$ some monomial from $\mu _{1}v_{1},\ldots ,\mu
_{t}v_{t}$ in all possible options. We can say more formal: for every function $\varphi :\left\{ 1,\ldots ,n\right\} \rightarrow \left\{ 1,\ldots
,t\right\} $ we obtain an addendum which is a results of substitution into monomial $u_{l}$ instead $j$-th from left entry of $x$ the monomial $\mu
_{\varphi \left( j\right) }v_{\varphi \left( j\right) }$ ($1\leq j\leq n$). Therefore all addenda, which we obtain after the opening of the brackets in $f_{n}\left( g_{m}\left( x\right) \right) $, distinct from the monomials discussed in **Lemma \[monomials\]** only by coefficients. All these addenda have degree $nm>1$, because $n>1$ or $m>1$. So addenda of $f_{n}\left( g_{m}\left( x\right) \right) $ can not cancel one another by **Lemma \[monomials\]**. These addenda can not be canceled by other addenda $f\left( g\left( x\right) \right) $, because all other addenda have degree $<nm$. Therefore all these addenda equal to $0$, because $f\left(
g\left( x\right) \right) \in F_{1}$. For $l\in \left\{ 1,\ldots ,r\right\} $ and $j\in \left\{ 1,\ldots ,t\right\} $ we take the addendum which is a results of substitution into monomial $\lambda _{l}u_{l}$ instead all entries of $x$ the monomial $\mu _{j}v_{j}$. The coefficient of this addendum is $\lambda _{l}\mu _{j}^{n}=0$. So $\lambda _{l}\mu _{j}=0$ for all $l\in \left\{ 1,\ldots ,r\right\} $ and all $j\in \left\{ 1,\ldots
,t\right\} $. It contradicts the fact that $f_{n}\left( x\right) \neq 0$ and $g_{m}\left( x\right) \neq 0$.
\[words\]The system of words$$W=\left\{ w_{0},w_{\lambda }\left( \lambda \in k\right)
,w_{-},w_{+},w_{\cdot }\right\} \label{words_list}$$satisfies the conditions Op1 and Op2 if and only if $w_{0}=0$, $w_{\lambda
}=\varphi \left( \lambda \right) x_{1}$, $w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$, $w_{\cdot }=ax_{1}x_{2}+bx_{2}x_{1}$, where $\varphi $ is an automorphism of the field $k$, $a,b\in k$, $a\neq \pm b$.
Let $W$ (see (\[words\_list\]) ) satisfies the conditions Op1 and Op2.
$w_{0}$ is an element of the $0$-generated free linear algebra. There is only one element in this algebra: $0$. This is the only one opportunity for $w_{0}$.
$w_{\lambda }\in F\left( x\right) $ for every $\lambda \in k$. Denote multiplications by scalars in $\left( F\left( x\right) \right) _{W}^{\ast }$ by $\ast $, i. e., $\lambda \ast f=w_{\lambda }\left( f\right) $ for every $f\in F\left( x\right) $ and every $\lambda \in k$. $\left( F\left( x\right)
\right) _{W}^{\ast }\in \Theta $, therefore, if $\lambda =0$ then $0\ast
x=w_{0}\left( x\right) =0$. If $\lambda \neq 0$ then$$1\ast x=\left( \lambda ^{-1}\lambda \right) \ast x=\lambda ^{-1}\ast \left(
\lambda \ast x\right) =w_{\lambda ^{-1}}\left( w_{\lambda }\left( x\right)
\right) =x.$$Hence $w_{\lambda }=\varphi \left( \lambda \right) x$ by **Corollary 1** from **Lemma \[monomials\]**, where $\varphi \left( \lambda \right)
\in k$. We can write $\varphi \left( 0\right) =0$. Also we have that for all $\lambda _{1},\lambda _{2}\in k$ fulfills$$\left( \lambda _{1}\lambda _{2}\right) \ast x=\varphi \left( \lambda
_{1}\lambda _{2}\right) x$$ and $$\left( \lambda _{1}\lambda _{2}\right) \ast x=\lambda _{1}\ast \left(
\lambda _{2}\ast x\right) =\lambda _{1}\ast \left( \varphi \left( \lambda
_{2}\right) x\right) =$$$$\varphi \left( \lambda _{1}\right) \left( \varphi \left( \lambda _{2}\right)
x\right) =\left( \varphi \left( \lambda _{1}\right) \varphi \left( \lambda
_{2}\right) \right) x.$$So $\varphi \left( \lambda _{1}\right) \varphi \left( \lambda _{2}\right)
=\varphi \left( \lambda _{1}\lambda _{2}\right) $. If $\mu \in k\setminus
\left\{ 0\right\} $, then the $1$-ary operation of multiplication by scalar $\mu $ is a verbal operation defined by some word $w_{\mu }^{\ast }\left(
x\right) \in \left( F\left( x\right) \right) _{W}^{\ast }$, written be the operations defined by system of words $W$ - see [TsurkovAutomEquiv]{}. Hence, $\mu f=w_{\mu }^{\ast }\left( f\right) $ holds for every $f\in F\left( x\right) $. Also there is $w_{\mu ^{-1}}^{\ast
}\left( x\right) \in \left( F\left( x\right) \right) _{W}^{\ast }$ such that $\mu ^{-1}f=w_{\mu ^{-1}}^{\ast }\left( f\right) $ for every $f\in F\left(
x\right) $. $x=\mu ^{-1}\left( \mu x\right) =w_{\mu ^{-1}}^{\ast }\left(
w_{\mu }^{\ast }\left( x\right) \right) $. There exists by Op2 an isomorphism $\sigma _{F\left( x\right) }:F\left( x\right) \rightarrow \left(
F\left( x\right) \right) _{W}^{\ast }$ such that $\sigma _{F\left( x\right)
}\left( x\right) =x$. So $\left( F\left( x\right) \right) _{W}^{\ast }$ is also $1$-generated free linear algebra of $\Theta $ with the free generator $x$. Hence there exists a decomposition $\left( F\left( x\right) \right)
_{W}^{\ast }=\bigoplus\limits_{i=1}^{\infty }F_{i}^{\ast }$, where $F_{i}^{\ast }$ are linear spaces of elements which are homogeneous according the degree of $x$ but in respect of operations defined by system of words $W$. Therefore $w_{\mu }^{\ast }\left( x\right) =\lambda \ast x$, where $\lambda \in k$, by **Corollary 1** from **Lemma \[monomials\]**. So $\mu x=\lambda \ast x=\varphi \left( \lambda \right) x$ and $\mu
=\varphi \left( \lambda \right) $, hence $\varphi :k\rightarrow k$ is a surjection.
$w_{+}\in F\left( x_{1},x_{2}\right) =F$. There exists $n\in
\mathbb{N}
$, such that$$w_{+}\left( x_{1},x_{2}\right) =p_{1}\left( x_{1},x_{2}\right) +p_{2}\left(
x_{1},x_{2}\right) +\ldots +p_{n}\left( x_{1},x_{2}\right) ,$$where $p_{i}\left( x_{1},x_{2}\right) \in F_{i}$, $1\leq i\leq n$. We have for every $\lambda \in k$ that $$w_{+}\left( \lambda \ast x_{1},\lambda \ast x_{2}\right) =\lambda \ast
w_{+}\left( x_{1},x_{2}\right) =\varphi \left( \lambda \right) w_{+}\left(
x_{1},x_{2}\right) =$$$$\varphi \left( \lambda \right) p_{1}\left( x_{1},x_{2}\right) +\varphi
\left( \lambda \right) p_{2}\left( x_{1},x_{2}\right) +\ldots +\varphi
\left( \lambda \right) p_{n}\left( x_{1},x_{2}\right)$$and$$w_{+}\left( \lambda \ast x_{1},\lambda \ast x_{2}\right) =p_{1}\left(
\lambda \ast x_{1},\lambda \ast x_{2}\right) +p_{2}\left( \lambda \ast
x_{1},\lambda \ast x_{2}\right) +\ldots +p_{n}\left( \lambda \ast
x_{1},\lambda \ast x_{2}\right) =$$$$p_{1}\left( \varphi \left( \lambda \right) x_{1},\varphi \left( \lambda
\right) x_{2}\right) +p_{2}\left( \varphi \left( \lambda \right)
x_{1},\varphi \left( \lambda \right) x_{2}\right) +\ldots +p_{n}\left(
\varphi \left( \lambda \right) x_{1},\varphi \left( \lambda \right)
x_{2}\right) =$$$$\varphi \left( \lambda \right) p_{1}\left( x_{1},x_{2}\right) +\left(
\varphi \left( \lambda \right) \right) ^{2}p_{2}\left( x_{1},x_{2}\right)
+\ldots +\left( \varphi \left( \lambda \right) \right) ^{n}p_{n}\left(
x_{1},x_{2}\right) .$$We can take $\lambda \in k$ such that $\varphi \left( \lambda \right) $ is not a solution of any equation $x^{i}=x$, where $2\leq i\leq n$. So, $p_{i}\left( x_{1},x_{2}\right) =0$ for $2\leq i\leq n$ by equality of the homogeneous components. Therefore $w_{+}=\alpha x_{1}+\beta x_{2}$, where $\alpha ,\beta \in k$. If we denote the operation defined by $w_{+}$ in $\left( F\left( x_{1},x_{2}\right) \right) _{W}^{\ast }$ by $\bot $, then $x_{1}\bot x_{2}=x_{2}\bot x_{1}$ holds, so $\alpha x_{1}+\beta x_{2}=\alpha
x_{2}+\beta x_{1}$ and $\alpha =\beta $. Also $x_{1}\bot 0=x_{1}$ holds and $\alpha x_{1}=x_{1}$, so $\alpha =\beta =1$. Now, by consideration of $F\left( x\right) $, we can conclude that for all $\lambda _{1},\lambda
_{2}\in k$ fulfills $$\varphi \left( \lambda _{1}+\lambda _{2}\right) x=\left( \lambda
_{1}+\lambda _{2}\right) \ast x=\lambda _{1}\ast x\bot \lambda _{2}\ast x=$$$$\lambda _{1}\ast x+\lambda _{2}\ast x=\varphi \left( \lambda _{1}\right)
x+\varphi \left( \lambda _{2}\right) x=\left( \varphi \left( \lambda
_{1}\right) +\varphi \left( \lambda _{2}\right) \right) x,$$so $\varphi \left( \lambda _{1}+\lambda _{2}\right) =$ $\varphi \left(
\lambda _{1}\right) +\varphi \left( \lambda _{2}\right) $ and $\varphi $ is an automorphism of the field $k$.
Its clear now that $w_{-}=-x\in F\left( x\right) $, because$$w_{-}\left( x\right) =-1\ast x=\varphi \left( -1\right) x=\left( -1\right)
x=-x.$$
$w_{\cdot }\in F\left( x_{1},x_{2}\right) $. We write $w_{\cdot }$ as sum of its homogeneous components according the degree of $x_{1}$:$$w_{\cdot }\left( x_{1},x_{2}\right) =p_{0}\left( x_{1},x_{2}\right)
+p_{1}\left( x_{1},x_{2}\right) +p_{2}\left( x_{1},x_{2}\right) +\ldots
+p_{n}\left( x_{1},x_{2}\right) .$$We denote the operation defined by $w_{\cdot }$ in $\left( F\left(
x_{1},x_{2}\right) \right) _{W}^{\ast }$ by $\times $. So we have for every $\lambda \in k$ that$$\left( \lambda \ast x_{1}\right) \times x_{2}=\lambda \ast \left(
x_{1}\times x_{2}\right) =\varphi \left( \lambda \right) w_{\cdot }\left(
x_{1},x_{2}\right) =$$$$\varphi \left( \lambda \right) p_{0}\left( x_{1},x_{2}\right) +\varphi
\left( \lambda \right) p_{1}\left( x_{1},x_{2}\right) +\varphi \left(
\lambda \right) p_{2}\left( x_{1},x_{2}\right) +\ldots +\varphi \left(
\lambda \right) p_{n}\left( x_{1},x_{2}\right) .$$and$$\left( \lambda \ast x_{1}\right) \times x_{2}=w_{\cdot }\left( \varphi
\left( \lambda \right) x_{1},x_{2}\right) =$$$$p_{0}\left( \varphi \left( \lambda \right) x_{1},x_{2}\right) +p_{1}\left(
\varphi \left( \lambda \right) x_{1},x_{2}\right) +p_{2}\left( \varphi
\left( \lambda \right) x_{1},x_{2}\right) +\ldots +p_{n}\left( \varphi
\left( \lambda \right) x_{1},x_{2}\right) =$$$$p_{0}\left( x_{1},x_{2}\right) +\varphi \left( \lambda \right) p_{1}\left(
x_{1},x_{2}\right) +\left( \varphi \left( \lambda \right) \right)
^{2}p_{2}\left( x_{1},x_{2}\right) +\ldots +\left( \varphi \left( \lambda
\right) \right) ^{n}p_{n}\left( x_{1},x_{2}\right) .$$We can take, as above, $\lambda \in k$ such that by equality of the homogeneous components we obtain that $w_{\cdot }\left( x_{1},x_{2}\right)
=p_{1}\left( x_{1},x_{2}\right) $. Now we write $w_{\cdot }\left(
x_{1},x_{2}\right) =p_{1}\left( x_{1},x_{2}\right) $ as sum of its homogeneous components according the degree of $x_{2}$:$$w_{\cdot }\left( x_{1},x_{2}\right) =r_{0}\left( x_{1},x_{2}\right)
+r_{1}\left( x_{1},x_{2}\right) +r_{2}\left( x_{1},x_{2}\right) +\ldots
+r_{m}\left( x_{1},x_{2}\right) .$$We have for every $\lambda \in k$ that$$x_{1}\times \left( \lambda \ast x_{2}\right) =\lambda \ast \left(
x_{1}\times x_{2}\right) =\varphi \left( \lambda \right) w_{\cdot }\left(
x_{1},x_{2}\right) =$$$$\varphi \left( \lambda \right) r_{0}\left( x_{1},x_{2}\right) +\varphi
\left( \lambda \right) r_{1}\left( x_{1},x_{2}\right) +\varphi \left(
\lambda \right) r_{2}\left( x_{1},x_{2}\right) +\ldots +\varphi \left(
\lambda \right) r_{m}\left( x_{1},x_{2}\right) .$$and$$x_{1}\times \left( \lambda \ast x_{2}\right) =w_{\cdot }\left( x_{1},\varphi
\left( \lambda \right) x_{2}\right) =$$$$r_{0}\left( x_{1},\varphi \left( \lambda \right) x_{2}\right) +r_{1}\left(
x_{1},\varphi \left( \lambda \right) x_{2}\right) +r_{2}\left( x_{1},\varphi
\left( \lambda \right) x_{2}\right) +\ldots +r_{m}\left( x_{1},\varphi
\left( \lambda \right) x_{2}\right) =$$$$r_{0}\left( x_{1},x_{2}\right) +\varphi \left( \lambda \right) r_{1}\left(
x_{1},x_{2}\right) +\varphi \left( \lambda \right) ^{2}r_{2}\left(
x_{1},x_{2}\right) +\ldots +\varphi \left( \lambda \right) ^{m}r_{m}\left(
x_{1},x_{2}\right) .$$And, as above, we can conclude that $w_{\cdot }\left( x_{1},x_{2}\right)
=r_{1}\left( x_{1},x_{2}\right) $ where $r_{1}\left( x_{1},x_{2}\right) $ is a homogeneous element of $F\left( x_{1},x_{2}\right) $ such that $\deg
_{x_{1}}r_{1}\left( x_{1},x_{2}\right) =1$ and $\deg _{x_{2}}r_{1}\left(
x_{1},x_{2}\right) =1$. Therefore $w_{\cdot }\left( x_{1},x_{2}\right)
=ax_{1}x_{2}+bx_{2}x_{1}$, where $a,b\in k$. If $a=b$ then the operation defined by $w_{\cdot }\left( x_{1},x_{2}\right) $ is commutative. If $a=-b$ then the operation defined by $w_{\cdot }\left( x_{1},x_{2}\right) $ is anticommutative. The isomorphisms $\sigma _{F}:F\rightarrow F_{W}^{\ast }$, where $F\in \mathrm{Ob}\Theta ^{0}$ can not exists in both these cases if $F$ is not a $0$-generated free algebra.
Therefore we prove that if the system of words (\[words\_list\]) satisfies the conditions Op1 and Op2 then $w_{0}=0$, $w_{\lambda }=\varphi \left(
\lambda \right) x_{1}$ for all $\lambda \in k$, $w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$, $w_{\cdot }=ax_{1}x_{2}+bx_{2}x_{1}$, where $\varphi $ is an automorphism of the field $k$, $a,b\in k$, $a\neq \pm b$.
Now we must prove that for all $\varphi \in \mathrm{Aut}k$ and all $a,b\in k$ such that $a\neq \pm b$ the system of words (\[words\_list\]) where $w_{0}=0 $, $w_{\lambda }=\varphi \left( \lambda \right) x_{1}$ for all $\lambda \in k $, $w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$, $w_{\cdot
}=ax_{1}x_{2}+bx_{2}x_{1}$ fulfills condition Op2. It means that we must build for every $F=F\left( X\right) \in \mathrm{Ob}\Theta ^{0}$ an isomorphism $\sigma _{F}:F\rightarrow F_{W}^{\ast }$ such that $\sigma
_{F}\mid _{X}=id_{X}$.
We will prove, first of all, that $H_{W}^{\ast }\in \Theta $ for every $H\in
\Theta $. Operations defined by $w_{0}$, $w_{-}$, $w_{+}$ coincide with $0$, $-$, $+$. So identities of the variety $\Theta $ (axioms of the linear algebra) relating to these operations fulfill in $H_{W}^{\ast }$. Hence we only need to check the axioms that involve the operations defined by $w_{\cdot }$ and $w_{\lambda }$ ($\lambda \in k$). As above we denote these operations by $\times $ and by $\lambda \ast $.$$\lambda \ast \left( x+y\right) =\varphi \left( \lambda \right) \left(
x+y\right) =\varphi \left( \lambda \right) x+\varphi \left( \lambda \right)
y=\lambda \ast x+\lambda \ast y,$$$$\left( \lambda \mu \right) \ast x=\varphi \left( \lambda \mu \right)
x=\varphi \left( \lambda \right) \varphi \left( \mu \right) x=\varphi \left(
\lambda \right) \left( \mu \ast x\right) =\lambda \ast \left( \mu \ast
x\right) ,$$$$\left( \lambda +\mu \right) \ast x=\varphi \left( \lambda +\mu \right)
x=\left( \varphi \left( \lambda \right) +\varphi \left( \mu \right) \right)
x=\varphi \left( \lambda \right) x+\varphi \left( \mu \right) x=\lambda \ast
x+\mu \ast x,$$$$1\ast x=\varphi \left( 1\right) x=1x=x,$$$$x\times \left( y+z\right) =ax\left( y+z\right) +b\left( y+z\right)
x=axy+axz+byx+bzx=x\times y+x\times z,$$$$\left( y+z\right) \times x=a\left( y+z\right) x+bx\left( y+z\right)
=ayx+azx+bxy+bxz=y\times x+z\times x,$$$$\lambda \ast \left( x\times y\right) =\varphi \left( \lambda \right) \left(
axy+byx\right) =a\left( \varphi \left( \lambda \right) x\right) y+by\left(
\varphi \left( \lambda \right) x\right) =\left( \varphi \left( \lambda
\right) x\right) \times y=$$$$\left( \lambda \ast x\right) \times y=x\times \left( \lambda \ast y\right)$$fulfills for every $x,y,z\in H$, $\lambda ,\mu \in k$.
Hence there exists a homomorphism $\sigma _{F}:F\rightarrow F_{W}^{\ast }$ such that $\sigma _{F}\mid _{X}=id_{X}$ for every $F=F\left( X\right) \in
\mathrm{Ob}\Theta ^{0}$. Our goal is to prove that these homomorphisms are isomorphisms. We will prove by induction by $i$ that $$\sigma _{F}\left( F_{i}\right) =F_{i}. \label{epi_homo}$$for every $i\in
\mathbb{N}
$. If $X=\left\{ x_{1},\ldots ,x_{n}\right\} $ then every element of $F_{1}$ has form $\lambda _{1}x_{1}+\ldots +\lambda _{n}x_{n}$, where $\lambda
_{1},\ldots ,\lambda _{n}\in k$.$$\sigma _{F}\left( \lambda _{1}x_{1}+\ldots +\lambda _{n}x_{n}\right)
=\lambda _{1}\ast \sigma _{F}\left( x_{1}\right) +\ldots +\lambda _{n}\ast
\sigma _{F}\left( x_{n}\right) =\varphi \left( \lambda _{1}\right)
x_{1}+\ldots +\varphi \left( \lambda _{n}\right) x_{n},$$so $\sigma _{F}\left( F_{1}\right) \subset F_{1}$.$$\sigma _{F}\left( \varphi ^{-1}\left( \lambda _{1}\right) x_{1}+\ldots
+\varphi ^{-1}\left( \lambda _{n}\right) x_{n}\right) =\lambda
_{1}x_{1}+\ldots +\lambda _{n}x_{n},$$so $\sigma _{F}\left( F_{1}\right) =F_{1}$.
Let (\[epi\_homo\]) proved for $i$ such that $1\leq i<r$. Every element of $F_{r}$ is a linear combination of the monomials of the form $uv$, where $u\in F_{i}$, $v\in F_{j}$, $i+j=r$.$$\sigma _{F}\left( uv\right) =\sigma _{F}\left( u\right) \times \sigma
_{F}\left( v\right) =a\sigma _{F}\left( u\right) \sigma _{F}\left( v\right)
+b\sigma _{F}\left( v\right) \sigma _{F}\left( u\right) ,$$so $\sigma _{F}\left( F_{r}\right) \subset F_{r}$, because, by our assumption, $\sigma _{F}\left( u\right) \in F_{i}$, $\sigma _{F}\left(
v\right) \in F_{j}$. Also, if $u=\sigma _{F}\left( \widetilde{u}\right) $, $v=\sigma _{F}\left( \widetilde{v}\right) $, where $\widetilde{u}\in F_{r}$, $\widetilde{v}\in F_{t}$, then$$\sigma _{F}\left( \widetilde{u}\widetilde{v}\right) =\sigma _{F}\left(
\widetilde{u}\right) \times \sigma _{F}\left( \widetilde{v}\right) =u\times
v=auv+bvu$$$$\sigma _{F}\left( \widetilde{v}\widetilde{u}\right) =\sigma _{F}\left(
\widetilde{v}\right) \times \sigma _{F}\left( \widetilde{u}\right) =v\times
u=avu+buv=buv+avu,$$fulfills. $a\neq \pm b$, so the matrix $\left(
\begin{array}{cc}
a & b \\
b & a\end{array}\right) $ is regular, hence there exist $\alpha ,\beta \in k$ such that $$uv=\alpha \sigma _{F}\left( \widetilde{u}\widetilde{v}\right) +\beta \sigma
_{F}\left( \widetilde{v}\widetilde{u}\right) =\sigma _{F}\left( \varphi
^{-1}\left( \alpha \right) \widetilde{u}\widetilde{v}+\varphi ^{-1}\left(
\beta \right) \widetilde{v}\widetilde{u}\right) .$$Therefore $\sigma _{F}\left( F_{r}\right) =F_{r}$. We can conclude that $\sigma _{F}$ is an epimorphism.
Now we will prove that $\ker \sigma _{F}=0$. Let $f\in \ker \sigma
_{F}\subset F\left( X\right) $. There exists $m\in
\mathbb{N}
$ such that $f\in \bigoplus\limits_{i=1}^{m}F_{i}$. $\sigma _{F}\left(
\bigoplus\limits_{i=1}^{m}F_{i}\right) =\bigoplus\limits_{i=1}^{m}F_{i}$ by (\[epi\_homo\]). $\sigma _{F}$ is a linear mapping from the linear space $\bigoplus\limits_{i=1}^{m}F_{i}$ with the original multiplication by scalars in $F$ to the $\left( \bigoplus\limits_{i=1}^{m}F_{i}\right) _{W}^{\ast }$ - the linear space $\bigoplus\limits_{i=1}^{m}F_{i}$ with the multiplication by scalars which we denote by $\ast $. From formulas $\sum\limits_{i=1}^{k}\left( \lambda _{i}\ast e_{i}\right) =\sum\limits_{i=1}^{k}\varphi \left(
\lambda _{i}\right) e_{i}$ and $\sum\limits_{i=1}^{k}\lambda
_{i}e_{i}=\sum\limits_{i=1}^{k}\left( \varphi ^{-1}\left( \lambda
_{i}\right) \ast e_{i}\right) $ we can conclude that if $E$ is a basis of the linear space $\bigoplus\limits_{i=1}^{m}F_{i}$ then $E$ is a basis of the linear space $\left( \bigoplus\limits_{i=1}^{m}F_{i}\right) _{W}^{\ast }$. So $\dim \bigoplus\limits_{i=1}^{m}F_{i}=\dim \left(
\bigoplus\limits_{i=1}^{m}F_{i}\right) _{W}^{\ast }<\infty $, therefore $\ker \left( \sigma _{F}\mid \bigoplus\limits_{i=1}^{m}F_{i}\right) =0$ and $f=0$.
Group $\mathfrak{A/Y}$.\[group\]
================================
From now on, $W$ is a system of words (\[words\_list\]) which fulfills conditions Op1 and Op2.
The decomposition (\[decomp\]) is not split in general case, i. e. $\mathfrak{S\cap Y\neq }\left\{ 1\right\} $ in general case. The strongly stable automorphism $\Phi $ of the category $\Theta ^{0}$ which corresponds to the system of words $W$ is inner, by [@PlotkinZhitAutCat Lemma 3], if and only if for every $F\in \mathrm{Ob}\Theta ^{0}$ there exists an isomorphism $c_{F}:F\rightarrow F_{W}^{\ast }$ such that $c_{F}\alpha
=\alpha c_{D}$ fulfills for every $\left( \alpha :D\rightarrow F\right) \in
\mathrm{Mor}\Theta ^{0}$ (by [@TsurkovAutomEquiv Remark 3.1] $\alpha $ is also a homomorphism from $D_{W}^{\ast }$ to $F_{W}^{\ast }$).
Hear we need to prove one technical lemma.
\[F/F\^2\]If $F=F\left( X\right) \in \mathrm{Ob}\Theta ^{0}$ and $c_{F}:F\rightarrow F_{W}^{\ast }$ is an isomorphism then there exists an isomorphism $c_{i}:F/F^{i}\rightarrow F_{W}^{\ast }/F^{i}$ such that $\chi
_{i}^{\ast }c_{F}=c_{i}\chi _{i}$, where $\chi _{i}:F\rightarrow F/F^{i}$ and $\chi _{i}^{\ast }:F_{W}^{\ast }\rightarrow F_{W}^{\ast }/F^{i}$ are natural homomorphisms, $i\in
\mathbb{N}
$.
If $H\in \Theta $ and $I$ is an ideal of $H$. If $\lambda \in k$, $y\in I$, $h\in H$, then $\lambda \ast y=\varphi \left( \lambda \right) y\in I$, $y\times h=ayh+bhy\in I$, analogously $h\times y\in I$. Therefore $I$ is an ideal of $H_{W}^{\ast }$. Hence $F^{i}$ is an ideal of $F_{W}^{\ast }$.
If $\sigma _{F}:F\rightarrow F_{W}^{\ast }$ is an isomorphism such that $\sigma _{F}\mid _{X}=id_{X}$, then by (\[epi\_homo\]) we have $c_{F}^{-1}\left( F^{i}\right) =c_{F}^{-1}\sigma _{F}\left( F^{i}\right)
=F^{i}$ because $c_{F}^{-1}\sigma _{F}:F\rightarrow F$ is an isomorphism. So $c_{F}\left( F^{i}\right) =F^{i}$. It finishes the proof.
\[stable\_inner\]The strongly stable automorphism $\Phi $ which corresponds to the system of words $W$ is inner if and only if $\varphi
=id_{k}$ and $b=0$.
We suppose that strongly stable automorphism $\Phi $ which corresponds to the system of words $W$ is inner. We assume that $\varphi \neq id_{k}$, i., e., there exists $\lambda \in k$ such that $\varphi \left( \lambda \right)
\neq \lambda $. We denote $F=F\left( x\right) $. We take $\alpha \in \mathrm{End}F$, such that $\alpha \left( x\right) =\lambda x$. We suppose that $c_{F}:F\rightarrow F_{W}^{\ast }$ is an isomorphism. $c_{2}$ is defined as in the Lemma \[F/F\^2\], and we by this Lemma we have:$$\chi _{2}^{\ast }c_{F}\left( x\right) =c_{2}\chi _{2}\left( x\right) =\mu
\ast \chi _{2}^{\ast }\left( x\right) =\chi _{2}^{\ast }\left( \mu \ast
x\right) =\chi _{2}^{\ast }\left( \varphi \left( \mu \right) x\right) ,$$where operations in algebra $F_{W}^{\ast }/F^{2}$ we denote by same symbols as operations in algebra $F_{W}^{\ast }$ and $\mu \in k\setminus \left\{
0\right\} $. Therefore $c_{F}\left( x\right) \equiv \varphi \left( \mu
\right) x\left( \func{mod}F^{2}\right) $. $\alpha \left( F^{2}\right)
\subset F^{2}$ fulfils, so$$\alpha c_{F}\left( x\right) =\alpha \left( \varphi \left( \mu \right)
x+f_{2}\right) \equiv \alpha \left( \varphi \left( \mu \right) x\right)
=\varphi \left( \mu \right) \alpha \left( x\right) =\varphi \left( \mu
\right) \lambda x\left( \func{mod}F^{2}\right) ,$$where $f_{2}\in F^{2}$. $$c_{F}\alpha \left( x\right) =c_{F}\left( \lambda x\right) =\lambda \ast
c_{F}\left( x\right) =\varphi \left( \lambda \right) c_{F}\left( x\right)
\equiv \varphi \left( \lambda \right) \varphi \left( \mu \right) x\left(
\func{mod}F^{2}\right) .$$$\mu \neq 0$, so $\varphi \left( \mu \right) \neq 0$, $\varphi \left(
\lambda \right) \neq \lambda $ hence $\alpha c_{F}\neq c_{F}\alpha $. This contradiction proves that $\varphi =id_{k}$.
Now we denote $F=F\left( x_{1},x_{2}\right) \in \mathrm{Ob}\Theta ^{0}$. By our assumption there exists an isomorphism $c_{F}:F\rightarrow F_{W}^{\ast }$ such that $c_{F}\alpha =\alpha c_{F}$ fulfills for every $\alpha \in \mathrm{End}F$. $c_{2}$ is defined as in the Lemma \[F/F\^2\]. $\alpha \left(
F^{2}\right) \subset F^{2}$ so we can define the homomorphism $\widetilde{\alpha }:F/F^{2}\rightarrow F/F^{2}$ such that $\widetilde{\alpha }\chi
_{2}=\chi _{2}\alpha $. From $c_{F}\alpha =\alpha c_{F}$ we can conclude $c_{2}\widetilde{\alpha }=\widetilde{\alpha }c_{2}$ fulfills. By Lemma [F/F\^2]{} $c_{2}$ is a regular linear mapping. We can take the endomorphisms $\alpha $ such that $\widetilde{\alpha }$ will be an arbitrary linear mapping from $k^{2}$ to $k^{2}$. Therefore $c_{2}$ must be a regular linear mapping from $k^{2}$ to $k^{2}$ which commutate with all linear mappings from $k^{2}$ to $k^{2}$. Hence $c_{2}$ must be a scalar mapping, i.e., $$\chi _{2}^{\ast }c_{F}\left( x_{i}\right) =c_{2}\chi _{2}\left( x_{i}\right)
=\lambda \chi _{2}^{\ast }\left( x_{i}\right) =\chi _{2}^{\ast }\left(
\lambda x_{i}\right) ,$$where $\lambda \in k\setminus \left\{ 0\right\} $, $i=1,2$. Therefore $c_{F}\left( x_{i}\right) =\lambda x_{i}+f_{i}$, where $f_{i}\in F^{2}$, $i=1,2$. We can remark that now we consider the case when $\varphi =id_{k}$, hence we need not distinguish between multiplication by scalar in $F$ and $F_{W}^{\ast }$.
Now we take $\alpha \in \mathrm{End}F$ such that $\alpha \left( x_{1}\right)
=x_{1}x_{2}$, $\alpha \left( x_{2}\right) =0$. If $u$ is a monomial which contain only entries of $x_{1}$, then $\deg _{x_{1}}\alpha \left( u\right)
+\deg _{x_{2}}\alpha \left( u\right) =2\deg _{x_{1}}u$. If a monomial $u$ contain at least one entry of $x_{2}$, then $\alpha \left( u\right) =0$. Hence $\alpha \left( F^{2}\right) \subset F^{3}$. So we have $$c_{F}\alpha \left( x_{1}\right) =c_{F}\left( x_{1}x_{2}\right) =c_{F}\left(
x_{1}\right) \times c_{F}\left( x_{2}\right) =ac_{F}\left( x_{1}\right)
c_{F}\left( x_{2}\right) +bc_{F}\left( x_{2}\right) c_{F}\left( x_{1}\right)
=$$$$a\left( \lambda x_{1}+f_{1}\right) \left( \lambda x_{2}+f_{2}\right)
+b\left( \lambda x_{2}+f_{2}\right) \left( \lambda x_{1}+f_{1}\right) \equiv
a\lambda ^{2}x_{1}x_{2}+b\lambda ^{2}x_{2}x_{1}\left( \func{mod}F^{3}\right)
.$$$$\alpha c_{F}\left( x_{1}\right) =\alpha \left( \lambda x_{1}+f_{1}\right)
\equiv \lambda x_{1}x_{2}\left( \func{mod}F^{3}\right) .$$Hence we conclude $b=0$ from $c_{F}\alpha =\alpha c_{F}$.
If $b=0$, i. e., $w_{\cdot }=ax_{1}x_{2}$, $a\neq 0$, then we take $c_{F}\left( f\right) =a^{-1}f$ for every $F\in \mathrm{Ob}\Theta ^{0}$ and every $f\in F$. It is obvious that $c_{F}$ is a regular linear mapping. $$c_{F}\left( f_{1}\right) \times c_{F}\left( f_{2}\right) =ac_{F}\left(
f_{1}\right) c_{F}\left( f_{2}\right) =a\left( a^{-1}f_{1}\right) \left(
a^{-1}f_{2}\right) =a^{-1}f_{1}f_{2}=c_{F}\left( f_{1}f_{2}\right) .$$for every $f_{1},f_{2}\in F$. So $c_{F}:F\rightarrow F_{W}^{\ast }$ is an isomorphism. It fulfils$$c_{F}\alpha \left( d\right) =a^{-1}\alpha \left( d\right) =\alpha \left(
a^{-1}d\right) =\alpha c_{F}\left( d\right)$$for every $\left( \alpha :D\rightarrow F\right) \in \mathrm{Mor}\Theta ^{0}$ and every $d\in D$.
\[str\_stb\_group\]The group $\mathfrak{S\cong }G\mathfrak{\leftthreetimes
}\mathrm{Aut}k$, where $G$ is the group of all regular $2\times 2$ matrices over field $k$, which have a form $\left(
\begin{array}{cc}
a & b \\
b & a\end{array}\right) $ and every $\varphi \in \mathrm{Aut}k$ acts on the group $G$ by this way: $\varphi \left(
\begin{array}{cc}
a & b \\
b & a\end{array}\right) =\left(
\begin{array}{cc}
\varphi \left( a\right) & \varphi \left( b\right) \\
\varphi \left( b\right) & \varphi \left( a\right)\end{array}\right) $.
We will define the mapping $\tau :G\mathfrak{\leftthreetimes }\mathrm{Aut}k\rightarrow \mathfrak{S}$. If $g\varphi \in G\mathfrak{\leftthreetimes }\mathrm{Aut}k$, where $g=\left(
\begin{array}{cc}
a & b \\
b & a\end{array}\right) $, then we define $\tau \left( g\varphi \right) =\Phi \in \mathfrak{S}$, where $\Phi $ corresponds to the system of words $W$ with $w_{\lambda
}=\varphi \left( \lambda \right) x_{1}$ for every $\lambda \in k$ and $w_{\cdot }=ax_{1}x_{2}+bx_{2}x_{1}$. By Section \[operations\] and Theorem \[words\] $\tau $ is bijection.
We consider $\tau \left( g_{1}\varphi _{1}\right) =\Phi _{1}$ and $\tau
\left( g_{2}\varphi _{2}\right) =\Phi _{2}$, where $g_{1}\varphi
_{1},g_{2}\varphi _{2}\in G\mathfrak{\leftthreetimes }\mathrm{Aut}k$ and $g_{1}=\left(
\begin{array}{cc}
a_{1} & b_{1} \\
b_{1} & a_{1}\end{array}\right) $, $g_{2}=\left(
\begin{array}{cc}
a_{2} & b_{2} \\
b_{2} & a_{2}\end{array}\right) $. Both these strongly stable automorphisms preserves all objects of $\Theta ^{0}$ and acts on morphisms of $\Theta ^{0}$ by theirs systems of bijections $\left\{ s_{F}^{\Phi _{i}}:F\rightarrow F\mid F\in \mathrm{Ob}\Theta ^{0}\right\} $, for $i=1,2$, according the formula ([biject\_action]{}). We have $\Phi _{2}\Phi _{1}\left( \alpha \right)
=s_{F}^{\Phi _{2}}s_{F}^{\Phi _{1}}\alpha \left( s_{D}^{\Phi _{1}}\right)
^{-1}\left( s_{D}^{\Phi _{2}}\right) ^{-1}$ for every $\left( \alpha
:D\rightarrow F\right) \in \mathrm{Mor}\Theta ^{0}$. So strongly stable automorphism $\Phi _{2}\Phi _{1}=\tau \left( g_{2}\varphi _{2}\right) \tau
\left( g_{1}\varphi _{1}\right) $ preserves all objects of $\Theta ^{0}$ and acts on morphisms of $\Theta ^{0}$ by system of bijections$$\left\{ s_{F}^{\Phi _{2}}s_{F}^{\Phi _{1}}:F\rightarrow F\mid F\in \mathrm{Ob}\Theta ^{0}\right\} .$$This system of bijections satisfies the conditions B1 and B2, so we can define the words $w_{\lambda }^{\Phi _{2}\Phi _{1}}$ for every $\lambda \in
k $ and $w_{\cdot }^{\Phi _{2}\Phi _{1}}$ which correspond to the automorphism $\Phi _{2}\Phi _{1}$ by formula (\[der\_veb\_opr\]). The words $w_{\lambda }^{\Phi _{i}}(\lambda \in k)$ and $w_{\cdot }^{\Phi _{i}}$ which correspond to the automorphism $\Phi _{i}$ have forms $w_{\lambda }^{\Phi
_{i}}=\varphi _{i}\left( \lambda \right) x_{1}(\lambda \in k)$ and $w_{\cdot
}^{\Phi _{i}}=a_{i}x_{1}x_{2}+b_{i}x_{2}x_{1}$ for $i=1,2$. So $$w_{\lambda }^{\Phi _{2}\Phi _{1}}=s_{F}^{\Phi _{2}}s_{F}^{\Phi _{1}}\left(
\lambda x_{1}\right) =s_{F}^{\Phi _{2}}\left( w_{\lambda }^{\Phi
_{1}}\right) =s_{F}^{\Phi _{2}}\left( \varphi _{1}\left( \lambda \right)
x_{1}\right) =\varphi _{2}\left( \varphi _{1}\left( \lambda \right) \right)
x_{1}=\left( \varphi _{2}\varphi _{1}\right) \left( \lambda \right) x_{1}$$for every $\lambda \in k$ and$$w_{\cdot }^{\Phi _{2}\Phi _{1}}=s_{F}^{\Phi _{2}}s_{F}^{\Phi _{1}}\left(
x_{1}x_{2}\right) =s_{F}^{\Phi _{2}}\left( w_{\cdot }^{\Phi _{1}}\right)
=s_{F}^{\Phi _{2}}\left( a_{1}x_{1}x_{2}+b_{1}x_{2}x_{1}\right) =$$$$\varphi _{2}\left( a_{1}\right) s_{F}^{\Phi _{2}}\left( x_{1}x_{2}\right)
+\varphi _{2}\left( b_{1}\right) s_{F}^{\Phi _{2}}\left( x_{2}x_{1}\right) =$$$$\varphi _{2}\left( a_{1}\right) \left(
a_{2}x_{1}x_{2}+b_{2}x_{2}x_{1}\right) +\varphi _{2}\left( b_{1}\right)
\left( a_{2}x_{2}x_{1}+b_{2}x_{1}x_{2}\right) =$$$$\left( \varphi _{2}\left( a_{1}\right) a_{2}+\varphi _{2}\left( b_{1}\right)
b_{2}\right) x_{1}x_{2}+\left( \varphi _{2}\left( a_{1}\right) b_{2}+\varphi
_{2}\left( b_{1}\right) a_{2}\right) x_{2}x_{1}.$$because $s_{F}^{\Phi _{i}}:F\rightarrow F_{W_{i}}^{\ast }$ is an isomorphism, $i=1,2$. Hence$$\Phi _{2}\Phi _{1}=\tau \left( g_{2}\varphi _{2}\right) \tau \left(
g_{1}\varphi _{1}\right) =\tau \left( g_{2}\varphi _{2}\left( g_{1}\right)
\varphi _{2}\varphi _{1}\right) =\tau \left( g_{2}\varphi _{2}\cdot
g_{1}\varphi _{1}\right) .$$
\[intersection\]Group $\mathfrak{S\cap Y}$ is isomorphic to the group $k^{\ast }I_{2}$ of the regular $2\times 2$ scalar matrices over field $k$.
By Propositions \[stable\_inner\] and \[str\_stb\_group\].
$\mathfrak{A/Y\cong }\left( G\mathfrak{/}k^{\ast }I_{2}\right) \mathfrak{\leftthreetimes }\mathrm{Aut}k$.
By Proposition \[str\_stb\_group\] and Corollary 1 we have that $\mathfrak{A/Y\cong }\left( G\mathfrak{\leftthreetimes }\mathrm{Aut}k\right) \mathfrak{/}k^{\ast }I_{2}$. And we have $\left( G\mathfrak{\leftthreetimes }\mathrm{Aut}k\right) \mathfrak{/}k^{\ast }I_{2}\mathfrak{\cong }\left( G\mathfrak{/}k^{\ast }I_{2}\right) \mathfrak{\leftthreetimes }\mathrm{Aut}k$ because $k^{\ast }I_{2}\vartriangleleft G$ and for every $\varphi \in \mathrm{Aut}k$ $\varphi \left( k^{\ast }I_{2}\right) \subset k^{\ast }I_{2}$ fulfills.
The symmetric group of the set which has $2$ elements - $\mathbf{S}_{\mathbf{2}}$ can be embedded in the multiplicative structure of the algebra $\mathbf{M}_{\mathbf{2}}\left( k\right) $ of the $2\times 2$ matrices over field $k$: $\mathbf{S}_{\mathbf{2}}\ni \left( 12\right) \rightarrow \left(
\begin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right) \in \mathbf{M}_{\mathbf{2}}\left( k\right) $, so $G\cong U\left( k\mathbf{S}_{\mathbf{2}}\right) $, where $U\left( k\mathbf{S}_{\mathbf{2}}\right) $ is the group of all invertible elements of the group algebra $k\mathbf{S}_{\mathbf{2}}$. Also $k^{\ast }I_{2}\cong U\left( k\left\{
e\right\} \right) $, where $e\in \mathbf{S}_{\mathbf{2}}$, $k\left\{
e\right\} $ is a subalgebra of $k\mathbf{S}_{\mathbf{2}}$, $U\left( k\left\{
e\right\} \right) $ is a group of all invertible elements of this subalgebra. Therefore $\mathfrak{A/Y\cong }\left( U\left( k\mathbf{S}_{\mathbf{2}}\right) \mathfrak{/}U\left( k\left\{ e\right\} \right) \right)
\mathfrak{\leftthreetimes }\mathrm{Aut}k$, where every $\varphi \in \mathrm{Aut}k$ acts on the algebra $k\mathbf{S}_{\mathbf{2}}$ by natural way: $\varphi \left( ae+b\left( 12\right) \right) =\varphi \left( a\right)
e+\varphi \left( b\right) \left( 12\right) $.
Example of two linear algebras which are automorphically equivalent but not** **geometrically equivalent.
=========================================================================================================
We take $k=\mathbb{Q}
$. $\Theta $ will be the variety of all linear algebras over $k$. $H$ will be the $2$-generated linear algebra, which is free in the variety corresponding to the identity $\left( x_{1}x_{1}\right) x_{2}=0$. We consider the strongly stable automorphism $\Phi $ of the category $\Theta
^{0}$ corresponding to the system of words $W$, where $b\neq 0$. Algebras $H$ and $H_{W}^{\ast }$ are automorphically equivalent by [TsurkovAutomEquiv]{}.
\[not\_geom\_equiv\]Algebras $H$ and $H_{W}^{\ast }$ are not geometrically equivalent.
Let $F=F\left( x_{1},x_{2}\right) $. The ideal $I=Id\left( H,\left\{
x_{1},x_{2}\right\} \right) $ of the all two-variables identities which are fulfill in the algebra $H$ will be the smallest $H$-closed set in $F$, because $I=\left( 0\right) _{H}^{\prime \prime }$, where $0\in F$. If algebras $H$ and $H_{W}^{\ast }$ are geometrically equivalent then the structures of the $H$-closed sets and of the $H_{W}^{\ast }$-closed sets in $F$ coincide. Hence $I$ must be the smallest $H_{W}^{\ast }$-closed set in $F$.
By [@TsurkovAutomEquiv Remark 5.1] $$T\rightarrow \sigma _{F}T \label{closed_bijection}$$is a bijection from the structure of the $H_{W}^{\ast }$-closed sets in $F$ to the structure of the $H$-closed sets in $F$. Hear $\sigma
_{F}:F\rightarrow F_{W}^{\ast }$ is an isomorphism from condition Op2. It is clear that the bijection (\[closed\_bijection\]) preserves inclusions of sets. So it transforms the smallest $H_{W}^{\ast }$-closed set to the smallest $H$-closed set, i. e. $I=\sigma _{F}I$ must fulfills.
It is obviously that $I\subset F^{3}$. By (\[epi\_homo\]) $\sigma
_{F}I\subset F^{3}$. We will compare the linear subspaces $I/F^{4}$ and $\left( \sigma _{F}I\right) /F^{4}$. $I=\left\langle \alpha \left( \left(
x_{1}x_{1}\right) x_{2}\right) \mid \alpha \in \mathrm{End}F\right\rangle $. Let $\alpha \left( x_{i}\right) \equiv \alpha _{1i}x_{1}+\alpha
_{2i}x_{2}\left( \func{mod}F^{2}\right) $, where $i=1,2$, $\alpha _{ji}\in k$. Then$$\alpha \left( \left( x_{1}x_{1}\right) x_{2}\right) \equiv \left( \left(
\alpha _{11}x_{1}+\alpha _{21}x_{2}\right) \left( \alpha _{11}x_{1}+\alpha
_{21}x_{2}\right) \right) \left( \alpha _{12}x_{1}+\alpha _{22}x_{2}\right)
\left( \func{mod}F^{4}\right) .$$We achieve after the extending of brackets that $I/F^{4}$ is a subspace of the linear space spanned by the elements of $F^{3}/F^{4}$ which have form $\left( x_{i}x_{j}\right) x_{k}+F^{4}$, where $i,j,k=1,2$. But$$\sigma _{F}I\ni \sigma _{F}\left( \left( x_{1}x_{1}\right) x_{2}\right)
=a\sigma _{F}\left( x_{1}x_{1}\right) \sigma _{F}\left( x_{2}\right)
+b\sigma _{F}\left( x_{2}\right) \sigma _{F}\left( x_{1}x_{1}\right) =$$$$=a\left( a+b\right) \left( x_{1}x_{1}\right) x_{2}+b\left( a+b\right)
x_{2}\left( x_{1}x_{1}\right) .$$We have that $a+b\neq 0$, $b\neq 0$, so $I/F^{4}\neq \left( \sigma
_{F}I\right) /F^{4}$ and $I\neq \sigma _{F}I$. This contradiction proves that algebras $H$ and $H_{W}^{\ast }$ are not geometrically equivalent.
[9]{} B. Plotkin, *Varieties of algebras and algebraic varieties. Categories of algebraic varieties.* Siberian Advanced Mathematics, Allerton Press, **7:2** (1997), pp. 64 – 97.
B. Plotkin, *Some notions of algebraic geometry in universal algebra,* Algebra and Analysis, **9:4** (1997), pp. 224 – 248, St. Petersburg Math. J., **9:4**, (1998) pp. 859 – 879.
B. Plotkin, *Algebras with the same (algebraic) geometry,* Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P.S. Novikov, Proceedings of the Steklov Institute of Mathematics, MIAN, [ v.242]{}, (2003), pp. 17 – 207.
B. Plotkin, G. Zhitomirski *On automorphisms of categories of free algebras of some varieties,* Journal of Algebra, **306:2**, (2006), 344 – 367.
A. Tsurkov, *Automorphic equivalence of algebras.* International Journal of Algebra and Computation. **17:5/6**, (2007), 1263–1271.
|
---
abstract: |
Homotopy type theory is an interpretation of Martin-Löf’s constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional systems of type theory as well as a computational approach to algebraic topology via type theory-based proof assistants such as Coq.
The present work investigates inductive types in this setting. Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined. Proofs of all results have been formally verified by the Coq proof assistant, and the proof scripts for this verification form an essential component of this research.
address:
- 'Department of Philosophy, Carnegie Mellon University'
- 'Dipartimento di Matematica e Informatica, Università degli Studi di Palermo'
- 'School of Computer Science, Carnegie Mellon University'
author:
- Steve Awodey
- Nicola Gambino
- Kristina Sojakova
date: 'May 2nd, 2012'
title: Inductive Types in Homotopy Type Theory
---
Introduction {#introduction .unnumbered}
============
The constructive type theories introduced by Martin-Löf are dependently-typed $\lambda$-calculi with operations for identity types ${{\mathsf{Id}}_{A}}(a,b)$, dependent products $(\Pi x {\, : \,} A)B(x)$ and dependent sums $(\Sigma x {\, : \,} A)B(x)$, among others [@MartinLofP:intttp; @MartinLofP:conmcp; @MartinLofP:inttt; @NordstromB:promlt; @NordstromB:marltt]. These are related to the basic concepts of predicate logic, *viz.* equality and quantification, via the familiar propositions-as-types correspondence [@HowardWH:foratn]. The different systems introduced by Martin-Löf over the years vary greatly both in proof-theoretic strength [@GrifforE:strsml] and computational properties. From the computational point of view, it is important to distinguish between the extensional systems, that have a stronger notion of equality, but for which type-checking is undecidable, and the intensional ones, that have a weaker notion of equality, but for which type-checking is decidable [@HofmannM:extcit; @MaiettiME:mintlf]. For example, the type theory presented in [@MartinLofP:inttt] is extensional, while that in [@NordstromB:marltt] is intensional.
The difference between the extensional and the intensional treatment of equality has a strong impact also on the properties of the various types that may be assumed in a type theory, and in particular on those of inductive types, such as the types of Booleans, natural numbers, lists and W-types [@MartinLofP:conmcp]. Within extensional type theories, inductive types can be characterized (up to isomorphism) as initial algebras of certain definable functors. The initiality condition translates directly into a recursion principle that expresses the existence and uniqueness of recursively-defined functions. In particular, W-types can be characterized as initial algebras of polynomial functors [@DybjerP:repids; @MoerdijkI:weltc]. Furthermore, within extensional type theories, W-types allow us to define a wide range of inductive types, such as the type of natural numbers and types of lists [@DybjerP:repids; @GambinoN:weltdp; @AbbottM:concsp]. Within intensional type theories, by contrast, the correspondence between inductive types and initial algebras breaks down, since it is not possible to prove the uniqueness of recursively-defined functions. Furthermore, the reduction of inductive types like the natural numbers to W-types fails [@DybjerP:repids; @GoguenH:inddtw].
In the present work, we exploit insights derived from the new models of intensional type theory based on homotopy-theoretic ideas [@AwodeyS:homtmi; @VoevodskyV:notts; @vandenBergB:topsmi] to investigate inductive types, thus contributing to the new area known as Homotopy Type Theory. Homotopical intuition justifies the assumption of a limited form of function extensionality, which, as we show, suffices to deduce uniqueness properties of recursively-defined functions up to homotopy. Building on this observation, we introduce the notions of *weak algebra homomorphism* and *homotopy-initial algebra*, which require uniqueness of homomorphisms up to homotopy. We modify the rules for W-types by replacing the definitional equality in the standard computation rule with its propositional counterpart, yielding a weak form of the corresponding inductive type. Our main result is that these new, weak W-types correspond precisely to homotopy-initial algebras of polynomial functors. Furthermore, we indicate how homotopical versions of various inductive types can be defined as special cases of the general construction in the new setting
The work presented here is motivated in part by the Univalent Foundations program formulated by Voevodsky [@VoevodskyV:unifp]. This ambitious program intends to provide comprehensive foundations for mathematics on the basis of homotopically-motivated type theories, with an associated computational implementation in the Coq proof assistant. The present investigation of inductive types serves as an example of this new paradigm: despite the fact that the intuitive basis lies in higher-dimensional category theory and homotopy theory, the actual development is strictly syntactic, allowing for direct formalization in Coq. Proof scripts of the definitions, results, and all necessary preliminaries are provided in a downloadable repository [@AwodeyS:indtht].
The paper is organized as follows. In section \[section:prelims\], we describe and motivate the dependent type theory over which we will work and compare it to some other well-known systems in the literature. The basic properties of the system and its homotopical interpretation are developed to the extent required for the present purposes. Section \[section:extW\] reviews the basic theory of W-types in extensional type theory and sketches the proof that these correspond to initial algebras of polynomial functors; there is nothing new in this section, rather it serves as a framework for the generalization that follows. Section \[section:extW\] on intensional W-types contains the development of our new theory; it begins with a simple example, that of the type ${\mathsf{2}}$ of Boolean truth values, which serves to indicate the main issues involved with inductive types in the intensional setting, and our proposed solution. We then give the general notion of weak W-types, including the crucial new notion of *homotopy-initiality*, and state our main result, the equivalence between the type-theoretic rules for weak W-types and the existence of a homotopy-initial algebra of the corresponding polynomial functor. Moreover, we show how some of the difficulties with intensional W-types are remedied in the new setting by showing that the type of natural numbers can be defined as an appropriate W-type. Finally, we conclude by indicating how this work fits into the larger study of inductive types in Homotopy Type Theory and the Univalent Foundations program generally.
Preliminaries {#section:prelims}
=============
The general topic of Homotopy Type Theory is concerned with the study of the constructive type theories of Martin-Löf under their new interpretation into abstract homotopy theory and higher-dimensional category theory. Martin-Löf type theories are foundational systems which have been used to formalize large parts of constructive mathematics, and also for the development of high-level programming languages [@MartinLofP:conmcp]. They are prized for their combination of expressive strength and desirable proof-theoretic properties. One aspect of these type theories that has led to special difficulties in providing semantics is the intensional character of equality. In recent work [@AwodeyS:homtmi; @VoevodskyV:notts; @vandenBergB:topsmi; @AwodeyS:typth], it has emerged that the topological notion of *homotopy* provides an adequate basis for the semantics of intensionality. This extends the paradigm of computability as continuity, familiar from domain theory, beyond the simply-typed $\lambda$-calculus to dependently-typed theories involving:
(i) dependent sums $(\Sigma x\colon\!{A})B(x)$ and dependent products $(\Pi x\colon\!{A})B(x)$, modelled respectively by the total space and the space of sections of the fibration modelling the dependency of $B(x)$ over $ x : A$;
(ii) and, crucially, including the identity type constructor ${{\mathsf{Id}}_{A}}(a,b)$, interpreted as the space of all *paths* in $A$ between points $a$ and $b$.
In the present work, we build on this homotopical interpretation to study inductive types, such as the natural numbers, Booleans, lists, and W-types. Within extensional type theories, W-types can be used to provide a constructive counterpart of the classical notion of a well-ordering [@MartinLofP:inttt] and to uniformly define a variety of inductive types [@DybjerP:repids]. However, most programming languages and proof assistants, such as Coq [@BertotY:inttpp], Agda [@NorellU:towppl] and Epigram [@McBrideC:viefl] use schematic inductive definitions [@CoquandT:inddt; @PaulinMorhringC:inddsc] rather than W-types to define inductive types. This is due in part to the practical convenience of the schematic approach, but it is also a matter of necessity; these systems are based on intensional rather than extensional type theories, and in the intensional theory the usual reductions of inductive types to W-types fail [@DybjerP:repids; @McBrideC:wtygnb]. Nonetheless, W-types retain great importance from a theoretical perspective, since they allow us to internalize in type theory arguments about inductive types. Furthermore, as we will see in Section \[section:intW\], a limited form of extensionality licensed by the homotopical interpretation suffices to develop the theory of W-types in a satisfactory way. In particular, we shall make use of ideas from higher category theory and homotopy theory to understand W-types as “homotopy-initial" algebras of an appropriate kind.
Extensional vs. intensional type theories
-----------------------------------------
We work here with type theories that have the four standard forms of judgement $$A : {\mathsf{type}}\, , \quad A = B : {\mathsf{type}}\, , \quad a : A \, , \quad a = b : A \, .$$ We refer to the equality relation in these judgements as *definitional equality*, which should be contrasted with the notion of *propositional equality* recalled below. Such a judgement $J$ can be made also relative to a *context* $\Gamma$ of variable declarations, a situation that we indicate by writing $\Gamma \vdash J$. When stating deduction rules we make use of standard conventions to simplify the exposition, such as omitting the mention of a context that is common to premisses and conclusions of the rule. The rules for identity types in intensional type theories are given in [@NordstromB:marltt Section 5.5]. We recall them here in a slighly different, but equivalent, formulation.
- ${\mathsf{Id}}$-formation rule. $$\begin{prooftree}
A : {\mathsf{type}}\quad
a : A \quad
b : A
\justifies
{{\mathsf{Id}}_{A}}(a,b) : {\mathsf{type}}\end{prooftree}$$
- ${\mathsf{Id}}$-introduction rule. $$\begin{prooftree}
a : A
\justifies
{\mathsf{refl}}(a) : {{\mathsf{Id}}_{A}}(a,a)
\end{prooftree}$$
- ${\mathsf{Id}}$-elimination rule. $$\begin{prooftree}
\begin{array}{l}
x, y : A, u : {{\mathsf{Id}}_{A}}(x,y) \vdash C(x,y,u) : {\mathsf{type}}\\
x : A \vdash c(x) : C(x,x,{\mathsf{refl}}(x))
\end{array}
\justifies
x, y : A, u : {{\mathsf{Id}}_{A}}(x,y) \vdash {\mathsf{idrec}}(x,y,u,c) : C(x,y,u)
\end{prooftree}$$
- ${\mathsf{Id}}$-computation rule. $$\begin{prooftree}
\begin{array}{l}
x, y : A, u : {{\mathsf{Id}}_{A}}(x,y) \vdash C(x,y,u) : {\mathsf{type}}\\
x : A \vdash c(x) : C(x,x,{\mathsf{refl}}(x))
\end{array}
\justifies
x : A \vdash {\mathsf{idrec}}(x,x,{\mathsf{refl}}(x), c) = c(x) : C(x, x, {\mathsf{refl}}(x)) \, .
\end{prooftree}$$
As usual, we say that two elements $a, b :A$ are *propositionally equal* if the type ${\mathsf{Id}}(a,b)$ is inhabited. Most work on W-types to date (*e.g.* [@DybjerP:repids; @MoerdijkI:weltc; @AbbottM:concsp]) has been in the setting of extensional type theories, in which the following rule, known as the *identity reflection rule*, is also assumed:
$$\label{equ:collapse}
\begin{prooftree}
p : {{\mathsf{Id}}_{A}}(a,b)
\justifies
a=b : A
\end{prooftree}$$
This rule collapses propositional equality with definitional equality, thus making the overall system somewhat simpler to work with. However, it destroys the constructive character of the intensional system, since it makes type-checking undecidable [@HofmannM:extcit]. For this reason, it is not assumed in the most recent formulations of Martin-Löf type theories [@NordstromB:marltt] or in automated proof assistants like Coq [@BertotY:inttpp].
In intensional type theories, inductive types cannot be characterized by standard category-theoretic universal properties. For instance, in this setting it is not possible to show that there exists a definitionally-unique function out of the empty type with rules as in [@NordstromB:marltt Section 5.2], thus making it impossible to prove that the empty type provides an initial object. Another consequence of this fact is that, if we attempt to define the type of natural numbers as a W-type in the usual way, then the usual elimination and computation rules for it are no longer derivable [@DybjerP:repids]. Similarly, it is not possible to show the uniqueness of recursively-defined functions out of W-types. When interpreted categorically, the uniqueness of such functions translates into the initiality property of the associated polynomial functor algebra, which is why the correspondence between W-types and initial algebras fails in the intensional setting.
Due to this sort of poor behaviour of W-types, and other constructions, in the purely intensional setting, that system is often augmented by other extensionality principles that are somewhat weaker than the Reflection rule, such as Streicher’s K-rule or the Uniqueness of Identity Proofs (UIP) [@StreicherT:invitt], which has recently been reconsidered in the context of Observational Type Theory [@AltenkirchT:obsen]. Inductive types in such intermediate systems are somewhat better behaved, but still exhibit some undesirable properties, making them less useful for practical purposes than one might wish [@McBrideC:wtygnb]. Moreover, these intermediate systems seem to lack a clear conceptual basis: they neither intend to formalize constructive sets (like the extensional theory) nor is there a principled reason to choose these particular extensionality rules, beyond their practical advantages.
The system ${\mathcal{H}}$
--------------------------
We here take a different approach to inductive types in the intensional setting, namely, one motivated by the homotopical interpretation. It involves working over a dependent type theory ${\mathcal{H}}$ which has the following deduction rules on top of the standard structural rules:
- rules for identity types as stated above;
- rules for $\Sigma$-types as in [@NordstromB:marltt Section 5.8];
- rules for $\Pi$-types as in [@GarnerR:strdpt Section 3.2];
- the propositional $\eta$-rule for $\Pi$-types, *i.e.* the axiom asserting that for every $f : (\Pi x : A) B(x)$, the type ${\mathsf{Id}}(f,
\lambda x. {\mathsf{app}}(f,x))$ is inhabited;
- the Function Extensionality axiom (FE), *i.e.* the axiom asserting that for every $f, g : A \rightarrow B$, the type $$(\Pi x : A){{\mathsf{Id}}_{B}}( {\mathsf{app}}(f, x), {\mathsf{app}}(g, x)) \rightarrow {{\mathsf{Id}}_{A \rightarrow B}}(f,g)$$ is inhabited.
Here, we have used the notation $A \rightarrow B$ to indicate function types, defined via $\Pi$-types in the usual way. Similarly, we will write $A \times B$ to denote the binary product of two types as usually defined via $\Sigma$-types.
### Remarks {#remarks .unnumbered}
(i) The rules for $\Pi$-types of ${\mathcal{H}}$ are derivable from those in [@NordstromB:marltt Section 5.4]. For simplicity, we will write $f(a)$ or $f a$ instead of ${\mathsf{app}}(f,a)$.
(ii) As shown in [@VoevodskyV:unifc], the $\eta$-rule for dependent functions and the function extensionality principle stated above imply the corresponding function extensionality principle for dependent functions, *i.e.* $$(\Pi x : A){{\mathsf{Id}}_{B(x)}}( f x, g x) \rightarrow {{\mathsf{Id}}_{(\Pi x : A) B(x)}}(f,g) \, .$$
(iii) The following form of the $\eta$-rule for $\Sigma$-types is derivable: $$\begin{prooftree}
c : (\Sigma x : A)B(x)
\justifies
\eta_{\Sigma}(c) : {\mathsf{Id}}(c, {\mathsf{pair}}( \pi_1 c \, , \pi_2 c)) \, ,
\end{prooftree}$$ where $\pi_1$ and $\pi_2$ are the projections. This can be proved by $\Sigma$-elimination, without FE.
(iv) ${\mathcal{H}}$ does *not* include the $\eta$-rules as definitional equalities, either for $\Sigma$-types or for $\Pi$-types (as is done in [@GoguenH:inddtw]).
(v) The type theory ${\mathcal{H}}$ will serve as the background theory for our study of inductive types and W-types. For this reason, we need not assume it to have any primitive types.
This particular combination of rules is motivated by the fact that ${\mathcal{H}}$ has a clear homotopy-theoretic sematics. Indeed, the type theory ${\mathcal{H}}$ is a subsystem of the type theory used in Voevodsky’s Univalent Foundations library [@VoevodskyV:unifc]. In particular, the Function Extensionality axiom is formally implied by Voevodsky’s Univalence axiom [@VoevodskyV:notts], which is also valid in homotopy-theoretic models, but will not be needed here. Note that, while the Function Extensionality axiom is valid also in set-theoretic models, the Univalence axiom is not. Although ${\mathcal{H}}$ has a straightforward set-theoretical semantics, we stress that it does not have any global extensionality rules, like the identity reflection rule, K, or UIP. This makes it also compatible with “higher-dimensional" interpretations such as the groupoid model [@HofmannM:gromtt], in which the rules of ${\mathcal{H}}$ are also valid.
Homotopical semantics
---------------------
The homotopical semantics of ${\mathcal{H}}$ is based on the idea that an identity term $p: {{\mathsf{Id}}_{A}}(a,b)$ is (interpreted as) a path $p: a\leadsto b$ between the points $a$ and $b$ in the space $A$. More generally, the interpretations of terms $a(x)$ and $b(x)$ with free variables will be continuous functions into the space $A$, and an identity term $p(x) : {{\mathsf{Id}}_{A}}\big(a(x),b(x)\big)$ is then a continuous family of paths, *i.e.* a homotopy between the continuous functions. Now, the main import of the ${\mathsf{Id}}$-elimination rule is that type dependency must respect identity, in the following sense: given a dependent type $$\label{equ:deptype}
x:A \vdash B(x) : {\mathsf{type}}\, ,$$ and $p: {{\mathsf{Id}}_{A}}(a,b)$, there is then a *transport* function $$p_{\, ! } : B(a) \rightarrow B(b),$$ which is defined by ${\mathsf{Id}}$-elimination, taking for $x : A$ the function ${\mathsf{refl}}(x)_{\, !} : B(x) \rightarrow B(x)$ to be the identity on $B(x)$. Semantically, given that an identity term $p: {{\mathsf{Id}}_{A}}(a,b)$ is interpreted as a path $p: a\leadsto b$, this means that a dependent type as in must be interpreted as a space $B\rightarrow A$, fibered over the space $A$, and that the judgement $$x,y:A \vdash{{\mathsf{Id}}_{A}}(x,y) : {\mathsf{type}}$$ is interpreted as the canonical fibration $A^I \rightarrow A\times A$ of the path space $A^I$ over $A \times A$. For a more detailed overview of the homotopical interpretation, see [@AwodeyS:typth].
Independently of this interpretation, each type $A$ can be shown to carry the structure of a weak $\omega$-groupoid in the sense of [@BataninM:mongcn; @LeinsterT:higohc] with the elements of $A$ as objects, identity proofs $p : {{\mathsf{Id}}_{A}}(a,b)$ as morphisms and elements of iterated identity types as $n$-cells [@vandenBergB:typwg; @LumsdaineP:weaci]. Furthermore, ${\mathcal{H}}$ determines a weak $\omega$-category $\mathcal{C}({\mathcal{H}})$ having types as 0-cells, elements $f : A \rightarrow B$ as 1-cells, and elements of (iterated) identity types as $n$-cells [@Lumsdaine:higcft]. The relation between the weak $\omega$-category structure of $\mathcal{C}({\mathcal{H}})$ and the homotopical interpretation of intensional type theories closely mirrors that between higher category theory and homotopy theory in modern algebraic topology, and some methods developed in the latter setting are also applicable in type theory. For instance, the topological notion of contractibility admits the following type-theoretic counterpart, originally introduced by Voevodsky in [@VoevodskyV:unifc].
A type $A$ is called *contractible* if the type $$\label{eq:contractible}
{\mathsf{iscontr}}(A) {=_{\mathrm{def}}}(\Sigma x:A)(\Pi y:A){{\mathsf{Id}}_{A}}(x,y)$$ is inhabited.
The type ${\mathsf{iscontr}}(A)$ can be seen as the propositions-as-types translation of the formula stating that $A$ has a unique element. However, its homotopical interpretation is as a space that is inhabited if and only if the space interpreting $A$ is contractible in the usual topological sense. The notion of contractibility can be used to articulate the world of types into different homotopical dimensions, or *h-levels* [@VoevodskyV:unifc]. This classification has proven to be quite useful in understanding intensional type theory. For example, it permits the definition of new notions of *proposition* and *set* which provide a useful alternative to the standard approach to formalization of mathematics in type theory [@VoevodskyV:unifc].
\[thm:idcontrcontr\] If $A$ is a contractible type, then for every $a, b : A$, the type ${{\mathsf{Id}}_{A}}(a,b)$ is again contractible. This can be proved by ${\mathsf{Id}}$-elimination [@AwodeyS:indtht].
Let us also recall from [@VoevodskyV:unifc] the notions of weak equivalence and homotopy equivalence. To do this, we need to fix some notation. For $f : A \rightarrow B$ and $y : B$, define the type $${\mathsf{hfiber}}(f,y) {=_{\mathrm{def}}}(\Sigma x : A) {{\mathsf{Id}}_{B}}(f x, y) \, .$$ We refer to this type as the *homotopy fiber* of $f$ at $y$.
\[thm:weq\] Let $f : A \rightarrow B$.
- We say that $f$ is a *weak equivalence* if the type $$\mathsf{isweq(f)} {=_{\mathrm{def}}}(\Pi y : B) \, {\mathsf{iscontr}}({\mathsf{hfiber}}(f,y))$$ is inhabited.
- We say that $f$ is a *homotopy equivalence* if there exist a function $g : B\rightarrow A$ and elements $$\begin{aligned}
\eta &: (\Pi x : A) {\mathsf{Id}}( g f x , x) \, ,\\
\varepsilon &: (\Pi y:B) {\mathsf{Id}}( f g y, y) \, .\end{aligned}$$ It is an *adjoint homotopy equivalence* if there are also terms $$\begin{aligned}
p &: (\Pi x : A) {\mathsf{Id}}( \varepsilon_{f x} \, , f \, \eta_x ) \, , \\
q &: (\Pi y : B) {\mathsf{Id}}( \eta_{g y} \, , g \, \varepsilon_y) \, ,\end{aligned}$$ where the same notation for both function application and the action of a function on an identity proof (which is easily definable by ${\mathsf{Id}}$-elimination), and we write $\alpha_x$ instead of $\alpha(x)$ for better readability.
The type $\mathsf{isweq(f)}$ can be seen as the propositions-as-types translation of the formula asserting that $f$ is bijective, while homotopy equivalence is evidently a form of isomorphism. Thus it is a pleasant fact that a function is a weak equivalence if and only if it is a homotopy equivalence [@VoevodskyV:unifc]. We also note that all type-theoretic constructions are homotopy invariant, in the sense that they respect this relation of equivalence, a fact which is exploited by the Univalence axiom [@VoevodskyV:notts].
In Section \[section:intW\] below, these and related homotopy-theoretic insights will be used to study inductive types, but first we must briefly review some basic facts about inductive types in the extensional setting.
Extensional W-types {#section:extW}
===================
We briefly recall the theory of W-types in fully extensional type theories. Let us begin by recalling the rules for W-types from [@MartinLofP:inttt]. To state them more conveniently, we sometimes write $W$ instead of $({\mathsf{W}}x : A) B(x)$.
- ${\mathsf{W}}$-formation rule. $$\begin{prooftree}
A : {\mathsf{type}}\qquad
x : A \vdash B(x) : {\mathsf{type}}\justifies
({\mathsf{W}}x : A) B(x) : {\mathsf{type}}\end{prooftree}$$
- ${\mathsf{W}}$-introduction rule. $$\begin{prooftree}
a:A \qquad
t : B(a) \rightarrow W
\justifies
{\mathsf{sup}}(a, t): W
\end{prooftree}$$
- ${\mathsf{W}}$-elimination rule. $$\begin{prooftree}
\hspace{-1ex} \begin{array}{l}
w : W \vdash C(w) : {\mathsf{type}}\\
x:A \, , u: B(x) \rightarrow W \, , v : (\Pi y : B(x)) C(u(y)) \vdash \\
\qquad c(x,u,v) : C({\mathsf{sup}}(x,u))
\end{array}
\justifies
w : W \vdash {\mathsf{wrec}}(w,c) : C(w)
\end{prooftree}$$
- ${\mathsf{W}}$-computation rule. $$\begin{prooftree}
\hspace{-1ex} \begin{array}{l}
w : W \vdash C(w) : {\mathsf{type}}\\
x:A \, , u: B(x) \rightarrow W \, , v : (\Pi y : B(x)) C(u(y)) \vdash \\
\qquad c(x,u,v) : C({\mathsf{sup}}(x,u))
\end{array}
\justifies
\begin{array}{l}
x : A, u : B(x) \rightarrow W \vdash
{\mathsf{wrec}}({\mathsf{sup}}(x,u), c) = \\
\qquad c(x,u, \lambda y. {\mathsf{wrec}}(u(y), c)) : C({\mathsf{sup}}(x,u)) \, .
\end{array}
\end{prooftree}$$
W-types can be seen informally as the free algebras for signatures with operations of possibly infinite arity, but no equations. Indeed, the premisses of the formation rule above can be thought of as specifying a signature that has the elements of $A$ as operations and in which the arity of $a : A$ is the cardinality of the type $B(a)$. Then, the introduction rule specifies the canonical way of forming an element of the free algebra, and the elimination rule can be seen as the propositions-as-types translation of the appropriate induction principle.
In extensional type theories, this informal description can easily be turned into a precise mathematical characterization. To do so, let us use the theory ${\mathcal{H}_{\mathrm{ext}}}$ obtained by extending ${\mathcal{H}}$ with the reflection rule in . Let $\mathcal{C}({\mathcal{H}_{\mathrm{ext}}})$ be the category with types as objects and elements $f : A \rightarrow B$ as maps, in which two maps are considered equal if and only if they are definitionally equal. The premisses of the introduction rule determines the *polynomial endofunctor* $P : \mathcal{C}({\mathcal{H}_{\mathrm{ext}}}) \rightarrow \mathcal{C}({\mathcal{H}_{\mathrm{ext}}})$ defined by $$P(X) {=_{\mathrm{def}}}(\Sigma x : A ) (B(x) \rightarrow X) \, .$$ A $P$-algebra is a pair consisting of a type $C$ and a function $s_C : PC \rightarrow C$, called the structure map of the algebra. The formation rule gives us an object $W {=_{\mathrm{def}}}({\mathsf{W}}x : A)
B(x)$ and the introduction rule (in combination with the rules for $\Pi$-types and $\Sigma$-types) provides a structure map $$s_W : PW \rightarrow W \, .$$ The elimination rule, on the other hand, states that in order for the projection $\pi_1 \colon C \rightarrow W$, where $C {=_{\mathrm{def}}}(\Sigma w {\, : \, } W) C(w)$, to have a section $s$, as in the diagram $$\xymatrix{
& C \ar[d]^{\pi_1} \\
W \ar[ru]^{s} \ar[r]_{1_W} & W,
}$$ it is sufficient for the type $C$ to have a $P$-algebra structure over $W$. Finally, the computation rule states that the section $s$ given by the elimination rule is also a $P$-algebra homomorphism.
The foregoing elimination rule implies what we call the *simple* ${\mathsf{W}}$-elimination rule: $$\begin{prooftree}
C : {\mathsf{type}}\qquad
x : A, v : B(x) \rightarrow C \vdash c(x,v) : C
\justifies
w : W \vdash {\mathsf{simp\textsf{-}wrec}}(w,c) : C
\end{prooftree}$$ This can be recognized as a recursion principle for maps from $W$ into $P$-algebras, since the premisses of the rule describe exactly a type $C$ equipped with a structure map $s_C
: PC \rightarrow C$. For this special case of the elimination rule, the corresponding computation rule again states that the function $$\lambda w. {\mathsf{simp\textsf{-}wrec}}(w,c) : W \rightarrow C \, ,$$ where $c(x,v) = s_C({\mathsf{pair}}(x,v))$ for $x : A$ and $v : B(x) \rightarrow C$, is a $P$-algebra homomorphism. Moreover, this homomorphism can then be shown to be definitionally unique using the elimination rule, the principle of function extensionality and the reflection rule. The converse implication also holds: one can derive the general ${\mathsf{W}}$-elimination rule from the simple elimination rule and the following $\eta$-rule $$\begin{prooftree}
\begin{array}{l}
C : {\mathsf{type}}\qquad w : W \vdash h(w) : C \\
x : A, v : B(x) \rightarrow C \vdash c(x,v) : C\\
x:A \, , u: B(x) \rightarrow W \vdash h\left({\mathsf{sup}}(x,u)) = c(x,\lambda y. hu(y)\right) : C
\end{array}
\justifies
w : W \vdash h(w) = {\mathsf{simp\textsf{-}wrec}}(w,c) : C
\end{prooftree}$$ stating the uniqueness of the ${\mathsf{simp\textsf{-}wrec}}$ term among algebra maps. Overall, we therefore have that in ${\mathcal{H}_{\mathrm{ext}}}$ induction and recursion are interderivable: $$\begin{array}{ccc}
\text{\underline{{\textbf{Induction}}}} & \Leftrightarrow & \text{\underline{{\textbf{Recursion}}}}\\[1ex]
\text{${\mathsf{W}}$-elimination} & & \text{Simple ${\mathsf{W}}$-elimination}\\
\text{${\mathsf{W}}$-computation} && \text{Simple ${\mathsf{W}}$-computation + $\eta$-rule}
\end{array}$$
Finally, observe that what we are calling recursion is equivalent to the statement that the type $W$, equipped with the structure map $s_W : PW \rightarrow W$ is the initial $P$-algebra. Indeed, assume the simple elimination rule, the simple computation rule and the $\eta$-rule; then for any $P$-algebra $s_C : PC\rightarrow C$, there is a function $f : W \rightarrow C$ by the simple elimination rule, which is a homomorphism by the computational rule, and is the unique such homomorphism by the $\eta$-rule. The converse implication from initiality to recursion is just as direct. Thus, in the extensional theory, to have an initial algebra for the endofunctor $P$ is the same thing as having a type $W$ satisfying the introduction, elimination and computation rules above. Section \[section:intW\] will be devoted to generalizing this equivalence to the setting of Homotopy Type Theory.
Inductive types as W-types
--------------------------
To conclude our review, recall that in extensional type theory, many inductive types can be reduced to W-types. We mention the following examples, among many others (see [@MartinLofP:inttt], [@DybjerP:repids], [@GoguenH:inddtw], [@MoerdijkI:weltc], [@GambinoN:weltdp], [@AbbottM:concsp]):
1. *Natural numbers*. \[extnatW\] The usual rules for ${\ensuremath{\mathsf{Nat}}}$ as an inductive type can be derived from its formalization as the following W-type. Consider the signature with two operations, one of which has arity $0$ and one of which has arity $1$; it is presented type-theoretically by a dependent type with corresponding polynomial functor (naturally isomorphic to) $$P(X) = \mathsf{1} + X \, ,$$ and the natural numbers ${\ensuremath{\mathsf{Nat}}}$ together with the canonical element $0:{\ensuremath{\mathsf{Nat}}}$ and the successor function $s : {\ensuremath{\mathsf{Nat}}}\rightarrow{\ensuremath{\mathsf{Nat}}}$ form an initial $P$-algebra $$(0, s) : \mathsf{1} + {\ensuremath{\mathsf{Nat}}}\rightarrow {\ensuremath{\mathsf{Nat}}}\, .$$
2. *Second number class.* As shown in [@MartinLofP:inttt], the second number class can be obtained as a W-type determined by the polynomial functor $$P(X) = \mathsf{1} + X + ({\ensuremath{\mathsf{Nat}}}\rightarrow X) \, .$$ This has algebras with three operations, one of arity $0$, one of arity $1$, and one of arity (the cardinality of) ${\ensuremath{\mathsf{Nat}}}$.
Intensional W-types {#section:intW}
===================
We begin with an example which serves to illustrate, in an especially simple case, some aspects of our theory. The type of Boolean truth values is not a W-type, but it can be formulated as an inductive type in the familiar way by means of formation, introduction, elimination, and computation rules. It then has an “up to homotopy" universal property of the same general kind as the one that we shall formulate in section \[subsection:main\] below for W-types, albeit in a simpler form.
Preliminary example {#subsection:prelimex}
-------------------
The standard rules for the type ${\mathsf{2}}$ given in [@NordstromB:marltt Section 5.1] can be stated equivalently as follows.
- ${\mathsf{2}}$-formation rule. $${\mathsf{2}}: {\mathsf{type}}\, .$$
- ${\mathsf{2}}$-introduction rules. $$0 : {\mathsf{2}}\, , \qquad 1 : {\mathsf{2}}\, .$$
- ${\mathsf{2}}$-elimination rule.$$\begin{prooftree}
x : {\mathsf{2}}\vdash C(x) : {\mathsf{type}}\qquad
c_0 : C(0) \qquad
c_1 : C(1)
\justifies
x: {\mathsf{2}}\vdash {\mathsf{2rec}}(x, c_0, c_1) : C(x)
\end{prooftree}$$
- ${\mathsf{2}}$-computation rules. $$\begin{prooftree}
x : {\mathsf{2}}\vdash C(x) : {\mathsf{type}}\qquad
c_0 : C(0) \qquad
c_1 : C(1)
\justifies
\left\{
\begin{array}{c}
{\mathsf{2rec}}(0, c_0, c_1) = c_0 : C(0) \, , \\
{\mathsf{2rec}}(1, c_0, c_1) = c_1 : C(1) \, .
\end{array}
\right.
\end{prooftree}$$
Although these rules are natural ones to consider in the intensional setting, they do not imply a strict universal property. For example, given a type $C$ and elements $c_0, c_1 :
C$, the function $\lambda x . {\mathsf{2rec}}(x, c_0,c_1) : {\mathsf{2}}\rightarrow C$ cannot be shown to be definitionally unique among the functions $f :
{\mathsf{2}}\rightarrow C$ with the property that $f(0)=c_0 : C$ and $f(1)=c_1 : C$. The best that one can do by using ${\mathsf{2}}$-elimination over a suitable identity type, and function extensionality, is to show that it is unique among all such maps up to an identity term, which itself is unique up to a higher identity, which in turn is unique up to …. This sort of weak $\omega$-universality, which apparently involves infinitely much data, can nonetheless be captured directly within the system of type theory (without resorting to coinduction) using ideas from higher category theory. To do so, let us define a *${\mathsf{2}}$-algebra* to be a type $C$ equipped with two elements $c_0 \, , c_1 : C$. Then, a *weak homomorphism* of ${\mathsf{2}}$-algebras $(f, p_0, p_1) : (C, c_0,c_1)\rightarrow (D,d_0,d_1)$ consists of a function $f : C\rightarrow D$ together with identity terms $$p_0 : {{\mathsf{Id}}_{D}}(f (c_0) ,d_0) \, , \qquad p_1 : {{\mathsf{Id}}_{D}}(f (c_1), d_1) \, .$$ This is a *strict homomorphism* when $f (c_0) = d_0 : D$, $f (c_1) = d_1 : D $ and the identity terms $p_0$ and $p_1$ are the corresponding reflexivity terms. We can then define the type of weak homomorphisms from $(C, c_0,c_1)$ to $(D,d_0,d_1)$ by letting $$\begin{gathered}
{\mathsf{2}\text{-}\mathsf{Alg}}[ (C, c_0,c_1), (D,d_0,d_1) \big] {=_{\mathrm{def}}}\\
(\Sigma f: C \rightarrow D) {\mathsf{Id}}(f (c_0), d_0) \times{{\mathsf{Id}}_{D}}(f (c_1), d_1) \, .\end{gathered}$$ The weak universality condition on the ${\mathsf{2}}$-algebra $({\mathsf{2}}, 0, 1)$ that we seek can now be determined as follows.
\[thm:boolhinitial\] A ${\mathsf{2}}$-algebra $(C, c_0,c_1)$ is *homotopy-initial* if for any ${\mathsf{2}}$-algebra $(D,d_0,d_1)$, the type $${\mathsf{2}\text{-}\mathsf{Alg}}\big[ (C, c_0,c_1), (D,d_0,d_1)\big]$$ is contractible.
The notion of homotopy initiality, or h-initiality for short, captures in a precise way the informal idea that there is essentially one weak algebra homomorphism $({\mathsf{2}}, 0, 1) \rightarrow (C,c_0,c_1)$. Moreover, h-initiality can be shown to follow from the rules of inference for ${\mathsf{2}}$ stated above. Indeed, the computation rules for ${\mathsf{2}}$ stated above evidently make the function $$\lambda x . {\mathsf{2rec}}(x, c_0,c_1) : {\mathsf{2}}\rightarrow C$$ into a *strict* algebra map, a stronger condition than is required for h-initiality. Relaxing these definitional equalities to propositional ones, we arrive at the following rules.
- Propositional ${\mathsf{2}}$-computation rules. $$\begin{prooftree}
x : {\mathsf{2}}\vdash C(x) : {\mathsf{type}}\qquad
c_0 : C(0) \qquad
c_1 : C(1)
\justifies
\left\{
\begin{array}{c}
{\mathsf{2comp}}_0(c_0, c_1) : {{\mathsf{Id}}_{C(0)}} \big({\mathsf{2rec}}(0, c_0, c_1), c_0) \, , \\
{\mathsf{2comp}}_1(c_0, c_1) : {{\mathsf{Id}}_{C(1)}} \big({\mathsf{2rec}}(1, c_0, c_1), c_1) \, .
\end{array}
\right.
\end{prooftree}$$
This variant is not only still sufficient for h-initiality, but also necessary, as we state precisely in the following.
\[prop:2hinitial\] Over the type theory ${\mathcal{H}}$, the formation, introduction, elimination, and propositional computation rules for ${\mathsf{2}}$ are equivalent to the existence of a homotopy-initial ${\mathsf{2}}$-algebra.
Suppose we have a type ${\mathsf{2}}$ satisfying the stated rules. Then clearly $({\mathsf{2}}, 0, 1)$ is a ${\mathsf{2}}$-algebra; to show that it is h-initial, take any ${\mathsf{2}}$-algebra $(C,c_0,c_1)$. By elimination with respect to the constant family $C$ and the elements $c_0$ and $c_1$, we have the map $\lambda x . {\mathsf{2rec}}(x, c_0,c_1) : {\mathsf{2}}\rightarrow C$, which is a weak algebra homomorphism by the propositional computation rules. Thus we obtain a term $h:{\mathsf{2}\text{-}\mathsf{Alg}}\big[ ({\mathsf{2}}, 0, 1), (C, c_0,c_1)\big]$. Now given any $k:{\mathsf{2}\text{-}\mathsf{Alg}}\big[ ({\mathsf{2}}, 0, 1), (C, c_0,c_1)\big]$, we need a term of type ${{\mathsf{Id}}_{}}(h,k)$. This term follows from a propositional $\eta$-rule, which is derivable by ${\mathsf{2}}$-elimination over a suitable identity type.
Conversely, let $({\mathsf{2}}, 0, 1)$ be an h-initial ${\mathsf{2}}$-algebra. To prove elimination, let $x:{\mathsf{2}}\vdash C(x):{\mathsf{type}}$ with $c_0 : C(0)$ and $ c_1 : C(1)$ be given, and consider the ${\mathsf{2}}$-algebra $(C', c'_0, c'_1)$ defined by: $$\begin{aligned}
C' &{=_{\mathrm{def}}}(\Sigma x: {\mathsf{2}})C(x) \, , \\
c'_0 &{=_{\mathrm{def}}}{\mathsf{pair}}(0, c_0) \, , \\
c'_1 &{=_{\mathrm{def}}}{\mathsf{pair}}(1, c_1)\, .\end{aligned}$$ Since ${\mathsf{2}}$ is h-initial, there is a map $r : {\mathsf{2}}\rightarrow C'$ with identities $p_0:{{\mathsf{Id}}_{}}(r 0, c'_0)$ and $p_1:{{\mathsf{Id}}_{}}(r 1, c'_1)$. Now, we would like to set $${\mathsf{2rec}}(x, c_0, c_1) = \pi_2 (r x) : C(x),$$ where $\pi_2$ is the second projection from $C'=(\Sigma x: {\mathsf{2}})C(x)$. But recall that in general $\pi_2(z) : C(\pi_1(z))$, and so (taking the case $x=0$) we have $\pi_2(r 0) : C(\pi_1(r 0))$ rather than the required $\pi_2(r 0) {\, : \, } C(0)$; that is, since it need not be that $\pi_1(r 0) = 0$, the term $\pi_2(r 0)$ has the wrong type to be ${\mathsf{2rec}}(0, c_0, c_1)$. However, we can show that $$\pi_1: (\Sigma x: {\mathsf{2}})C(x)\rightarrow {\mathsf{2}}$$ is a weak homomorphism, so that the composite $\pi_1\circ r : ({\mathsf{2}}, 0, 1)\rightarrow ({\mathsf{2}}, 0, 1)$ must be propositionally equal to the identity homomorphism $1_{\mathsf{2}}: ({\mathsf{2}}, 0, 1)\rightarrow ({\mathsf{2}}, 0, 1)$, by the contractibility of ${\mathsf{2}\text{-}\mathsf{Alg}}\big[ ({\mathsf{2}}, 0, 1), ({\mathsf{2}}, 0, 1)\big]$. Thus there is an identity term $p : {{\mathsf{Id}}_{}}(\pi_1\circ r, 1_{\mathsf{2}})$, along which we can transport using $p_! : C(\pi_1( r 0)) \rightarrow C(0)$, thus taking $\pi_2(r 0 ) : C(\pi_1 (r 0))$ to the term $p_! ( \pi_2( r 0) ) :C(0)$ of the correct type. We can then set $${\mathsf{2rec}}(x, c_0, c_1) = p_! (\pi_2 (r x)) : C(x)$$ to get the required elimination term. The computation rules follow by a rather lengthy calculation.
Proposition \[prop:2hinitial\] is the analogue in Homotopy Type Theory of the characterization of ${\mathsf{2}}$ as a strict coproduct $1+1$ in extensional type theory. It makes precise the rough idea that, in intensional type theory, ${\mathsf{2}}$ is a kind of homotopy coproduct or weak $\omega$-coproduct in the weak $\omega$-category $\mathcal{C}({\mathcal{H}})$ of types, terms, identity terms, higher identity terms, …. It is worth emphasizing that h-initiality is a purely type-theoretic notion; despite having an obvious semantic interpretation, it is formulated in terms of inhabitation of specific, definable types. Indeed, Proposition \[prop:2hinitial\] and its proof have been completely formalized in the Coq proof assistant [@AwodeyS:indtht].
A development entirely analogous to the foregoing can be given for the type ${\ensuremath{\mathsf{Nat}}}$ of natural numbers. In somewhat more detail, one introduces the notions of a ${\ensuremath{\mathsf{Nat}}}$-algebra and of a weak homomorphism of ${\ensuremath{\mathsf{Nat}}}$-algebras. Using these, it is possible to define the notion of a homotopy-initial ${\ensuremath{\mathsf{Nat}}}$-algebra, analogue to that of a homotopy-initial ${\mathsf{2}}$-algebra in Definition \[thm:boolhinitial\]. With these definitions in place, one can prove an equivalence between the formation, introduction, elimination and propositional computation rules for ${\ensuremath{\mathsf{Nat}}}$ and the existence of a homotopy-initial ${\ensuremath{\mathsf{Nat}}}$-algebra. Here, the propositional computation rules are formulated like those above, *i.e.* by replacing the definitional equalities in the conclusion of the usual computation rules [@NordstromB:marltt Section 5.3] with propositional equalities. We do not pursue this further here, however, since ${\ensuremath{\mathsf{Nat}}}$ can also be presented as a W-type, as we discuss in section \[subsec:define\] below.
The main theorem {#subsection:main}
----------------
Although it is more elaborate to state (and difficult to prove) owing to the presence of recursively generated data, our main result on W-types is analogous to the foregoing example in the following respect: rather than being strict initial algebras, as in the extensional case, weak W-types are instead homotopy-initial algebras. This fact can again be stated entirely syntactically, as an equivalence between two sets of rules: the formation, introduction, elimination, and propositional computation rules (which we spell out below) for W-types, and the existence of an h-initial algebra, in the appropriate sense. Moreover, as in the simple case of the type ${\mathsf{2}}$, the proof of the equivalence is again entirely constructive.
The required definitions in the current setting are as follows. Let us assume that $$x:A \vdash B(x) : {\mathsf{type}}\, ,$$ and define the associated polynomial functor as before: $$\label{eq:polyfunc}
PX = (\Sigma x : A) (B(x) \rightarrow X) \, .$$ (Actually, this is now functorial only up to propositional equality, but this change makes no difference in what follows.) By definition, a $P$-algebra is a type $C$ equipped a function $s_C : PC \rightarrow C$. For $P$-algebras $(C,s_C)$ and $(D,s_D)$, a *weak homomorphism* between them $(f, s_f) : (C, s_C) \rightarrow (D, s_D)$ consists of a function $f : C \rightarrow D$ and an identity proof $$s_f : {{\mathsf{Id}}_{PC \rightarrow D}}\big( f \circ s_C \, , s_{D} \circ Pf \big) \, ,$$ where $Pf : PC\rightarrow PD$ is the result of the easily-definable action of $P$ on $f: C \rightarrow D$. Such an algebra homomorphism can be represented suggestively in the form: $$\xymatrix{
PC \ar[d]_{s_C} \ar[r]^{Pf} \ar@{}[dr]|{s_f} & PD \ar[d]^{s_D}\\
C \ar[r]_{f} & D }$$ Accordingly, the type of weak algebra maps is defined by $$\begin{gathered}
{P\text{-}\mathsf{Alg}}\big[ (C,s_C), (D, s_D) \big]
{=_{\mathrm{def}}}\\
(\Sigma f: C \rightarrow D) \, {\mathsf{Id}}(f\circ s_C, s_D\circ Pf) \, .\end{gathered}$$
A $P$-algebra $(C, s_C)$ is *homotopy-initial* if for every $P$-algebra $(D, s_D)$, the type $${P\text{-}\mathsf{Alg}}\big[ (C, s_C), (D, s_D) \big]$$ of weak algebra maps is contractible.
The notion of h-initiality captures a universal property in which the usual conditions of existence and uniqueness are replaced by conditions of existence and uniqueness up to a system of higher and higher identity proofs. To explain this, let us fix a $P$-algebra $(C,s_C)$ and assume that it is homotopy-initial. Then, given any $P$-algebra $(D,s_D)$, there is a weak homomorphism $(f,s_f) : (C,s_C) \rightarrow (D,s_D)$, since the type of weak maps from $(C,s_C)$ to $(D,s_D)$, being contractible, is inhabited. Furthermore, for any weak map $(g,s_g) : (C,s_C) \rightarrow (D,s_D)$, the contractibility of the type of weak maps implies that there is an identity proof $$p : {\mathsf{Id}}\big( (f,s_f), (g, s_g) \big) \, ,$$ witnessing the uniqueness up to propositional equality of the homomorphism $(f,s_f)$. But it is also possible to prove that the identity proof $p$ is unique up to propositional equality. Indeed, since $(f,s_f)$ and $(g,s_g)$ are elements of a contractible type, the identity type ${\mathsf{Id}}( (f,s_f), (g, s_g) )$ is also contractible, as observed in Remark \[thm:idcontrcontr\]. Thus, if we have another identity proof $q : {\mathsf{Id}}( (f,s_f), (g, s_g) )$, there will be an identity term $\alpha : {\mathsf{Id}}(p,q)$, which is again essentially unique, and so on. It should also be pointed out that, just as strictly initial algebras are unique up to isomorphism, h-initial algebras are unique up to weak equivalence. It then follows from the Univalence axiom that two h-initial algebras are propositionally equal, a fact that we mention only by the way. Finally, we note that there is also a homotopical version of Lambek’s Lemma, asserting that the structure map of an h-initial algebra is itself a weak equivalence, making the algebra a *homotopy fixed point* of the associated polynomial functor. The reader can work out the details from the usual proof and the definition of h-initiality, or consult [@AwodeyS:indtht].
The deduction rules that characterize homotopy-initial algebras are obtained from the formation, introduction, elimination and computation rules for W-types stated in Section \[section:extW\] by simply replacing the ${\mathsf{W}}$-computation rule with the following rule, that we call the propositional ${\mathsf{W}}$-computation rule.
- Propositional ${\mathsf{W}}$-computation rule. $$\begin{prooftree}
\begin{array}{l}
w : W \vdash C(w) : {\mathsf{type}}\\
\hspace{-1ex}
\begin{array}{c}
x : A, u : B(x) \rightarrow W, v : (\Pi y : B(x)) C(u(y)) \vdash \\
c(x,u,v) : C({\mathsf{sup}}(x,u))
\end{array}
\end{array}
\justifies
\begin{array}{l}
x : A, u : B(x) \rightarrow W \vdash
{\mathsf{wcomp}}(x,u,c) : \\
\qquad {\mathsf{Id}}\big(
{\mathsf{wrec}}({\mathsf{sup}}(x,u), c), c(x,u,\lambda y.{\mathsf{wrec}}( u(y), c )
\big)
\end{array}
\end{prooftree}$$
\[thm:wtypesinvariance\] One interesting aspect of this group of rules, to which we shall refer as the *rules for homotopical W-types*, is that, unlike the standard rules for W-types, they are invariant under propositional equality. To explain this more precisely, let us work in a type theory with a type universe ${\mathsf{U}}$ closed under all the forms of types of ${\mathcal{H}}$ and W-types. Let $A : {\mathsf{U}}$, $B : A
\rightarrow {\mathsf{U}}$ and define $W {=_{\mathrm{def}}}({\mathsf{W}}x : A) B(x)$. The invariance of the rules for homotopy W-types under propositional equality can now be expressed by saying that if we have a type $W' : {\mathsf{U}}$ and an identity proof $p : {{\mathsf{Id}}_{U}}(W, W')$, then the ${\mathsf{Id}}$-elimination rule implies that $W'$ satisfies the same rules as $W$, in the sense that there are definable terms playing the role of the primitive constants that appear in the rules for $W$.
We can now state our main result. Its proof has been formalized in the Coq system, and the proof scripts are available at [@AwodeyS:indtht]; thus we provide only an outline of the proof.
\[theorem:main\] Over the type theory ${\mathcal{H}}$, the rules for homotopical W-types are equivalent to the existence of homotopy-initial algebras for polynomial functors.
The two implications are proved separately. First, we show that the rules for homotopical W-types imply the existence of homotopy-initial algebras for polynomial functors. Let us assume that $x : A \vdash B(x) : {\mathsf{type}}$ and consider the associated polynomial functor $P$, defined as in . Using the ${\mathsf{W}}$-formation rule, we define $W {=_{\mathrm{def}}}({\mathsf{W}}x : A) B(x)$ and using the ${\mathsf{W}}$-introduction rule we define a structure map $s_W : PW \rightarrow W$, exactly as in the extensional theory. We claim that the algebra $(W, s_W)$ is h-initial. So, let us consider another algebra $(C,s_C)$ and prove that the type $T$ of weak homomorphisms from $(W, s_W)$ to $(C,s_C)$ is contractible. To do so, observe that the ${\mathsf{W}}$-elimination rule and the propositional ${\mathsf{W}}$-computation rule allow us to define a weak homomorphism $(f, s_f) : (W, s_W) \rightarrow (C, s_C)$, thus showing that $T$ is inhabited. Finally, it is necessary to show that for every weak homomorphism $(g, s_g) : (W, s_W) \rightarrow (C, s_C)$, there is an identity proof $$\label{equ:prequired}
p : {\mathsf{Id}}( (f,s_f), (g,s_g) ) \, .$$ This uses the fact that, in general, a type of the form ${\mathsf{Id}}( (f,s_f), (g,s_g) )$, is weakly equivalent to the type of what we call *algebra $2$-cells*, whose canonical elements are pairs of the form $(e, s_e)$, where $e : {\mathsf{Id}}(f,g)$ and $s_e$ is a higher identity proof witnessing the propositional equality between the identity proofs represented by the following pasting diagrams: $$\xymatrix{
PW \ar@/^1pc/[r]^{Pg} \ar[d]_{s_W} \ar@{}[r]_(.52){s_g} & PD \ar[d]^{s_D} \\
W \ar@/^1pc/[r]^g \ar@/_1pc/[r]_f \ar@{}[r]|{e} & D } \qquad
\xymatrix{
PW \ar@/^1pc/[r]^{Pg} \ar[d]_{s_W} \ar@/_1pc/[r]_{Pf} \ar@{}[r]|{Pe}
& PD \ar[d]^{s_D} \\
W \ar@/_1pc/[r]_f \ar@{}[r]^{s_f} & D }$$ In light of this fact, to prove that there exists a term as in , it is sufficient to show that there is an algebra 2-cell $$(e,s_e) : (f,s_f) \Rightarrow (g,s_g) \, .$$ The identity proof $e : {\mathsf{Id}}(f,g)$ is now constructed by function extensionality and ${\mathsf{W}}$-elimination so as to guarantee the existence of the required identity proof $s_e$.
For the converse implication, let us assume that the polynomial functor associated to the judgement $x : A \vdash B(x) : {\mathsf{type}}$ has an h-initial algebra $(W,s_W)$. To derive the ${\mathsf{W}}$-formation rule, we let $({\mathsf{W}}x {\, : \, } A) B(x) {=_{\mathrm{def}}}W$. The ${\mathsf{W}}$-introduction rule is equally simple to derive; namely, for $a : A$ and $t \colon B(a) \rightarrow W$, we define ${\mathsf{sup}}(a,t) : W$ as the result of applying the structure map $s_W \colon PW \rightarrow W$ to ${\mathsf{pair}}(a,t) : PW$. For the ${\mathsf{W}}$-elimination rule, let us assume its premisses and in particular that $w : W \vdash C(w) : {\mathsf{type}}$. Using the other premisses, one shows that the type $C {=_{\mathrm{def}}}(\Sigma w : W) C(w)$ can be equipped with a structure map $s_C : PC \rightarrow C$. By the h-initiality of $W$, we obtain a weak homomorphism $(f, s_f) : (W, s_W) \rightarrow (C, s_C)$. Furthermore, the first projection $\pi_1 : C \rightarrow W$ can be equipped with the structure of a weak homomorphism, so that we obtain a diagram of the form $$\xymatrix{
PW \ar[r]^{Pf} \ar[d]_{s_W} & PC \ar[d]^{s_C} \ar[r]^{P \pi_1} & PW \ar[d]^{s_W} \\
W \ar[r]_f & C \ar[r]_{\pi_1} & W \, .}$$ But the identity function $1_W : W \rightarrow W$ has a canonical structure of a weak algebra homomorphism and so, by the contractibility of the type of weak homorphisms from $(W,s_W)$ to itself, there must be an identity proof between the composite of $(f,s_f)$ with $(\pi_1, s_{\pi_1})$ and $(1_W, s_{1_W})$. This implies, in particular, that there is an identity proof $p : {\mathsf{Id}}( \pi_1 \circ f, 1_W)$. Since $(\pi_2 \circ f) w : C( (\pi_1 \circ f) w)$, we can define $${\mathsf{wrec}}(w,c) {=_{\mathrm{def}}}p_{\, ! \,}( ( \pi_2 \circ f) w ) : C(w)$$ where the transport $p_{\, ! \,}$ is defined via ${\mathsf{Id}}$-elimination over the dependent type $$u : W \rightarrow W \vdash C ( u (w)) : {\mathsf{type}}\, .$$ The verification of the propositional ${\mathsf{W}}$-computation rule is a rather long calculation, involving several lemmas concerning the naturality properties of operations of the form $p_{\, ! \,}$.
Definability of inductive types {#subsec:define}
-------------------------------
We conclude this section by indicating how the limited form of extensionality that is assumed in the type theory ${\mathcal{H}}$, namely the principle of function extensionality, allows us to overcome the obstacles in defining various inductive types as W-types mentioned at the end of Section \[section:extW\], provided that both are understood in the appropriate homotopical way, *i.e.* with all types being formulated with propositional computation rules.
Consider first the paradigmatic case of the type of natural numbers. To define it as a W-type, we work in an extension of the type theory ${\mathcal{H}}$ with
- formation, introduction, elimination and propositional computation rules for types $\mathsf{0}$, $\mathsf{1}$ and $\mathsf{2}$ that have zero, one and two canonical elements, respectively;
- the rules for homotopy W-types, as stated above;
- rules for a type universe ${\mathsf{U}}$ reflecting all the forms of types of ${\mathcal{H}}$, W-types, and $\mathsf{0}$, $\mathsf{1}$ and $\mathsf{2}$.
In particular, the rules for $\mathsf{2}$ are those given in Section \[subsection:prelimex\]. We then proceed as follows. We begin by setting $A =\mathsf{2}$, as in the extensional case. We then define a dependent type $$x : A \vdash B(x) : {\mathsf{U}}$$ by $\mathsf{2}$-elimination, so that the propositional $\mathsf{2}$-computation rules give us propositional equalities $$p_0 : {{\mathsf{Id}}_{U}}( \mathsf{0}, B(0)) \, , \qquad
p_1 : {{\mathsf{Id}}_{U}}( \mathsf{1}, B(1)) \, .$$ Because of the invariance of the rules for $\mathsf{0}$ and $\mathsf{1}$ under propositional equalities (as observed in Remark \[thm:wtypesinvariance\]), we can then derive that the types $B(0)$ and $B(1)$ satisfy rules analogous to those for $\mathsf{0}$ and $\mathsf{1}$, respectively. This allows us to show that the type $${\ensuremath{\mathsf{Nat}}}{=_{\mathrm{def}}}({\mathsf{W}}x : A) B(x)$$ satisfies the introduction, elimination and propositional computation rules for the type of natural numbers. The proof of this fact proceeds essentially as one would expect, but to derive the propositional computation rules it is useful to observe that for every type $X : {\mathsf{U}}$, there are adjoint homotopy equivalences, in the sense of Definition \[thm:weq\], between the types $\mathsf{0} \rightarrow X$ and $\mathsf{1}$, and between $\mathsf{1} \rightarrow X$ and $X$. Indeed, the propositional identities witnessing the triangular laws are useful in the verification of the propositional computation rules for ${\ensuremath{\mathsf{Nat}}}$. For details, see the formal development in Coq provided in [@AwodeyS:indtht]. Observe that as a W-type, ${\ensuremath{\mathsf{Nat}}}$ is therefore also an h-initial algebra for the equivalent polynomial functor $P(X) = \mathsf{1}+ X$, as expected.
Finally, let us observe that the definition of a type representing the second number class as a W-type, as discussed in [@MartinLofP:inttt], carries over equally well. Indeed, one now must represent type-theoretically a signature with three operations: the first of arity zero, the second of arity one, and the third of arity ${\ensuremath{\mathsf{Nat}}}$. For the first two we can proceed exactly as before, while for the third there is no need to prove auxiliary results on adjoint homotopy equivalences. As before, the second number class supports an h-initial algebra structure for the corresponding polynomial functor $P(X) = \mathsf{1} + X + ({\ensuremath{\mathsf{Nat}}}\rightarrow X)$. Again, the formal development of this result in Coq can be found in [@AwodeyS:indtht].
Future work {#section:future}
===========
The treatment of W-types presented here is part of a larger investigation of general inductive types in Homotopy Type Theory. We sketch the projected course of our further research.
1. In the setting of extensional type theory, Dybjer [@DybjerP:repids] showed that every strictly positive definable functor can be represented as a polynomial functor, so that all such inductive types are in fact W-types. This result should generalize to the present setting in a straightforward way.
2. Also in the extensional setting, Gambino and Hyland [@GambinoN:weltdp] showed that general tree types [@PeterssonK:setcis] [@NordstromB:promlt Chapter 16], viewed as initial algebras for general polynomial functors, can be constructed from W-types in locally cartesian closed categories, using equalizers. We expect this result to carry over to the present setting as well, using ${{\mathsf{Id}}_{}}$-types in place of equalizers.
3. In [@VoevodskyV:notts] Voevodsky has shown that all inductive types of the Predicative Calculus of Inductive Constructions can be reduced to the following special cases:
- $\mathsf{0}$, $\mathsf{1}$, $A+B$, $(\Sigma x : A)B(x)$,
- ${{\mathsf{Id}}_{A}}(a,b)$,
- general tree types.
Combining this with the foregoing, we expect to be able to extend our Theorem \[theorem:main\] to the full system of predicative inductive types underlying Coq.
Finally, one of the most exciting recent developments in Univalent Foundations is the idea of Higher Inductive Types (HITs), which can also involve identity terms in their signature [@LumsdaineP:higit; @ShulmanM:higit]. This allows for algebras with equations between terms, like associative laws, coherence laws, etc.; but the really exciting aspect of HITs comes from the homotopical interpretation of identity terms as paths. Viewed thus, HITs should permit direct formalization of many basic geometric spaces and constructions, such as the unit interval $I$; the spheres $S^n$, tori, and cell complexes; truncations, such as the \[bracket\] types [@AwodeyS:prot]; various kinds of quotient types; homotopy (co)limits; and many more fundamental and fascinating objects of geometry not previously captured by type-theoretic formalizations. Our investigation of conventional inductive types in the homotopical setting should lead to a deeper understanding of these new and important geometric analogues.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Andrej Bauer, Frank Pfenning, Robert Harper, Vladimir Voevodsky and Michael Warren for helpful discussions on the subject of this paper. In particular, Vladimir Voevodsky suggested a simplification of the proof that the rules for homotopical W-types imply h-initiality.
Steve Awodey gratefully acknowledges the support of the National Science Foundation, Grant DMS-1001191 and the Air Force OSR, Grant 11NL035. Nicola Gambino is grateful for the support and the hospitality of the Institute for Advanced Study, where he worked on this project. This work was supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Kristina Sojakova is grateful for the support of CyLab at Carnegie Mellon under grants DAAD19-02-1-0389 and W911NF-09-1-0273 from the Army Research Office, as well as for the support of the Qatar National Research Fund under grant NPRP 09-1107-1-168.
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abstract: |
We studied the Active Galactic Nuclei (AGN) radio emission from a compilation of hard X-ray selected samples, all observed in the 1.4 GHz band. A total of more than 1600 AGN with 2-10 keV de-absorbed luminosities higher than 10$^{42}$ [erg s$^{-1}$ cm$^{-2}$]{} were used.
For a sub-sample of about 50 $z$$\lesssim$0.1 AGN it was possible to reach a $\sim$80% fraction of radio detections and therefore, for the first time, it was possible to almost completely measure the probability distribution function of the ratio between the radio and the X-ray luminosity R$_X$= log(L$_{1.4}$/L$_X$), where L$_{1.4}$/L$_X$= $\nu$L$_\nu$(1.4 GHz)/L$_X$(2-10 keV). The probability distribution function of R$_X$ was functionally fitted as dependent on the X-ray luminosity and redshift, $P(R_X| L_X, z)$. It roughly spans over 6 decades ($-7$$<$R$_X$$<$$-1$), and does not show any sign of bi-modality. It resulted that the probability of finding large values of the R$_X$ ratio increases with decreasing X-ray luminosities and (possibly) with increasing redshift. No statistical significant difference was found between the radio properties of the X-ray absorbed (N$_H$$>$$10^{22}$ cm$^{-2}$) and unabsorbed AGN.
The measure of the probability distribution function of R$_X$ allowed us to compute the kinetic luminosity function and the kinetic energy density which, at variance with what assumed in many galaxy evolution models, is observed to decrease of about a factor of five at redshift below 0.5. About half of the kinetic energy density results to be produced by the more radio quiet (R$_X$$<$-4) AGN.
In agreement with previous estimates, the AGN efficiency $\epsilon_{kin}$ in converting the accreted mass energy into kinetic power ($L_K$$=$$\epsilon_{kin}\dot{m} c^2$) is, on average, $\epsilon_{kin}$$\simeq$$5\times 10^{-3}$. The data suggest a possible increase of $\epsilon_{kin}$ at low redshifts.
author:
- |
F. La Franca$^1$, G. Melini$^1$ and F. Fiore$^2$\
\
$^1$ Dipartimento di Fisica, Università Roma Tre, via della Vasca Navale 84, I-00146 Roma, Italy\
email: [lafranca@fis.uniroma3.it, melini@fis.uniroma3.it]{}\
$^2$ INAF-Osservatorio Astronomico di Roma, via Frascati 33, I-00040, Monteporzio Catone, Italy
title: |
Tools for computing the AGN feedback: radio-loudness distribution\
and the kinetic luminosity function
---
Introduction
============
One of the main questions in galaxy formation evolution studies is the role of AGN feedback. According to popular AGN/galaxy co-evolutionary scenarios, once central super massive black holes (SMBHs) reaches masses $>$$10^{7}-10^8$ M$_\odot$, the AGN can heat efficiently the galaxy interstellar matter (ISM) through winds, shocks, and high-energy radiation (Silk & Rees 1998, Fabian 1999), inhibiting further accretion and star-formation and making the galaxy colors redder (see e.g. Cattaneo et al. 2009 for a review).
Unfortunately there are still relatively few direct observations of AGN feedback, and its inclusion in galaxy evolution models is often performed using adjustable parameters to obtain the observed galaxy colors. Indeed, the results of the hydrodynamic N-body simulations (see, e.g. Di Matteo, Springel & Hernquist 2005; Springel 2005; Hopkins et al. 2005, 2006) and of the semi-analytical models of galaxy formation and evolution (SAMs, Monaco, Salucci & Danese 2000; Kauffmann & Haenhelt 2000; Volonteri, Haardt & Madau 2003; Granato et al. 2004; Menci et al. 2006, 2008; Croton et al. 2006; Bower et al. 2006; Marulli et al. 2008) depend on the AGN triggering mechanism and on the AGN feedback description. Is the feedback at work mainly during luminous AGN phases (the so called “quasar” or “radiative” mode, see e.g. Menci et al. 2008)? Or rather AGN feedback proceeds continuously during the Cosmic time at a low rate (the so called “radio” or “kinetic” mode, see e.g. Croton et al. 2006, Marulli et al. 2008)?
In the first case the AGN feedback is associated to the the main radiative, and then X-ray, activity of the AGN which is thought to occur during episodes of accretion of cold gas coming from the galaxy ISM following galaxy encounters (e.g. Barnes & Hernquist 1992, Cavaliere & Vittorini 2000, Menci et al. 2008). In this scenario the AGN feedback is usually associated to the radiation and its total efficiency must be proportional to the AGN fraction (the AGN luminosity function vs. the galaxy luminosity function) and to the efficiency in releasing the AGN power in the galaxy ISM.
In the second case the radio feedback is assumed to be related to a low, uninterrupted and constant matter accretion rate onto the central SMBH ($\sim$10$^{-5}$ M$_\odot$/yr), coming from a quiescent inflow of gas cooling from the halo’s hot atmosphere (e.g. Monaco, Salucci & Danese 2000; Croton et al. 2006; Bower et al. 2006). This accretion rate is too small to contribute significantly to the bolometric output of the AGN. At variance, the mechanical effects of accelerated particles (jets), which are observed to be responsible for the large cavities on the intra-cluster medium revealed in the X-rays (e.g. McNamara et al. 2000), are believed to significantly perturb the ISM into radio galaxies (Saxton et al. 2005, Sutherland & Bicknell 2007, Tortora et al. 2009; Krause & Gaibler 2009 and references therein). In this scenario, radio mode feedback total efficiency is thought to be proportional to the total accreted mass and then to the SMBH mass function and to the way the energy is channelled by the SMBH into its host galaxy and it is released in the ISM and intergalactic medium (IGM).
In the Croton et al. (2006) and Marulli et al. (2008) models, the radio-mode feedback is more effective in suppressing the cooling flows in the massive galaxies at late times (low redshifts). In the Cattaneo et al. (2006) model the cooling and star-formation are efficiently suppressed by the AGN radio feedback for haloes above a critical mass of $\sim$10$^{12}$ M$_\sun$ below $z$$\sim$3. Bower et al. (2006) assume that the AGN energy injection is determined by a self-regulating feedback loop that starts when the luminosity exceeds some fraction of the Eddington luminosity. Recently, Shabala & Alexander (2009) have presented a galaxy formation and evolution model where the radio feedback occurs when the AGN accretion rate falls below a certain value and enters the advection dominated accretion flow (ADAF) regime. In their model the radio (and then kinetic) power is assumed to scale linearly with the accretion rate.
To support the above studies several authors have compared the mechanical (kinetic) luminosity function (LF) of radio sources to the bolometric AGN LFs (e.g. Best et al. 2006; Merloni & Heinz 2008; Shankar et al. 2008; Kording, Jester & Fender 2008; Cattaneo & Best 2009; Smolčić et al. 2009). All these works are based on the convolution of some empirical relation between the AGN kinetic power and its radio luminosity (see e.g. Willott et al. 1999; Best et al. 2006; Merloni & Heinz 2007; Birzan et al. 2008) with the AGN radio LF. As a consequence, these works deal with the AGN as a population and do not allow the inclusion in the models of the kinetic power of each single source during its evolution.
In the Croton et al. (2006) and Bower et al. (2006) SAMs it is not implemented any direct relation between the radio activity (feedback) and the X-ray luminosity (main episodes of mass accretion). However, a strong correlation between the X-ray (L$_X$) and radio luminosity of AGN is observed (e.g. Brinkmann et al. 2000), and L$_X$ is a good proxy of the SMBH accretion rate $\dot{m}$, via the knowledge of the X-ray bolometric correction K$_X$ and the efficiency $\epsilon$ of conversion of mass accretion into radiation $$L_X = {L_{bol}\over K_X}= {\epsilon \dot{m}{c^2}\over{(1-\epsilon) K_X}},$$ where L$_{bol}$ is the bolometric luminosity and it is assumed $\epsilon \simeq 0.1$ (see e.g. Marconi et al. 2004; Vasudevan & Fabian 2009).
The aim of this paper is therefore to estimate the AGN kinetic power linking the AGN radio emission to the accretion rate related to the AGN activity (the luminous phase). Our approach is then different from most of the above described models, and it can be considered, in some sort, as a quasar mode feedback (i.e. related to the luminous-accreting phases), but associated to the emitted radio-kinetic power. This can provide a robust quantitative root to a different kind of radio feedback, thus guiding its self-consistent inclusion in SAMs. In this framework, a very useful ingredient is the measure of the probability distribution function P(R$_X$) of the ratio, R$_X$, between the AGN radio and hard X-ray luminosity \[R$_X$=$\nu$L$_\nu$(1.4 GHz)/L$_X$(2-10 keV)\]. To this purpose we used a data-set of more than 1600 X-ray selected AGN, observed in the radio band at 1.4 GHz, to measure P(R$_X$) as a function of both luminosity and redshift: P(R$_X |$ L$_X$, $z$).
To estimate the AGN kinetic LF and its evolution, we first computed the radio LF by convolving P(R$_X |$ L$_X$, $z$), with the AGN 2-10 keV LF. We then convolved this radio LF with some of the relations (available from the literature) between the AGN radio and kinetic luminosities. As sanity tests, we first compared our results with previous studies on the relationship between the AGN X-ray and radio luminosities, and then we checked if previous measures of the radio LF and counts were correctly reproduced.
We adopted a flat cosmology with H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_M$=0.30 and $\Omega_\Lambda$=0.70. Unless otherwise stated, uncertainties are quoted at the 68% (1$\sigma$) confidence level.
![Distribution of the off-sets between the cross-correlated radio and X-ray sources.[]{data-label="Fig_radec"}](lafranca_fig01.ps){width="7.5cm"}
The data
========
As our objective was to use the R$_X$ distribution in order to estimate the kinetic (radio) luminosity of the X-ray selected AGN, we had to measure a radio emission which was as much as possible causally linked (contemporary) with the observed X-ray activity (accretion). We then decided to measure the radio fluxes in a region as much close as possible to the AGN, therefore minimizing the contribution of objects like the radio lobes in FRII sources (Fanaroff & Riley 1974). For this reason we built up a large data-set of X-ray selected AGN (where redshift and N$_H$ column densities estimates were available) observed at 1.4 GHz with a $\sim$1 typical spatial resolution (in our cosmology 1 corresponds, at maximum, to about 8 Kpc at $z$$\sim$2). The cross correlation of the X-ray and radio catalogues was carried out inside a region with 5of radius (almost less than or equal to the size of the central part of a galaxy like ours), following a maximum likelihood algorithm as described by Sutherland and Saunders (1992) and Ciliegi et al. (2003). In Figure \[Fig\_radec\] we show the off-sets between the X-ray and radio positions of the whole sample. These resulted to have a root mean square (rms) of 1.4. Therefore we expect to have (properly) preferentially included compact FRI radio sources in our cross-correlation and have excluded most of the contribution of the (especially bright) extended FRII radio lobes from our analysis (see sect 3.4.2 for a discussion on the contribution of the excluded FRII sources to the radio counts).
The total AGN sample was built up from a compilation of complete (i.e. with almost all redshift and N$_H$ measures available) hard (mostly 2-10 keV) X-ray selected samples, with unabsorbed 2-10 keV luminosities higher than 10$^{42}$ erg/s[^1], as explained below.
The bright sample. The [*SWIFT*]{}, [*INTEGRAL*]{} and [*HEAO*]{}-1 missions
----------------------------------------------------------------------------
In order to build up a large unbiased bright AGN (low redshift) sample, we joined the AGN samples recently generated from the [*SWIFT*]{} and [*INTEGRAL*]{} missions with the sample published by Grossan (1992) using the [*HEAO*]{}-1 data. In the case of sources in common, priority was given first to the [*SWIFT*]{} data and then to the [*INTEGRAL*]{} ones and lastly to the sample of Grossan (1992).
The radio information was obtained via cross-correlation with the 1.4 GHz radio data taken from the FIRST VLA survey (Becker et al. 1995). The FIRST images have 1.8 large pixels, a typical rms sensitivity of 0.15 mJy and a resolution of 5. In the case of no radio detection a 5$\sigma$ upper limit of 0.75 mJy was adopted. The large area NVSS (Condon et al. 1998) and SUMSS/ATCA (Mauch et al. 2003) radio surveys have positional uncertainties significantly larger than FIRST and therefore were not used in our analysis.
- [*SWIFT.*]{} The [*SWIFT*]{} sample we used is composed by 121 sources with high galactic latitude ($| b |$$ >$$15\degr$) and detected with 14-195 keV fluxes brighter than 10$^{-11}$ [erg s$^{-1}$ cm$^{-2}$]{}(Tueller et al. 2008). All but one of the 121 sources have a redshift and an optical spectroscopic classification available (Tueller et al. 2008). 44 out of the 121 sources have been observed by [*FIRST*]{}. We limited our analysis to the 40 sources with 2-10 keV unabsorbed X-ray luminosity higher than 10$^{42}$ erg/s: there are 18 broad optical emission line type 1 and 1.5 AGN (AGN1/1.5), 13 narrow emission line type 2 AGN (AGN2), 2 galaxies and 7 BL Lac in this sub-sample. In this and in all the other samples, all the BL Lac were excluded from our analysis as their radio fluxes are strongly amplified by the boosting of the relativistic radio jets. Therefore the AGN sample we used is composed by 33 sources in total (28 detected by FIRST). [N$_{H}$]{} column densities measures were provided by Tueller et al. (2008).
- [*INTEGRAL.*]{} We tried to complement the [*SWIFT*]{} data with the catalogue of 46 sources detected by [*INTEGRAL*]{} at a 5$\sigma$ significancy level by Beckmann et al. (2006). However, after removing two sources without N$_H$ measurements (from Sazonov et al. 2007), and 21 sources already included in the [*SWIFT*]{} sample, we ended with 18 sources which were not covered by the [*FIRST*]{} radio observations. Therefore, no source from the [*INTEGRAL*]{} catalogue from Beckmann et al. (2006) was included in our analysis.
- [*GROSSAN.*]{} The bright sample was eventually complemented with the sample of the [*HEAO*]{}-1 sources described by Grossan (1992) as revised by Brusadin (2003). Brusadin et al. (2003) investigated, from the total sample of Grossan (1992), those 74 sources with 2-10 keV fluxes brighter than 2$\times$10$^{-11}$ [erg s$^{-1}$ cm$^{-2}$]{} . For 52 of these 74 sources the original optical counterparts were observed in the hard X-rays by at least one among the [*ASCA*]{}, [*Beppo-SAX*]{} and [*XMM-Newton*]{} satellites. All the counterparts resulted to be real hard X-ray sources and a new estimate of the [N$_{H}$]{} column densities were derived. We used in this work a sub-sample of 66 objects, from the sample of Brusadin (2003), for which reliable measure of the [N$_{H}$]{} column densities were available.
Forty-one out of these 66 sources are not included in the [*SWIFT*]{} sample, and ten (10 AGN1/1.5) of them were observed by FIRST and have 2-10 keV X-ray unabsorbed luminosities higher than 10$^{42}$ erg/s. Nine out of these last ten sources were also detected in the radio band by FIRST.
In summary, the bright sample contains 43 X-ray sources (28 AGN1, 13 AGN2, 2 galaxies; once the BL Lac were excluded) observed in the 1.4 GHz radio band by FIRST, with 37 detections.
AMSS
----
We selected those 87 sources of the [*ASCA*]{} Medium Sensitivity Survey (AMSS; as in Akiyama et al. 2003) having a S/N$>$5.5 significant detection in the X-rays, $|b|$$>$30$\degr$ and $\delta$$>$-20$\degr$; 76 of these sources were identified as AGN: seven clusters, 3 BL Lac and one star were excluded. Forty-three of these AGN were observed by FIRST (33 AGN1, 10 AGN2), and 11 were detected.
HBSS
----
We used the 67 sources of the [*XMM-Newton*]{} Hard Bright Sensitivity Survey (HBSS; Della Ceca et al. 2004) with 4.5-7.5 keV fluxes brighter than 7$\times$10$^{-14}$ [erg s$^{-1}$ cm$^{-2}$]{} (which corresponds to a 2-10 keV limit of 3.5$\times$10$^{-13}$ [erg s$^{-1}$ cm$^{-2}$]{} if a spectral index $\alpha=0.7$, where $F_\nu\propto \nu^{-\alpha}$, is assumed). A subsample of 62 sources were used in our analysis after the exclusion of 2 stars, one cluster and 2 sources without a spectroscopic identification. Thirty-two sources were observed by FIRST (23 AGN1, 9 AGN2), while 6 were detected.
ALSS
----
The [*ASCA*]{} Large Sky Survey (ALSS) is a contiguous 7 $\deg^{2}$ strip in the North Galactic Pole region (Ueda et al. 1999); we selected a sample with a limiting 2-10 keV flux of 1$\times$10$^{-13}$ [erg s$^{-1}$ cm$^{-2}$]{} from Akiyama et al. (2000). This sample contains 30 AGN (25 AGN1, 5 AGN2), as well as two clusters, one star and one object without spectroscopic identification, which were excluded from our analysis. All the 30 AGN were observed by FIRST, while nine were detected.
COSMOS
------
In order to use the COSMOS survey, we used the XMM-Newton X-ray catalogue by Cappelluti et al. (2009) with a limiting 2-10 keV flux of $\sim$3$\times$10$^{-15}$ [erg s$^{-1}$ cm$^{-2}$]{} and cross-correlated with the spectroscopic identifications by Trump et al. (2009) and the photometric redshift estimates from Ilbert et al. (2009) and Salvato et al. (2009). The [N$_{H}$]{} measures were derived by our analysis of the hardness ratios (HR). The radio data were obtained from Schinnerer et al. (2007). In order to allow a uniform radio coverage (with 1.4 GHz rms of 15 $\mu$Jy) the central 1 deg$^2$ squared area with limits 9$^h$5840$^s$$<$$\alpha$$<$10$^h$240$^s$ and 142$<$$\delta$$<$242 was used. In this area the X-ray catalogue contains 712 sources. 677 are extragalactic with a redshift measure available and 2-10 keV luminosity higher than 10$^{42}$ erg/s. 389 have a spectroscopic redshift (186 AGN1, 52 AGN2, 71 emission line galaxies, ELG, 32 normal passive galaxies, GAL, 48 no class), while 288 have only a photometric redshift estimate available. We used all the radio detections with 1.4 GHz flux limits brighter than 75 $\mu$Jy (5$\sigma$) and used the same threshold as an upper limit for all the remaining sources even though lower flux detections were available in some cases. In total 141 out of the 677 sources, contained in our selected region of the COSMOS field, were detected in the 1.4 GHz radio band.
[lrrcccc]{} Sample & N$_{1.4}$& N$_X$& $N_{1.4}\over N_X$& F$_{1.4}$& F$_X$& $F_{1.4}\over F_X$\
& (1) & (2)& (3)& (4)& (5) & (6)\
Bright & 37 & 43 &0.86& 750 & 2$\times$10$^{-11}$ & -6.3\
AMSS & 11 & 43 & 0.26 & 750& 3$\times$10$^{-13}$& -4.5\
ALSS & 9 & 30 & 0.30 &750 & 1$\times$10$^{-13}$ &-4.0\
HBSS & 6 & 32& 0.19 & 750 & 3.5$\times$10$^{-13}$ &-4.5\
COSMOS & 141 & 677 & 0.21& 75&3$\times$10$^{-15}$ &-3.5\
CLANS & 69 & 139 & 0.50&19 & 3$\times$10$^{-15}$ & -4.1\
ELAIS-S1 & 45 & 421 & 0.11& 150&2$\times$10$^{-15}$ & -3.0\
CDF-S & 12 & 94 & 0.13&70& 2.6$\times$10$^{-16}$&-2.4\
CDF-N & 45 & 162 & 0.28& 45 & 1.4$\times$10$^{-16}$&-2.3\
\
Total & 375 & 1641 & 0.23 & ... & ...& ...\
\[tab\_samp\]
ELAIS-S1
--------
In the European Large Area [*ISO*]{} Survey field S1 (ELAIS-S1) we used the catalogue of the XMM-Newton sources published by Puccetti et al. (2006), which reaches a 2-10 keV flux limit of 2$\times$10$^{-15}$ [erg s$^{-1}$ cm$^{-2}$]{}. The spectroscopic identifications and classifications provided by Feruglio et al. (2008) and Sacchi et al. (2009) were used, while the 1.4 GHz radio data (with a 5$\sigma$ limit of 150 $\mu$Jy) were taken from Middelberg et al. (2009). The whole sample contains 421 extragalactic sources with a redshift measure available and 2-10 keV luminosity higher than 10$^{42}$ erg/s: 240 have been identified and classified spectroscopically (116 AGN1, 34 AGN2, 68 ELG, 22 GAL), while 181 have a photometric redshift available. Forty-five out of these 421 AGN were detected at 1.4 GHz above the 150 $\mu$Jy limit.
![2-10 keV X-ray de-absorbed luminosity as a function of the redshift for all the AGN observed at 1.4 GHz used in this work. [*Top*]{}. Crosses indicate the sources observed in the radio band, while filled squares show those objects which have been detected in the radio. The dashed lines show the two $z$-L$_X$ regions used to measure the distribution of R$_X$ at R$_X$$<$-4. [*Bottom*]{}. Crosses indicate the sources with log [N$_{H}$]{} $<$ 22 cm$^{-2}$, while filled squares show those AGN with log [N$_{H}$]{} $\geq$ 22 cm$^{-2}$. The dashed lines show the five $z$-L$_X$ regions used to measure the distribution of R$_X$ at R$_X$$\geq$-4.[]{data-label="Fig_LZ"}](lafranca_fig03.ps){width="8cm"}
CLANS
-----
In the Chandra Lockman Area North Survey (CLANS) field the X-ray data were taken from Trouille et al. (2008), which also publish the spectroscopic and photometric redshift measures. Our sample consists of the sources with S/N$>$3 detections in the hard 2-8 keV band and includes in the circular area with radius of 0.32 $\deg$ and center in: $\alpha$ 10$^h$46$\arcmin$, $\delta$ $59\arcdeg01\arcmin$ (J2000). This area contains 139 extragalactic sources with a redshift measure available and 2-10 keV luminosity higher than 10$^{42}$ erg/s: 113 sources were spectroscopically identified (58 AGN1, 34 AGN2, 16 ELG, 5 galaxies), while 29 sources have a photometric redshift available. The [N$_{H}$]{} measures were derived from the analysis of the HR. Radio data were obtained from Owen et al. (2008). We conservatively modeled the spatial dependence of the (5$\sigma$) radio flux limits, which vary from 18.5 $\mu$Jy in the central region up to 59.2 $\mu$Jy near the edges. Sixty-nine out of the 139 sources were radio-detected.
CDFS
----
In the [*Chandra*]{} Deep Field South (CDFS) the subsample of the GOODS-S X-ray sources from the catalogue of Alexander et al. (2003) was used. The whole sample consists of the 94 point like extragalactic sources over a 2-10 keV flux limit (at the aim point) of 2.6$\times$10$^{-16}$ [erg s$^{-1}$ cm$^{-2}$]{} with a redshift measure available and a 2-10 keV luminosity higher than 10$^{42}$ erg/s. Spectroscopic identifications for 69 sources, as well as 25 photometric redshifts, were obtained from (Brusa et al. 2009b). Twenty-nine sources were identified as AGN (17 AGN1, 12 AGN2), 39 as ELG and one as a galaxy. The radio data were taken from Miller et al. (2008) and have a 1.4 GHz 5$\sigma$ flux limit of 70 $\mu$Jy. Twelve sources were radio-detected.
CDFN
----
In the [*Chandra*]{} Deep Field North (CDFN) we used the 296 2-8 keV X-ray sources detected by [*Chandra*]{} within the GOODS-N area with a 2-10 keV flux limit (at the aim point) of 1.4$\times$10$^{-16}$ [erg s$^{-1}$ cm$^{-2}$]{} by Alexander et al. (2003). To convert the 2-8 keV fluxes in the 2-10 keV band a spectral index $\alpha$$=$$0.4$ were assumed. The spectroscopic identifications were taken from Trouille et al. (2008). The sample consists of 162 extragalactic sources with a redshift measure available and 2-10 keV luminosity higher than 10$^{42}$ erg/s: 104 sources were spectroscopically identified (16 AGN1, 20 AGN2, 46 ELG, 22 galaxies), while 50 sources have a photometric redshift; eight other spectroscopical redshift were retrieved from literature. [N$_{H}$]{} values were obtained from the analysis of the HR. The radio information were obtained from the new data reduction from Biggs & Ivison (2006) of the 1.4 GHz VLA observation of Richards (2000). Forty-five out of the 162 extragalactic sources were identified in the 1.4 GHz band over a (5$\sigma$) flux limit of 45 $\mu$Jy.
The whole X-ray sample
----------------------
In summary, our radio observed hard X-ray selected extragalactic total sample contains 1641 sources with both redshifts (either spectroscopic or photometric) and [N$_{H}$]{} column densities measured, and with un-absorbed 2-10 keV luminosities higher than 10$^{42}$ erg/s[^2]. 1003 sources have [N$_{H}$]{} higher than 10$^{22}$ cm$^{-2}$ (hereafter defined “X-ray absorbed”). 375 (23%) sources were detected in the 1.4 GHz radio band. See Table \[tab\_samp\] for a summary of the main properties of all samples used. In Figure \[Fig\_FxFr\] we show the (5$\sigma$) radio 1.4 GHx flux limits of each survey as a function of their deepest 2-10 keV X-ray flux limits. The distribution of the de-absorbed X-ray luminosity of all sources (distinguished according to radio detection and X-ray absorption classes) as a function of redshift is shown in Figure \[Fig\_LZ\].
The distribution of the R=L$_R$/L$_X$ ratio
===========================================
The method
----------
We searched for a functional fit of the probability distribution function of R$_X$, as a function of the X-ray luminosity, L$_X$, and the redshift: $P(R_X | L_X, z)$. The method is based on the comparison, through $\chi^2$ estimators, of the observed and expected numbers of AGNs (in the L$_X$-$z$-R$_X$ space) obtained by taking into account the observational selection effects (i.e. the radio flux limits) of each sample. Once a probability distribution function $P(R_X | L_X, z)$ is assumed, the number of expected AGNs in a given bin of the L$_X$-$z$-R$_X$ space is the result of the sum, over the number of all AGN contained in the L$_X$-$z$ bin, of the expected number of AGNs contained in that bin of R$_X$, by taking into account the radio flux limits on each source. This method reproduces the observations and consequently properly takes into account both the radio detections and the upper limits (see La Franca et al. 1994, 1997, 2002, 2005 for similar applications).
The fit
-------
[lcccccccc]{} Model & N& $\gamma_R$ & $\gamma_L$ & $R_C$ & $\alpha_L$ & $\alpha_z$ & $\chi^2/$d.o.f. & P$(\chi^2)$\
1 - box & $...$ & $...$& $...$& $...$& $...$& $...$& 670.17/12 & 0\
2 - no dep & 1.0620 & 0.476 & 1.93 & -4.313 & $...$ & $...$ & 46.73/20 & $6.4\times 10^{-4}$\
3 - dep z & 1.0899 & 0.467 & 1.69 & -4.319 & $...$ & 0.028 & 41.98/19 & 0.020\
4 - dep L & 1.0652 & 0.429 & 1.70 & -4.386 & 0.056 & $...$ & 30.19/19 & 0.049\
5 - dep L,z & 1.0230 & 0.369 & 1.69 & -4.578 & 0.109 & -0.066& 22.77/18 & 0.200\
\
1$\sigma$ errors & & $^{+0.040}_{-0.031}$ & $^{+0.18}_{-0.31}$ & $^{+0.110}_{-0.086}$ & $^{+0.019}_{-0.025}$ & $^{+0.024}_{-0.016}$ & &\
\
\
\[tab\_fit\]
As a first test we assumed a constant (flat) probability distribution function of R$_X$ in the range -7$<$R$_X$$<$0 (see Figure \[Fig\_box\]). This distribution, although different from the true one, allows to see which is the shape of the average true distribution via the analysis, in each bin, of the deviations of the observed numbers of AGN from the expected ones. Figure \[Fig\_box\] shows that the average distribution function is a-symmetrical with a long tail at large R$_X$ values.
We then divided the L$_X$-$z$-R$_X$ space into 23 independent bins, which were chosen in order to look for possible dependences of the $P(R_X| L_X, z)$ shape on the luminosity and redshift (see Figure \[Fig\_LZ\] for a representation of the L$_X$-$z$ regions) while ensuring that about, at least, 10 AGN were observed in each bin (see Figure \[Fig\_LogDistr5\]).
After several trials, we found that the probability distribution function distribution of R$_X$ is, indeed, a-symmetrical, showing a maximum at $R_X$$=$$R_0$, where the median is located. At R$_X$ larger than $R_0$ the distribution is fairly well represented by a Lorentz function having width $\gamma_r$, which provides a shallow decline at large R$_X$ values, while at R$_X$ smaller than $R_0$ the exponent 2 of the Lorentz function is better substituted by an exponent 4 which gives a wider shoulder at $R_X$$ \lesssim$$ R_0$ and then a steep decline at even lower R$_X$. The width of the left ($R_X$$<$$R_0$) part of the distribution is controlled by the $\gamma_l$ parameter. Therefore the probability distribution function $P(R_X| L_X, z)$ is expressed by the following formula: $$\begin{aligned}
P(R_X)=\left\{\begin{array}{l l}\frac{N}{A\pi\gamma_l \left[ 1 + \left(\frac{R_0(L_X,z)-R_X}{\gamma_l} \right)^4 \right]}& \mbox{($R_X<R_0$)}\\
\\
\frac{A\ N}{\pi\gamma_r \left[ 1 + \left(
\frac{R_X-R_0(L_X,z)}{\gamma_r} \right)^2 \right]}
& \mbox{($R_X\geq R_0$),}\\ \end{array} \right. \\end{aligned}$$
where, in order to obtain a continuous function at $R_X$$=$$R_0$ results $A = \sqrt{\gamma_r/\gamma_l}$, and the parameter $N$ is constrained by the probability normalization requirement: $\int P(R_X| L_X, z)dR_X=1$. For $0.3$$\leq$$z$$\leq3.0$ and $42.2$$\leq$$LogL_X$$\leq$$47.0$, we allowed to vary, as a function of $L_X$ and $z$, the position of the maximum (median) $R_0$ of the distribution, according to the following formula: $$R_0 = R_C \left[ \alpha_L (LogL_X-44)+ 1 \right] \left[ \alpha_z (z-0.5) + 1 \right].
\label{Eq_dipLZ}$$ At redshifts and luminosities outside these ranges, $R_0$ was kept constant, equal to the values assumed at the limits of the ranges.
In Table \[tab\_fit\] the results of the fits carried out using this parameterization are reported. Confidence regions of each parameter were obtained by computing $\chi^2$ at a number of values around the best-fit solution, while leaving the other parameters free to float (see Lampton et al. 1976). The 68% confidence regions quoted correspond to $\Delta \chi^2$$=$$1$.
The solution without dependences on both the luminosity and the redshift is rejected by the $\chi^2$ test, while the solutions either depending only on luminosity or redshift provide barely acceptable fits to the data. A fairly good fit to the data (20% $\chi^2$ probability) is instead provided by the solution \#5 where both a dependence on the luminosity and the redshift is allowed. However, it should be noted that the parameter of the redshift dependence, $\alpha_z$, is different from zero only at 3$\sigma$ confidence level.
The data and the shape of the best fit \#5 probability distribution function are shown in Figure \[Fig\_LogDistr5\], while the corresponding dependences of $R_0$ on L$_X$ and $z$ are shown in Figure \[Fig\_LZ5\]. In Figure \[Fig\_distr5\] we show the shape of the best fit \#5 probability distribution function in different bins of X-ray luminosity and redshift, with evidence (the continuous lines) to the part which is actually constrained by the data.
![Probability distribution function of R$_X$ as a function of L$_X$ and $z$. The dashed line shows our best fit solution \#5 (see Table \[tab\_fit\]). The limits in the L$_X$-$z$ space of the seven regions used are shown in Figure \[Fig\_LZ\].[]{data-label="Fig_LogDistr5"}](lafranca_fig05.ps){width="8.5cm"}
![Probability distribution function in different bins of L$_X$ and $z$, as computed in our best fit solution \#5 (see Table \[tab\_fit\]). The continuous line shows the range of R$_X$ where the fit is constrained by the data, while the dashed line shows where the distribution is extrapolated.[]{data-label="Fig_distr5"}](lafranca_fig07.ps){width="8cm"}
![1.4 GHz luminosity as a function of the intrinsic 2-10 keV luminosity of all AGN contained in our sample (only the AGN with logL$_X$$\geq$42 have been used in our analysis). Radio detection are shown by green dots, while radio upper limits are shown by arrows. The dependence, according to fit \#5, of the position of the median R$_0$ of the $P(R_X | L_X, z)$ distribution (see eq. \[Eq\_dipLZ\]) at redshift 0 and 3 is shown by red continuos lines. For comparison the relations derived by Brinkmann et al. (2000; orange long dashed), Panessa et al. (2007; blue dotted), and Ballantyne (2009; magenta dashed) are shown. The relations derived by the measure of the AGN fundamental plane by Merloni et al. (2003) and Li et al. (2008) are shown (green dot-dashed and cyan dot-dashed lines, respectively) assuming a fixed BH mass of 10$^8$ M$_\sun$ (see text).[]{data-label="Fig_LRLX"}](lafranca_fig08.ps){width="8.3cm"}
Dependence on N$_H$
-------------------
We did not find any significant dependence of the $P(R_X| L_X, z)$ distribution on the N$_H$ column densities. The sample was divided into absorbed (N$_H$$>$10$^{22}$ cm$^{-2}$) and un-absorbed AGN. For both sub-samples the $\chi^2$ test on the best fit solution \#5 (even with a different sampling in order to always observe at least 10 objects in each bin) provided probabilities larger than 10% (up to 70%), and was then not able to reject our best fit distribution. This result is in agreement with the analysis of the X-ray absorption properties of the faint radio sources of the CDFS by Tozzi et al. (2009).
Sanity checks
-------------
### The L$_R$-L$_X$ relation
We compared our measure of the probability distribution function of R$_X$ with previous measures on the relationship between the AGN radio and X-ray luminosities. In Figure \[Fig\_LRLX\] we show the 1.4 GHz luminosity as a function of the intrinsic 2-10 keV luminosity for all the sources of our sample. Our analysis differs from many previous studies where single general log-log linear relations were derived. In these studies, even when the presence of censored data, or data with errors in both axes (see e.g. La Franca et al. 1995), were taken into account, it was assumed the presence of a symmetrical (usually Gaussian) distribution of the deviations from the best fit relation, which were attributed to an intrinsic scatter. Our method, instead, by taking into account all the censored data, allows to measure the [*shape*]{} of the distribution of the intrinsic scatter and its possible dependences on other variables (such as the luminosity and redshift in our case). In Figure \[Fig\_LRLX\] we compare our best fit solution \#5 with the relations derived by other authors such as Brinkmann et al. (2000) in the soft 0.5-2 keV band[^3], Merloni et al. (2003), Panessa et al. (2007), Li et al. (2008) and Ballantyne (2009; for the radio quiet AGN) in the 2-10 keV band. The relations by Merloni et al. (2003) and Li et al. (2008) were derived from their measure of the AGN fundamental plane (i.e. including also a dependence on the BH mass), assuming a fixed BH mass of 10$^8$ M$_\sun$.
As our measured distribution function is a-symmetrical and depends on the redshift, the comparison is not straightforward (in Figure \[Fig\_LRLX\] we plot the position of the median R$_0$ of the distribution at redshifts 0 and 3). It results that our best fit solution \#5 of the dependence of the 1.4 GHz luminosity on the 2-10 keV luminosity is flatter than obtained in previous works. Our fitted relation corresponds to a power-law L$_R$$\propto$L$_X^\alpha$ with indexes $\alpha$=0.48 and $\alpha$=0.58 at redshifts 0 and 3, respectively. The relations fitted by the other authors have instead a wide range of power-law indexes (0.6$\lesssim\alpha\lesssim1.5$), systematically steeper than our result. This difference is partly caused by our new method used to measure the L$_R$-L$_X$ relation but it is also caused by our introduction (and measure) of a dependence on the redshift of the average R$_X$ values. In fact, as shown in Figure \[Fig\_LRLX\], our solution \#4, obtained without the inclusion of a redshift dependence, results in a steeper slope, having a power-law index $\alpha=0.75$. Figure \[Fig\_LRLX\] also shows that the fundamental plane by Merloni et al. (2003) has a power-law index ($\alpha$=0.60) close to our best fit estimate ($\alpha$=0.48-0.58), and for BH masses of 10$^8$ M$_\sun$ is located between our best fit relations at redshift 0 and 3. In the fundamental plane estimate by Merloni et al. (2003) R$_X$ has also a dependence on the BH mass of the type $\propto$$0.78\rm{log}(\rm{M}_{BH})$. This implies that our estimates at redshift 0 and 3 are roughly similar to the fundamental plane measures for BH masses of 10$^7$ and 10$^9$ M$_\sun$, respectively. Therefore, as in our (and all flux limited) samples, high redshift AGN are on average more luminous, and thus probably host on average more massive BH, we can infer that our measure of the increase with the redshift of the median of the R$_X$ distribution is qualitatively in agreement with the AGN fundamental plan measures where an increase of the average R$_X$ with the BH masses is observed.
![Predicted 1.4 GHz radio LF according to our fit \#5, at redshifts 0.2, 0.4, 0.7 and 1.0 (red continuous lines). The dashed red lines show the part of the LF reproduced by the P(R$_X$) distribution with R$_X$$>$-4. The black long dashed lines show the radio luminosity function as estimated by Smolčic et al. (2009).[]{data-label="Fig_LF5"}](lafranca_fig09.ps){width="8.3cm"}
### The 1.4 GHz luminosity function and counts
We then verified if the measured probability distribution function of R$_X$, once convolved with the 2-10 keV LF, properly reproduces (taking into account the uncertainties) independent previous measures of the 1.4 GHz LF, $\Phi_R(L_{1.4},z)$, and integral counts, $N(>S)$. We used the 2-10 keV LF, $\Phi_X(L_X, z)$, as measured by La Franca et al. (2005), and modified by allowing a steep exponential decline of the AGN density at redshifts larger than $z=2.7$, as measured by Brusa et al. (2009a). As discussed in La Franca et al. (2005), the density of Compton thick AGN with 24$<$logN$_H$$\leq$26 cm$^{-2}$ was assumed to be equal to the density of the Compton thin AGN with 22$<$logN$_H$$\leq$24 cm$^{-2}$ (this assumption resulted to properly reproduce the cosmic X-ray background). The X-ray LF can be converted into a radio LF by the formula: $$\Phi_R(L_{1.4} ,z) = \int P(R_X | L_X, z)\Phi_X(L_X, z)d{\rm log}L_X.
\label{eq_rlf}$$ The X-ray LF was integrated starting from an X-ray luminosity logL$_X$=41 erg/s. The predicted radio LF is shown in Figure \[Fig\_LF5\] and compared with the 1.4 GHz radio LF measured by Smolčić et al. (2009). As discussed in Smolčić et al. (2009), this radio LF includes (like in our measure of the distribution of R$_X$) mostly the FRI sources. Although also affected by the uncertainties on the measure of the X-ray LF by La Franca et al. (2005), our best fit \#5 of the probability distribution function of R$_X$ provides a fairly good reproduction of the FRI radio luminosity function.
As a result of this computation, our estimate of the distribution of R$_X$ allows to predict the AGN FRI radio LF at luminosities smaller than L$_{1.4}$=10$^{21}$ W Hz$^{-1}$, never probed before. However at these low radio luminosities the AGN could be overwhelmed by the radio emission of the hosting galaxy because during periods of strong star formation activity the supernovae remnants accelerate cosmic rays which radiate synchrotron emission in local magnetic fields. According to Ranalli et al. (2003), strong star forming galaxies ($\sim$10$^2$ M$_\sun$/yr) have about 10$^{42}$ erg/s 2-10 keV luminosities, and, in general, in all star forming galaxies a linear relation between the 1.4 GHz radio and the 2-10 keV luminosites is observed, which corresponds to a value R$_X$$\simeq-$2.0. At face value this would imply that for the lowest luminosity AGN (logL$_X$$\simeq$42-43 erg/s) most of our measure of the P(R$_X$) distribution (which spans in the range $-$7$<$R$_X$$<-$1) could be contaminated if the hosting galaxies are undergoing a strong star formation activity. However, as shown in Figure \[Fig\_distr5\], at low R$_X$ values the P(R$_X$) distribution is mostly measured from AGN with 43$<$logL$_X$$<$45 erg/s, and redshift lower than $z$$\sim$0.5. At these low redshifts it is very unlikely for galaxies to harbor a star formation stronger than $\sim$10 M$_\sun$/yr (Elbaz et al. 2007; Noeske et al. 2007), which roughly corresponds to an emission in the 2-10 keV band of logL$_X$$\sim$41 erg/s (Ranalli et al. 2003). Therefore, for example, in a low redshift AGN with logL$_X$$\sim$44 erg/s, having an hosting galaxy whose star formation emits a 2-10 keV luminosity of logL$_X$$\sim$41 erg/s, the corresponding radio emission (having an intrinsic R$_X$$\simeq$-2.0) can contaminate our measure of the R$_X$ distribution only for R$_X$ values smaller than -5. We can then conclude that only at the lowest R$_X$ values ($\lesssim-5$) our measure of the P(R$_X$) distribution is potentially affected by a contamination from the radio emission due to the star formation activity of the AGN hosting galaxies.
We used the above computed radio LF (eq. \[eq\_rlf\]) to derive the expected integral counts N($>$S) from the following equation: $$N(>{\rm S})={1\over{4\pi}} \int {{dv}\over{dz}}dz \int^{L_{max}} _{Sk(z)4\pi d^2_l(z)}
\Phi(L_{1.4}, z)d{\rm log}L_{1.4},$$ where $k(z)=(1+z)^{\alpha_R-1}$ (with $F_\nu \propto\nu^{-\alpha_R}$ and $\alpha_R=0.5$) is the radio k-correction, $d_l(z)$ the luminosity distance, and the integral counts are measured in sr$^{-1}$ units. The predicted radio counts are shown in Figure \[Fig\_Counts5\]. As our measured distribution of R$_X$ represents only the FRI population (see discussion in sect. 2), in order to reproduce the total radio counts, the contribution derived from the LF of the FRII population (as measured by Wilman et al. 2008) was added. The reproduced counts are in good agreement with the observations. At 1.4 GHz fluxes below 1 mJy the euclidean radio counts flatten, due to the appearance of the population of the star forming galaxies. In this context, it is matter of discussion which is the fraction of the AGN at these fluxes. Our results agree with recent estimates of the AGN contributions to the sub-milliJansky radio counts from Seymour et al. (2008) and Padovani et al. (2009). Similar results (at these fluxes) have also been obtained from semi-empirical simulations of the extragalactic radio counts by Jarvis & Rawlings (2004), Wilman et al. (2008) and Ballantyne (2009).
The kinetic luminosity function
===============================
Once the radio LF of the FRI sources is measured (Equation \[eq\_rlf\] and Figure \[Fig\_LF5\]), in order to derive the kinetic mechanical LF and its evolution, we should convolve the 1.4 GHz LF with a relation which converts the radio luminosity L$_{1.4}$ into a mechanical power L$_K$. In the last decade, several authors have worked on the estimate of this relation (Willott et al. 1999; Birzan et al. 2004, 2008; Best et al. 2006; Heinz et al. 2007; Merloni & Heinz 2008). Following the discussion of Cattaneo and Best (2009) we used two different estimates which are representative of two different luminosity regimes. At high radio luminosities (L$_{1.4}$$\ga$10$^{25}$ W Hz$^{-1}$), Willott et al. (1999) used the minimum entropy density that the plasma radio lobes must have in order to emit the observed synchrotron radiation and obtained (see Figure \[Fig\_LRLK\]): $$L_{\rm K} = 1.4 \times 10^{37} \left(\frac{L_{\rm
1.4\,GHz}}{10^{25}{\rm W\,Hz}^{-1}}\right)^{0.85}{\rm W}.
\label{w2}$$
A second approach is to infer $L_{\rm K}$ from the mechanical work that the lobes do on the surrounding hot gas. The expanding lobes of relativistic synchrotron-emitting plasma open cavities (of volume $V$) in the ambient thermal X-ray emitting plasma. The minimum work in inflating these cavities is done for reversible (quasi-static) inflation and equals $pV$, where $p$ is the pressure of the ambient gas. Best et al. (2006) derived a relation between radio and mechanical luminosity based upon this estimate for the energy associated with these cavities, combined with an estimate of the cavity ages from the buoyancy timescale (from Birzan et al. 2004). Comparing the mechanical luminosities of 19 nearby radio sources that have associated X-ray cavities with their 1.4GHz monochromatic radio luminosities leads to a relation $$L_{\rm K} = 1.2 \times 10^{37}
\left(\frac{L_{\rm 1.4\,GHz}}{10^{25}{\rm W\,Hz}^{-1}}\right)^{0.40}
{\rm W},
\label{b2}$$ which is better suited for low luminosities (L$_{1.4}$$\la$ 10$^{25}$ W Hz$^{-1}$) and is close to the estimate by Birzan et al. (2008; see Figure \[Fig\_LRLK\]). Using a similar method, Merloni & Heinz (2008) obtained at high radio luminosities, a relation similar to the one by Willott et al. (1999) but with about 0.5-1 dex higher kinetic luminosities (see Figure \[Fig\_LRLK\]).
In this work (as in Cattaneo and Best 2009) we used Equation \[w2\] at high luminosities (L$_{1.4}$$\ga$10$^{25}$ W Hz$^{-1}$) and Equation \[b2\] at lower luminosities. Using these relations, once derived the radio LF from the X-ray LF and the R$_X$ distribution according to Equation 4, we can estimate the kinetic LF, $\Phi_K(L_K, z)$, by the formula $$\Phi_K(L_K, z)= { d N(L_K, z) \over{ dV d{\rm log} L_K}}= \Phi_{1.4}(L_{1.4}(L_K), z) { d {\rm log}L_{1.4} \over{ d{\rm log} L_K}},$$ while the bolometric radiative LF, $\Phi_{rad}(L_B, z)$, is derived from the X-ray LF, via the bolometric correction $L_B = K_X(L_X)L_X$, by $$\Phi_{rad}(L_B, z)= { d N(L_B, z) \over{ dV d{\rm log} L_B}}= \Phi_X(L_X(L_B), z) { d{\rm llog} L_{X} \over{ d{\rm log} L_B}},$$ where for K$_X$(L$_X$) we used the relation from Marconi et al. (2004). Both the, so derived, kinetic and radiative (bolometric) LF are shown in Figure \[Fig\_KLF\], while in Figure \[Fig\_LKLF\] the kinetic power density, L$_K$$\Phi_K$(L$_K$,$z$), as function of the kinetic luminosity is shown.
![Bolometric radiative (green continuous line) and kinetic AGN LF. The magenta and blue lines show the kinetic LF predicted using the relations linking the radio and kinetic luminosities of Willott et al. (1999) and Best et al. (2006) respectively. The red line is the result obtained using the combination of the relations of Willott et al. (1999) and Best et al. (2006) at high and low luminosities respectively (see text and Figure \[Fig\_LRLK\]). []{data-label="Fig_KLF"}](lafranca_fig12.ps){width="8.5cm"}
![Kinetic power density, L$_K$$\Phi_K$(L$_K$,$z$), as function of the kinetic luminosity according to the fit solution \#5 (red continuos line). The results by Merloni & Heinz (2008) are shown by a cyan shaded area.[]{data-label="Fig_LKLF"}](lafranca_fig13.ps){width="8.5cm"}
By integrating in luminosity the above derived kinetic and radiative (bolometric) LF it is possible to estimate the dependence of the AGN mechanical and radiative power per unit cosmic volume as a function of redshift,
$$\Omega_K(z)= \int L_K(L_{1.4}) \Phi_{1.4}(L_{1.4},z) d{\rm log}L_{1.4},
\label{Eq_Ok}$$
and $$\Omega_{rad}(z)= \int L_{bol}(L_X)\Phi_X(L_x,z)d{\rm log}L_X,
\label{Eq_Or}$$ which are shown in Figure \[Fig\_Kin1\]. As both the above equations depends on the X-ray luminosity function (see eq. \[eq\_rlf\]) the integral lower limit was logL$_X$=41 erg/s which corresponds to a lower limit of of logL$_B$$\simeq$35.5 W for eq. \[Eq\_Or\] and, according to the R$_X$ distribution and the relation between L$_{1.4}$ and L$_K$ (eq. \[b2\]) to a lower limit of logL$_K$$\simeq$34.5 W for eq. \[Eq\_Ok\]. The resulting kinetic power density would be similar if, at all luminosities, the conversion of the radio into kinetic luminosity by Best et al. (2006), eq. \[b2\], would be used, while a factor 4-6 lower values would be obtained using the relation from Willott et al. (1999), eq. \[w2\], only. As shown in Figure \[Fig\_KLF\] this difference is caused by the steep drop off of the 1.4 GHz radio LF at luminosities higher than LogL$_R$$\sim$25 W (see Figure \[Fig\_LF5\]); as a consequence the resulting power density (the integrated kinetic LF) depends mainly on which conversion of the radio luminosity into kinetic power is used at low luminosities.
![[*Bottom*]{}. Power density as a function of redshift. The kinetic power density derived from the best fit solution \#5 is shown by a continuous red line. The red dashed line shows the result if the more radio loud objects (R$_X$$>$-4) only are used (see text and Figure \[Fig\_LF5\]). The magenta and orange dashed lines show the kinetic luminosity density as estimated by Kording et al. (2008) and Cattaneo & Best (2009), respectively. The cyan and blue areas show the estimates by Merloni & Heinz (2008) and Smolčić et al. (2010), respectively. The radiative and kinetic power densities, as used in the model of Croton et al. (2006), are shown by continuous and dashed black lines, respectively. [*Top*]{}. Ratio between the kinetic and radiative power densities. The red and black continuous lines are the results by our fit \#5 and Croton et al. (2006), respectively.[]{data-label="Fig_Kin1"}](lafranca_fig14.ps){width="8.5cm"}
Discussion
==========
The probability distribution function of R$_X$, $P(R_X|
L_X, z)$ estimated in Section 3 depends on both luminosity and redshift: the average R$_X$ increases with decreasing luminosity and (possibly) increasing redshift (see best fit \#5 in Table \[tab\_fit\] and Figure \[Fig\_LZ5\]). The observed increase of the average R$_X$ value with decreasing luminosity is similar to previous results and models where (in analogy with X-ray binaries) low luminosity AGN are expected to be more likely radio loud [^4] (see e.g. Merloni & Heinz 2008; Körding, Jester & Fender 2008).
The knowledge of the R$_X$ distribution, once convolved with the X-ray LF and the relations between the kinetic and radio luminosity, allows to estimate the kinetic LF and its evolution. At luminosities higher than the break of the bolometric LF (L$_K$$\sim$10$^{39}$ W) the kinetic LF results to be more than two orders of magnitude smaller than the bolometric LF (see Figure \[Fig\_KLF\]), while at lower luminosities the relevance of the kinetic LF increases, reaching values comparable to the bolometric one at L$_K$$\sim$10$^{36}$ W, which roughly corresponds to the minimum luminosity experimentally probed by the X-ray LF (logL$_X$$\sim$42 erg/s; see e.g. La Franca et al. 2005). The kinetic LF shows a maximum in the range 10$^{35}$$<$$L_{K}$$<$10$^{37}$ W, where most ($\sim$90%) of the kinetic power density (shown in Figure \[Fig\_LKLF\]) is produced.
In Figure \[Fig\_Kin1\] we show the kinetic and radiative power density as a function of the redshift. We also show the kinetic power density corresponding to the more radio emitting AGN having $R_X$ larger than -4, and then corresponding to the population which is typically represented by the radio LF (see in Figure \[Fig\_LF5\] the comparison of the reproduced radio LF with R$_X$$>$-4 with the radio LF of FRI sources as measured by Smolčić et al. 2009). It is then possible to see that the kinetic power density could be underestimated by up to a factor of about two, if the radio LF alone is used, without taking into account the low radio luminosity AGN population.
Our estimates are in qualitative agreement with the trends of the radiative and kinetic power density with redshift used by Croton et al (2006) at $z>0.5$ (see Figure \[Fig\_Kin1\]). However, at lower redshifts we find a sharp (a factor of five) decrease of both radiative and kinetic power densities from $z\sim$0.5 (i.e. about 5 billion years ago) to $z=0$. This result is quite robust, and comes from the strong negative evolution of the AGN LF from $z\sim2$ down to $z=0$, which has been observed since the first studies of the QSO evolution in the optical (see e.g. Marshall et al. 1983; Croom et al. 2009 for recent results), and measured by many other authors in the hard X-rays (e.g. Ueda et al. 2003; La Franca et al. 2005; Hasinger 2008). Conversely, Croton et al. (2006) assume an almost continuous increase of both the kinetic and radiative powers, due to the assumption that both phenomena are related to an almost constant accretion onto the SMBH. Under this assumption Croton et al. (2006) overestimate both the AGN radiative and kinetic power densities at low ($z$$\lesssim$0.5) redshift, allowing only for a much shallower decrease of the kinetic feedback (a factor of 30%) and of the AGN radiative power.
In Figure \[Fig\_Kin1\] ([*top*]{}) we show the ratio of the $\Omega_K/\Omega_{rad}$ plotted as function of redshift. According to our best fit \#5, at $z$$>$0.5 the kinetic power density is $\sim5\%$ of the radiative density, and increases up to about 30% at decreasing redshifts. This increase at low redshifts of the ratio of the kinetic power to the radiative power density could help in modeling the quenching of the star formation at low redshift. In Croton et al. (2006) a milder increase is assumed, which should be attributed to the, above discussed, overestimate of the low redshift AGN radiative density.
Many previous results on the AGN kinetic LF are based on the convolution of some relations between the kinetic and radio power with direct measures of the AGN radio LF (e.g. Shankar et al. 2008; Merloni & Heinz 2008; Körding, Jester & Fender 2008; Cattaneo & Best 2009; Smolčić et al. 2009). As already discussed in the introduction, the measure of the R$_X$ distribution is useful in order to allow a detailed implementation of the AGN feedback within the galaxy formation and evolution models because it gives the opportunity to predict the radio luminosity (and thus feedback) of each AGN as a function of its luminosity (accretion rate) and redshift.
According to our best fit \#5, in the redshift range 0.5$<$$z$$<$3 the integrated kinetic power density is $\sim$1-2$\times 10^{33}$ W Mpc$^{-3}$ (see Figure \[Fig\_Kin1\]). This is in rough agreement with the previous estimates by Merloni & Heinz (2008), Körding, Jester and Fender (2008) and Smolčić et al. (2009). At lower redshift ($z$$<$0.5) we observe a drop by a factor of five, similar to what observed by Körding, Jester and Fender (2008), and in agreement, within the uncertainties, with Merloni & Heinz (2008). On the contrary our results are, at any redshift, 2-8 times greater than that reported by Cattaneo & Best (2009).
Merloni & Heinz (2008) found that their kinetic LF roughly corresponds to a constant overall efficiency in converting the accreted mass energy into kinetic power $\epsilon_{kin}$$\simeq$$ 3-5
\times10^{-3}$ (where $L_K=\epsilon_{kin}\dot{m}c2$). Their results are similar to ours where, according to equations \[Eq\_Ok\] and \[Eq\_Or\], on average, we have $\epsilon_{kin}$$\simeq$$
(\Omega_K/\Omega_R) \epsilon_R\simeq 5\times 10^{-3}$, as we measure $\Omega_K/\Omega_R\simeq 0.05$ (see the $\Omega_K/\Omega_R$ ratio as a function of redshift in Figure \[Fig\_Kin1\], [*top*]{}), and assuming a radiative efficiency $\epsilon_R=0.1$ (Marconi et al. 2004). However, we observe an increase of the $\Omega_K/\Omega_R$ ratio (i.e. of the kinetic efficiency) up to a value 0.3 at decreasing redshifts, which Merloni & Heinz (2008) observe in the most massive objects only ($> 10^{8}-10^{9} M_\sun$; see e.g. their Figure 13). It should also be noted that, although the integrated kinetic power density of Merloni & Heinz (2008) is in agreement with our estimate, their kinetic LF is similar to our measure only at L$_K$$\sim$10$^{36}$ W (see Figure \[Fig\_LKLF\]), while it is definitely larger (by about an order of magnitude) at higher kinetic luminosities. Once integrated in luminosity, the computed power densities are similar (at $z$$\lesssim$3) because our low luminosity limit (L$_K$=10$^{34}$ W) is significantly lower than that used by Merloni & Heinz (2008; L$_K$=10$^{36}$ W).
Shankar et al. (2008), found that the ratio of the kinetic to bolometric luminosity, defined as $g_k=L_K/L_B$, is constant and equal to $g_k=0.10$ with a scatter of $\sigma=0.38$. According to equations \[Eq\_Ok\] and \[Eq\_Or\], $g_k$ corresponds roughly to the ratio $\Omega_K/\Omega_{rad}$ (plotted as function of redshift in Figure \[Fig\_Kin1\], [*top*]{}), which, as discussed above, levels at $\sim$0.05 at $z>1$ while increases up to 0.3 at $z=0$.
Conclusions
===========
We used a sample of more than 1600 X-ray selected AGN observed at 1.4 GHz to measure the probability distribution function, $P(R_X| L_X, z)$, of the ratio $R_X$ of the radio to intrinsic X-ray luminosity, as a function of the X-ray luminosity and redshift.
The knowledge of the $P(R_X| L_X, z)$ distribution is necessary to estimate the AGN kinetic (radio) feedback into the hosting galaxies by allowing to couple it with the luminous, accreting, phases of the AGN activity.
The average value of R$_X$ increases with decreasing X-ray luminosities and (possibly) increasing redshift. At variance, we did not find a statistical significant difference between the radio properties of the X-ray absorbed (N$_H$$>$$10^{22}$ cm$^{-2}$) and un-absorbed AGN.
We were able to better measure the densities of the more radio quiet (R$_X$$<$-4) AGN which resulted to be responsible of about half of the derived kinetic power density.
According to our analysis the value of the kinetic energy density is in qualitative agreement with the last generation galaxy evolution scenarios, where radio mode AGN feedback is invoked to quench the star formation in galaxies and slow down the cooling flows in galaxy clusters. However at redshifts below 0.5, similarly to what observed by Körding, Jester and Fender (2008), we find a sharp (about a factor of five) decrease of the kinetic energy density, which is strictly related the AGN density evolution, but which is not included in many of the galaxy/AGN formation and evolution models where, instead, the radio mode feedback is assumed to continuously increase (or only smoothly decrease) at low redshift.
We thank A. Lamastra, N. Menci, R. Morganti, P. Ranalli, V. Smol[č]{}i[ć]{} and G. Zamorani for discussions. We are grateful to L. Trouille and A. Barger for the support in allowing us to use the CLANS data, and to D. Ballantyne, R. Della Ceca and A. Merloni for providing data in machine readable format. We thanks the anonymous referee for his useful comments. We acknowledge financial contribution from contract ASI-INAF I/088/06/0 and PRIN-MIUR grant 2006-02-5203.
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[^1]: Throughout this work we assumed that all the X-ray sources having 2-10 keV unabsorbed luminosities higher than 10$^{42}$ erg/s are AGN (see e.g. Ranalli et al. 2003 for a study of the typical X-ray luminosities of star forming galaxies).
[^2]: Other X-ray samples, such as HELLAS2XMM (Fiore et al. 2003; Cocchia et al. 2007), the XMM/Lockman Hole (Brunner et al. 2008) and the XMM Medium Survey (Barcons et al. 2007), were not included in our work because unbiased, homogeneous and dedicated radio 1.4 GHz observations were not available.
[^3]: An X-ray spectral index $\alpha$=0.7 was assumed to convert the 0.5-2 keV luminosities into 2-10 keV luminosities.
[^4]: As discussed in sect. 2 and 3, these trends do not include the most luminous (radio loud; mostly FRII) sources. As far as the more radio loud population is concerned, it has been observed that the fraction of more radio loud AGN increases with increasing optical (or X-ray) luminosity and decreasing redshift (Miller et al. 1990; Visnovsky et al. 1992; Padovani 1993; La Franca et al. 1994, Goldschmidt et al. 1999, Ivezi[ć]{} et al. 2002; Cirasuolo et al. 2003; Jiang et al. 2007).
|
---
author:
- 'A. Jorissen[^1]'
- 'M. Mayor'
- 'S. Udry'
bibliography:
- '/home/bibtex/ajorisse\_articles.bib'
date: 'Received date / Accepted date'
title: The distribution of exoplanet masses
---
Introduction
============
@Han-2001:a suggested that most of the exoplanet candidates discovered so far have masses well above the lower limit defined by $\sin i = 1$ (where $i$ is the inclination of the orbital plane on the sky) and should therefore be considered as brown dwarfs or even stars rather than planets. The present paper shows that this claim is not consistent with the distribution of masses extracted from the observed $M_2 \sin i$ distribution (where $M_2$ is the mass of the companion) under the reasonable assumption that the orbits are oriented at random in space. Although the distributions of $M_2$ and $M_2 \sin i$ are related through an integral equation of Abel’s kind [@Chandrasekhar-1950; @Lucy-74], its numerical solution is ill-behaved. Two different methods are used here to overcome that difficulty. In the first method (Sect. \[Sect:Abel\]), the formal solution of Abel’s equation is implemented numerically on an input $M_2 \sin i$ distribution that has been optimally smoothed with an adaptive kernel procedure [@Silverman-86] to remove the high-frequency fluctuations caused by small-number statistics. The other method (Sect. \[Sect:LR\]) is based on the Lucy-Richardson inversion algorithm [@Richardson-72; @Lucy-74].
The basic reason why the $M_2$ distribution obtained in Sect. \[Sect:results\] differs from that of @Han-2001:a is because these authors concluded that most of the systems containing exoplanet candidates are seen nearly pole-on. This conclusion, based on the analysis of the Hipparcos [*Intermediate Astrometric Data*]{} (IAD), has however been shown to be incorrect [e.g. @Halbwachs-2000:a; @Pourbaix-2001:a; @Pourbaix-2001:b], as summarized in Sect. \[Sect:discussion\].
While this paper was being referred, @Zucker-Mazeh-2001 and @Tabachnik-Tremaine-2001 proposed other interesting approaches to derive the exoplanet mass distribution. Zucker & Mazeh derive the binned true mass distribution by using a maximum likelihood approach to retrieve the average values of the mass distribution over the selected bins. Their results are in very good agreement with ours. Tabachnik & Tremaine suppose that the period and mass distributions follow power laws, and derive the corresponding power-law indices from a maximum likelihood method. On the contrary, the methods used in the present paper (and in Zucker & Mazeh’s) are [*non-parametric in nature,*]{} since they do not require to make any [*a priori*]{} assumptions on the functional form of the mass distribution. This is especially important since the comparison of the shapes of the mass distributions for exoplanets and low-mass stellar companions may provide clues to identify the process by which they formed. By imposing a power-law function like that observed for low-mass stellar companions, @Tabachnik-Tremaine-2001 somehow implicitly assume that these processes must be similar.
The integral equation of Abel’s kind relating the distributions of $M_2 \sin i$ and $M_2$ {#Sect:Abel}
=========================================================================================
The $M_2 \sin i$ values for low-mass companions of main sequence stars may be extracted from the spectroscopic mass function and from the primary mass as derived through e.g., isochrone fitting. Let $\Phi(Y)$ be the observed distribution of $Y \equiv M_2 \sin i$ which is easily derived from the observed spectroscopic mass functions provided that $M_2 << M_1$ as it is expected to be the case for the systems under consideration. Then, the seeked distribution $\Psi(M_2)$ obeys the relation $$\label{Eq:integral}
\Phi(Y) = \int_0^\infty \Psi(M_2)\; \Pi(Y | M_2)\; {\rm d}M_2.$$ The kernel $\Pi(Y | M_2)$ corresponds to the conditional probability of observing the value $Y$ given $M_2$. Under the assumption that the orbits are oriented at random in space, the inclination angle $i$ distributes as $\sin i$, and the following expression is obtained for the kernel: $$\Pi(Y | M_2) = \frac{\sin i_0} {M_2 \cos i _0},$$ where $i_0$ satisfies the relations $M_2 \sin i_0 - Y = 0$ and $0 \le
i_0 \le 90$. Eliminating the inclination $i_0$ in the above relation yields $$\label{Eq:Pi}
\Pi(Y | M_2) = \frac{Y}{M_2} \frac{1}{(M_2^2 - Y^2)^{1/2}}\quad\quad\mbox{\rm with}\quad Y \le M_2.$$ Eq.(\[Eq:integral\]) thus rewrites $$\Phi(Y) = Y \int_Y^\infty \Psi(M_2) \; \frac{1}{M_2 (M_2^2 - Y^2)^{1/2}}
\; {\rm d}M_2 .
\label{Eq:integral2}$$ Eq.(\[Eq:integral2\]) is the integral equation to be solved for $\Psi(M_2)$. It can be reduced to Abel’s integral equation by the substitutions [@Chandrasekhar-1950] $$Y^2 = 1/\eta \hspace{6pt}{\rm and} \hspace{6pt} M_2^2 = 1/t.$$ With these substitutions, Eq.(\[Eq:integral2\]) becomes $$\label{Eq:Abel}
\phi(\eta) = \int_0^{\eta} \frac{\psi(t)}{(\eta - t)^{1/2}} \; {\rm
d}t,$$ where $$\phi(\eta) \equiv \Phi(\frac{1}{\sqrt{\eta}}) \hspace{6pt} {\rm and}
\hspace{6pt} \psi(t) \equiv \frac{1}{2\;\sqrt{t}}
\Psi(\frac{1}{\sqrt{t}}).$$ It is well known that the solution of Abel’s equation (Eq. \[Eq:Abel\]) is given by $$\label{Eq:Abelsolution}
\psi(t) = \frac{1}{\pi} \int_0^t
\frac{\partial\phi}{\partial \eta}\frac{1}{(t - \eta)^{1/2}} \; {\rm
d} \eta + \frac{1}{\pi} \frac{\phi(0)}{\sqrt{t}},$$ where $\phi(0) = {\rm lim}_{Y \rightarrow \infty} \Phi(Y) = 0$.
While Eq. (\[Eq:Abelsolution\]) represents the formal solution of the problem, it is difficult to implement numerically, since it requires the differentiation of the observed frequency distribution $\Phi(Y)$. Unless the observations are of high precision, it is well known that this process can lead to misleading results. To overcome that difficulty, the observed frequency distribution has been smoothed in an optimal way (see Appendix) before being used in Eq. (\[Eq:Abelsolution\]). The solution $\Psi(t)$ is then computed numerically using standard differentiation and integration schemes.
The Lucy-Richardson inversion algorithm applied to Abel’s integral equation {#Sect:LR}
===========================================================================
The Lucy-Richardson algorithm provides another robust way to invert Eq. (\[Eq:integral2\]) [see also @Cerf-Boffin-94]. The method starts from the Bayes theorem on conditional probability in the form $$\Psi(M_2) \; \Pi(Y| M_2) = \Phi(Y) \; R(M_2 | Y),$$ where $R(M_2 | Y)$ is the reciprocal kernel corresponding to the integral equation inverse to the one that needs to be solved (Eq. \[Eq:integral\]): $$\label{Eq:integralinv}
\Psi(M_2) = \int_0^{M_2} \Phi(Y)\; R( M_2| Y)\; {\rm d}Y.$$
The reciprocal kernel represents the conditional probability that the binary system has a companion mass $M_2$ when the observed $M_2 \sin
i$ value amounts to $Y$. Thus, one has: $$\begin{aligned}
R(M_2|Y) &=& \frac{\Psi(M_2) \; \Pi(Y|M_2)}{\Phi(Y)} \\
&=&
\frac{\Psi(M_2) \; \Pi(Y|M_2)}{\int_0^{\infty} \Psi(M_2)\; \Pi(Y|M_2)\; {\rm d}M_2},\end{aligned}$$ which obviously satisfies the normalization condition $\int_0^{\infty}
R(M_2|Y)\; {\rm d}M_2 = 1$. The problem in solving Eq. (\[Eq:integralinv\]) arises because $R(M_2|Y)$ also depends on $\Psi(M_2)$, so that an iterative procedure must be used. If $\Psi_r(M_2)$ is the $r$th estimate of $\Psi(M_2)$, it can be used to obtain the $(r+1)$th estimate in the following way: $$\label{Eq:1}
\Psi_{r+1}(M_2) = \int_0^{M_2} \Phi(Y)\; R_r(M_2|Y)\; {\rm d}Y$$ with $$\label{Eq:2}
R_r(M_2|Y) = \frac{\Psi_r(M_2)\; \Pi(Y|M_2)}{\Phi_r(Y)}$$ and $$\Phi_r(Y) = \int_0^{\infty} \Psi_r(M_2)\; \Pi(Y|M_2)\; {\rm d}M_2.$$ Thus, $\Phi_r(Y)$ represents the corresponding $r$th estimate of the observed distribution $\Phi(Y)$. Eqs.(\[Eq:1\]) and (\[Eq:2\]) together yield the recurrence relation for the $\Psi_r$’s, $$\label{Eq:recurrence}
\Psi_{r+1}(M_2) = \Psi_r(M_2) \; \int_0^{M_2} \frac{\Phi(Y)}{\Phi_r(Y)}\; \Pi(Y| M_2)\; {\rm d}Y,$$ with $\Pi(Y|M_2)$ given by Eq. (\[Eq:Pi\]) for the problem under consideration. The conditions for convergence of this recurrence relation are discussed by @Lucy-74 and @Cerf-Boffin-94. It needs only be remarked here that (i) the iterative scheme converges if $\Phi_r(Y)$ tends to $\Phi(Y)$, given the normalization of the probability $\Pi(Y|M_2) {\rm d}Y$, and (ii) the full convergence of the method is not necessarily desirable, as the successive estimates $\Phi_r(Y)$ will tend to match $\Phi(Y)$ on increasingly smaller scales, but the small-scale structure in $\Phi(Y)$ is likely to be dominated by the noise in the input data. This is well illustrated in Fig. \[Fig:2\] below.
When the number of data points is small [typically $N <
100$; @Cerf-Boffin-94], it is advantageous to express $\Phi(Y)$ as $$\Phi(Y) = \frac{1}{N}\sum_{n = 1}^{N} \delta(Y - y_n)$$ where the $y_n \; (n=1,\dots N)$ are the $N$ individual measured $M_2
\sin i$ values and $\delta(x)$ is the Dirac ‘function’ such that $x_0
= \int \delta (x - x_0)\; {\rm d}x$. Substitution in Eq. (\[Eq:1\]) then yields $$\Psi_{r+1}(M_2) = \frac{1}{N}\sum_{n=1}^N R_r(M_2|y_n)$$ where $R_r(M_2|y_n)$ is defined as in Eq. (\[Eq:2\]). The sample size should nevertheless be large enough for the functions $R_r(M_2|y_n)$ to have sufficient overlaps so as to produce a smooth $\Psi_{r+1}(M_2)$ function.
In the application of the method described in Sect. \[Sect:results\], the initial mass distribution $\Psi_0(M_2)$ was taken as a uniform distribution, but it has been checked that this choice has no influence on the final solution.
The frequency distribution $\Psi(M_2)$ {#Sect:results}
======================================
The cumulative frequency distribution of the $M_2 \sin i$ values smaller than 17M$_{\rm J}$, where M$_{\rm J}$ is the mass of Jupiter (= M$_\odot$ /1047.35), available in the literature (as of April 4, 2001) is presented in Fig. \[Fig:1\]. It appears to be sufficiently well sampled to attempt the inversion procedure. The corresponding frequency distributions smoothed with two different smoothing lengths, locally self-adapting around $h_{\rm
opt}=1$M$_{\rm J}$ and 2$h_{\rm opt}$ (see Appendix) are presented as well for comparison.
The sample includes 60 main-sequence stars hosting 67 companions with $M_2 \sin i < 17$ M$_{\rm J}$. Among those, 6 stars are orbited by more than one companion, namely and [^2], [0.16, 0.38 M$_{\rm J}$; @Mayor-2000], [7.22, 16.2 M$_{\rm
J}$; @Udry-2000:b], [0.71, 2.20 and 4.45 M$_{\rm J}$; @Butler-99] and [0.56, 1.88 M$_{\rm J}$; @Marcy-2001]. The inversion process is only able to treat these systems under the hypothesis that the orbits of the different planets in a given system are [*not*]{} coplanar, since Eq. (\[Eq:integral2\]) to hold requires random orbital inclinations. The case of coplanar and non-coplanar orbits are discussed separately in the remaining of this section.
Non-coplanar orbits
-------------------
Figure \[Fig:2\] compares the solutions $\Psi(M_2)$ obtained from the Lucy-Richardson algorithm (after 2 and 20 iterations, denoted $\Psi_2$ and $\Psi_{20}$, respectively) and from the formal solution of Abel’s integral equation with smoothing lengths $h_{\rm opt}$ and 2 $h_{\rm opt}$ on $\Phi(Y)$ (the corresponding solutions are denoted $\Psi_{h_{\rm opt}}$ and $\Psi_{2h_{\rm opt}}$). The solutions from the two methods basically agree with each other, although solutions with different degrees of smoothness may be obtained with each method. On the one hand, $\Psi_{20}$ and $\Psi_{h_{\rm opt}}$ exhibit high-frequency fluctuations that may be traced back to the statistical fluctuations in the input data. This can be seen by noting that the peaks present in $\Psi_{h_{\rm opt}}$ correspond in fact to the high-frequency fluctuations already present in $\Phi_{h_{\rm opt}}$ (Fig. \[Fig:1\]). These fluctuations should thus not be given much credit. The same explanation holds true for $\Psi_{20}$, since it was argued in Sect. \[Sect:LR\] that the solutions $\Psi_r$ resulting from a large number of iterations tend to match $\Phi$ at increasingly small scales (i.e., higher frequencies) where statistical fluctuations become dominant. On the other hand, $\Psi_{2}$ and $\Psi_{2h_{\rm opt}}$ are much smoother, and are probably better matches to the actual distribution. The local maximum around $M_2 \sim 1$ M$_{\rm J}$ is very likely however an artifact of the strong detection bias against low-mass companions.
The most striking feature of the $\Psi(M_2)$ distribution displayed in Fig. \[Fig:2\] is its decreasing character, reaching zero for the first time around $M_2 = 10$ M$_{\rm J}$, and in any case well before 13.6 $M_{\rm J}$. The latter value, corresponding to the minimum stellar mass for igniting Deuterium, does not in any way mark the transition between giant planets and brown dwarfs, as sometimes proposed. That transition, which is thus likely to occur at smaller masses, must rely instead on the different mechanisms governing the formation of planets and brown dwarfs. Another argument favouring a giant-planet/brown-dwarf transition mass smaller than 13.6 M$_{\rm J}$ is provided e.g., by the observation of [*free-floating*]{} (and thus most likely [*stellar*]{}) objects with masses probably smaller than 10 M$_{\rm J}$ in the $\sigma$ Orionis star cluster [@Zapatero-2000]. The $\Psi(M_2)$ distribution nevertheless clearly exhibits a tail of objects clustering around $M_2 \sim
15$ M$_{\rm J}$, due to ($M_2 \sin i =
11.5$ M$_{\rm J}$), (14.3 M$_{\rm J}$), (15.0 M$_{\rm J}$) and (16.2 M$_{\rm J}$). It would be interesting to investigate whether these systems differ from those with smaller masses in some identifiable way (periods, eccentricities, metallicities,...), so as to assess whether or not they form a distinct class (Udry et al., in preparation).
The jackknife method [e.g., @Lupton-93] has been used to estimate the uncertainty on the $\Psi_{2h_{\rm opt}}$ solution. In a first step, 67 input $\Phi_{2h_{\rm opt}}$ distributions are computed, corresponding to all 67 possible sets with one data point removed from the original set. Eq. \[Eq:Abelsolution\] is then applied to these 67 different input distributions. The resulting distributions are displayed in Fig. \[Fig:jackknife\], which shows that the threshold observed at 10 M$_{\rm J}$ is a robust result not affected by the uncertainty on the solution.
Coplanar orbits in multi-planets systems
----------------------------------------
All the results discussed so far are obtained under the assumption that orbits of planets belonging to a planetary system are [*not*]{} coplanar. To evaluate the impact of this hypothesis, the following procedure has been applied. In a first step, the Lucy-Richardson algorithm is applied on the data set excluding the 13 planets belonging to planetary systems. That mass distribution obtained after 2 iterations is then completed by mass estimates for the remaining 13 planets. For each of the 6 different systems, an inclination $i$ is drawn from a $\sin i$ distribution. This is done through the expression $i = {\rm acos} \; x$, where $x$ is a random number with uniform deviate. The same value of $i$ is then applied to all planets in a given system to extract $M_2$ from the observed $M_2 \sin i$ value. The distributions of exoplanet masses obtained with and without the hypothesis of coplanarity are compared in Fig. \[Fig:coplanar\], and it is seen that planetary systems are not yet numerous enough for the coplanarity hypothesis to alter significantly the resulting $\Psi(M_2)$ distribution.
In any case, the main result of the present paper is that [*the statistical properties of the observed $M_2 \sin i$ distribution coupled with the hypothesis of randomly oriented orbital planes confine the vast majority of planetary companion masses below about 10 $M_{\rm J}$*]{}. @Zucker-Mazeh-2001 reach the same conclusion.
It should be remarked that the above conclusion cannot be due to detection biases, since the high-mass tail of the $M_2$ distribution is not affected by the difficulty of finding low-amplitude, long-period orbits.
Discussion {#Sect:discussion}
==========
Although the assumption of random orbital inclinations seems reasonable, it is at variance with the conclusion of @Han-2001:a that most of the systems containing exoplanet candidates are seen nearly pole-on. These authors reached this conclusion by trying to extract the astrometric orbit, hence the orbital inclination, from the Hipparcos IAD. @Halbwachs-2000:a had already cautioned that this approach is doomed to fail for systems with apparent separations on the sky that are below the Hipparcos sensitivity (i.e. $\la 1$ mas). In those cases, the solution retrieved from the fit of the IAD residuals is spurious, since the true angular semi-major axis $a$ is simply too small to be seen by Hipparcos. Since @Halbwachs-2000:a have shown that $a$ actually follows a Rayleigh probability distribution, the fit of the IAD residuals will yield a solution larger than the true value, in fact of the order of the residuals. But since $a \sin i$ is constrained by the spectroscopic orbital elements, the too-large astrometric $a$ value will force $i$ to be close to $0$ to match the spectroscopic value of the product $a \sin i$, as convincingly shown by @Pourbaix-2001:a. Hence, this approach gives the impression that all orbits are seen nearly face-on. As an illustrative example, @Pourbaix-2001:b have shown that such an approach leads to a stellar mass for the companion of that, on another hand, has been proven to be a 0.69M$_{\rm J}$ planet by the photometric observation of the planet transit in front of the star [@Charbonneau-2000].
The @Han-2001:a result is moreover statistically very unlikely if the orbital planes are oriented at random in space [@Pourbaix-2001:b]. @Han-2001:a have tried to justify this unlikely statistical occurrence by invoking biases against high-amplitude orbits in the selection process of the radial-velocity-monitoring samples. To the contrary, the planet-search surveys were specifically devised to avoid such biases, as they aim at finding not only giant planets but also brown dwarfs so as to constrain the substellar secondary mass function of solar-type stars. Furthermore, @Han-2001:a argument is totally invalid in the case of volume-limited, statistically well-defined samples like that of the CORALIE planet-search programme in the southern hemisphere [@Udry-2000:a]. This sample has been specifically designed to detect companions of solar-type stars all the way from $q=M_2/M_1 = 1$ down to $q\leq 0.001$.
Appendix: Non-parametric treatment of the data {#appendix-non-parametric-treatment-of-the-data .unnumbered}
==============================================
To decrease the noise and allow a tractable use of the information present in small data samples, heavy smoothing techniques are often required. A common practice consists in converting a set of discrete positions into binned “counts”. Binning is a crude sort of smoothing and many studies in statistical analysis have shown that, unless the smoothing is done in an optimum way, some, or even most, of the information content of the data could be lost. This is especially true when a large amount of smoothing is necessary, which then changes the “shape” of the resulting function. In statistical terms, the smoothing process not only decreases the noise (i.e., the function’s variance), but at the same time introduces a bias in the estimate of the function.
The variance-bias trade off is unavoidable but, for a given level of variance, the bias increase can be minimized. The correct manner of achieving that task is provided by the so-called [*non-parametric density estimate*]{} methods for the determination of the “frequency” function of a given parameter or by the [*non-parametric regression*]{} methods for the determination of a smooth function $g$ inferred from direct measurements of $g$ itself. Moreover, [*adaptive*]{} non-parametric methods are designed to filter the data in some local, objective way minimizing the bias, in order to get the smooth desired function without constraining its form in any way. The theory and algorithms related to those methods, originally built to handle ill-conditioned problems (either under-determined or extremely sensitive to errors or incompleteness in the data), are widely discussed in the specialized literature and summarized in easy-to-read textbooks [e.g., @Silverman-86; @Hardle-90; @Scott-92].
The simplest of the available algorithms is provided by the kernel estimator leading to the following form of the normalized “frequency” function $$\label{Eq:frequency}
\hat f_K(x) = {1\over N h}\sum_{n=1}^N K\left({x-X_n\over h}\right)$$ with the normalization $$\int_{-\infty}^{\infty} K(u)\;{\rm d}u = 1,$$ where $X_n$ ($n=1,...N$) are the $n$ available data points. Each data point is thus simply replaced by a ‘bump’. The [*kernel function*]{} $K$ determines the shape of the bumps while the [*bandwidth*]{} $h$ (also called [*smoothing parameter*]{}) determines their width[^3].
In the [*adaptive kernel*]{} version, a local bandwidth $h_n=h(X_n,f)$ is defined and used in Eq. \[Eq:frequency\]. In order to follow the “true” underlying function in the best possible way, the amount of smoothing should be small when $f$ is large whereas more smoothing is needed where $f$ takes lower values. A convenient method to do so consists in deriving first a pilot estimate $\tilde f$ of $f$, e.g. by using an histogram or a kernel with fixed bandwidth $h_{opt}$, and then by defining the local bandwidths $$h_n=h(X_n)=h_{opt}[\tilde f(X_n)/s]^{-\alpha},$$ where $$\log{s}=\frac{1}{N}\sum_{n=1}^N{\log{\tilde f(X_n)}}.$$ It may be shown [@Abramson-82] that $\alpha=1/2$ leads to a bias of smaller order than for the fixed $h$ estimate.
The optimum kernel $K$ may be taken as the one minimizing the integrated mean square error beween $f$ and $\hat f$ (MISE), where $$MISE(\hat{f}) = E\int\!\left[ \hat{f}(x) - f(x) \right]^2 {\rm d}x$$ is usually taken as an indicator of efficiency. In the above expression, $E$ denotes the statistical expectation. This optimum kernel is the so-called Epanechnikov ($K_e$) or quadratic kernel: $$K_e(u) = {3\over 4}(1-u^2),\ \ \ |u|<1.$$ Other choices of $K$ differ only slightly in their asymptotic efficiencies and could be more adapted to particular purposes as e.g., for bi-dimensional data.
The pilot smoothing length ($h_{opt}$) is the only subjective parameter required by the method. It relates to the quality of the sampling of the variable under consideration. There are several ways for automatically estimating an optimum value of $h_{opt}$ [see e.g. @Silverman-86 for an extensive review]. A simple approach based on the data variance gives in our case $h_{opt}\simeq 1.0$M$_{\rm
J}$. As the derivative of the frequency function rather than the function itself is actually used in Eq. (\[Eq:Abel\]), a larger pilot smoothing length ($2h_{opt}$) was also considered in order to remove spurious small-statistics fluctuations of the density estimate.
[^1]: Research Associate, F.N.R.S., Belgium
[^2]: see also [http://www.eso.org/outreach/press-rel/pr2001/pr-07-01.html]{}
[^3]: In the regression problem, the Nadaraya-Watson estimator [@Nadaraya-64; @Watson-64] is commonly used: $$g(x) = \frac{\displaystyle \sum_{n=1}^N{g(X_n) K(\frac{x-X_n}{h})}}
{\displaystyle \sum_{n=1}^N{K(\frac{x-X_n}{h})}}
%g(x) = {\displaystyle \sum_{i=1}^n{g(X_i) K(\frac{x-X_i}{h})}/}
%{\displaystyle \sum_{i=1}^n{K(\frac{x-X_i}{h})}} .$$
|
---
abstract: |
We present a homogeneous kinematic analysis of red giant branch stars within 18 of the 28 Andromeda dwarf spheroidal (dSph) galaxies, obtained using the Keck I LRIS and Keck II DEIMOS spectrographs. Based on their $g-i$ colors (taken with the CFHT MegaCam imager), physical positions on the sky, and radial velocities, we assign probabilities of dSph membership to each observed star. Using this information, the velocity dispersions, central masses and central densities of the dark matter halos are calculated for these objects, and compared with the properties of the Milky Way dSph population. We also measure the average metallicity (\[Fe/H\]) from the co-added spectra of member stars for each M31 dSph and find that they are consistent with the trend of decreasing \[Fe/H\] with luminosity observed in the Milky Way population. We find that three of our studied M31 dSphs appear as significant outliers in terms of their central velocity dispersion, And XIX, XXI and XXV, all of which have large half-light radii ($\gta700$pc) and low velocity dispersions ($\sigma_v<5\kms$). In addition, And XXV has a mass-to-light ratio within its half-light radius of just $[M/L]_{\rm
half}=10.3^{+7.0}_{-6.7}$, making it consistent with a simple stellar system with no appreciable dark matter component within its $1\sigma$ uncertainties. We suggest that the structure of the dark matter halos of these outliers have been significantly altered by tides.
author:
- 'Michelle L. M. Collins, Scott C. Chapman, R. Michael Rich, Rodrigo A. Ibata, Nicolas F. Martin, Michael J. Irwin, Nicholas F. Bate, Geraint F. Lewis, Jorge Peñarrubia, Nobuo Arimoto, Caitlin M. Casey, Annette M. N. Ferguson, Andreas Koch, Alan W. McConnachie, Nial Tanvir'
title: A kinematic study of the Andromeda dwarf spheroidal system
---
Introduction
============
The underlying nature of the dark matter halos of dwarf spheroidal galaxies (dSphs) has garnered significant attention from the scientific community over the past decade. The goal of recent observational studies of these objects has been to make critical tests of structure formation scenarios, particularly focusing on the viability of the canonical $\Lambda$CDM model. There is the long standing issue of the relative dearth of these faintest of galaxies observed surrounding nearby galaxies when compared with the number of dark matter subhalos produced in $N-$body simulations, which is referred to as the “missing satellite” problem [@klypin99; @moore99]. The extent to which this mismatch is considered problematic has decreased over recent years as both theorists and observers have sought to reconcile the simulated and observable Universe. From a modelling point of view, one does not expect stars to be able to form within all dark matter subhalos seen in simulations, and at a certain mass limit ($V_{\rm max}\lta15\kms$, see @penarrubia08a [@koposov09]), star formation is unable to proceed. Thus, there is a lower limit placed on galaxy formation. This mass limit is also tied to feedback processes that can remove the baryonic reservoirs required for star formation (e.g., @bullock00 [@somerville02; @kravtsov10; @bullock10; @nickerson11; @kazantzidis11]). This would imply that only the most massive subhalos seen in simulations are able to form and retain luminous populations. Observers have also attempted to quantify current survey completeness and radial selection effects to account for the number of satellites we are not currently able to detect (e.g., @koposov08 [@tollerud08; @walsh09]). These studies suggest that there are of order a few hundred satellites within the Milky Way’s virial radius that we have yet to detect.
The high dark matter dominance of dSph galaxies also singles them out as objects of interest. With total dynamic mass-to-light ratios of $[M/L]\sim10-1000$s and half-light radii of $r_{\rm half}\sim10-1000$ pc, they are ideal systems with which to probe the inner density profiles of dark matter halos. Recent imaging and spectroscopic observations of these objects within the Local Group have shown that, despite spanning approximately 5 decades in luminosity, the dSphs of the MW share a common mass scale and a universal density profile [@strigari08; @walker09b; @wolf10]. With a kinematic resolution of a few 10s of parsecs, these objects allow us to start addressing the question of whether the central regions of these halos follow cuspy density profiles as predicted by cosmological simulations [@navarro97], or constant density cores, similar to what is observed in low surface brightness galaxies [@blaise01; @deblok02; @deblok03; @deblok05; @swaters03; @kassin06; @spano08]. From studies of brighter dSphs, such as Sculptor and Fornax [@walker11; @amorisco12; @jardel12] it appears that their halo density profiles are also inconsistent with hosting central cusps. It is possible that these objects originally formed with cuspy density profiles, and that these have been subsequently modified by baryonic feedback. If this is truly the case, one should be able to observe cuspy profiles in the fainter dSph population (@zolotov12) as these do not contain enough baryons to drive this change in the dark matter density profile. Perhaps the only way to gain further insight into this contentious issue is by measuring the kinematics for large numbers of stellar tracers within these objects and analysing them with detailed models that do not make [*ab initio*]{} assumptions about the underlying density profiles, or the velocity anisotropy of both the dark matter and stars, such as those employed by @walker11 and @jardel12.
To date, the majority of studies involving the detailed kinematics of dSphs have revolved largely around those belonging to the Milky Way, as these are nearby enough that we can measure the velocities of their member stars to a high degree of accuracy. However, there are currently only $\sim25$ known MW dSphs, with luminosities ranging from $10^2-10^7\lsun$. For the very faintest, some controversy remains as to whether they are massively dark-matter dominated (see e.g., @niederste09 [@simon11]), but almost all of them have been shown to be consistent with the universal mass profiles of @walker09a and @wolf10. One notable exception to this is the Hercules object [@aden09], which some have argued is currently undergoing significant tidal disruption [@martin10]. Andromeda represents the only other system for which comparable kinematic analyses can be performed. M31 now has 28 dSph companions known, whose luminosities range from $\sim10^4 - 10^8\lsun$, the majority of which have been discovered by the CFHT Pan-Andromeda Archaeological Survey (PAndAS @martin06 [@ibata07; @irwin08; @mcconnachie08; @martin09; @richardson11]). The relatively brighter lower bound for the luminosities of M31 dSphs compared to the MW is a detection limit issue, rather than a sign of differing stellar populations (Martin et al. 2013, in prep). It has been noted by a number of authors (e.g., @mcconnachie06a [@tollerud12; @mcconnachie12]) that for the brighter dSphs ($M_V<-8$), those belonging to M31 are 2–3 times more extended in terms of their half-light ($r_{\rm half}$) and tidal ($r_t$) radii compared with the MW. In these papers, the underlying cause of this discrepancy was not identified, but it has been argued that it could be an effect of environment, with the mass distribution of the host playing an important role. Subsequent work by @brasseur11b, who included the fainter, non-classic M31 dSphs for the first time, showed that statistically, the relationship between size and luminosity for dSphs in the MW and Andromeda are actually largely consistent with one another, however there remain a number of significantly extended outliers within the Andromedean system (e.g., And II, And XIX, $r_{\rm
half}\sim1.2\kpc$ and $1.5\kpc$ respectively), and the scatter in this relationship is large (up to an order of magnitude at a given luminosity, @mcconnachie12).
Working from the @mcconnachie06a results, @penarrubia08a modelled the expected velocity dispersions for the M31 dSphs, assuming that all dSph galaxies are embedded within similar mass dark matter halos. A robust prediction of their modeling was that, given the larger radial extents, the dSphs of M31 should be [*kinematically hotter*]{} than their MW counterparts by a factor of $\sim2$. At the time of writing, they had only 2 measured velocity dispersions for the M31 dSphs, those of And II and And IX [@cote99; @chapman05]. New studies of the kinematics of M31 dSphs [@collins10; @collins11b; @kalirai10; @tollerud12; @chapman12] have dramatically increased the number of systems with a measured velocity dispersion, and have shown that instead of being kinematically hotter, these systems are either very similar to, or in a number of cases (e.g., And II, And XII, And XIV, And XV and And XXII), [*colder*]{} than their MW counterparts. In particular, a significant recent kinematic study of 15 M31 dSph companions using the Keck II DEIMOS spectrograph by the Spectroscopic and Photometric Landscape of Andromeda’s Stellar Halo (SPLASH, @tollerud12) concluded that the M31 dSph system largely obeys very similar mass-size-luminosity scalings as those of the MW. However, they also identified 3 outliers (And XIV, XV and XVI) that appear to possess much lower velocity dispersions, and hence maximum circular velocities, than would be expected for these systems. Such a result suggests that there are significant differences in the formation and/or evolution of the M31 and MW dSph systems.
To investigate this further, our group has been systematically surveying the known dSphs of M31 with the Keck I LRIS and Keck II DEIMOS spectrographs, and have obtained kinematic data for 18 of the 28 galaxies. In this paper, we present new spectroscopic analysis for the 11 dSphs, Andromeda (And) XVII, And XVIII, And XIX, And XX, And XXI, And XXIII, And XXIV, And XXV, And XXVI, the tidally disrupting And XXVII, And XXVIII and And XXX (Cassiopeia II) using an algorithm we have developed that implements a probabilistic method of determining membership for each galaxy. In addition we re-analyze the kinematics of 6 dSphs that our group has previously observed (And V, VI, XI, XII, XIII, XXII) using this method with the aim of providing a homogeneous analysis of all dSphs observed by our group to date. We also provide the individual stellar velocities and properties for every star observed in our dSphs survey, allowing us to present a large catalog of stellar kinematics that will be of interest to those studying dSph systems and Milky Way-like galaxies, whether observationally or theoretically.
The outline of this paper is as follows. In § 2 we discuss the relevant observations, data reduction techniques. In § 3 we outline our algorithm for the classification of member stars. In § 4 we present an analysis of our new kinematic datasets. In § 5 we report on the masses and densities of the dark matter halos of the M31 dSphs, comparing them to those of the MW dSphs. In §6 we report on the metallicities, \[Fe/H\], of the M31 dSphs as measured from the co-added spectra of their member stars. Finally, we conclude in § 7.
Observations {#sect:obs}
============
Photometry and target selection {#sect:photobs}
-------------------------------
The PAndAS survey [@mcconnachie09], conducted using the 3.6 metre Canada France Hawaii Telescope (CFHT), maps out the stellar density of the disc and halo regions of the M31–M33 system over a projected area of $\sim350\rm{deg}^2$ ($\sim55,000\kpc^2$), resolving individual stars to depths of $g=26.5$ and $i=25.5$ with a signal to noise ratio of 10, making this survey the deepest, highest resolution, contiguous map of the majority of the extended stellar halo of an L$_*$ galaxy to date. Each of the $411$ fields in this survey ($0.96\times0.94$ deg$^2$) has been observed for at least 1350s in both MegaCam $g$ and $i$ filters, in $<0.8^{\prime\prime}$ seeing. This survey was initiated following two precursor surveys of the M31 system, the first of which surveyed the central $\sim40$ deg$^2$ conducted with the 2.5 metre Isaac Newton Telescope [@ferguson02; @irwin05], and revealed a wealth of substructure in the Andromeda stellar halo, including the giant southern stream [@ibata01c]. To better understand this feature, and to probe deeper into the M31 (and M33) stellar halo, a survey of the south west quadrant of the M31 halo was initiated using the CFHT [@ibata07], and revealed yet more substructure, including the arc like stream Cp and Cr [@chapman08] and a number of dwarf spheroidal satellites [@martin06]. This CFHT survey was then extended into the full PAndAS project. For details of the processing and reduction of these data, see @richardson11. This survey has introduced us to a wealth of stellar substructure, debris and globular clusters within the Andromeda–Triangulum system. In addition, it has led to the discovery of 17 dSphs. These objects were detected in the PAndAS survey maps as over-densities in matched-filter surface density maps of metal poor red giant branch (RGB) stars and were presented in @martin06 [@ibata07; @irwin08; @mcconnachie08; @martin09] and @richardson11. We briefly summarise the photometric properties of all dSphs discussed within this paper in Table \[tab:photobs\].
[lccccc]{} And V & 01:10:17.1 & +47:37:41.0 & -10.2 & 302$\pm44$ & 742$^{+21}_{-22}$\
And VI & 23:51:39.0 & +24:35:42.0 & -10.6 & 524$\pm49$ & 783$\pm28$\
And XI & 00:46:20.0 & +33:48:05.0 & -6.9 & 158$^{+9}_{-23}$ & 763$^{+29}_{-106}$\
And XII & 00:47:27.0 & +34:22:29.0 & -6.4 & 324$^{+56}_{-72}$ & 928$^{+40}_{-136}$\
And XIII & 00:51:51.0 & +33:00:16.0 & -6.7 & 172$^{+34}_{-39}$ & 760$^{+126}_{-154}$\
And XVII & 00:37:07.0 & +44:19:20.0 & -8.5 & 262$^{+53}_{-46}$ & 727$^{+39}_{-25}$\
And XVIII & 00:02:14.5 & +45:05:20.0 & -9.7 & 325$\pm24$ & 1214$^{+40}_{-43}$\
And XIX & 00:19:32.1 & +35:02:37.1 & -9.6 & 1481$^{+62}_{-268}$& 821$^{+32}_{-148}$\
And XX & 00:07:30.7 & +35:07:56.4 & -6.3 & 114$^{+31}_{-12}$ & 741$^{+42}_{-52}$\
And XXI & 23:54:47.7 & +42:28:15.0 & -9.8 & 842$\pm77$ & 827$^{+23}_{-25}$\
And XXII & 01:27:40.0 & +28:05:25.0 & -6.5 & 252$^{+28}_{-47}$ & 920$^{+32}_{-139}$\
And XXIII & 01:29:21.8 & +38:43:08.0 & -10.2 & 1001$^{+53}_{-52}$& 748$^{+31}_{-21}$\
And XXIV & 01:18:30.0 & +46:21:58.0 & -7.6 & 548$^{+31}_{-37}$ & 898$^{+28}_{-42}$\
And XXV & 00:30:08.9 & +46:51:07.0 & -9.7 & 642$^{+47}_{-74}$ & 736$^{+23}_{-69}$\
And XXVI & 00:23:45.6 & +47:54:58.0 & -7.1 & 219$^{+67}_{-52}$ & 754$^{+218}_{-164}$\
And XXVII & 00:37:27.2 & +45:23:13.0 & -7.9 & 657$^{+112}_{-271}$ &1255$^{+42}_{-474}$\
And XXVIII & 22:32:41.2 & +31:12:51.2 & -8.5 & 210$^{+60}_{-50}$ &650$^{+150}_{-80}$\
And XXX (Cass II) & 00:36:34.9 & +49:38:48.0 & -8.0 & 267$^{+23}_{-36}$ & 681$^{+32}_{-78}$\
For the majority of these objects, the PAndAS dataset formed the basis for our spectroscopic target selection. Using the color selection boxes presented in @mcconnachie08 [@martin09] and @richardson11, we isolated the RGBs of each dSph, then prioritised each star on this sequence depending on their color, $i$-band magnitude, and distance from the centre of the dSph. Stars lying directly on the RGB, with $20.3<i_0<22.5$ and distance, $d< 4r_{\rm half}$ (where $r_{\rm half}$ is the half-light radius, measured on the semi-major axis of the dSph) were highly prioritised (priority A), followed by stars on the RGB within the same distance from the centre with $22.5<i_0<23.5$ (priority B). The remainder of the mask was then filled with stars in the field with $20.3<i_0<23.5$ and $0.5<g-i< 4$ (priority C). In general, it is the brighter, higher priority A stars that ultimately show the highest probability of membership. We also use the PAndAS photometry to help us determine membership of the dSph (discussed in § \[sect:membership\]).
And XXVIII is the one dSph in our sample that is not covered by the PAndAS survey. For this object, target selection followed an identical methodology tot hat detailed above, but used photometry from the 8th data release of the Sloan Digital Sky Survey (SDSS-III). Details of the observations and analysis of this photometry can be found in @slater11.
Keck Spectroscopic Observations {#sect:specobs}
-------------------------------
The DEep-Imaging Multi-Object Spectrograph (DEIMOS), situated on the Nasmyth focus of the Keck II telescope is an ideal instrument for obtaining medium resolution (R$\sim1.4$Å) spectra of multiple, faint stellar targets in the M31 dSphs. The data for the dSphs within this work were taken between Sept 2004 and Sept 2012 in photometric conditions, with typical seeing between $0.5-1^{\prime\prime}$. Our chosen instrumental setting covered the observed wavelength range from 5600–9800Å and employed exposure times of 3x20 minute integrations. The majority of observations employed the 1200 line/mm grating, although for 4 dSphs (And XI, XII, XIII and XXIV) the lower resolution 600 line/mm grating ($R\sim3.8$Å FWHM) was used. The spectra from both setups typically possess signal-to-noise (S:N) ratios of $>3$Å$^{-1}$ for our bright targets ($i\lta22.0$), though some of our fainter targets fall below this level. Information regarding the spectroscopic setup and observations for each dSph are displayed in Table \[tab:specobs\].
The resulting science spectra are reduced using a custom built pipeline, as described in @ibata11. Briefly, the pipeline identifies and removes cosmic rays, corrects for scattered light, performs flat-fielding to correct for pixel-to-pixel variations, corrects for illumination, slit function and fringing, wavelength calibrates each pixel using arc-lamp exposures, performs a 2-dimensional sky subtraction, and finally extracts each spectra – without resampling – in a small spatial region around the target. This results in a large set of pixels for each target, each of which carries a flux and wavelength (with associated uncertainties) plus the value of the target spatial profile at that pixel. We then derive velocities for all our stars with a Bayesian approach, using the Ca II Triplet absorption feature. Located at rest wavelengths of 8498, 8542 and 8662 Å, these strong features are ideal for determining the velocities of our observed stars. We determine the velocities by using an Markov Chain Monte Carlo procedure where a template Ca II spectrum was cross-correlated with the non-resampled data, generating a most-likely velocity for each star, and a likely uncertainty based on the posterior distribution that incorporates all the uncertainties for each pixel. Typically our velocity uncertainties lie in the range of $5-15\kms$. Finally, we also correct these velocities to the heliocentric frame.
[lcccccccccc]{} And V & 16 Aug 2009 & LRIS &831/8200&3.0Å& 01:10:18.21 & +47:37:53.3 & 0$\deg$ & 3600 & 50&15\
And VI & 17–19 Sept 2009& DEIMOS & 1200 & 1.3Å& 23:51:51.49 & +24:34:57.0 & 0$\deg$ & 5400 & 113& 43\
And XI & 23 Sept 2006 & DEIMOS & 600 & 3.5Å& 00:46:28.08 & +33:46:28.8 & 0$\deg$ & 3600 &33 &5\
And XII & 21–23 Sept 2006& DEIMOS & 600 & 3.5Å& 00:47:32.89 & +34:22:28.6 & 0$\deg$ & 3600 &49 &8\
And XIII & 23 Sept 2006 & DEIMOS & 600 & 3.5Å& 00:52:00.22 &+32:59:16.2 & 0$\deg$ & 3600 & 46& 4\
And XVII & 26 Sept 2011 & DEIMOS & 1200 & 1.3Å& 00:37:51.09 & +44:17:51.9 & 280$\deg$ & 3600 &149& 8\
And XVIII & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:02:14.50 &+45:05:20.0&90$\deg$& 3600 & 73 & 4\
And XIXa & 26 Sept 2011 & DEIMOS & 600 & 3.5Å& 00:19:45.04 & +35:05:28.8 & 270$\deg$& 3600 & 107&15\
And XIXb & 26 Sept 2011 & DEIMOS & 1200 & 1.3Å& 00:19:30.88 & +35:07:34.1 & 0$\deg$ & 3600 & 108& 9\
And XX & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:07:30.69 & +35:08:02.4 & 90$\deg$ &3600 & 85 &4\
And XXI & 26 Sept 2011 & DEIMOS & 1200 & 1.3Å& 23:54:47.70 & +42:28:33.6 & 180$\deg$ &3600& 157&32\
And XXIIa & 23 Sept 2009 & DEIMOS & 1200 & 1.3Å& 01:27:52.37 &+28:05:22.3 & 90$\deg$&1200 &93 &4\
And XXIIb & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:27:52.37 & +28:05:22.3 & 0$\deg$&3600& 73&6\
And XXIIIa & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:29:18.18 & +38:43:50.4 & 315$\deg$ & 3600 &196 & 24\
And XXIIIb & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:29:21.87 & +38:44:58.7 & 245$\deg$ & 3600 & 189&18\
And XXIVa & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:18:32.90 & +46:22:50.0 & 30$\deg$ &3600 & 192 & 1\
And XXIVb & 31 May 2011 & DEIMOS & 600 & 3.5Å& 01:18:32.90 & +46:22:50.0 & 0$\deg$ &2700 & 115 & 11\
And XXV & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:30:01.88 &+46:50:31.0 & 90$\deg$ &3600& 183& 26\
And XXVI & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:23:41.42 & +47:54:56.8 & 90$\deg$ &3600 & 179 &6\
And XXVII & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:37:31.93 & +45:23:55.4 & 45$\deg$ & 3600 & 131&8\
And XXVIII& 21 Sept 2012 & DEIMOS & 1200 & 1.3Å& 22:32:32.71 & +31:13:13.2 & 140$\deg$ & 3600 & 102&17\
Cass II & 28 Sept 2011 & DEIMOS & 1200 & 1.3Å& 00:37:00.84 & +49:39:12.0 & 270$\deg$ & 3600 & 156& 8\
### Telluric velocity corrections {#sect:telluric}
With slit-spectroscopy, systematic velocity errors can be introduced if stars are not well aligned within the centre of their slits. Such misalignments can result from astrometric uncertainties or a slight offset in the position angle of the mask on the sky. For our astrometry, we take the positions of stars from PAndAS photometry, which have an internal accuracy of $\sim0.1^{\prime\prime}$ and a global accuracy of $\sim0.^{\prime\prime}25$ [@segall07]. This can translate to velocity uncertainties of up to $\sim15\kms$ for our DEIMOS setup. In previous studies, authors have tried to correct for this effect by cross correlating their observed spectra with telluric absorption features (e.g., @sohn07 [@simon07; @kalirai10; @collins10; @tollerud12]). These atmospheric absorption lines are superimposed on each science spectrum, and should always be observed at their rest-frame wavelengths. Thus, if one is able to determine the offset of these features, shifts caused by misalignment of the science star within the slit can be corrected for, and this can be applied on a slit by slit basis. The strongest of these features is the Fraunhofer A-band, located between 7595–7630Å. An example of this feature is shown in the left panel of Fig. \[fig:telluric\].
{width="0.45\hsize"} {width="0.45\hsize"}
While we believe this correction is robust in the high S:N regime, we argue against applying this correction in studies of Andromedean satellites, where S:N is often quite low (typically less than 8–10Å$^{-1})$ for 1 hour observations of faint ($i\gta21$) RGB stars. We find that when we compute this offset for all stars within our sample, those with high S:N tend to cluster within a few $\kms$ of the average telluric correction found for the mask. As the S:N decreases below about 10Å$^{-1}$, the scatter about this mean value increases dramatically, as do the uncertainties computed for each individual correction. This is because the telluric feature is a single, very broad and asymmetric feature. It is therefore easy in the noisy regime for the cross correlation routine to misalign the template and science spectrum whilst still producing a high confidence cross-correlation maximum. We show this effect explicitly in the right hand panel of Fig. \[fig:telluric\] where we plot the deviation of the telluric correction for every star within our sample from the average correction determined for the spectroscopic mask it was observed with ($v_{tel}-<v_{tel}>$) as a function of S:N. The cyan points represent the individual data, and the large black points represent the median of all points within 1Å bins in S:N. The error bars represent the dispersion within each bin. It is plainly seen that the median value for each bin is consistent with zero (i.e., the median), and that the dispersion increases with decreasing S:N. If we were to apply these velocity corrections to all our stars, it is probable that we would merely increase the velocity uncertainties rather than reducing them.
For this reason, we take a different approach. Using solely the telluric velocity corrections of stars from each observed mask whose spectra have S:N$>7$, we measure (a) the average telluric correction for the mask and (b) the evolution of the telluric correction as a function of mask position. In this way, we can track any gradient in our measurements that could be caused by e.g., rotational offsets in our mask. In general, we find these corrections to be slight. The average measured offset across all our masks is $3.8\kms$ (ranging from between $-3.4\kms$ and $+10.6\kms$). The measured gradients are very slight, resulting in an average end-to-end mask difference of 2.6$\kms$, with a range of $0.1-7.2\kms$, typically within our measured velocity uncertainties.
A probabilistic determination of membership {#sect:membership}
===========================================
Determining membership for Andromedean dwarf spheroidals is notoriously difficult in the best of cases. We only possess information about the velocity, CMD position, distance from the centre of the dSph and spectroscopic metallicity (although this carries large uncertainties of $>0.3$ dex for individual stars). Depending on the systemic velocity of the dSph, we must try to use these properties to distinguish the likely members from either Milky Way halo K-dwarfs ($v_{hel}\ga-150\kms$) or M31 halo giants ($v_{halo}\approx-300\kms$, $\sigma_{v,halo}\approx90\kms$, @chapman06). In the case of Galactic contamination, our spectra also cover the region of the Na I doublets ($\sim 8100$Å). As this feature is dependent on the stellar surface gravity, it is typically stronger in dwarf stars than in giants. However, there is a significant overlap between the two, especially in the CMD color region of interest for Andromedean RGB stars. In the past, groups have focused on making hard cuts on likely members in an attempt to weed out likely contaminants (e.g., based on their distance from the centre of the galaxy or their velocity, @chapman05 [@collins10; @collins11b; @kalirai10]), but such ‘by eye’ techniques are not particularly robust. @tollerud12 recently presented an analysis of a number of M31 dSphs where they used a more statistical method to ascertain likely membership, using the distance from the centre of, and position in the CMD of stars targeted within their DEIMOS masks. Here we employ a similar technique that will assess the probability of stars being members of a dSph based on (i) their position on the CMD; (ii) their distance from the centre of the dSph and, in addition; (iii) their position in velocity space, giving the likelihood of membership as:
$$\label{eqn:probtotal}
P_i\propto P_{CMD}\times P_{dist}\times P_{vel}$$
Below, we fully outline our method, and implement a series of tests to check it can robustly recover the kinematics of the M31 dSphs.
Probability based on CMD position, $P_{CMD}$
--------------------------------------------
For the first term, $P_{CMD}$, we are interested in where a given star observed in our mask falls with respect to the RGB of the observed dSph. @tollerud12 use the distance of a given star from a fiducial isochrone fit to the dwarf photometry to measure this probability. Here, we determine this value from the data itself, rather than using isochrones. Using the PAndAS CFHT photometry, we construct a normalised Hess diagram for the central region (i.e., within $2\times r_{\rm half}$) of the dSph, and one of a surrounding ‘field’ comparison region. By combining these two Hess diagrams, we can then map both the color distribution of the dSph and that of our contaminating populations. We use both directly as probability maps, where the densest region would have a value of 1.
{width="0.45\hsize"} {width="0.45\hsize"}
So we are not dominated by shot noise of sparsely populated regions of the CMD, we use only the region of the RGB. We do this by assigning a generous bounding box around the RGB as seen in the left hand panel of Fig. \[hess\] where we display the PAndAS CMD for the well populated And XXI RGB. We have zoomed in on the region for which DEIMOS observations with reliable velocities can be obtained, e.g., $i<23.5$. The bounding box is shown with red dashed lines. Anything that falls outside this region is therefore assigned a probability of $P_{CMD}=0$. The resulting probability map for And XXI is shown in the right hand panel. Red points represent all DEIMOS stars that have $P_{CMD}>10^{-6}$, while the blue stars show stars from the DEIMOS mask that are far removed from the And XXI RGB, and thus not considered to be members.
Probability based on distance position, $P_{dist}$
--------------------------------------------------
The second term in our probability function, $P_{dist}$ can be easily determined from the known radial profile of the dSphs. The half-light radii of all these objects are known and can be found in @mcconnachie06b [@zucker04; @zucker07; @mcconnachie08; @martin09; @collins10; @collins11b; @richardson11]. We also know that their density profiles are well represented by a Plummer profile with a scale radius of $r_p\equiv r_{\rm half}$. Therefore, we can define the probability function as a normalised Plummer profile [@plummer11], i.e.,:
$$P_{dist}=\frac{1}{\pi r_p^2[1+(r/r_p)^{2}]^2}$$
The above equation assumes that the systems we are studying are perfectly spherical. While the majority of these systems are not significantly elliptical, it is important to consider the effect of any observed deviations from sphericity. We therefore modify $r_p$ based on a given stars angular position with respect to the dwarfs major axis, $\theta_i$, such that:
$$r_p=\frac{r_{\rm half}(1-\epsilon)}{1+\epsilon{\rm cos}\theta_i}$$
where $\epsilon$ is the measured ellipticity of the dSph as taken from @mcconnachie12.
Probability based on velocity, $P_{vel}$ {#sect:velprob}
----------------------------------------
The final term, $P_{vel}$, contains information about the likelihood of a given star belonging to a kinematic substructure that is not well described by the velocity profiles of either the MW halo dwarfs or the Andromeda halo giants, both of which are determined empirically from our DEIMOS database of $>20000$ stars. From an analysis of these stars (selecting out any obvious substructures and the Andromedean disc) we find that the M31 halo is well approximated by a single Gaussian with a systemic velocity of $v_{r,halo}=-308.8\kms$ and $\sigma_{v,halo}=96.3\kms$, giving a probability density function for a given star with a velocity $v_i$ and velocity uncertainty of $v_{err,i}$ of:
$$\begin{aligned}
P_{halo}=\frac{1}{\sqrt{2\pi(\sigma_{v,halo}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,halo}-v_{r,i}}{\sqrt{\sigma_{v,halo}^2+v_{err,i}^2}}\right)^2\Big] \end{aligned}$$
The MW halo population is well approximated by 2 Gaussians with $v_{r,MW 1}=-81.2\kms$, $\sigma_{v,MW 1}=36.5\kms$ and $v_{r,MW 2}=-40.2\kms$ and $\sigma_{v,MW 2}=48.5\kms$, resulting in a probability density function for a given star with a velocity $v_i$ and velocity uncertainty of $v_{err,i}$ of: $$\begin{aligned}
P_{MW}=\frac{R}{\sqrt{2\pi(\sigma_{v,MW
1}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,MW
1}-v_{r,i}}{\sqrt{\sigma_{v,MW1}^2+v_{err,i}^2}}\right)^2\Big] \\+ \frac{(1-R)}{\sqrt{2\pi(\sigma_{v,MW
2}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,MW 2}-v_{r,i}}{\sqrt{\sigma_{v,MW
2}^2+v_{err,i}^2}}\right)^2\Big]
\end{aligned}$$
where $R$ is the fraction of stars in the first MW peak, and $(1-R)$ is the fraction of stars in the second peak. The value of $R$ is determined empirically from our DEIMOS data set.
A strong kinematic peak outside of these two populations can then be searched for using a maximum likelihood technique, based on the approach of @martin07. We search for the maximum in the likelihood function that incorporates the two contamination populations plus an additional Gaussian structure with systemic velocity $v_{r,substr}$ and a dispersion of $\sigma_{v,substr}$, defined as:
$$\begin{aligned}
\label{eq:mlpeta}
P_{substr}=\frac{1}{\sqrt{2\pi([\eta\sigma_{v,substr}]^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,substr}-v_{r,i}}{\sqrt{[\eta\sigma_{v,substr}]^2+v_{err,i}^2}})]\right)^2\Big]
\end{aligned}$$
Here, to ensure we haven’t biased our $P_{substr}$ strongly against stars that lie within the wings of the Gaussian distribution of dwarf spheroidal velocities, we have included a multiplicative free parameter, $\eta$, to our derived value of $\sigma_v$. To determine the ideal value of $\eta$, we ran our algorithm over all our datasets, changing the value of $\eta$ from $0.5-10.5$ to see its effect on the final derived systemic velocities and velocity dispersion. We find that in all cases, the solutions converge at values of $\eta\sim2-4$. For dSphs whose kinematics are well separated from contaminants, the derived kinematics can remain stable up to much larger values of $\eta$, however for those with systemic velocities within the velocity regime of the Milky Way, the solution quickly destabilizes as more contaminants are included as probable members. We show this implicitly in Fig. \[fig:etatest\], where we present the effect of modifying $\eta$ on 6 dSphs, And XII, XIX, XXI, XXII, XXIII and XXV. These objects were selected as they nicely probe our datasets with low numbers of probable member stars ($\sim8$), to those where we have 10s of probable members, as well as sampling dSphs from highly contaminated to well isolated kinematic regimes. The value of $\eta$ is therefore independently determined for each dataset separately, and we report its final value in Table \[tab:kprops\].
$$\begin{aligned}
\label{eq:mlp}
P_{substr}=\frac{1}{\sqrt{2\pi(\sigma_{v,substr}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,substr}-v_{r,i}}{\sqrt{\sigma_{v,substr}^2+v_{err,i}^2}})]\right)^2\Big]
\end{aligned}$$
The likelihood function can then be simply written as:
$$\rm{log}[{\mathcal{L}}(v_r,\sigma_v)]=\sum_{i=1}^{N}\rm{log}\Big(
\alpha P_{i,{\rm halo}}+\beta P_{i,{\rm MW}}+\gamma P_{i,{\rm substr}}\Big)$$
where $\alpha$, $\beta$ and $\gamma$ represent the Bayesian priors, i.e., the expected fraction of stars to reside in each population. These are determined by starting with arbitrary fractions (for example, 0.2, 0.5 and 0.3 respectively) and are then adjusted to the posterior distribution until priors and posteriors match. This technique will therefore identify an additional kinematic peak, independent of the MW and M31 halo populations, if it exists. We stress that these are not the final systemic velocity and dispersion of the dSph, but merely indicate a region in velocity space in which an excess of stars above the two contaminant populations is seen. In Fig. \[fig:vtest\], we show the result of this process for the And XXI dSph. Here, the substructure is clearly visible as a cold spike at $\sim-400\kms$.
![A velocity histogram for all observed stars in the field of And XXI. Our empirical Gaussian fits to the full Keck II data set for the M31 and MW halos are overlaid in blue and green respectively. A cold, kinematic peak at $-400\kms$ is also seen, and this is the likely signature of the dSph. Our coarse, initial ML procedure identifies this peak, and the values of $v_{sys}$ and $\sigma_v$ it measures are used to derive our kinematic probability, $P_{vel}$.[]{data-label="fig:vtest"}](fig2.eps){width="0.85\hsize"}
Now that we have a velocity profile for our three components (MW, M31 halo and the dSph), we can assign probabilities for each star within our sample belonging to each population using simple Bayesian techniques, i.e., the probability that a given star belongs to the substructure, $P_{vel}$, is:
$$P_{vel}=\frac{\gamma P_{substr}}{\alpha P_{halo}+\beta P_{MW}+\gamma P_{substr}}$$
and the probability of being a contaminant is:
$$P_{nvel}=\frac{\alpha P_{halo}+\beta P_{MW}}{\alpha P_{halo}+\beta P_{MW}+\gamma P_{substr}}$$
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"}
Measuring $v_r$ and $\sigma_v$
------------------------------
Upon applying this to our data, we can identify the most probable members of each dSph, without having to apply any additional constraints or cuts.
Having established the membership probability for each observed star (as detailed above) we now calculate the kinematic properties of each dSph; namely their systemic velocities ($v_r$) and velocity dispersions ($\sigma_v$). We use the maximum likelihood technique of @martin07, modified to include our probability weights for each star. We sample a coarse grid in $(v_r,\sigma_v)$ space and determining the parameter values that maximise the likelihood function (ML), defined as:
$$\label{eq:ml}
{\rm log}[{\mathcal{L}}(v_r,\sigma_v)]=-\frac{1}{2}\sum_{i=1}^{N}\Big[P_i{\rm
log}(\sigma_{\mathrm{tot}}^2)+P_i\frac{v_r-v_{r,i}}{\sigma_{\mathrm{tot}}}^2\\+P_i{\rm
log}(2\pi)\Big]
$$
where $N$ is the number of stars in the sample, $v_{r,i}$ is the radial velocity measured for the $i^\mathrm{th}$ star, $v_{err,i}$ is the corresponding uncertainty and $\sigma_{\mathrm{tot}}=\sqrt{\sigma_v^2+v_{err,i}^2}$. In this way, we aim to separate the intrinsic dispersion of a system from the dispersion introduced by the measurement uncertainties.
Testing our probabilistic determination of membership and calculations of kinematic properties {#sect:test}
----------------------------------------------------------------------------------------------
Having developed the above technique, it is important for us to rigorously test that it is robust enough to accurately determine the global kinematic properties for each of our datasets. In Appendix A, we examine in detail a number of potential issues that could cause our algorithm to return biased or incorrect results. These are the inclusion of a velocity dependent term in our probability calculation, the effect of including low S:N data (S:N$<5$Å) in our analysis and the effect of small sample sizes ($N_*<8$) on our measurements of kinematic properties. We briefly summarize our findings here, and refer the reader to Appendix A for a more detailed description.
This work has introduced the concept of assigning a probability of membership for a given star to a dSph based on the prior knowledge of the velocity profiles of our expected contaminant populations, $P_{vel}$, a technique that has not previously been used in the study of M31 dSphs. To test that this is not biasing our results, we can simply remove this term from Eqn. \[eqn:probtotal\], and follow the technique of T12, where they use only $P_{CMD}$ and $P_{dist}$ terms and then cut all stars with $P_{i}<0.1$ and velocities that lie greater than $3\sigma$ from the mean of this sample from their final analysis. We find that both techniques produce very similar results, however our algorithm is more robust in regimes where the systemic velocity of the dSph is close to that of the MW, and in dSphs where our number of probable member stars is low ($N_*<10$).
To test the effect of low S:N data on our calculations of $v_r$ and $\sigma_v$, we use our datasets for And XXI, XXIII and XXV, all of which have $\geq25$ associated members. For each dataset, we apply a series of cuts to the sample based on S:N (at levels of S:N$>2,3,4$ and 5Å) and rerun our algorithm. In all cases we find that the derived probabilities do not significantly differ when the low S:N data are included, justifying our inclusion of all stars for which velocities are calculated by our pipeline.
We also test the ability of our algorithm to measure $v_r$ and $\sigma_v$ in the small $N_*$ regime. For some of our datasets, we are only able to identify a handful of stars as probable members. In theory, one can calculate velocity dispersions accurately from only 3 stars if one is confident of ones measurement uncertainties, as is demonstrated by @aaronson83 measurement of the velocity dispersion for Draco from only 3 stars, which remains consistent with modern day measurements from significantly larger datasets [@walker09b]. We can test if our results are similarly robust using our larger datasets (such as And XXI, XXIII and XXV) by randomly selecting 4, 6, 8, 10, 15, 20 and 25 stars from these datasets and rerunning our algorithm to determine $v_r$ and $\sigma_v$ from these subsets. We repeat this exercise 1000 times for each sample size, and examine the mean and standard deviations for the computed quantities. We find that on average, for all sample sizes, our routine measures systemic velocities and velocity dispersions that are entirely consistent with those measured from the full sample, with a spread that is very comparable to typical errors produced by our ML routine in these low $N_*$ regimes. As such, we conclude that our technique is able to place sensible limits on these values, even when dealing with as few as four member stars.
Finally, as the individual positions, velocities and velocity uncertainties for all the stars analyzed in T12 are publicly available (with the data from the non-member stars having been kindly passed on to us by the SPLASH team), we can check that our algorithm is able to reproduce the values they measure for their M13 dSph sample. We find that, in all cases, we calculate systemic velocities and velocity dispersions that agree with their measured values to well within their $1\sigma$ uncertainties.
These tests demonstrate that our method is robust enough to accurately determine the global kinematic properties of M31 dSphs across a wide range of sample sizes and data quality. We therefore proceed to apply it to the datasets of all of the dSphs for which our group has acquired Keck II DEIMOS observations to date.
{width="0.95\hsize"}
The kinematics of M31 dSphs {#sect:results2}
===========================
With this vast dataset of dSph kinematics now in hand, we can begin to statistically probe their structures more fully. In Fig. \[fig:vels\] we display a summary of the velocities and positions of 26 of the 28 dSphs for which kinematic data are available, where the values are taken from this work, T12 and @kalirai10. In the following sections, we will discuss the individual stellar kinematics, masses and chemistries of the dSphs analyzed within this work.
Andromeda XVII {#sect:and17}
--------------
And XVII was discovered by @irwin08, and it is located at a projected distance of $\sim40$ kpc to the North West of Andromeda. A detailed study of deep imaging obtained with the Large Binocular Camera on the LBT was also performed by @brasseur11a, and throughout, we use the structural properties as determined from this work. It is a faint, compact galaxy ($M_V=-8.61, r_{\rm half}=1.24'$ or 262$^{+53}_{-46}$ pc). In the left panel of Fig. \[fig:And17\] we display the PAndAS color magnitude diagram for And XVII. Over-plotted we show the observed DEIMOS stars, color-coded by their probabilities of membership. The open symbols represent stars for which $P_i<10^{-6}$. We employ this cut solely to make clarify which stars have the highest probability of belonging to And XVII. In the right hand panel, we display the basic kinematic information for And XVII. In the top panel of this subplot, we show a velocity histogram for all stars observed within the LRIS mask, and stars with $P_{i}>10^{-6}$ are highlighted with a filled red histogram. The centre panel shows the velocities as a function of distance from the centre of And XVII (and the red dashed lines indicate $1,2,3$ and $4\times r_{\rm half}$), and the lower panel shows the photometric metallicities for all stars, as determined using @dart08 CFHT isochrones. Again, all points are color-coded by their probability of membership. Finally, the two lower panels show the resulting, one dimensional, probability weighted, marginalized maximum likelihood distributions for $v_r$ and $\sigma_v$ for this data set. From the kinematics presented in Fig. \[fig:And17\], which represent the first spectroscopic observations of this object, we see the signature of the dwarf galaxy as a cold spike at $v_r\sim-250\kms$. From the lower panels of this figure and the accompanying CMD we see that there is a cluster of 7 stars sitting within this spike that are centrally concentrated and are consistent with the RGB of the dwarf itself, leading us to believe that our algorithm has cleanly detected the signature of the galaxy. Interestingly, we also see 3 stars that, kinematically, are indistinguishable from the stars that have been dubbed as probable members in our analysis. However, they all sit at large distances from the centre of And XVII, equivalent to greater than 6 times the half-light radius of the dwarf, and hence the routine has classified them as likely members of the M31 halo rather than And XVII members. But, given their tight correlation in velocity to the systemic velocity of And XVII the possibility exists that these are extra-tidal stars of And XVII. No sign of extra-tidal features were cited in either the discovery paper of And XVII or the LBT followup, but given its position in the north M31 halo where contamination from the MW becomes increasingly problematic, and its relatively low luminosity ($M_V=-8.5$), such features may be difficult to see within the imaging. However, at present they are considered unlikely members by our routine, and do not factor into our calculation of global kinematic properties for this object. We find $v_r=-251.6^{+1.8}_{-2.0}\kms$ and $\sigma_v=2.9^{+2.2}_{-1.9}\kms$.
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Andromeda XVIII {#sect:and18}
---------------
Andromeda XVIII (And XVIII) was detected by @mcconnachie08 in the PAndAS CFHT maps. Located at a projected distance of $\sim110$ kpc to the North-West of M31, it is one of the most distant of its satellites, sitting $\sim600$ kpc behind the galaxy, making spectroscopic observations of its individual RGB stars taxing, as they are all relatively faint ($i\gta22.2$). Thus, from our 1 hour DEIMOS observation, we were only able to confirm 4 stars as probable members (see Fig. \[fig:And18\]). We determine the global systemic velocity to be $v_r=-346.8\pm2.0\kms$, and we are unable to resolve a velocity dispersion, finding $\sigma_v=0.0^{+2.7}\kms$ where the upper bound is determined from the formal $1\sigma$ confidence interval produced by our maximum likelihood analysis. This suggests that the 4 stars we are able to confirm as members do not adequately sample the underlying velocity profile. The systemic velocity we measure is different to that presented in T12 of $v_r=-332.1\pm2.7$ at a level of 3.4$\sigma$. Our 1$\sigma$ limit of 2.7$\kms$ is also at odds with the dispersion determined by T12 ($\sigma_v=9.7\pm2.3$). They were able to measure velocities for significantly more probable member stars (22 vs. 4) owing to their longer integration of 3 hours. The faintness of our targets and shorter exposure time could mean that strong sky absorption lines have systematically skewed our velocity measurements for these stars, and this could explain the discrepancy of our measurements with respect to those of T12. To check against this, we again perform weighted cross-correlations to each of the 3 Ca II lines individually. We find that the values we obtain, and their average, are fully consistent with that derived from the technique described in § \[sect:specobs\], differing by less than $3\kms$ from those values. The true systemic velocity of And XVIII therefore remains unclear. However, given their larger sample size, the T12 systemic properties are more statistically robust than those we present here.
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Andromeda XIX {#sect:and19}
-------------
Andromeda XIX (And XIX) was first reported in @mcconnachie08, and is a relatively bright, very extended ($M_V=-9.3$, $r_{\rm half}=1.5$ kpc) dSph, located at a projected distance of $\sim180$ kpc to the south west of M31. Its unusual morphology, very low surface brightness $\Sigma_v=30.2$mag/arcsec$^2$, and evidence in the photometry for a possible link to the major axis substructure reported in @ibata07 caused @mcconnachie08 to question whether And XIX was truly a dynamically relaxed system, or whether it had experienced a significant tidal interaction. Here, we present the first spectroscopic observations of the And XIX satellite in Fig. \[fig:And19\] from two DEIMOS masks placed at different position angles. These data allow us to comment on its dark matter content, and on the likelihood of a tidal origin for its unique structure. We identify 27 stars where $P_i>10^{-6}$ within the system. These measurements were made increasingly challenging as the systemic velocity we measure is $v_r=-111.6^{+1.6}_{-1.4}\kms$, placing it within the regime of Galactic contamination. However, we are confident that our algorithm is robust to this unfortunate location of And XIX in velocity space (see discussion in § \[sect:test\] and Appendix A). As a further check that none of the stars we define as probable members are actually foreground contaminants, we measure the equivalent widths of the Na I doublet lines (located at $\sim8100$Å). These gravity-sensitive absorption lines are typically significantly stronger in foreground dwarf stars than M31 RGB stars, although there is some overlap between the two populations. For the stars tagged as probable members by our algorithm, we find no evidence of strong absorption in the region of the Na I doublet, indicating that we are not selecting foreground stars as members. We measure a relatively cold velocity dispersion for this object of $\sigma_v=4.7^{+1.6}_{-1.4}\kms$, which is surprising given the radial extent of this galaxy. This result will be discussed further in § \[sect:mass\], and in a follow-up paper (Collins et al. in prep).
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Andromeda XX {#sect:and20}
------------
And XX was the third of three dSphs discovered by @mcconnachie08, and is notable for being one of the faintest dSph companions detected surrounding Andromeda thus far. With $M_V=-6.3$ and $r_{\rm half}=114^{+31}_{-12}$ pc, it is a challenging object to study spectroscopically as there are very few stars available to target on its RGB, as shown in the top left subplot in Fig. \[fig:And20\]. As a result, our algorithm is only able to find 4 stars for which $P_i>10^{-6}$. These are found to cluster around $v_r=-456.2^{+3.1}_{-3.6}\kms$, with a dispersion of $\sigma_v=7.1^{+3.9}_{-2.5}\kms$. Despite the low number of stars, we are confident in this detection, as the systemic velocity places it in the outer wings of the velocity profile of the M31 halo. And XX is also located at a large projected distance from M31 of $\sim130$ kpc, where we expect the density of the M31 halo to be very low. As such, seeing 4 halo stars so tightly correlated in velocity in the wings of the halo velocity profile within such a small area of the sky (all stars are within 1 arcmin of the centre of And XX) is highly unlikely. We caution the reader that, while we are confident that our algorithm is able to measure velocity dispersions for sample sizes as small as 4 stars, as we are not probing the full velocity profile of this object this measurement ideally needs to be confirmed with larger numbers of member stars.
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Andromeda XXI {#sect:and21}
-------------
Andromeda XXI (And XXI) was identified within the PAndAS imaging maps by @martin09. It is a relatively bright dSph ($M_V=-9.9$), located at a projected distance of $\sim150$ kpc from M31, and it has a half-light radius of $r_{\rm half}=842\pm77$ pc. We present our spectroscopic observations for this object in Fig. \[fig:And21\], and in the top right subplot, we can clearly see the signature of And XXI as a cold spike in velocity with 32 probable member stars, located at $v_r=-362.5\pm0.9\kms$, with a curiously low velocity dispersion of only $\sigma_v=4.5^{+1.2}_{-1.0}\kms$. These results are completely consistent with those of T12, where they measured $v_r=-361.4\pm5.8\kms$ and $\sigma_v=7.2\pm5.5\kms$. As their sample contained only 6 likely members compared with the 29 we identify here, ours constitute a more statistically robust measurement of the global kinematics for this object than those presented in T12.
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Andromeda XXII {#sect:and22}
--------------
Andromeda XXII (And XXII) was identified within the PAndAS imaging maps by @martin09, and is a relatively faint dSph, with $M_V=-6.5$. Its physical position in the halo, located at a distance of 224 kpc in projection from M31, but only 42 kpc in projection from M33, led the authors to postulate that it could be the first known dSph satellite of M33. Subsequent work analysing the kinematics of And XXII by T12 measured a systemic velocity for And XXII of $-126.8\pm3.1\kms$ from 7 stars, more compatible with the systemic velocity of M33 ($-178\kms$, @mateo98) than that of M31. Another study by @chapman12 using the same data and the same method we present here concluded the same, measuring a systemic velocity for the satellite of $-129.8\pm2.0$ from 12 probable member stars, consistent with the T12 value. @chapman12 also compare the position and kinematics of And XXII with a suite of $N-$body simulations of the M31-M33 system, concluding that And XXII was a probable M33 satellite.
In Fig. \[fig:And22\], we present the same data as analyzed by @chapman12 for completeness. The velocity dispersion of And XXII is just resolved at $\sigma_v=2.8^{+1.9}_{-1.4}\kms$, completely consistent with the value of $\sigma_v=3.5^{+4.2}_{-2.5}\kms$ from T12. As our values are calculated from a 50% greater sample size, we posit that they are the more statistically robust.
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Andromeda XXIII {#sect:and23}
---------------
Andromeda XXIII (And XXIII) was the first of five M31 dSphs identified by @richardson11. Located at a projected distance of $\sim130$ kpc to the east of Andromeda, it is relatively bright, with $M_V=-10.2$, and extended, with $r_{\rm half}=1001^{+53}_{-52}$ pc. Our routine clearly detects a strong cold kinematic peak for And XXIII located around $-230\kms$ and calculates a systemic velocity of $v_r=-237.7\pm1.2\kms$, and a velocity dispersion of $\sigma_v=7.1\pm1.0\kms$ from 40 probable member stars, as show in Fig. \[fig:And23\]. This small, positive velocity relative to M31, combined with its large projected distance from the host suggests that And XXIII is not far past the apocentre of its orbit, heading back towards M31.
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Andromeda XXIV {#sect:and24}
--------------
And XXIV was also first reported in @richardson11. Relatively faint and compact ($M_V=-7.6$, $r_{\rm half}=548^{+31}_{-37}$ pc), spatially it is located $\sim200\kpc$ from M31, along its northern major axis. And XXIV was observed on two separate occasions as detailed in Table \[tab:specobs\]. For the first mask, there was an error in target selection, and as a result, only one star that lay on the RGB of And XXIV was observed. The second mask was observed in May 2011, however owing to target visibility, only a short integration of 45 minutes was obtained, which resulted in higher velocity uncertainties than typically expected ($\sim8\kms$ vs. $\sim5\kms$). For this reason, we have only included stars from this mask with $i<22.0$, as the spectra for fainter stars were too noisy to determine reliable velocities from. The systemic velocity of And XXIV also unsatisfactorily coincides with that of the MW halo contamination, as can be seen in Fig. \[fig:and24\]. As for And XIX, we check the strength of the Na I doublet of all the stars classified as potential members for And XXIV, and find no significant absorption, making them unlikely foreground contaminants. But, given the lower quality of this dataset, this check is far from perfect, and it is possible that we have included contaminants from the MW within our sample. Owing to the larger velocity uncertainties of the And XXIV dataset, and the overlap of And XXIV with the MW, the determination of probability of membership for stars within this dataset is based largely on their position in the color magnitude diagram (e.g., location on the RGB) and their distance from the centre of And XXIV.
When we run our machinery over the data acquired from both masks, we identify only 3 probable members and determine a systemic velocity of $v_r=-128.2\pm5.2\kms$ and we resolve a velocity dispersion of $\sigma_v=0.0^{+7.3}\kms$. Given the lower quality of this dataset in comparison to the remainder of those we present in this work, and the overlap of And XXIV in velocity space with contamination from the MW, a robust kinematic detection and characterisation of this galaxy is made incredibly challenging. As such, we present these results as a tentative identification of And XXIV, and do not include its measured properties in the remainder of our analysis. Further kinematic follow up of And XXIV is required to understand this system. We present the velocities of all bright stars for which velocity measurements were possible in Table \[tab:members\] so that they may be helpful for any future kinematic analysis of this system.
[lccccccccc]{} And V & 9 & 1:10:2.38 &47:37:48.5 & 23.640 & 22.402 & -371.270 & 5.080 & 1.700 & 0.014\
And V & 12 & 1:10:5.540 & 47:36:41.6 & 23.100 & 21.530 & -406.500 & 3.600& 2.000 & 0.111\
...&...&...&...&...&...&...&...&...&...\
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Andromeda XXV {#sect:and25}
-------------
And XXV was identified in @richardson11 as a relatively bright ($M_V=-9.7$), extended ($r_{\rm half}=642^{+47}_{-74}$ pc) dwarf spheroidal, located at a projected distance of $\sim90$ kpc to the north west of M31. As with And XXIII, we present here a kinematic analysis of And XXV. The results are displayed in Fig. \[fig:And25\]. We see that the systemic velocity of And XXV ($v_r=-107.8\pm1.0\kms$), places it in the regime of the Galactic foreground. However, given the strong over-density of stars with this velocity relative to the expected contribution of MW stars, we are confident that our routine has detected 25 likely members for this object. We check the strength of the Na I doublet in these likely members, and find no significant absorption, making them unlikely foreground contaminants. As for And XIX and XXI, we find that And XXV has a curiously low velocity dispersion for its size, with $\sigma_v=3.0^{+1.2}_{-1.1}\kms$. We discuss the significance of this further in § \[sect:mass\] and Collins et al (2013, in prep).
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Andromeda XXVI {#sect:and26}
--------------
And XXVI is a relatively faint ($M_V=-7.1$) dSph with $r_{\rm half}=219^{+67}_{-52}$ pc, also first reported in @richardson11. Its low luminosity makes observing large numbers of member stars difficult, owing to the paucity of viable targets on the RGB that can be observed with DEIMOS. As a result, our routine has identified only 6 stars as potential members, highlighted in Fig. \[fig:And26\]. The dwarf has a systemic velocity of $v_r=-261.6^{+3.0}_{-2.8}\kms$, and a fairly typical velocity dispersion of $\sigma_v=8.6^{+2.8}_{-2.2}\kms$. As with And XX, while we believe our routine can robustly measure the velocity dispersions of systems with only 6 confirmed members, to be truly confident of this value, follow up of And XXVI to increase the number of likely members is required.
In @conn12, from an analysis of the photometry of And XXVI, they determined a distance modulus to the object of $(m-M)_0=24.39^{+0.55}_{-0.53}$ from a Markov-Chain-Monte-Carlo analysis of the PAndAS photometry of And XXVI. This value corresponds to an $i-$band magnitude for the TRGB of And XXVI of $m_{i,0}=21.1^{+0.55}_{-0.53}$. Our CMDs for the dwarfs are not extinction corrected, but using the extinction values from @richardson11 of $E(B-V)=0.110$ [@schlegel98], this would correspond to an $i-$band magnitude of $m_{i,TRGB}=21.3^{+0.55}_{-0.53}$. Three targets were observed with magnitudes and colors that should be consistent with their belonging to And XXVI. However. we find that all these objects have velocities that are consistent with being Galactic foreground contaminants. Given the position of And XXVI in the northern M31 halo, where contamination from the MW increases, this is not unexpected. The brightest star we observe that is likely associated with And XXVI has $m_i=21.9$ ($m_{i,0}=21.7$). Assuming that this brightest confirmed member star of And XXVI sits at the RGB tip, the distance estimate becomes 1.1 Mpc. While this value is higher than that of @conn11, it is still within their upper distance estimate. Additionally, as not every star brighter than the member with $m_{i,0}=21.7$, with colors consistent with the And XXVI RGB was observed, this value merely represents an upper limit, on the distance to And XXVI and highlights the difficulty of calculating distances to these faint galaxies where RGB stars are sparse.
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Andromeda XXVII {#sect:and27}
---------------
Andromeda XXVII (And XXVII) is a somewhat unusual object as it is currently undergoing tidal disruption, spreading its constituent stars into a large stellar stream, named the northwestern arc, discovered in the PAndAS survey by @richardson11. As such, it is unlikely to be in virial equilibrium, if it remains bound at all.
When determining the kinematics of And XXVII, we find the results somewhat unsatisfactory. Our routine determines $v_r=-539.6^{+4.7}_{-4.5}\kms$ and $\sigma_v=14.8^{+4.3}_{-3.1}\kms$ from 11 stars. However, from an inspection of Fig. \[fig:And27\], we see that there is significant substructure around $v_{hel}\sim-500\kms$, much of which is considered to be unassociated with And XXVII in this analysis as it does not fall within a cold, well-defined Gaussian velocity peak. Given the disrupting nature of And XXVII, it is likely that a different analysis is required for this object, and we shall discuss this further in a future analysis, where the kinematics of the northwestern arc itself are also addressed. From this first pass however, it would appear that And XXVII may no longer be a gravitationally bound system.
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Andromeda XXVIII {#sect:and28}
----------------
And XXVIII was recently discovered in the 8th data release of the SDSS survey [@slater11]. It has $M_v=-8.5$ and $r_{\rm half}=210^{+60}_{-50}$ pc. It is also potentially one of Andromeda’s most distant satellites, with a host-satellite projected separation of $365^{+17}_{-1}$ kpc. The And XXVIII satellite is not covered by the PAndAS footprint, so we must instead use the original SDSS photometry for our analysis. A CMD with the SDSS $i-$band and $r-i$ colors for And XXVIII is shown in Fig. \[fig:And28\], where all targets brighter that $i\sim23.5$ within $3r_{\rm half}$ are shown. The photometry here do not show an RGB that is as convincing as those from the PAndAS survey, so to guide the eye, we also overplot an isochrone from @dart08 with a metallicity of $\feh=-2.0$, corrected for the distance of And XXVIII as reported in @slater11.
In a recent paper, @tollerud13 discussed the kinematics of this object as derived from 18 members stars. They find $v_r=-328.0\pm2.3\kms$ and $\sigma_v=8.1\pm1.8\kms$ from their full sample. They then remove two stars that they categorize as outliers based on their distance from the centre of And XXVIII, which alters their measurements to $v_r=-331.1\pm1.8\kms$ and $\sigma_v=4.9\pm1.6\kms$. Analyzing our own DEIMOS dataset for this object, we find $v_r=-326.1\pm2.7\kms$ and $\sigma_v=6.6^{+2.9}_{-2.2}\kms$ based on 17 probable members. This is fully consistent with the results from the full sample in @tollerud13. However, the systemic velocity we measure is offset at a level of $\sim1\sigma$ from their final value (calculated after excluding 2 outliers). This offset is small, and is probably attributable to our differing methodologies for classifying stars as members. As we believe our method is more robust (as discussed in § \[sect:test\] and Appendix A), we will use our derived parameters for this object in the remainder of our analysis.
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And XXX/Cassiopeia II {#sect:and30}
---------------------
And XXX – also known as Cass II owing to its spatial location, overlapping the Cassiopeia constellation – is a recently discovered dSph from the PAndAS survey (Irwin et al. in prep). It has $M_v=-8.0$ and $r_{\rm
half}=267^{+23}_{-36}$ pc. Located to the north west of Andromeda, it sits within 60 kpc of the two close dwarf elliptical M31 companions, NGC 147 and NGC 185. With these 3 objects found so close together in physical space, it is tempting to suppose them a bound system within their own right, but this can only be borne out by comparing their kinematics.
Conspiring to confound us, we find that Cass II has kinematics that place it well within the regime of Galactic foreground, as can be seen in Fig. \[fig:Cass2\]. However, our analysis is able to detect the dSph as a cold spike consisting of 8 likely members. As for And XIX, we check the strength of the Na I doublet in these likely members, and find no significant absorption, making them unlikely foreground contaminants. We measure $v_r=-139.8^{+6.0}_{-6.6}\kms$, and a fairly typical velocity dispersion of $\sigma_v=11.8^{+7.7}_{-4.7}\kms$.
The systemic velocity of Cass II ($v_r=-139.8^{+6.0}_{-6.6}\kms$) puts it within $\sim50\kms$ of those of NGC 147 and NGC 185 ($v_r=-193\pm3\kms$ and $v_r=-210\pm7\kms$, @mateo98), lending further credence to the notion that these 3 systems are associated with one another. This will be discussed in more detail in Irwin et al. (2013, in prep).
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A note on previous work
-----------------------
Finally, we also use our new algorithm to reanalyze all our previously published M31 dSph datasets. These include And V, VI [@collins11b], XI, XII and XIII [@chapman07; @collins10]. Details of the results of this reanalysis can be found in Appendix B. In summary, we find that our algorithm measures systemic velocities and velocity dispersions that are fully consistent with our previous work. We present these results in Table \[tab:kprops\]. And V, XI, XII and XIII are also analyzed by T12, so we compare our findings with theirs. For And XI and XII, our results are based on two and four times the number of stars respectively, and as such, supercede those presented in T12. In the case of And V and XIII, the T12 measurements are based sample sizes with four times the number of stars as our datasets, making their findings more robust.
In previous studies by our group [@chapman05; @letarte09; @collins10], we also published kinematic analyses for three additional M31 dSphs; And IX, And XV and And XVI. In T12, it was noted that the values presented in these works for systemic velocities and velocity dispersions were not consistent with those measured in their analyses. We revisited these datasets in light of this discrepancy, to see if our new technique could resolve this issue. We found that these discrepancies remained. For And IX, we measure a systemic velocity of $v_r=-204.8\pm2.1\kms$ cf. $v_r=-209.4\pm2.5\kms$ in T12 and a velocity dispersion of $\sigma_v=2.0^{+2.7}_{-2.0}\kms$ cf. $\sigma_v=10.9\pm2.0\kms$. Not only are their measurements determined from 4 times the member stars that we possess, we also experienced problems with our radial velocity measurements for the stars observed with this mask, due to the use of the minislitlet approach pioneered by @ibata05. This setup resulted in poor sky subtraction for many of the science spectra, lowering the quality of our radial velocity measurements. As such, the T12 results supercede those of our previous work [@chapman05; @collins10].
For And XV and XVI, we measure a systemic velocities of $v_r=-354.6\pm4.9\kms$ and $v_r=-374.1\pm6.8\kms$ cf. $v_r=-323.0\pm1.4\kms$ and $v_r=-367.3\pm2.8\kms$ from T12. We also note offsets in our velocity dispersions for And XV and XVI, where we measure $\sigma_v=9.6^{+4.1}_{-2.6}\kms$ and $\sigma_v=17.3^{+6.4}_{-4.4}\kms$ cf. $\sigma_v=3.8\pm2.9\kms$. In this instance, the data for both And XV and XVI were taken in poor conditions, with variable seeing that averaged at $1.8^{\prime\prime}$ and patchy cirrus. These conditions significantly deteriorated the quality of our spectra, and made the measurement of reliable radial velocities extremely difficult. Again, this leads us to conclude that the measurements made in T12 supercede those presented by our group in @letarte09.
The masses and dark matter content of M31 dSphs
===============================================
Measuring the masses and mass-to-light ratios of our sample {#sect:mass}
-----------------------------------------------------------
As dSph galaxies are predominantly dispersion supported systems, we can use their internal velocity dispersions to measure masses for these systems, allowing us to infer how dark matter dominated they are. There are several methods in the literature for this (e.g., @illingworth76 [@richstone86]), but recent work by @walker09a has shown that the mass contained within the half-light radius ($M_{\rm half}$)of these objects can be reliably estimated using the following formula:
$$M_{\rm half}=\mu r_{\rm half}\sigma_{v,{\rm half}}^2$$
where $\mu=580\msun{\rm pc}^{-1}$km$^{-2}$s$^2$, $r_{\rm half}$ is the spherical half-light radius in pc and $\sigma_{v,{\rm half}}$ is the luminosity- averaged velocity dispersion. This mass estimator is independent of the (unknown) velocity anisotropy of the tracer population, however, it is sensitive to the embeddedness of the stellar component within the DM halo. Particularly, the mass tends to be slightly over-estimated the more embedded the stars are [@walker11], especially if the dark matter halo follows a cored density profile.
As numerous authors have shown that the velocity dispersion profiles of dSphs are constant with radius (e.g., @walker07 [@walker09b]), we assume our measured values of $\sigma_v$ are representative of the luminosity-averaged velocity dispersion ($\sigma_{v,{\rm half}}$) used by @walker09b. However, if it transpired that the velocity dispersion profiles of the Andromedean dSphs were not flat, but declined or increased with radius, this would no longer true. We see no evidence for this behaviour in our dataset, although low-number statistics means we are unable to completely rule out this possibility. We calculate this for all our observed dSphs (including those we reanalyzed from previous works, see Appendix B) using results from the Keck LRIS and DEIMOS dataset, and report their masses within $r_{\rm half}$ ($M_{\rm half}$) in Table \[tab:kprops\].
[lcccccc]{} And V & 2.0 & $-391.5\pm2.7$ & $12.2^{+2.5}_{-1.9}$ & $2.6^{+0.66}_{-0.56}$& $88.4^{+22.3}_{-18.9}$ & $-2.0\pm0.1$\
And VI & 2.5 & $-339.8\pm1.8$ & $12.4^{+1.5}_{-1.3}$ & $4.7\pm0.7$& $27.5^{+4.2}_{-3.9}$ & $-1.5\pm0.1$\
And XI & 2.5 & $-427.5^{+3.5}_{-3.4}$ & $7.6^{+4.0(*)}_{-2.8}$ & $0.53^{+0.28}_{-0.21}$& $216^{+115}_{-87}$ & $-1.8\pm0.1$\
And XII & 2.5 & $-557.1\pm1.7$ & $0.0^{+4.0}$ & $0.0^{+0.3}$ & $0.0^{+194}$ & $-2.2\pm0.2$\
And XIII & 2.5 & $-204.8\pm4.9$ & $0.0^{+8.1(*)}$ & $0.0^{+0.7}$ & $0.0^{+330}$ & $-1.7\pm0.3$\
And XVII & 2.5 & $-251.6^{+1.8}_{-2.0}$ & $2.9^{+2.2}_{-1.9}$ & $0.13^{+0.22}_{-0.13}$& $12^{+22}_{-12}$& $-1.7\pm0.2$\
And XVIII & 2.5 & $-346.8\pm2.0$ & $0.0^{+2.7}$ & $0.0^{+0.14}$ & $0^{+5}$&$-1.4\pm0.3$\
And XIX & 2.0 & $-111.6^{+1.6}_{-1.4}$ & $4.7^{+1.6}_{-1.4}$ &$1.9^{+0.65}_{-0.66}$& $84.3^{+37}_{-38}$ & $-1.8\pm0.3$\
And XX & 2.5 & $-456.2^{+3.1}_{-3.6}$ & $7.1^{+3.9(*)}_{-2.5}$ & $0.33^{+0.20}_{-0.12}$& $238.1^{+147.6}_{-90.2}$& $-2.2\pm0.4$\
And XXI & 5.0 & $-362.5\pm0.9$ & $4.5^{+1.2}_{-1.0}$ & $0.99^{+0.28}_{-0.24}$ & $25.4^{+9.4}_{-8.7}$& $-1.8\pm0.1$\
And XXII & 2.0 & $-129.8\pm2.0$ & $2.8^{+1.9}_{-1.4}$ & $0.11^{+0.08}_{-0.06}$& $76.4^{+58.4}_{-48.1}$& $-1.8\pm0.6$\
And XXIII & 4.0 & $-237.7\pm1.2$ & $7.1\pm1.0$ & $2.9\pm4.4$& $58.5\pm36.2$& $-2.2\pm0.3$\
And XXIV & 1.5 & $-128.2\pm5.2$ & $0.0^{+7.3(*)}$ & $0.4^{+0.7}_{-0.4}$ & $82^{+157}_{-82}$ & $-1.8\pm0.3$\
And XXV & 2.5 & $-107.8\pm1.0$ & $3.0^{+1.2}_{-1.1}$ & $0.34^{+0.14}_{-0.12}$& $10.3^{+7.0}_{-6.7}$ & $-1.9\pm0.1$\
And XXVI & 3.0 & $-261.6^{+3.0}_{-2.8}$ & $8.6^{+2.8(*)}_{-2.2}$ & $0.96^{+0.43}_{-0.34}$ & $325^{+243}_{-225}$ & $-1.8\pm0.5$\
And XXVII & 1.5 & $-539.6^{+4.7}_{-4.5}$ & $14.8^{+4.3}_{-3.1}$ & $8.3^{+2.8}_{-3.9}$& $1391^{+1039}_{-1128}$ & $-2.1\pm0.5$\
And XXVIII & 2.5 & $-326.2\pm2.7$ & $6.6^{+2.9}_{-2.1}$ & $0.53^{+0.28}_{-0.21}$& $51^{+30}_{-25}$ & $-2.1\pm0.3$\
And XXX (Cass II) & 2.0 & $-139.8^{+6.0}_{-6.6}$ & $11.8^{+7.7}_{-4.7}$ & $2.2^{+1.4}_{-0.9}$ & $308^{+269}_{-219}$ & $-1.7\pm0.4$\
From these masses, it is trivial to estimate the dynamical central mass-to-light ratios for the objects, $[M/L]_{\rm half}$. We list these values for each dSph in Table \[tab:kprops\], where the associated uncertainties also take into account those from the measured luminosities and distances to these dSphs [@mcconnachie12; @conn12], as well as those on the masses measured in this work.
Comparing the mass-to-light ratios of M31 and Milky Way dSphs
-------------------------------------------------------------
By combining our measurements of the kinematic of M31 dSphs in this work with those from T12 and @tollerud13, we find ourselves with a set of kinematic properties as measured for 27 of the 28 Andromeda dSphs (owing to the difficulties experienced with the And XXIV dataset, we do not include this object in our subsequent analysis). This near-complete sample allows us to fully compare the masses and mass-to-light ratios for the M31 satellite system with those measured in the Milky Way satellites. Before beginning this analysis, we compile Table \[tab:summary\] which presents the kinematics for the full M31 satellite system, which combines the results from this work, T12, @kalirai10, and @tollerud13. In cases where two measurements for a dSph exist, we use those that were calculated from larger numbers of likely members, as these are the more robust. We begin by comparing the mass-to-light ratios (which indicate the relative dark matter dominance of these objects) of the two populations as a function of luminosity. In Fig \[fig:ml\] we show these values for all MW (red triangles, with values taken from @walker09b), and M31 (blue circles) dSphs as a function of their luminosity. We can see that all these objects are clearly dark matter dominated, excluding And XII and And XVII where we are unable to resolve the mass with current datasets. We also see that they follow the trend of increasing $[M/L]_{\rm half}$ with decreasing luminosity, as is seen in their MW counterparts.
The one tentative exception to this is the And XXV dSph. From our dataset, we measure a value of $[M/L]_{\rm half}=10.3^{+7.0}_{-6.7}$ for this object, making it consistent with a stellar population with no dark matter within its $1\sigma$ uncertainties. This result is surprising and would be of enormous importance if confirmed with a larger dataset than our catalog of 26 likely members as it would be the first dSph to be observed with a negligible dark matter component. And XXV is also one of the members of the recently discovered thin plane of satellites in Andromeda [@ibata13], and so the presence or absence of dark matter in And XXV might tell us more about the origins of this plane which are currently poorly understood.
![Dynamical mass-to-light ratio within the half-light radius, $[M/L]_{\rm half}$, as a function of half-light radius for all M31 (blue circles), MW (red triangles) and isolated dSphs (cyan squares).[]{data-label="fig:ml"}](fig19.eps){width="0.95\hsize"}
Comparing the masses of M31 and Milky Way dSphs
-----------------------------------------------
Finally, we discuss how the masses for the full sample of Andromeda dSphs for which kinematic data are available compare with those of the MW dSphs. For the M31 dSph population, we again use our compilation of kinematic properties assembled in Table \[tab:summary\]. We plot the velocity dispersions, mass within the half-light radius, and central densities for all M31 (blue circles) and MW (red triangles, @walker09b [@aden09; @koposov11; @simon11]) dSphs as a function of radius. We then overplot the best-fit NFW and cored mass profiles for the MW, taken from @walker09b. In general, we see that the M31 and MW objects are similarly consistent with these profiles, an agreement that was also noted by T12. However, there are 3 objects which are clear outliers to these relations. These are And XIX, XXI and XXV, with velocity dispersions of $\sigma_v=4.7^{+1.6}_{-1.4}\kms,
\sigma_v=4.5^{+1.2}_{-1.0}\kms$ and $\sigma_v=3.0^{+1.2}_{-1.1}\kms$, as derived in this work. Given their half-light radii ($r_{\rm
half}=$1481$^{+62}_{-268}$ pc, $r_{\rm half}=$842$\pm77$ pc and $r_{\rm
half}=$642$^{+47}_{-74}$ pc), one would expect them to have dispersions of closer to $9\kms$ in order to be consistent with the MW mass profile. As they stand, these three objects are outliers at a statistical significance of $2.5\sigma$, $3.0\sigma$ and $3.4\sigma$ (calculated directly from their likelihood distributions as presented in Figs. \[fig:And19\], \[fig:And21\] and \[fig:And25\]). Similarly, in T12 they noted that And XXII and And XIV were outliers in the same respect as And XIX, XXI and XXV, albeit at a lower significance. These difference can also be observed in terms of the enclosed masses and densities within $r_{\rm half}$.
{width="0.9\hsize"}
In @collins11b we argued that the low velocity dispersion seen in some Andromeda dwarfs were a result of tidal forces exerted on their halos by the host over the course of their evolution, and that this effect was predominantly seen in dSphs where their half-light radii were more extended for a given luminosity than expected, such is the case our three outliers, And XIX, XXI and XXV. This result therefore adds weight to the trend presented in that work. A number of recent works trying to account for the lower than predicted central masses of dSph galaxies within the Local Group also support this notion. For example, @penarrubia10 demonstrated that the presence of a massive stellar disk in the host galaxy (such as those of the MW and M31) can significantly reduce the total masses of its associated satellites. In addition, recent, papers by @zolotov12 and @brooks12, where the effect of baryons within dark matter only simulations was measured also find that tidal forces exerted by host galaxies where a massive disk is present will serve to reduce the masses of its satellite population at a far greater rate than hosts without baryons. And XIX, XXI and XXV may thus represent a population of dSph satellites whose orbital histories about M31 have resulted in substantial fractions of their central mass being removed by tides. It should be noted, however, that tides not only reduce the central masses and densities of dSph halos, they also reduce the spatial size of the luminous component [@penarrubia08b; @penarrubia10], albeit at a slower rate. The tidal scenario is therefore slightly difficult to reconcile with these outlying M31 dSphs having the largest sizes, unless they were both more massive and spatially larger in the past.
Other recent theoretical works have also shown that the removal of baryons from the very centres of dark matter halos by baryonic feedback (from star formation and supernovae, for example) can also help to lower the central masses and densities of satellite galaxies (e.g., @pontzen12 [@zolotov12; @brooks12]). For this method to work effectively, however, very large ‘blow outs’ of gas are required, of the order $\sim10^8-10^9\msun$, equivalent to $\sim40000$ SNe. This would require a minimum initial satellite luminosity of $M_V<-12$ [@zolotov12; @garrison13], significantly brighter than the current luminosities of our outliers ($M_V\sim-10$). Therefore, if feedback has indeed played a role in the shaping of the dark matter halos of And XIX, XXI and XXV, one assumes it would have to have operated in tandem with tidal stripping. We will discuss the implications and interpretation of these result further in a companion paper (Collins et al. in prep).
Metallicities {#sect:metals}
=============
Our observational setup was such that we cover the calcium triplet region (Ca II) of all our observed stars. This strong, absorption feature is useful not only for calculating velocities for each star, but also metallicities. For RGB stars, such as we have observed, there is a well known relation between the equivalent widths (EWs) of the Ca II lines, and the iron abundance, $\feh$, of the object. The calibration between these two values has been studied and tested by numerous authors, using both globular clusters and dSphs, and is valid down to metallicities as low as $\feh\sim-4$ (see e.g., @battaglia08 [@starkenburg10]). Following the @starkenburg10 method, which extends the sensitivity of this method down to as low as $\feh\sim-4$, we fit Gaussian functions to the three Ca II peaks to estimate their equivalent widths (EWs), and calculate \[Fe/H\] using equation \[eqn:cat\]:
$$\begin{aligned}
\feh=-2.87+0.195(V_{RGB}-V_{HB})+0.48\Sigma \rm{Ca}\\-0.913\Sigma
\rm{Ca}^{-1.5}+0.0155\Sigma \rm{Ca}(V_{RGB}-V_{HB})
\end{aligned}
\label{eqn:cat}$$
where $\Sigma$Ca=0.5EW$_{8498}$+EW$_{8542}$+0.6EW$_{8662}$, $V_{RGB}$ is the magnitude (or, if using a composite spectrum, the average, S:N weighted magnitude) of the RGB star, and $V_{HB}$ is the mean $V$-magnitude of the horizontal branch (HB). Using $V_{HB}-V_{RGB}$ removes any strong dependence on distance or reddening in the calculated value of \[Fe/H\], and gives the Ca II line strength at the level of the HB. For M31, we set this value to be $V_{HB}=$25.17 [@holland96][^1]. As the dSphs do not all sit at the same distance as M31, assuming this introduces a small error into our calculations, but it is at a far lower significance than the dominant uncertainty introduced by the noise within the spectra themselves. For individual stars, these measurements carry large uncertainties ($\gta0.4$ dex), but these are significantly reduced when stacking the spectra into a composite in order to measure an average metallicity for a given population.
Uncertainties on the individual measurements of \[Fe/H\] from our stellar spectra are typically large ($\ge0.5$ dex), so for a more robust determination of the average metallicities we co-add the spectra for each dSph (weighting by the S:N of each individual stellar spectrum, which is required to be a minimum of 2.5Å$^{-1}$) and measure the resulting EWs. In a few cases, not all 3 Ca II lines are well resolved. For And V, IX, XVII, XVIII XXVI and XXVIII, the third Ca II line is significantly affected by skylines, whilst for And XXIV, the first Ca II line is distorted. In the case of And XIII, only the second Ca II line appears well resolved. In these cases, we neglect the affected lines in our estimate of \[Fe/H\], and derive reduced equivalent widths from the unaffected lines. Where the third line is affected, this gives $\Sigma$Ca=1.5EW$_{8498}$+EW$_{8542}$. Where the second line is affected, we find $\Sigma$Ca=EW$_{8542}$+EW$_{8662}$. Finally, where only the second line seems reliable we use $\Sigma$Ca=1.7EW$_{8542}$. These coefficients are derived empirically from high S:N spectra where the absolute values of \[Fe/H\] are well known. We test these variations of $\Sigma$Ca by applying them to our high S:N co-added spectra where all three lines are well resolved, such as And XXI and XXV, and we find that all three formulae produce consistent values of \[Fe/H\]. The composite spectra for each satellite are shown in Figs. \[fig:sumspec\] and \[fig:sumspec2\]. In all cases, we find that our results are consistent with photometric metallicities derived in previous works.
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"}
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"}
In the MW, it has been observed that the average metallicities of the dSph population decrease with decreasing luminosity (e.g., @kirby08 [@kirby11]). In Fig. \[fig:mvfeh\], we plot the spectroscopic metallicities of the M31 dSphs (blue dSphs) as a function of absolute magnitude. We plot the MW dSphs as red triangles (@martin07 [@kirby08; @kirby11; @belokurov09; @koch09]). We also include those M31 dSphs for which only photometric measurements of \[Fe/H\] are available (And I, II, III, VII and XIV, @kalirai10 [@tollerud12]), and these are highlighted as encircled blue points. The dashed line represents the best-fit to the MW dSph population from @kirby11. The three relatively metal rich (\[Fe/H\]$\sim-1.5$ to $-2.0$), faint ($L\sim1000\lsun$) MW points are the three ultra faint dSphs Willman I, Boötes II and Segue 2, and these three were not included in the @kirby11 analysis, where the best fit MW relation was determined. We see that the metallicities for a given luminosity in the M31 dSphs also loosely define a relationship of decreasing metallicity with decreasing luminosity, and they agree with that defined by their MW counterparts within their associated uncertainties. However, it is also noteworthy that for dSphs with $L<10^6\lsun$, the Andromeda satellites are also consistent with having a constant metallicity of $\sim-1.8$. The same levelling off of average metallicity at lower luminosities was noted by @mcconnachie12, where they note that this break occurs at the same luminosity as a break in the luminosity-surface brightness relation for faint galaxies. As such, it could imply that the denisty of baryons in these systems, rather than the total number of baryons, could be the most important facor in determining their chemical evolution. The error bars we present here are still significant, so it is hard to fully interpret this result, but the hint of a metallicity floor in these lower luminosity systems is intriguing.
In Fig. \[fig:mvfeh\], we highlight the positions of our three kinematic outliers, And XIX, XXI and XXV, and we see that they fall almost exactly on the MW relation. In this figure, systems that have experienced extreme tidal stripping would move horizontally to the left, as their luminosity would gradually decrease as stars are stripped, but their chemistry would remain unaffected. One would expect to see such behaviour only after the stellar component began to be removed in earnest, after the majority of the dark matter halo had been removed. If their central densities were lowered by some active feedback mechanism, such as SNII explosions (e.g., @zolotov12), one would expect the objects to become more enriched and perhaps brighter, moving them up and to the right, potentially allowing them to remain on the MW relation. To confirm that this was the case for And XIX, XXI and XXV, we would require more information on the abundances of these objects and their star formation histories, which we do not currently possess.
![Spectroscopically derived \[Fe/H\] vs. luminosity for all MW (red triangles, taken from @kirby11, with additional measurements taken from @martin07 [@belokurov09; @koch09]) and M31 dSphs (blue circles, this work). The solid line represents the best fit relationship between these two parameters as taken from @kirby11. The dashed lines represent the $1\sigma$ scatter about this relationship. We see that the M31 dSphs follow this relationship very well within their associated uncertainties. As we discuss in § \[sect:metals\], those galaxies with $L<10^6\lsun$ are also consistent with having a constant metallicity, which could indicate a metallicity floor in these fainter systems.[]{data-label="fig:mvfeh"}](fig22.eps){width="0.95\hsize"}
[lccccccccccc]{} And I& -11.8 & 80 & -1.45$\pm$0.37 & N/A &-376.3$\pm2.2$ & 10.2$\pm1.9$&$656^{+68}_{-67}$ & 727$^{+18}_{-17}$ &(1),(2),(3)\
And II& -12.6 & 95 &-1.64$\pm$0.34 & N/A &-193.6$\pm1.0$ & 7.3$\pm$ 0.8& 1136$\pm46$ & 630$\pm15$ &(1),(3),(4)\
And III& -10.2 & 43 & -1.78$\pm$0.27 & N/A &-344.3$\pm1.7$ & 9.3$\pm1.4$& $463^{+44}_{-45}$ & 723$^{+18}_{-24}$&(1),(2),(3)\
And V& -9.6 & 85 & -1.6$\pm0.3$ & -1.8$\pm$ 0.2& -397.3$\pm1.5^{(b)}$ & 10.5$\pm1.1^{(b)}$& $302\pm44$ & 742$^{+21}_{-22}$ &(2),(3),(5)\
And VI& -11.5 & 38 & -1.3$\pm0.3$ & -1.5$\pm$ 0.3& -339.8$\pm1.9$ & 12.4$^{+1.5}_{-1.3}$& $524\pm49$ & 783$\pm28$ & (3),(5),(6)\
And VII& -13.3 & 18 & -1.4$\pm$0.3 & N/A &-307.2$\pm1.3$ & 13.0$\pm$ 1.0& $776\pm42$ & $762\pm35$ &(1),(2),(3)\
And IX& -8.1 & 32 & -2.2$\pm0.2$ & -1.9$\pm$ 0.6&-209.4$\pm2.5^{(b)}$ & 10.9$\pm2.0^{(b)}$& $436^{+68}_{-24}$ &$600^{+91}_{-23}$ & (1),(2),(3)\
And X& -8.1 & 22 &-1.93$\pm$0.48 & N/A & -164.1$\pm1.7$ & 6.4$\pm1.4$& $253^{+21}_{-65}$ &$670^{+24}_{-39}$ & (1),(2),(3)\
And XI& -6.9 & 5 &-2.0$\pm0.2$ & -2.0$\pm$ 0.3 & -427.5$^{+3.4}_{-3.5}$ & 7.6$^{+4.0}_{-2.8}$&$158^{+9}_{-23}$ & $763^{+29}_{-106}$& (3),(6),(7)\
And XII&-6.4 & 8 & -1.9$\pm0.2$ & -2.0$\pm$ 0.3 & -557.1$\pm1.7$ & 0.0$^{+4.0}$&$324^{+56}_{-72}$ & $928^{+40}_{-136}$ & (3),(6),(7)\
And XIII& -6.7 & 12 & -2.0$\pm0.2$ & -1.9$\pm$ 0.7 & -185.4$\pm2.4^{(b)}$ &5.8$\pm2.0^{(b)}$ & $172^{+34}_{-39}$ & $760^{+126}_{-154}$ &(2),(3),(7)\
And XIV & -8.3 & 38 & -2.26$\pm$0.3 & N/A & -480.6$\pm1.2$ &5.3$\pm$ 1.0 & $392^{+185}_{-205}$ & $793^{+23}_{-179}$ &(1),(2),(3)\
And XV & -9.4 & 29 & -1.1 & N/A &-323$\pm1.4^{(b)}$ & 4.0$\pm1.4^{(b)}$ & $220^{+29}_{-15}$ & $626^{+79}_{-35}$& (1),(2),(3)\
And XVI & -9.4 & 7 & -1.7 & -2.0$\pm$0.5 &-367.3$\pm2.8^{(b)}$ & 3.8$\pm2.9^{(b)}$& $123^{+13}_{-10}$ &$476^{+44}_{-29}$ & (1),(2),(3)\
And XVII& -8.5 & 7 & -1.9 &-1.7$\pm0.3$ & -251.6$^{+1.8}_{-2.0}$ & 2.9$^{+2.2}_{-1.9}$ &$262^{+53}_{-46}$ &$727^{+39}_{-25}$ & (1),(3),(6)\
And XVIII&-9.7 & 22 & -1.8$\pm0.5$ & N/A &-332.1$\pm2.7^{(b)}$ & 9.7$\pm2.3^{(b)}$ &$325\pm24$ & $1214^{+40}_{-43}$ & (1),(2),(3)\
And XIX & -9.3& 27 & -1.9$\pm0.4$ & -1.9$\pm0.6$ & -111.6$^{+1.6}_{-1.4}$ & 4.7$^{+1.6}_{-1.4}$ &$1481^{+62}_{-268}$ & $821^{+32}_{-148}$ & (1),(3),(6)\
And XX& -6.3 & 4 & -1.5$\pm0.5$ & -2.3$\pm0.8$ & -456.2$^{+3.1}_{-3.6}$ &7.1$^{+3.9}_{-2.5}$ &$114^{+31}_{-12}$ & $741^{+42}_{-52}$&(1),(3),(6)\
And XXI & -9.9 & 32 & -1.8 & -1.8$\pm0.4$ &-362.5$\pm0.9$ & 4.5$^{+1.2}_{-1.0}$ & $842\pm77$ & $827^{+23}_{-25}$ &(1),(3),(6)\
And XXII& -6.5 & 12 & -1.8 &-1.85$\pm$ 0.1&-129.8$\pm2.0$& 2.8$^{+1.9}_{-1.4}$ & $252^{+28}_{-47}$ &$920^{+32}_{-139}$ & (1),(2),(3),(8)\
And XXIII &-10.2 & 42&-1.8$\pm0.2$ & -2.3$\pm0.7$ & -237.7$\pm1.2$ & 7.1$\pm1.0$&$1001^{+53}_{-52}$ & $748^{+31}_{-21}$ & (1),(3),(6)\
AndXXIV& -7.6 & 3 &-1.8$\pm0.2$ & -1.8$\pm0.3$ & $-128.2\pm5.2^{(c)}$ &$0.0^{+7.3(c)}$ & $548^{+31}_{-37}$ & $898^{+28}_{42}$ &(1),(3),(6)\
And XXV & -9.7 & 25 &-1.8$\pm0.2$ &-2.1$\pm0.2$ &-107.8$\pm1.0$ & 3.0$^{+1.2}_{-1.1}$& $642^{+47}_{-74}$ &$736^{+23}_{-69}$ & (1),(3),(6)\
And XXVI& -7.1 & 6&-1.9$\pm0.2$ & -1.8$\pm0.5$ & -261.6$^{+3.0}_{-2.8}$ & 8.6$^{+2.8}_{-2.2}$&$219^{+67}_{-52}$ & $754^{+218}_{-164}$ & (1),(3),(6)\
AndXXVII & -7.9 & 11 &-1.7$\pm0.2$ & -1.5$\pm0.28$ & -539.6$^{+4.7}_{-4.5}$ & 14.8$^{+4.3}_{-3.1}$& $657^{+112}_{-271}$ &$1255^{+42}_{-474}$ & (1),(3),(6)\
AndXXVIII & -8.5 & 17 &-2.0$\pm0.2$ & -2.1$\pm0.3$ & -326.2$\pm2.7$ & 6.6$^{+2.9}_{-2.1}$& $210^{+60}_{-50}$ &$650^{+150}_{-80}$ & (6),(9)\
AndXXIX & -8.3 & 24 &-1.8$\pm0.2$ & N/A & $-194.4\pm1.5$ & $5.7\pm1.2$& $360\pm60$ &$730\pm75$ & (10),(11)\
And XXX (Cass II)& -8.0 & 8 &-1.6$\pm0.4$ & -2.2$\pm0.4$ &-139.8$^{+6.0}_{-6.6}$ & 11.8$^{+7.7}_{-4.7}$& $267^{+23}_{-36}$ & $681^{+32}_{-78}$ & (3),(6),(12)\
Conclusions {#sect:conc}
===========
Using new and existing spectroscopic data from the Keck I LRIS and Keck II DEIMOS spectrographs, we have homogeneously derived kinematic properties for 18 of the 28 known Andromeda dSph galaxies. Using a combination of their $g-i$ colors, positions on the sky and radial velocities, we determine the likelihood of each observed star belonging to a given dSph, thus filtering out MW foreground or M31 halo contaminants. We have measured both their systemic velocities and their velocity dispersions, with the latter allowing us to constrain the mass and densities within their half-light radii. For the first time, we confirm that And XVII, XIX, XX, XXIII, XXVI and Cass II are dark matter dominated objects, with dynamical mass-to-light ratios within the half-light radius of $[M/L]_{\rm half}>10\msun/\lsun$.
For And XXV, a bright M31 dSph ($M_V=-9.7$) we measure a mass-to-light ratio of only $[M/L]_{\rm half}=10.3^{+7.0}_{-6.7}\msun/\lsun$ from a sample of 26 stars, meaning that it is consistent with a simple stellar system with no appreciable dark matter component within its $1\sigma$ uncertainties. If this were confirmed with larger datasets, it would prove to be a very important object for our understanding of the formation and evolution of galaxies.
We compare our computed velocity dispersions and mass estimates with those measured for MW dSphs, and find that the majority of the M31 dSphs have very similar mass-size scalings to those of the MW. However, we note 3 significant outliers to these scalings, namely And XIX, XXI and XXV, who possess significantly lower velocity dispersions than expected for their size. These results builds on the identification of three potential outliers in the @tollerud12 dataset (And XIV, XV and XVI). We suggest that the lower densities of the dark matter halos for these outliers could be an indication that they have encountered greater tidal stresses from their host over the course of their evolution, decreasing their masses. However, these bright systems still fall on the luminosity-metallicity relation established for the dSph galaxies of the Local Group. If these objects had undergone significant tidal disruption, we would expect them to lie above this relation. As such, this remains puzzling, and requires dedicated follow up studies to fully map out the kinematics of these unusual systems.
We measure the metallicities of all 18 dSphs from their co-added spectra and find that they are consistent with the established MW trend of decreasing metallicity with decreasing luminosity.
This work represents a significant step forward in understanding the mass profiles of dwarf spheroidal galaxies. Far from residing in dark matter halos with identical mass profiles, we show that the halos of these objects are complex, and differ from one to the next, with their environment and tidal evolution imprinting themselves upon the dynamics of their stellar populations. The Andromeda system of dSphs presents us with an opportunity to better understand these processes, and our future work will further illuminate the evolutionary paths taken by these smallest of galaxies.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Hans-Walter Rix for helpful discussions regarding this manuscript. We are also grateful to the referee for their helpful and detailed suggestions for improving this work. We thank the SPLASH collaboration for providing us with details of their observations of dSphs as presented in T12.
Most of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.
Based in part on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.
Based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
R.I. gratefully acknowledges support from the Agence Nationale de la Recherche though the grant POMMME (ANR 09-BLAN-0228).
G.F.L. gratefully acknowledges financial support for his ARC Future Fellowship (FT100100268) and through the award of an ARC Discovery Project (DP110100678).
N.B. gratefully acknowledges financial support through the award of an ARC Discovery Project (DP110100678).
A.K. thanks the Deutsche Forschungsgemeinschaft for funding from Emmy-Noether grant Ko 4161/1.
[90]{} , M., 1983 , 266, L11
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Testing the membership probability algorithm
============================================
The inclusion of a velocity term in the calculation of $P_i$ {#sect:vptest}
------------------------------------------------------------
In T12, the authors do not impose a velocity probability criterion for their membership calculations. Instead they require all member stars to have a total probability, based on their positions and colors, of $P_{member}>0.1$, and then apply a $3\sigma$ clipping to this final sample to prevent any outliers from significantly inflating their calculated velocity dispersions. In our analysis, we have avoided making any hard cuts to our sample by also utilising prior information on the velocities of our expected contaminant populations and member stars. In § \[sect:velprob\], we tested our velocity probability criteria was not overly biasing our final measurements of velocity dispersion to artificially lower values with the introduction of an extra parameter, $\eta$, that allows us to add additional weight to stars in the tails of the Gaussian velocity distribution. However, we can further test our velocity criterion by removing it entirely from the probabilistic determination, and instead implementing the same cuts presented by T12. This involves cutting stars where $P_i<0.1$ (as determined from $P_{CMD}$ and $P_{dist}$), and also by iteratively removing all stars that have velocities that do not lie within $3\sigma$ of the mean of the remaining sample. In Table \[tab:vpcuts\], we present the results of this on our measured values of $v_r$ and $\sigma_v$ for our full sample of dSphs. For all objects (bar And XXVII, which is a unique case, as described in § \[sect:and27\]) the systemic velocities derived are within $\sim2-3\kms$ of one another. The velocity dispersions we measure from our full algorithm tend to be slightly higher on average, and this is to be expected as we do not cut any stars from our analysis, and therefore outliers in the velocity profile may be assigned non-negligible membership probabilities that will allow them to increase this measurement. By and large, these differences are not significant, with the final values agreeing to well within their $1\sigma$ uncertainties.
It is interesting that our algorithm appears to perform better when dealing with dSphs where the number of member stars is low. This is best demonstrated by And XI (see Fig. \[fig:And11\]). Our algorithm identifies 5 stars with non-negligible probabilities of membership, clustered around $v_r\sim-430\kms$. Our full algorithm measures a systemic velocity of $v_r=-427.5^{+3.5}_{-3.4}\kms$ and a velocity dispersion of $\sigma_v=7.5^{+4.0}_{-2.8}\kms$. One of these stars is slightly offset from the other 4 with a more negative velocity of $v_r=-456.8\kms$. Although this star has a reasonably high probability of being a member based on its distance from the centre of And XI, and its position in the CMD, it does not survive the $3\sigma$ velocity clipping procedure of T12. As the number of member stars is so low, cutting one star from the sample can have a significant effect, and as such, while the T12 procedure determines a very similar systemic velocity of $v_r=-425.0\pm3.1\kms$ it is unable to resolve a velocity dispersion. This effect is also seen other systems (such as And XIII, XVII, XXII and XXVI), although it is typically less pronounced.
Another regime where our algorithm performs better than that of T12 is where the systemic velocity of the system in question is within the regime of the contaminating Milky Way K-dwarfs. An example of this is the unusual system, And XIX, where our algorithm measures a systemic velocity of $v_r=-111.6^{+1.6}_{-1.4}\kms$ and a velocity dispersion of $\sigma_v=4.7^{+1.6}_{-1.4}$. However, the procedure of T12 is less able to resolve the kinematics of the system, measuring $v_r=-109.3\pm5.3\kms$ and a velocity dispersion of $\sigma_v=1.8^{+9.1}_{-1.8}$. The much larger uncertainty on the dispersion is a result of including Milky Way contaminants in the sample which can be difficult to cut out without applying prior knowledge of the velocity profile of this population. $3\sigma$ clipping allows outliers to contribute more significantly to the measured profile in this instance, increasing the uncertainty. A similar effect is seen in the And XXIV and And XXX (Cass II) objects, which also have systemic velocities in the Milky Way contamination regime.
These results lead us to conclude that the inclusion of a $P_{vel}$ term in our analysis allows us to more effectively determine the true kinematics of the systems we are studying. Further, as no cuts to the sample are required using this method, it allows for a more unbiased study of the kinematics of dSphs than that of T12.
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.9\hsize"}
[lcccc]{} And V & $-391.1\pm2.9$ & $10.8^{+3.0}_{-2.3}$ & $-391.5\pm2.7$ & $12.2^{+2.5}_{-1.9}$\
And VI & $-339.0\pm3.0$ & $11.9^{+2.9}_{-2.3}$ & $-339.8\pm1.8$ & $ 12.4^{+1.5}_{-1.3}$\
And XI & $-425.0\pm3.1$ & $0^{+3.5}$ & $-427.5^{+3.5}_{-3.4}$ & $7.6^{+4.0}_{-2.8}$\
And XII & $-558.8\pm3.7$ & $0^{+6.8}$ & $-557.1\pm1.7$ & $0^{+4.0}$\
And XIII & $-203.8\pm8.4$ & $0^{+16.2}$ & $-204.8\pm4.9$& $0.0^{+8.1}$\
And XVII & $-260.0^{+8.0}_{-7.8}$ & $1.8^{+9.1}_{-1.8}$ &$-254.3^{+3.3}_{-3.7}$ &$2.9^{+5.0}_{-2.9}$\
And XVIII &$-345.1\pm3.3$ & $0^{+4.4}$ & $-346.8\pm2.0$& $0^{+2.7}$\
And XIX & $-109.3\pm5.3$ & $1.5^{+6.8}_{-1.5}$ & $-111.6^{+1.6}_{-1.4}$& $4.7^{+1.6}_{-1.4}$\
And XX & $-454.6^{+4.6}_{-5.7}$ & $7.7^{+8.4}_{-3.9}$ &$-456.2^{+3.1}_{-3.6}$ &$7.1^{+3.9}_{-2.5}$\
And XXI &$-363.4^{+2.0}_{-1.8}$ & $3.2^{+2.3}_{-2.1}$ & $-362.5\pm0.9$&$4.5^{+1.2}_{-1.0}$\
And XXII &$-131.4\pm2.7$ & $0^{+3.1}$ &$-129.8\pm2.0$ &$2.8^{+1.9}_{-1.4}$\
And XXIII & $-236.9\pm2.1$ & $8.4^{+1.9}_{-1.5}$ &$-237.7\pm1.2$ &$7.1\pm1.0$\
And XXIV &$-129.2\pm3.6$ & $0^{+6.1}$ & $-129.9^{+4.3}_{-4.4}$& $3.5^{+6.6}_{-3.5}$\
And XXV & $-107.7^{+1.9}_{-1.8}$ & $3.3^{+2.2}_{-1.8}$ &$-107.8\pm1.0$ & $3.0^{+1.2}_{-1.1}$\
And XXVI & $-264.1\pm4.5$ & $0^{+4.8}$ & $-261.7^{+3.1}_{-2.8}$& $8.7^{+2.9}_{-2.3}$\
And XXVII & $-517.6^{+42.8}_{-43.2}$ & $19.3^{+17}_{-19}$ & $-539.6^{+4.7}_{-4.5}$& $14.8^{+4.3}_{-3.1}$\
And XXX (CassII) & $-140.1^{+8.6}_{-9.3}$ & $14.1^{+12.9}_{-6.1}$ &$-139.8^{+6.0}_{-6.6}$ &$11.8^{+7.7}_{-4.7}$\
The effect of low signal-to-noise data on measuring $v_r$ and $\sigma_v$
------------------------------------------------------------------------
For our brightest targets ($i\lta22.5$) the S:N of our spectra is typically $>3$Å$^{-1}$. However, as our targets become fainter, so too their S:N falls. For spectra with S:N$\gta1.5$Å$^{-1}$, our pipeline is still able to measure velocities based on the Ca II triplet, with reasonable measurement uncertainties. However, it is prudent to check whether the inclusion of these velocities, calculated from significantly noisier spectra, has a detrimental effect on our ability to measure the kinematic properties of our dSph sample.
Such a test is straightforward to implement. We have a number of dSphs within our sample for which our probabilistic analysis identifies $\sim30$ likely members (such as And XXI, XXIII and XXV). We can therefore use these samples to impose S:N cuts on our data to see the effect of this on our measurements of $v_r$ and $\sigma_v$. We present the results of this test in Table \[tab:sntest\], and our finding is that, as the level of our imposed S:N cut increases (and so the number of included stars decreases), the systemic velocity remains more or less constant. The measured velocity dispersion, however, shows some variation. In the case of And XXI and XXV, the dispersion increases with increased S:N, however not significantly. In both cases the dispersion calculated from the higher S:N data lies well within $1\sigma$ of that calculated from the lower S:N data. Intuitively, this makes sense as the spectra with higher S:N are likely to have lower velocity uncertainties, and so our maximum-likelihood analysis will attribute more of the spread in measured velocities to an intrinsic dispersion, rather than to our measurement errors. In the case of And XXIII, we find the opposite to be true. As our S:N cut increases, we find that our measured dispersion decreases. This may be because the number of member stars in subsequent quality cuts drops off more rapidly for And XXIII than And XXI and XXV. This suggests that we should be extra cautious when interpreting our measured velocity dispersions for dSphs where both the average S:N of member stars, and the number of member stars, is low.
[lccccccccc]{} S:N$>2$ & 20 & $-362.9\pm0.9$& $3.5^{+0.9}_{-0.7}$ & 22 & $-238.0\pm1.2$ & $6.6\pm1.1$ & 27 & $-107.7\pm0.9$ & 2.7$\pm1.1$\
S:N$>3$ & 11 & $-364.2\pm0.9$ & $3.1\pm0.8$ & 10 & $-238.3\pm1.4$ & $5.1^{+1.4}_{-1.2}$ & 24 & $-107.7\pm1.0$ & $3.0\pm1.2$\
S:N$>4$ & 5 & $-363.9\pm1.5$ & $4.0^{+1.5}_{-1.1}$ & 5 & $-239.4\pm1.1$ & $5.7^{+1.5}_{-1.3}$ & 16 & $-108.2\pm1.2$ & $3.0^{+1.4}_{-1.2}$\
S:N$>5$ & 3 & $-362.6^{+2.2}_{-2.3}$ & $4.5^{+2.6}_{-1.5}$ & 2 & $-239.5\pm1.9$ & $0.0^{+4.9}$ & 13 & $-109.0\pm1.2$ & $2.8^{+1.5}_{-1.3}$\
The effect of small sample sizes on determining kinematic properties of dwarf galaxies
--------------------------------------------------------------------------------------
Obtaining reliable velocities for member stars of faint and distant systems is a difficult task that can only be achieved with the largest optical telescopes, such as Keck. Given the demand for facilities such as this, any observing time awarded must be used as effectively as possible, and this often means compromising between deep pointings for a few objects, and shallower pointings for a number of objects. With longer or multiple exposures on a single target, one can build up impressive samples of member stars for an individual dSph. For example, the SPLASH collaboration observed a total of 95 members in And II, one of the brightest M31 dSph companions [@kalirai10] by taking 2 separate exposure fields over this large object. However, multiple exposures such as these produce diminishing returns as you move down the luminosity scale to fainter, more compact dSphs. This is both because of their smaller size with respect to the DEIMOS field of view, and the fewer number of bright stars available on the RGB to target. In this case, the only way to identify more members is by integrating for longer, but given the paucity of stars, the trade-off between time spent exposing and additional members observed can be quite expensive. Such difficulties inevitably lead to the inference of dynamical properties for an entire system from a handful of stars. It is important for us to understand the effect this bias has on our results, and how reliable the quoted values are. We test this using our datasets for which we identify $>25$ member stars, namely And XIX, XXI, XXIII and XXV using the following method. We select 4, 6, 8, 10, 15, 20, 25 and 30 stars at random from each dataset and then measure the systemic velocity and velocity dispersion using our probability algorithm. This was repeated 1000 times for each sample size. In cases where the algorithm is unable to resolve a velocity dispersion, we throw out the result and resimulate, as null results here will affect our averages and will not inform us whether the instances in which we are able to resolve a velocity dispersion from small numbers of stars are producing valid, reliable result. We display the resulting values in Table \[tab:sampsz\], with the true value recovered from the full sample shown in bold in the final row for comparison. We show that in all these cases, the systemic velocity and velocity dispersion are recovered well within the scatter of the 1000 simulations even when dealing with sample sizes as small as 4 stars, so long as the measurement is resolved. In cases where we are unable to resolve a dispersion, we find that our resulting uncertainties are not meaningful. This is shown explicitly in the case of And XVIII, where we can compare our upper limit for the velocity dispersion as determined from our algorithm with the dispersion calculated in T12 from a much larger dataset. We see that our uncertainty is not consistent with their result. As such, we advise that in all cases where we calculate velocity dispersions from small samples ($N_*<8$, And XI, XII, XX XXIV and XXVI), the dispersion measurements should be treated as indications of the likely dispersion, and need to be confirmed with follow-up studies.
[ccccccc]{} 4 & $-362.6\pm3.1$ & $5.0\pm3.3$ & $-237.2\pm3.8$ & $7.5\pm3.6$ & $-108.8\pm2.4$ & 5.4$\pm4.3$\
6 & $-363.0\pm2.0$ & $4.0\pm1.9$ & $-237.3\pm3.0$ & $6.9\pm2.6$& $-108.0\pm2.7$ & $4.0\pm2.3$\
8 & $-362.9\pm2.1$ & $4.2\pm2.0$ & $-236.8\pm2.8$ & $6.9\pm2.3$ & $-107.6\pm2.0$ & $2.7\pm1.1$\
10 & $-362.7\pm1.8$ & $4.1\pm1.8$ & $-237.3\pm2.5$ & $6.5\pm2.1$ & $-107.9\pm1.5$ & $3.1\pm1.0$\
15 & $-362.9\pm1.5$ & $4.3\pm1.2$ & $-237.4\pm2.0$ & $6.8\pm1.4$ & $-107.9\pm1.5$ & $3.1\pm1.0$\
20 & $-362.9\pm1.3$ & $4.3\pm1.2$ & $-237.6\pm1.7$ & $6.9\pm1.4$ & $-107.8\pm1.2$ & $3.0\pm0.9$\
25 & $-363.0\pm1.0$ & $4.4\pm1.0$ & $-237.4\pm1.4$ & $7.1\pm1.7$ & $-107.8\pm1.0$ & $3.1\pm0.8$\
30 & $-362.8\pm1.1$ & $4.4\pm1.1$ & $-237.7\pm1.3$ & $7.2\pm1.4$ & – & –\
[**Full sample**]{} & $\boldsymbol{-362.5\pm0.9}$ &$\boldsymbol{4.5^{+1.2}_{-1.1}}$ & $\boldsymbol{-237.7\pm1.2}$ & $\boldsymbol{7.1\pm1.0}$ & $\boldsymbol{-107.8\pm1.0}$& $\boldsymbol{3.0^{+1.2}_{-1.1}}$\
Testing our algorithm on the SPLASH sample of M31 dSphs
-------------------------------------------------------
In T12, the authors reported on the kinematic properties of 15 M31 dSphs, And I, III, V, VII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVIII, XXI and XXII, and the positions, and measured velocities (plus uncertainties) for each likely member stars were published as part of that work. The authors were kind enough to also give us access to these properties for their non-member stars so that we might run our algorithm over the full samples to see if we reproduce their results. Our technique for assigning membership probability differs from theirs in that we use the velocities of stars as an additional criterion for membership, whereas they use a cut on both the resulting probability ($P({\mathrm member}>0.1$), and a $3\sigma$ clipping on the velocity. In addition, where we use PAndAS CFHT MegaCam $g-$ and $i-$band photometry for our membership analysis,the SPLASH team use their own Washington-DDO51 filter photometric dataset, in membership classification. As such, small differences might be expected, but if our technique is robust our results should well mirror those of T12. In Table \[tab:splashcomp\] we compare our calculated values of $v_r$ and $\sigma_v$ to those published in T12. As the measurements made in T12 for And XI and XII are made from only 2 stars, we do not include these in this test. In general, the results from both analyses agree to within $1\sigma$ of one another, with the majority of them being well within this bound. Typically, we find that our procedure measures slightly larger values for $\sigma_v$ than that of T12 (with the exception of And IX, And XVIII and And XXII). This is to be expected, as we do not cut stars from our analysis based on their velocity, instead we down-weight their probability of membership. As such, those stars considered as outliers would naturally inflate our dispersions above those measured by T12, but the effect is marginal. These results demonstrate that our technique for assigning probability of membership of individual stars within M31 dSphs based on their photometric properties and velocities is robust, and comparable to that of T12. However, as discussed in § \[sect:vptest\], we find that our technique is superior as it requires no cuts to the final dataset to be made, reducing the bias in these measurements.
[lcccc]{} And I & -376.3$\pm2.2$ & 10.2$\pm1.9$ & $-376.3\pm1.3$ & $10.6\pm1.0$\
And III & -344.3$\pm1.7$ & 9.3$\pm1.4$& $-344.2\pm1.2$ & $10.1\pm1.9$\
And V & -397.3$\pm1.5$ & 10.5$\pm1.1$ & $-396.0\pm1.0$ & $11.4\pm1.2$\
And VII & -307.2$\pm1.3$ & 13.0$\pm$ 1.0& $-307.1\pm1.1$ & $13.1\pm0.9$\
And IX & -209.4$\pm2.5$ & 10.9$\pm2.0$ & $-210.3\pm1.9$ & $10.2^{+1.9}_{-1.7}$\
And X & -164.1$\pm1.7$ & 6.4$\pm1.4$ & $-165.3\pm1.5$ & $6.0^{+1.3}_{-1.2}$\
And XIII& -185.4$\pm2.4$ & 5.8$\pm2.0$ & $-183.0^{+2.4}_{-2.3}$ & $8.6^{+2.1}_{-1.7}$\
And XV & -323$\pm1.4$ & 4.0$\pm1.4$ & $-322.6\pm1.1$ & $6.0^{+2.0}_{-1.8}$\
And XVI & -367.3$\pm2.8$ & 3.8$\pm2.9$& $-366.1^{+4.0}_{-3.1}$ & $4.2^{+4.8}_{-4.2}$\
And XVIII &-332.1$\pm2.7$ & 9.7$\pm2.3$ & $-330.7^{+3.9}_{-4.1}$ & $7.5^{+4.5}_{-3.1}$\
And XXI & $-361.4\pm5.8$ & $7.2\pm5.5$ & $-358.9^{+5.1}_{-5.6}$ & $8.5^{+6.3}_{-5.1}$\
And XXII & $-126.8\pm3.1$ & $3.54^{+4.16}_{-2.49}$ & $-124.2^{+4.6}_{-4.5}$ & $0.0^{+5.7}$\
Reanalyzing our previously published results
============================================
To ensure our analysis of the global properties of the M31 dSph population in § \[sect:mass\] is homogenous, we reanalyzed our previously published datasets using our new probability algorithm. Here we briefly summarize the results of this analysis and compare the new results to the published works. The objects we discuss here are And V, VI (first published in @collins11b), XI, XII and XIII (published in @chapman05 [@chapman07; @collins10]). And V was observed with the LRIS instrument, while the remaining objects were observed with the DEIMOS instrument.
Andromeda V {#sect:and5}
-----------
Andromeda V (And V) was observed using the LRIS instrument on Keck I rather than the DEIMOS instrument on Keck II. LRIS has a lower resolution than DEIMOS, and a smaller field of view, which lowers the accuracy of velocity measurement and limits us to only $\sim50$ targets within a mask compared with $100-200$ for a DEIMOS mask. The raw data were also not reduced using our standard pipeline, owing to problems with the arc-lamp calibrations, and were instead analyzed using the NOAO.ONEDSPEC and NOAO.TWODSPEC packages in IRAF. The results from this reduction, plus an analysis of the data using hard cuts in velocity, distance and color to determine membership were first published in @collins11b. Our full probabilistic analysis identifies 17 stars with a non-negligible probability of belonging to And V. Our technique determines a systemic velocity of $v_r=-391.5\pm2.7\kms$ and $\sigma_v=12.2^{+2.5}_{-1.9}\kms$. Comparing these values to our previously published results ($v_r=-393.1\pm4.2\kms$ and $\sigma_v=11.5^{+5.3}_{-4.3}\kms$, @collins11b) we find them to be consistent within the quoted uncertainties. We also compare our results to those of T12, who measured $v_r=−397.3\pm1.5\kms$ and $\sigma_v=10.5\pm1.1\kms$ from a larger sample of stars (85 members cf. 17) using the higher resolution DEIMOS spectrograph. The velocity dispersions of both are consistent within their $1\sigma$ uncertainties, as are the systemic velocities. Given the difference of a factor of 5 in number of probable member stars between our study and that of T12, this consistency is reassuring, and demonstrates the ability of our technique to accurately determine the kinematics of M31 dSph galaxies from small sample sizes.
Andromeda VI {#sect:and6}
------------
As And VI sits at a large projected distance from the centre of M31 ($\sim270$ kpc), it was not observed as part of the PAndAS survey and we were unable to use CFHT data for our $P_{CMD}$ determination. Instead, we use Subaru Suprime-cam data (PI. N. Arimoto, see @collins11b for a full discussion of this data). Using our full probabilistic analysis, we identify 45 stars with $P_i>10^{-6}$. Our technique determines a most-likely $v_r=-339.8\pm1.8\kms$, with $\sigma_v=12.4^{+1.5}_{-1.3}\kms$. Comparing these values with the results of @collins11b who measured $v_r=-344.8\pm2.5\kms$ and $\sigma_v=9.4^{+3.2}_{-2.4}\kms$, we see that both the systemic velocities and the velocity dispersions are consistent within quoted uncertainties. The slight differences between our previous study and this work are simply a result of the application of our new technique.
Andromeda XI {#sect:and11}
------------
The kinematic properties for Andromeda XI as measured from this DEIMOS data set were first published in @collins10. Here we identify 5 stars as probable members. We determine most-likely parameters of $v_r=-427.5^{+3.5}_{-3.4}\kms$ and $\sigma_v=7.6^{+4.0}_{-2.8}\kms$. In @collins10 we measured $v_r=-419.4^{+4.4}_{-3.8}\kms$ and we were unable to resolve the velocity dispersion for the dSph, measuring $\sigma_v=0.0^{+4.6}\kms$ (where the upper bound represents the formal 1$\sigma$ uncertainty on the unresolved dispersion), which implied a higher systemic velocity and lower velocity dispersion for And XI. However, in that analysis, one star with a velocity of $\sim-440\kms$ was considered to be an outlier based on its velocity, and thus excluded from the kinematic analysis. Here, our algorithm gives this star a non-zero probability of membership, which likely decreases the systemic velocity and increases the dispersion. T12 also presented observations for And XI, but they were not able to cleanly detect the galaxy. They identified 2 stars with highly negative velocities ($\sim-460\kms$), which are offset from our systemic velocity by $\sim30\kms$. Given the very negative velocities of their 2 stars, the probability of them both being M31 contaminants seems low, and some other explanation may be more suitable. Between observations, there is one star common to both ($\alpha=$00:46:19.10,$\delta=$+33:48:4.1), for which we measure a velocity of $-427.16\kms$ compared with $-461.6\kms$. This amounts to a statistical difference at the level of $5\sigma$. One obvious avenue to check is that there has been no velocity offset introduced by a rogue skyline that falls within the region of one of the three Ca II lines. We have carefully checked the spectra of each of our 5 probable members to see if this has been the case. We also rederive the velocity based on cross-correlations with each of the three lines individually, rather than with the full triplet. We find these results, and their average to be entirely consistent within the associated errors from the velocities derived using the technique discussed in § \[sect:specobs\]. This large discrepancy is puzzling, particularly as the methods used to measure velocities in this work are almost identical to those of T12. Without further data it is not possible for either team to pin down the exact issue, or which of the datasets gives the true systemic velocity. This argues for taking further observations within these faint systems, in the hopes of better understanding both the systems themselves, and any systematics introduced by the DEIMOS instrument. It is comforting that, in all cases where this offset is observed, the velocity dispersions measured by each team are consistent with one another, suggesting the problem affects all observations identically.
In this case, we have identified a greater number of potential members stars than in the work of T12, the spectra for all of which have relatively high S:N (S:N$\sim5-10$Å$^{-1}$). Therefore, our results should be considered as more robust than those of T12.
Andromeda XII {#sect:and12}
-------------
Kinematic properties for Andromeda XII as determined from the DEIMOS data set presented here were previously published in @chapman07 and @collins10, and in both cases, membership was largely determined using hard cuts in velocity. This object has been of particular interest as it possesses an extremely negative systemic velocity with respect to Andromeda, suggesting that it is on its first infall into the local group (see @chapman07 for a full discussion). Using our new algorithm, we measure $v_r=-557.1\kms$ and an unresolved velocity dispersion of $\sigma_v=0.0^{+4.0}\kms$, where the upper bound represents the formal $1\sigma$ uncertainty on the measurement. As the dispersion is unresolved, the lower error bound is undefined, as having a negative velocity dispersion is unphysical. These values are completely consistent with the results of @chapman07 and @collins10 ($v_r=-558.4\pm3.2\kms$ and $\sigma_v=2.6^{+5.1}_{-2.6}\kms$). T12 also presented observations for And XII, but as for And XI, they were not able to cleanly detect the galaxy. They identified 2 stars with highly negative velocities ($\sim-530\kms$), which are offset from our systemic velocity by $\sim30\kms$. In this instance, both the stars they observed overlap with two of our likely members, situated at $\alpha$=00:47:28.63,$\delta$=+34:22:43.1 and $\alpha=$00:47:24.69,$\delta=$+34:22:23.9, and these velocities are offset from those that we measure at a statistical level of $3.8\sigma$. This suggests that the self-same calibration effect that causes an offset between our results for And XI and those of T12, is present here also. In the case of And XII, our mask was observed over two separate nights, giving us two velocity measurements for each star (as discussed in @chapman07 and @collins10), and we saw no evidence for systematic offsets of this magnitude in the night to night velocities, making a calibration error within our dataset seem unlikely, though not impossible. We therefore conclude that, owing to our larger sample of members and repeat observations, our measurements for the kinematic properties of And XII are more robust than those of T12.
Andromeda XIII {#sect:and13}
--------------
The kinematic properties of Andromeda XIII were also presented in @collins10, And XIII sits at a large projected distance from Andromeda $(~\sim120$ kpc) in the southern M31 halo, so we expect contamination from the Milky Way and Andromeda halo to be low. It is surprising then, that we see significant structure within our DEIMOS field. This is also seen within the 3 fields observed by T12, who attribute this over-density of stars located at $v_r\sim-120\kms$ to an association with the TriAnd over-density within the Galactic halo. We too see a number of stars between $-140\kms$ and $-100\kms$. From their positions within the CMD of And XIII, they appear more consistent with MW foreground K-dwarfs than M31 RGB stars. As such, these are also likely associated to this MW substructure.
Using our full probabilistic analysis, we identify the most probable And XIII stars as those 4 that cluster around $v\sim-200\kms$, and we determine $v_r=-204.8\pm4.9\kms$ and are unable to resolve a velocity dispersion, with $\sigma_v=0.0^{+8.1}$, where the upper limit indicates the formal $1\sigma$ uncertainty on the likelihood distribution. Given the very large uncertainties on these values (mostly a factor of the low number of member stars) it comes as no surprise perhaps that these results are consistent with the results in @collins10 ($v_r=-195.0^{+7.4}_{-8.4}\kms$ and $\sigma_v=9.7^{+8.9}_{-4.5}$), although not with those of T12 ($v_r=-185.4\pm2.4\kms$ and $\sigma_v=5.6\pm2.0\kms$), where they derived parameters from three times the number of member stars that we present here. Given the highly substructured nature of the And XIII field, and the fact that our detection is at a very low significance (only 4 stars), we find that the T12 measurements for the kinematics of And XIII are more robust than ours.
[^1]: This assumed value is sensitive to age and metallicity effects, see @chen09 for a discussion, however owing to the large distance of M31, small differences in this value within the M31 system will have a negligible effect on metallicity calculations
|
---
abstract: 'We define analytic torsion of ${\mathbb{Z}}_2$-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analytic and twisted holomorphic torsions, etc. The definition uses pseudo-differential operators and residue traces. We also study properties of analytic torsion for ${\mathbb{Z}}_2$-graded elliptic complexes, including the behavior under variation of the metric. For compact odd dimensional manifolds, the analytic torsion is independent of the metric, whereas for even dimensional manifolds, a relative version of the analytic torsion is independent of the metric. Finally, the relation to topological field theories is studied.'
address:
- 'Department of Mathematics, University of Adelaide, Adelaide 5005, Australia'
- 'Department of Mathematics, University of Colorado, Boulder, CO 80309, USA and Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong'
author:
- Varghese Mathai
- Siye Wu
title: 'Analytic Torsion of ${\mathbb{Z}}_2$-graded Elliptic Complexes'
---
[arXiv:1001.3212v2\[math.DG\]]{}\
revised: April, 2010
[^1]
Introduction {#introduction .unnumbered}
============
In [@MW], we investigated the analytic torsion for the twisted de Rham complex $({\mathit{\Omega}}^\bullet(X,{{\EuScript{E}}}),d_{{\EuScript{E}}}+H\wedge\cdot)$, where ${{\EuScript{E}}}$ is a vector bundle with a flat connection $d_{{\EuScript{E}}}$ and $H$ is a closed differential form of odd degree on a closed compact oriented manifold $X$. The novel feature of our construction was the necessary use of pseudo-differential operators and residue traces in defining the torsion. When $X$ is odd dimensional, we showed that it was independent of the choice of metric. In this paper, we generalize this construction, by defining analytic torsion for an arbitrary ${\mathbb{Z}}_2$-graded elliptic complex as an element in the graded determinant line of the cohomology of the complex. The definition again uses pseudo-differential operators and residue traces. We also study properties of analytic torsion for ${\mathbb{Z}}_2$-graded elliptic complexes, including its behavior under variation of the metric. For compact odd dimensional manifolds, the analytic torsion is independent of the metric, whereas for even dimensional manifolds, only a relative version of the analytic torsion is independent of the metric.
We specialize this construction to several new situations where the analytic torsion can be defined. This includes the case of flat superconnection complexes and the analytic torsion of the twisted Dolbeault complex $({\mathit{\Omega}}^{0,\bullet}(X,{{\EuScript{E}}}),\bar\partial_{{{\EuScript{E}}}}+H\wedge\cdot)$, where ${{\EuScript{E}}}$ is a holomorphic vector bundle and $H$ is a $\bar\partial$-closed differential form of type $(0,{\rm odd})$ on a closed connected complex manifold $X$. When $H$ is zero, this was first studied by Ray and Singer in [@RS3]. Although the definition depends on a choice of Hermitian metric, we deduce from our general theory that a relative version of torsion, defined as a ratio of the twisted holomorphic torsions, is independent of the metric. (Of course they do depend on the complex structure.) Twisted holomorphic torsion is defined in several natural situations including, for Calabi-Yau manifolds, or whenever there is a holomorphic gerbe.
Finally, we explain how twisted analytic torsion appears in topological field theory with a twisted abelian Chern-Simons action functional.
For a more detailed literature review on analytic torsion and its variants, we refer to the introduction in [@MW].
We briefly summarize the contents of the paper. §\[sect:z2\] is on the definition of ${\mathbb{Z}}_2$-graded elliptic complexes. §\[sect:defn\] provides the definition of the analytic torsion of a ${\mathbb{Z}}_2$-graded elliptic complex as an element in the graded determinant line of the cohomology of the complex. §\[sect:prop\] contains functorial properties of the analytic torsion. §\[sect:inv\] estabilshes the invariance of the analytic torsion under deformation of metrics in the odd dimensional case. §\[sect:rel\] shows the invariance of the relative analytic torsion under deformation of metrics in the even dimensional case. §\[sect:superconn\] contains the definition and properties of analytic torsion of flat superconnections. §\[sect:dolbeault\] contains the definition and properties of the analytic torsion of twisted Dolbeault complexes. §\[sect:tft\] relates the twisted analytic torsion to topological field theories.
${\mathbb{Z}}_2$-graded elliptic complexes {#sect:z2}
==========================================
Let $X$ be a smooth closed manifold of dimension $n$ and ${{\EuScript{E}}}={{\EuScript{E}}}^{\bar0}\oplus{{\EuScript{E}}}^{\bar1}$, a ${\mathbb{Z}}_2$-graded vector bundle over $X$. (We use $\bar k$ to denote the integer $k$ modulo $2$.) Suppose $D\colon{\mathit{\Gamma}}(X,{{\EuScript{E}}})\to{\mathit{\Gamma}}(X,{{\EuScript{E}}})$ is an elliptic differential operator which is odd with respect to the grading in ${{\EuScript{E}}}$ and satisfies $D^2=0$. Then $D$ is of the form $D={\quad\;\;D_{\bar1}\choose D_{\bar0}\quad\;\;}$ on ${\mathit{\Gamma}}(X,{{\EuScript{E}}})={\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar0})\oplus{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar1})$, where $D_{\bar0}\colon{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar0})\to{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar1})$ and $D_{\bar1}\colon{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar1})\to{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar0})$ are differential operators such that $D_{\bar1}D_{\bar0}=0$ and $D_{\bar0}D_{\bar1}=0$. Furthermore, $$\cdots\to{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar0})\stackrel{D_{\bar0}}{\longrightarrow}
{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar1})\stackrel{D_{\bar1}}{\longrightarrow}{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar0})
\stackrel{D_{\bar0}}{\longrightarrow}{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar1})\to\cdots$$ is a ${\mathbb{Z}}_2$-graded elliptic complex, which we denote by $({{\EuScript{E}}},D)$ for short. Its cohomology groups are $$H^{\bar0}(X,{{\EuScript{E}}},D)=\ker D_{\bar0}/{\operatorname{im}}D_{\bar1}, \quad
H^{\bar1}(X,{{\EuScript{E}}},D)=\ker D_{\bar1}/{\operatorname{im}}D_{\bar0}.$$ It follows from the Hodge theorem for elliptic complexes as will be explained shortly that they are finite dimensional. We call $$b_{\bar0}(X,{{\EuScript{E}}},D)=\dim H^{\bar0}(X,{{\EuScript{E}}},D),\quad
b_{\bar1}(X,{{\EuScript{E}}},D)=\dim H^{\bar1}(X,{{\EuScript{E}}},D)$$ the Betti numbers of the ${\mathbb{Z}}_2$-graded elliptic complex. Its index or Euler characteristic is $\chi(X,{{\EuScript{E}}},D)=b_{\bar0}(X,{{\EuScript{E}}},D)-b_{\bar1}(X,{{\EuScript{E}}},D)$.
We choose a Riemannian metric $g$ on $X$ and an Hermitian form of type $h={h_{\bar0}\quad\;\;\choose\quad\;\;h_{\bar1}}$ on ${{\EuScript{E}}}={{\EuScript{E}}}^{\bar0}\oplus{{\EuScript{E}}}^{\bar1}$. Then there is a scalar product ${\langle}\cdot,\cdot{\rangle}$ on ${\mathit{\Gamma}}(X,{{\EuScript{E}}})$. The Laplacian $L=D^\dagger D+DD^\dagger$ on ${\mathit{\Gamma}}(X,{{\EuScript{E}}})={\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar0})\oplus{\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar1})$ is, in graded components, $L={L_{\bar0}\quad\;\;\choose\quad\;\;L_{\bar1}}$, where $$L_{\bar0}=D_{\bar0}^\dagger D_{\bar0}+D_{\bar1}D_{\bar1}^\dagger,\quad
L_{\bar1}=D_{\bar1}^\dagger D_{\bar1}+D_{\bar0}D_{\bar0}^\dagger.$$ They are self-adjoint elliptic operators with positive-definite leading symbols. By the Hodge theorem for elliptic complexes, one has $$H^{\bar0}(X,{{\EuScript{E}}},D)\cong\ker L_{\bar0},\quad
H^{\bar1}(X,{{\EuScript{E}}},D)\cong\ker L_{\bar1}.$$ By ellipticity, these spaces are finite dimensional, and hence $b_{\bar0}$, $b_{\bar1}$ are finite. Let $K(t,x,y)=\Big(\substack{K_{\bar0}(t,x,y)\qquad\qquad\\
\qquad\qquad K_{\bar1}(t,x,y)}\Big)$, where $t>0$, $x,y\in X$, be the kernel of $e^{-tL}={e^{-tL_{\bar0}}\qquad\quad\choose\qquad\quad e^{-tL_{\bar1}}}$. Suppose the order of $L$ (or that of $L_{\bar0}$ and $L_{\bar1}$) is $d>1$. By Lemma 1.7.4 of [@Gil], when restricted to the diagonal, the heat kernel has the asymptotic expansion $$K(t,x,x)\sim\sum_{l=0}^\infty t^{\frac{2l-n}{d}}a_l(x),$$ where $a_l(x)=\Big(\substack{a_{l,\bar0}(x)\qquad\quad\\ \qquad\quad
a_{l,\bar1}(x)}\Big)$ can be computed locally as an combinatorial expression in the jets of the symbols. We set $a_{\frac{n}{2}}(x)=0$ if $n$ is odd. Denote by $a_{\frac{n}{2}}=\Big(\substack{a_{\frac{n}{2},\bar0}\quad\quad\;\\
\quad\quad\;a_{\frac{n}{2},\bar1}}\Big)$ the operator acting on ${\mathit{\Gamma}}(X,{{\EuScript{E}}})$ point-wisely by $a_{\frac{n}{2}}(x)=\Big(\substack{a_{\frac{n}{2},\bar0}(x)
\qquad\quad\\ \qquad\quad a_{\frac{n}{2},\bar1}(x)}\Big)$. Then the index density of the elliptic complex is ${\operatorname{str}}a_{\frac{n}{2}}(x)$ and the index is $\chi(X,{{\EuScript{E}}})={\operatorname{Str}}(a_{\frac{n}{2}})$. Here and subsequently, ${\operatorname{str}}$ is the point-wise supertrace whereas ${\operatorname{Str}}$ is the supertrace taken on the space of sections.
Definition of the analytic torsion {#sect:defn}
==================================
We generalize the construction in §2 of [@MW]. Recall that the zeta-function of a semi-positive definite self-adjoint operator $A$ (whenever it is defined) is $$\zeta(s,A)={\operatorname{Tr}}'A^{-s},$$ where ${\operatorname{Tr}}'$ stands for the trace restricted to the subspace orthogonal to $\ker(A)$. If $\zeta(s,A)$ can be extended meromorphically in $s$ so that it is holomorphic at $s=0$, then the zeta-function regularized determinant of $A$ is $${\operatorname{Det}}' A=e^{-\zeta'(0,A)}.$$ If $A$ is an elliptic differential operator of order $d$ on a compact manifold $X$ of dimension $n$, then $\zeta(s,A)$ is holomorphic when $\mathrm{Re}(s)>n/d$ and can be extended meromorphically to the entire complex plane with possible simple poles at $\{\frac{n-l}{m},l=0,1,2,\dots\}$ only [@Se]. Moreover, the extended function is holomorphic at $s=0$ and therefore the determinant ${\operatorname{Det}}'A$ is defined for such an operator.
We return to the setting of the ${\mathbb{Z}}_2$-graded elliptic complex $({{\EuScript{E}}},D)$ in §\[sect:z2\]. As in [@MW], we consider the partial Laplacian $D^\dagger D=\Big(\substack{D_{\bar0}^\dagger D_{\bar0}\qquad\;\\
\qquad\;D_{\bar1}^\dagger D_{\bar1}}\Big)$.
For $k=0,1$, $\zeta(s,D_{\bar k}^\dagger D_{\bar k})$ is holomorphic in the half plane for $\mathrm{Re}(s)>n/d$ and extends meromorphically to ${\mathbb{C}}$ with possible simple poles at $\{\frac{n-l}{d},l=0,1,2,\dots\}$ and possible double poles at the negative integers only, and is holomorphic at $s=0$.
Let $P={P_{\bar0}\quad\;\choose\quad\;P_{\bar1}}$ be the projection onto the closure of ${\operatorname{im}}D^\dagger={\operatorname{im}}D^\dagger_{\bar0}\oplus{\operatorname{im}}D^\dagger_{\bar1}$. As $DD^\dagger$ and $L$ are equal and invertible on (the closure of) ${\operatorname{im}}D$, we have $$P=D^\dagger(DD^\dagger)^{-1}D=D^\dagger L^{-1}D,$$ which implies that $P$ (and hence $P_{\bar0}$, $P_{\bar1}$) is a pseudo-differential operator of order $0$. Moreover, for $k=0,1$, $$\zeta(s,D_{\bar k}^\dagger D_{\bar k})={\operatorname{Tr}}(P_{\bar k}L_{\bar k}^{-s}).$$ By general theory [@GrSe; @Gr06], $\zeta(s,D_{\bar k}^\dagger D_{\bar k})$ is holomorphic in the half plane $\mathrm{Re}(s)>n/d$ and extends meromorphically to ${\mathbb{C}}$ with possible simple poles at $\{\frac{n-l}{d}$, $l=0,1,2,\dots\}$ and possible double poles at the negative integers only. The Laurent series of $\zeta(s,D_{\bar k}^\dagger D_{\bar k})$ at $s=0$ is $${\operatorname{Tr}}(P_{\bar k}L_{\bar k}^{-s})=
\frac{c_{-1}(P_{\bar k},L_{\bar k})}{s}+c_0(P_{\bar k},L_{\bar k})
+\sum_{l=1}^\infty c_l(P_{\bar k},L_{\bar k})\,s^l.$$ Here $c_{-1}(P_{\bar k},L_{\bar k})=\frac{1}{d}{\operatorname{res}}(P_{\bar k})$, where ${\operatorname{res}}(P_{\bar k})$ is known as the non-commutative residue of $P_{\bar k}$ [@Wo; @Gui]. Since $P_{\bar k}$ is a projection, ${\operatorname{res}}(P_{\bar k})=0$ [@Wo; @BrLe; @Gr03]. Therefore $\zeta(s,D_{\bar k}^\dagger D_{\bar k})$ is regular at $s=0$.
The scalar product on ${\mathit{\Gamma}}(X,{{\EuScript{E}}}^{\bar k})$ restricts to one on the null space of the Laplacian, $\ker(L_{\bar k})\cong H^{\bar k}(X,{{\EuScript{E}}},D)$. For $k=0,1$, let $\{\nu_{\bar k,i}\}_{i=1}^{b_{\bar k}}$ be an oriented orthonormal basis of $H^{\bar k}(X,{{\EuScript{E}}},D)$ and let $\eta_{\bar k}=\nu_{\bar k,1}\wedge\cdots\wedge\nu_{\bar k,b_{\bar k}}$, the unit volume element. Then $\eta_{\bar0}\otimes\eta_{\bar1}^{-1}\in\det H^\bullet(X,{{\EuScript{E}}},D)$.
The analytic torsion of the ${\mathbb{Z}}_2$-graded elliptic complex $({{\EuScript{E}}},D)$ is $$\tau(X,{{\EuScript{E}}},D)=({\operatorname{Det}}'D_{\bar0}^\dagger D_{\bar 0})^{1/2}
({\operatorname{Det}}'D_{\bar1}^\dagger D_{\bar1})^{-1/2}
\eta_{\bar0}\otimes\eta_{\bar1}^{-1}\in\det H^\bullet(X,{{\EuScript{E}}},D).$$
Functorial properties of the analytic torsion {#sect:prop}
=============================================
We summarize some properties of the analytic torsion of ${\mathbb{Z}}_2$-graded elliptic complexes, generalizing those of the Ray-Singer torsion [@RS] and of the torsion of the twisted de Rham complex [@MW]. We omit the proofs as they are similar.
Suppose $X$ is a compact, closed, oriented Riemannian manifold and ${{\EuScript{E}}}_1,{{\EuScript{E}}}_2$ are two ${\mathbb{Z}}_2$-graded Hermitian vector bundles over $X$. Then ${{\EuScript{E}}}_1\oplus{{\EuScript{E}}}_2$ is also a ${\mathbb{Z}}_2$-graded vector bundle with $({{\EuScript{E}}}_1\oplus{{\EuScript{E}}}_2)^{\bar k}={{\EuScript{E}}}_1^{\bar k}\oplus{{\EuScript{E}}}_2^{\bar k}$ for $k=0,1$. If $({{\EuScript{E}}}_1,D_1)$ and $({{\EuScript{E}}}_2,D_2)$ are two ${\mathbb{Z}}_2$-graded elliptic complexes on $X$, then so is the direct sum $({{\EuScript{E}}}_1\oplus{{\EuScript{E}}}_2,D_1\oplus D_2)$ defined in the obvious way. We have the following
Under the natural identification of determinant lines, $$\tau(X,{{\EuScript{E}}}_1\oplus{{\EuScript{E}}}_2,D_1\oplus D_2)
=\tau(X,{{\EuScript{E}}}_1,D_1)\otimes\tau(X,{{\EuScript{E}}}_2,D_2).$$
Now suppose $p\colon X\to X'$ is a covering of compact, closed, oriented manifolds (with finite index). Choose a Riemannian metric on $X'$, which pulls back to one on $X$. Let ${{\EuScript{E}}}\to X$ be a ${\mathbb{Z}}_2$-graded Hermitian vector bundle. Then the vector bundle $p_*{{\EuScript{E}}}\to X'$ defined by $(p_*{{\EuScript{E}}})_{x'}=\bigoplus_{x\in p^{-1}(x')}{{\EuScript{E}}}_x$ (for $x'\in X'$) is also ${\mathbb{Z}}_2$-graded and has an induced Hermitian form. There is a natural isometry ${\mathit{\Gamma}}(X,{{\EuScript{E}}})\cong{\mathit{\Gamma}}(X',p_*{{\EuScript{E}}})$. If $D$ is a differential operator on ${\mathit{\Gamma}}(X,{{\EuScript{E}}})$, the operator $p_*D$ on ${\mathit{\Gamma}}(X',p_*{{\EuScript{E}}})$ given by the above isomorphism is a differential operator on $X'$. If $({{\EuScript{E}}},D)$ is a ${\mathbb{Z}}_2$-graded elliptic complex, then so is $(p_*{{\EuScript{E}}},p_*D)$. We have
Under the natural identification of determinant lines, $$\tau(X,{{\EuScript{E}}},D)=\tau(X',p_*D).$$
Finally, suppose $X_i$ ($i=1,2$) are closed, oriented Riemannian manifolds and ${{\EuScript{E}}}_i\to X_i$ ($i=1,2$) are ${\mathbb{Z}}_2$-graded Hermitian vector bundles. Denote by $\pi_i\colon X_1\times X_2\to X_i$ ($i=1,2$) the projections. Set ${{\EuScript{E}}}_1\boxtimes{{\EuScript{E}}}_2=\pi_1^*{{\EuScript{E}}}_1\otimes\pi_2^*{{\EuScript{E}}}_2$; it is also a ${\mathbb{Z}}_2$-graded vector bundle with $({{\EuScript{E}}}_1\boxtimes{{\EuScript{E}}}_2)^{\bar0}=\pi_1^*{{\EuScript{E}}}_1^{\bar0}\otimes\pi_2^*{{\EuScript{E}}}_2^{\bar0}
\oplus\pi_1^*{{\EuScript{E}}}_1^{\bar1}\otimes\pi_2^*{{\EuScript{E}}}_2^{\bar1}$ and $({{\EuScript{E}}}_1\boxtimes{{\EuScript{E}}}_2)^{\bar1}=\pi_1^*{{\EuScript{E}}}_1^{\bar0}\otimes\pi_2^*{{\EuScript{E}}}_2^{\bar1}
\oplus\pi_1^*{{\EuScript{E}}}_1^{\bar1}\otimes\pi_2^*{{\EuScript{E}}}_2^{\bar0}$. If $({{\EuScript{E}}}_1,D_1)$ and $({{\EuScript{E}}}_2,D_2)$ are two ${\mathbb{Z}}_2$-graded elliptic complexes, then so is $({{\EuScript{E}}}_1\boxtimes{{\EuScript{E}}}_2,D_1\boxtimes D_2)$, where the operator $D_1\boxtimes D_2$ acts on ${\mathit{\Gamma}}(X_1\times X_2,{{\EuScript{E}}}_1\boxtimes{{\EuScript{E}}}_2)$ according to $$(D_1\boxtimes D_2)(\pi_1^*s_1\otimes\pi_2^*s_2)
=\pi_1^*(D_1s_1)\otimes\pi_2^*s_2
+(-1)^{|s_1|}\pi_1^*s_1\otimes\pi_2^*(D_2s_2)$$ for any $s_i\in{\mathit{\Gamma}}(X_i,{{\EuScript{E}}}_i)$, $i=1,2$. We have
Under the natural identification of determinant lines, $$\tau(X_1\times X_2,{{\EuScript{E}}}_1\boxtimes{{\EuScript{E}}}_2,D_1\boxplus D_2)
=\tau(X_1,{{\EuScript{E}}}_1,D_1)^{\otimes\chi(X_2,{{\EuScript{E}}}_2,D_2)}\otimes
\tau(X_2,{{\EuScript{E}}}_2,D_2)^{\otimes\chi(X_1,{{\EuScript{E}}}_1,D_1)}.$$
Invariance of the torsion under deformation of metrics: the odd dimensional case {#sect:inv}
================================================================================
We note that the operator $D$, or $D_{\bar0}$ and $D_{\bar1}$ in the ${\mathbb{Z}}_2$-graded elliptic complex $({{\EuScript{E}}},D)$ are not dependent on the metric. However, the corresponding partial Laplacians $D^\dagger D$ or $D_{\bar0}^\dagger D_{\bar0},D_{\bar1}^\dagger D_{\bar1}$ do depend on metric, and therefore a priori, so does the analytic torsion $\tau(X,{{\EuScript{E}}},D)$. In this section, we study the dependence of the analytic torsion on the metrics and prove that for closed, oriented odd-dimensional manifolds $X$, the torsion $\tau(X,{{\EuScript{E}}},D)$ is independent of the choice of metric.
Suppose we change the metric $g$ on $X$ and the Hermitian form $h$ on ${{\EuScript{E}}}$ to $g_u$ and $h_u$, respectively, along a path parameterized by $u\in{\mathbb{R}}$. Although the torsion $\tau(X,{{\EuScript{E}}},D)$ is an element of the determinant line $\det H^\bullet(X,{{\EuScript{E}}})$, its variation $$\frac{\partial}{\partial u}\log\tau(X,{{\EuScript{E}}},D)=
\tau(X,{{\EuScript{E}}},D)^{-1}\frac{\partial}{\partial u}\tau(X,{{\EuScript{E}}},D)$$ makes sense as a function of $u$. For any $u$, the Hermitian structure ${\langle}\cdot,\cdot{\rangle}_u$ on ${\mathit{\Gamma}}(X,{{\EuScript{E}}})$ is related to the undeformed one by $${\langle}\,\cdot\,,\cdot\,{\rangle}_u={\langle}{\mathit{\Gamma}}_u(\cdot),\cdot\,{\rangle}$$ for some invertible operator ${\mathit{\Gamma}}_u={{\mathit{\Gamma}}_{\bar0}\quad\;\,\choose\quad\;\,{\mathit{\Gamma}}_{\bar1}}$. Let ${\alpha}_u={\mathit{\Gamma}}^{-1}_u\frac{\partial}{\partial u}{\mathit{\Gamma}}_u$. We have the following
\[thm:inv\] Under the above deformation of $g$ and $h$, we have $$\frac{\partial}{\partial u}\log\tau(X,{{\EuScript{E}}},D)
={\operatorname{Str}}\big({\alpha}\,a_{\frac{n}{2}}\big).$$ In particular, the above is zero if $\dim X=n$ is odd. In this case, the analytic torsion $\tau(X,{{\EuScript{E}}},D)$ is independent of the choice of metric.
We generalize the proof of Lemma 3.1 in [@MW]. The adjoint of $D$ with respect to ${\langle}\cdot,\cdot{\rangle}_u$ is $D^\dagger_u={\mathit{\Gamma}}_u^{-1}D^\dagger{\mathit{\Gamma}}_u$. Its variation is given by $$\frac{\partial}{\partial u}D^\dagger_u=-[{\alpha}_u,D^\dagger_u].$$ In graded components, this is $$\frac{\partial D_{\bar0}}{\partial u}=-{\alpha}_{\bar1}D_{\bar0}+D_0{\alpha}_{\bar0},
\quad
\frac{\partial D_{\bar1}}{\partial u}=-{\alpha}_{\bar0}D_{\bar1}+D_1{\alpha}_{\bar1},$$ where ${\alpha}_{\bar0}={\mathit{\Gamma}}_{\bar0}^{-1}\frac{\partial}{\partial u}{\mathit{\Gamma}}_{\bar0}$, ${\alpha}_{\bar1}={\mathit{\Gamma}}_{\bar1}^{-1}\frac{\partial}{\partial u}{\mathit{\Gamma}}_{\bar1}$. Following [@RS; @MW], we set $$\begin{aligned}
f(s,u)&=\int_0^\infty t^{s-1}{\operatorname{Str}}(e^{-tD^\dagger D}P)\,dt \\
&=\Gamma(s)\big(\zeta(s,D_{\bar0}^\dagger D_{\bar0})
-\zeta(s,D_{\bar1}^\dagger D_{\bar1})\big),\end{aligned}$$ omitting the dependence on $u$ on the right-hand side. Then, as $P\frac{\partial P}{\partial u}=0$, $$\begin{aligned}
\frac{\partial f}{\partial u}
&=\int_0^\infty\!t^{s-1}{\operatorname{Str}}\Big(t[{\alpha},D^\dagger]De^{-tD^\dagger D}
+Pe^{-tD^\dagger D}\frac{\partial P}{\partial u}\Big)\,dt \\
&=\int_0^\infty\!t^{s-1}{\operatorname{Str}}\Big(t{\alpha}\big(e^{-tD^\dagger D}D^\dagger D
+e^{-tDD^\dagger}DD^\dagger\big)
+e^{-tD^\dagger D}P\frac{\partial P}{\partial u}\Big)dt \\
&=\int_0^\infty\!t^s{\operatorname{Str}}\big({\alpha}e^{-tL}L\big)\,dt \\
&=-\int_0^\infty\!t^s\frac{\partial}{\partial t}
{\operatorname{Str}}\big({\alpha}(e^{-tL}-Q)\big)\,dt \\
&=s\int_0^\infty\!t^{s-1}{\operatorname{Str}}\big({\alpha}(e^{-tL}-Q)\big)\,dt.\end{aligned}$$ By the asymptotic expansion of ${\operatorname{Str}}({\alpha}e^{-tL})$ as $t\downarrow 0$, $\int_0^1 t^{s-1}{\operatorname{Str}}({\alpha}e^{-tL})\,dt$ has a possible first order pole at $s=0$ with residue ${\operatorname{Str}}({\alpha}a_{\frac{n}{2}})$. On the other hand, because of the exponential decay of ${\operatorname{Str}}\big(\alpha(e^{-tL}-Q)\big)$ for large $t$, $\int_1^\infty t^{s-1}{\operatorname{Str}}\big({\alpha}(e^{-tL}-Q)\big)\,dt$ is an entire function in $s$. So $$\frac{\partial f}{\partial u}\Big|_{s=0}
=-{\operatorname{Str}}\big({\alpha}(a_{\frac{n}{2}}-Q)\big)$$ is finite and hence $$\frac{\partial}{\partial u}\big(\zeta(0,D_{\bar0}^\dagger D_{\bar0})
-\zeta(0,D_{\bar1}^\dagger D_{\bar1})\big)=0.$$ Since $$\log\Big(\frac{{\operatorname{Det}}'D_{\bar 0}^\dagger D_{\bar 0}}
{{\operatorname{Det}}'D_{\bar 1}^\dagger D_{\bar 1}}\Big)
=-\lim_{s\to0}\Big[f(s,u)-\frac{1}{s}\big(\zeta(0,D_{\bar0}^\dagger D_{\bar0})
-\zeta(0,D_{\bar1}^\dagger D_{\bar1})\big)\Big],$$ we get $$\frac{\partial}{\partial u}\log\Big(\frac{{\operatorname{Det}}'D_{\bar 0}^\dagger
D_{\bar 0}}{{\operatorname{Det}}'D_{\bar 1}^\dagger D_{\bar 1}}\Big)
={\operatorname{Str}}\big({\alpha}(a_{\frac{n}{2}}-Q)\big).$$ Finally, along the path of deformation, the volume elements $\eta_{\bar0}$, $\eta_{\bar1}$ can be chosen so that (cf. Lemma 3.3 of [@MW]) $$\frac{\partial}{\partial u}(\eta_{\bar0}\otimes\eta_{\bar1}^{-1})
=-\frac{1}{2}{\operatorname{Str}}({\alpha}Q)\,\eta_{\bar0}\otimes\eta_{\bar1}^{-1}.$$ The results then follow.
When the elliptic complex is the (${\mathbb{Z}}$-graded) de Rham complex of differential forms with values in a flat vector bundle, the variation of the torsion can be integrated to an anomaly formula [@BZ].
Invariance of relative torsion under deformation of metrics: the even dimensional case {#sect:rel}
======================================================================================
When $n=\dim X$ is even, the torsion does depend on the metrics $g$ on $X$ and $h$ on ${{\EuScript{E}}}$ (Theorem \[thm:inv\]). However, we will prove that the relative analytic torsion defined below is independent of the choice of metric.
We first explain extension by flat bundles. Let ${\mathit{\Pi}}=\pi_1(X)$ be the fundamental group of $X$ and $\rho\colon{\mathit{\Pi}}\to{\operatorname{GL}}(m,{\mathbb{C}})$, a representation of ${\mathit{\Pi}}$. Then $\rho$ determines a flat bundle ${{\EuScript{F}}}_\rho$ over $X$ given by $${{\EuScript{F}}}_\rho=\big(\widetilde X\times{\mathbb{C}}^m\big)/\sim,\qquad
(x{\gamma},v)\sim(x,\rho({\gamma}) v),$$ where $\widetilde X$ is the universal cover of $X$. Smooth sections of ${{\EuScript{F}}}_\rho$ are smooth maps $s\colon\widetilde X\to{\mathbb{C}}^m$ that are ${\mathit{\Pi}}$-equivariant, i.e., $s\circ{\gamma}=\rho({\gamma})s$ for all ${\gamma}\in{\mathit{\Pi}}$. We want to extend $D$ to an action on the sections of ${{\EuScript{E}}}_\rho={{\EuScript{E}}}\otimes{{\EuScript{F}}}_\rho$. Since $D$ is a differential operator, it lifts to the universal cover $\widetilde X$ as a ${\mathit{\Pi}}$-periodic operator $\widetilde D\colon
{\mathit{\Gamma}}(\widetilde X,\widetilde{{\EuScript{E}}})\to{\mathit{\Gamma}}(\widetilde X,\widetilde{{\EuScript{E}}})$, where $\widetilde{{\EuScript{E}}}$ is the pull-back of ${{\EuScript{E}}}$ to $\widetilde X$. By tensoring with the identity operator on ${\mathbb{C}}^m$, we can extend it to $\widetilde D\colon{\mathit{\Gamma}}(\widetilde X,\widetilde{{\EuScript{E}}}\otimes{\mathbb{C}}^m)\to
{\mathit{\Gamma}}(\widetilde X,\widetilde{{\EuScript{E}}}\otimes{\mathbb{C}}^m)$. Since for any ${\mathit{\Pi}}$-equivariant section $s\in{\mathit{\Gamma}}(\widetilde X,\widetilde{{\EuScript{E}}}\otimes{\mathbb{C}}^m)$, $$(\widetilde Ds)\circ{\gamma}=\widetilde D(s\circ{\gamma})
=\widetilde D(\rho(\gamma)s)=\rho(\gamma)(\widetilde Ds),$$ the operator $\widetilde D$ descends to a differential operator $D_\rho\colon{\mathit{\Gamma}}(X,{{\EuScript{E}}}_\rho)\to{\mathit{\Gamma}}(X,{{\EuScript{E}}}_\rho)$. If $({{\EuScript{E}}},D)$ is a ${\mathbb{Z}}_2$-graded elliptic complex, then so is $({{\EuScript{E}}}_\rho,D_\rho)$.
Now suppose $X$ is a closed, compact, oriented Riemannian manifold and ${{\EuScript{E}}}$ is a ${\mathbb{Z}}_2$-graded Hermitian vector bundle. Let $\rho_1,\rho_2$ be unitary representations of ${\mathit{\Pi}}$ of the same dimension $m$. Then the flat bundles ${{\EuScript{F}}}_{\rho_i}$ are Hermitian bundles and so are ${{\EuScript{E}}}_{\rho_i}={{\EuScript{E}}}\otimes{{\EuScript{F}}}_{\rho_i}$ ($i=1,2$). Furthermore, if $({{\EuScript{E}}},D)$ is a ${\mathbb{Z}}_2$-graded elliptic complex as in §\[sect:z2\], then so are $({{\EuScript{E}}}_{\rho_i},D_{\rho_i})$ for $i=1,2$.
The [*relative analytic torsion*]{} is the quotient $$\tau(X,{{\EuScript{E}}}_{\rho_1},D_{\rho_1})\otimes\tau(X,{{\EuScript{E}}}_{\rho_2},D_{\rho_2})^{-1}
\in\det H^\bullet(X,{{\EuScript{E}}}_{\rho_1},D_{\rho_1})\otimes
\det H^\bullet(X,{{\EuScript{E}}}_{\rho_2},D_{\rho_2})^{-1}.$$
To show its independence of the metric, let $K_{\rho_i}(t,x,y)$ denote, for $i=1,2$, the heat kernel of the Laplacians $L_{\rho_i}=D_{\rho_i}^\dagger D_{\rho_i}+D_{\rho_i}D_{\rho_i}^\dagger$. Since the Hermitian bundles ${{\EuScript{E}}}_{\rho_1}$ and ${{\EuScript{E}}}_{\rho_2}$, together with the differential operators $D_{\rho_1}$ and $D_{\rho_2}$ are locally identical, the difference in the two heat kernels, when restricted to the diagonal, is exponentially small for small $t$. More precisely, we have
\[prop:rel\] In the notation above, there are positive constants $C,C'$ such that as $t\to 0$, one has for all $x\in X$, $$|K_{\rho_1}(t,x,x)-K_{\rho_2}(t,x,x)|\le
Ct^{-n/d}\exp[-C't^{-\frac{d}{d-1}}],$$ where $d$ is the order of the Laplacians.
Let $\widetilde K(t,x,y)$ denote the heat kernel of the Laplacian $\widetilde L=\widetilde D^\dagger\widetilde D
+\widetilde D\widetilde D^\dagger$ on $\widetilde X$. Then by the Selberg principle, one has for $x,y\in\widetilde X$, $$K_{\rho_j}(t,\bar x,\bar y)=
\sum_{{\gamma}\in{\mathit{\Pi}}}\widetilde K(t,x,y{\gamma})\rho_j({\gamma}),$$ where $\bar x\in X$ stands for the projection of $x\in\widetilde X$. It follows that $$K_{\rho_1}(t,\bar x,\bar y)-K_{\rho_2}(t,\bar x,\bar y)
=\sum_{{\gamma}\in{\mathit{\Pi}}\setminus\{1\}}\widetilde K(t,x,y{\gamma})
(\rho_1({\gamma})-\rho_2({\gamma})).$$ Since $\rho_i$ ($i=1,2$) are unitary representations, one has $$|K_{\rho_1}(t,\bar x,\bar y)-K_{\rho_2}(t,\bar x,\bar y)|\le
\sum_{{\gamma}\in{\mathit{\Pi}}\setminus\{1\}}2|\widetilde K(t,x,y\gamma)|.$$ The off-diagonal Gaussian estimate for the heat kernel on $\widetilde X$ is [@BrSu] $$|\widetilde K(t,x,y)|\le C_1t^{-n/d}
\exp\Big[-C_2\Big(\frac{d(x,y)}{t}\Big)^{\frac{d}{d-1}}\Big],$$ where $d(x,y)$ is the Riemannian distance between $x,y\in\widetilde X$. Therefore $$|K_{\rho_1}(t,\bar x,\bar x)-K_{\rho_2}(t,\bar x,\bar x)|
\le2C_1t^{-n/d}\sum_{{\gamma}\in{\mathit{\Pi}}\setminus\{1\}}
\exp\Big[-C_2\Big(\frac{d(x,x{\gamma})}{t}\Big)^{\frac{d}{d-1}}\Big].$$ By Milnor’s theorem [@Mi], there is a positive constant $C_3$ such that $d(x,x{\gamma})\ge C_3\ell({\gamma})$, where $\ell$ denotes a word metric on ${\mathit{\Pi}}$. Moreover, the number of elements in the sphere $S_l$ of radius $l$ in ${\mathit{\Pi}}$ satisfies $\#S_l\le C_4\,e^{C_5l}$ for some positive constants $C_4,C_5$. Therefore $$\begin{aligned}
&\sum_{{\gamma}\in{\mathit{\Pi}}\setminus\{1\}}
\exp\Big[-C_2\Big(\frac{d(x,x{\gamma})}{t}\Big)^{\frac{d}{d-1}}\Big] \\
\le&\sum_{{\gamma}\in{\mathit{\Pi}}\setminus\{1\}}
\exp\big[-C'(\ell({\gamma})/t)^{\frac{d}{d-1}}\big] \\
\le&\;\sum_{l=1}^\infty\exp\big[-C'(l/t)^{\frac{d}{d-1}}\big]\,C_4\,e^{C_5l} \\
\le&\;C_4\exp[-C't^{-\frac{d}{d-1}}]\sum_{l=1}^\infty
\exp\big[-C'(l^{\frac{d}{d-1}}-1)+C_5l\big] \end{aligned}$$ for all $t$ such that $0<t\le1$ for some positive constant $C'$. Since $\frac{d}{d-1}>1$, the infinite sum over $l$ converges and hence the result.
\[thm:rel\] Let $X$ be a closed oriented manifold of even dimension. Let $\rho_1,\rho_2$ be unitary representations of $\pi_1(X)$ of the same dimension. Then the relative analytic torsion $\tau(X,{{\EuScript{E}}}_{\rho_1},D_{\rho_1})\otimes\tau(X,{{\EuScript{E}}}_{\rho_2},D_{\rho_2})^{-1}$ is independent of the choice of metric.
By Theorem \[thm:inv\], under a one-parameter deformation of the metric, $$\frac{\partial}{\partial u}\log\tau(X,{{\EuScript{E}}}_{\rho_i},D_{\rho_i})
={\operatorname{Str}}\big({\alpha}\,a_{\frac{n}{2}}^{\rho_i})\big)$$ for $i=1,2$. By Proposition \[prop:rel\], we have $a_{\frac{n}{2}}^{\rho_1}=a_{\frac{n}{2}}^{\rho_2}$. Therefore the change in relative torsion is zero.
Analytic torsion of flat superconnections {#sect:superconn}
=========================================
The concept of superconnection was initiated by Quillen, cf. [@Q; @MQ; @BGV]. Let $X$ be a smooth manifold and ${{\EuScript{F}}}={{\EuScript{F}}}^{\bar0}\oplus{{\EuScript{F}}}^{\bar1}$, a ${\mathbb{Z}}_2$-graded vector bundle over $X$. Then the space ${\mathit{\Omega}}(X,{{\EuScript{F}}})$ of ${{\EuScript{F}}}$-valued differential forms has a ${\mathbb{Z}}_2$-grading with $${\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar0}={\mathit{\Omega}}^{\bar0}(X,{{\EuScript{F}}}^{\bar0})\oplus{\mathit{\Omega}}^{\bar1}(X,{{\EuScript{F}}}^{\bar1}),
\quad
{\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar1}={\mathit{\Omega}}^{\bar0}(X,{{\EuScript{F}}}^{\bar1})\oplus{\mathit{\Omega}}^{\bar1}(X,{{\EuScript{F}}}^{\bar0}).$$ A [*superconnection*]{} is a first-order differential operator ${\mathbb{A}}$ on ${\mathit{\Omega}}(X,{{\EuScript{F}}})$ that is odd with respect to the ${\mathbb{Z}}_2$-grading and satisfies $${\mathbb{A}}({\alpha}\wedge s)=d{\alpha}\wedge s+(-1)^{|{\alpha}|}{\alpha}\wedge{\mathbb{A}}s$$ for any ${\alpha}\in{\mathit{\Omega}}(X)$ and $s\in{\mathit{\Omega}}(X,{{\EuScript{F}}})$. The bundle ${{\mathrm{End}}}({{\EuScript{F}}})$ is also ${\mathbb{Z}}_2$-graded and ${\mathbb{A}}$ extends to ${\mathit{\Omega}}(X,{{\mathrm{End}}}({{\EuScript{F}}}))$. The curvature of the superconnection is $F_{\mathbb{A}}={\mathbb{A}}^2\in{\mathit{\Omega}}(X,{{\mathrm{End}}}({{\EuScript{F}}}))^{\bar0}$. It satisfies the Bianchi identity ${\mathbb{A}}F_{\mathbb{A}}=0$. A superconnection ${\mathbb{A}}$ is of the form ${\mathbb{A}}=\nabla+{{A}}$, where $\nabla$ is a usual connection on ${{\EuScript{F}}}$ preserving the grading and ${{A}}\in{\mathit{\Omega}}(X,{{\mathrm{End}}}({{\EuScript{F}}}))^{\bar1}$. Thus the superconnections form an affine space modeled on the vector space ${\mathit{\Omega}}(X,{{\mathrm{End}}}({{\EuScript{F}}}))^{\bar1}$.
The superconnection is flat if $F_{\mathbb{A}}=0$. In this case, writing ${\mathbb{A}}={\quad\;{\mathbb{A}}_{\bar1}\choose{\mathbb{A}}_{\bar0}\quad\;}$, there is a ${\mathbb{Z}}_2$-graded elliptic complex $$\cdots\to{\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar 0}\stackrel{{\mathbb{A}}_{\bar0}}{\longrightarrow}
{\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar1}\stackrel{{\mathbb{A}}_{\bar1}}{\longrightarrow}{\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar0}
\stackrel{{\mathbb{A}}_{\bar0}}{\longrightarrow}{\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar1}\to\cdots,$$ We can define the cohomology groups $H^{\bar k}(X,{{\EuScript{F}}},{\mathbb{A}})$, $k=0,1$. In fact, this is a special case of §\[sect:z2\] with ${{\EuScript{E}}}={\mbox{\fontsize{12pt}{0pt}\selectfont $\wedge$}}TX\otimes{{\EuScript{F}}}$ and $D={\mathbb{A}}$. If $X$ is a closed, compact, oriented Riemannian manifold and ${{\EuScript{F}}}$ is an Hermitian vector bundle, then we can define the analytic torsion of a flat superconnection as $\tau(X,{{\EuScript{F}}},{\mathbb{A}})=\tau(X,{{\EuScript{E}}},D)\in\det H^\bullet(X,{{\EuScript{F}}},{\mathbb{A}})$ with the above choice of $({{\EuScript{E}}},D)$. The functorial properties (§\[sect:prop\]) and invariance under metric deformations (§\[sect:inv\], \[sect:rel\]) hold in this case.
We consider a special case when ${{\EuScript{F}}}={{\EuScript{F}}}^{\bar0}$ and ${{\EuScript{F}}}^{\bar1}=0$. Then ${\mathit{\Omega}}(X,{{\EuScript{F}}})^{\bar k}={\mathit{\Omega}}^{\bar k}(X,{{\EuScript{F}}})$ for $k=0,1$. A superconnection is of the form $\nabla+{{A}}$, where $\nabla$ is a usual connection on ${{\EuScript{F}}}$ and ${{A}}\in{\mathit{\Omega}}^{\bar1}(X,{{\mathrm{End}}}({{\EuScript{F}}}))$. Suppose ${{A}}$ is of degree $3$ or higher. Then the superconnection is flat if and only if $\nabla$ is flat and $\nabla{{A}}+{{A}}^2=0$. When ${{A}}$ is of the form ${{A}}=H{\operatorname{id}}_{{\EuScript{F}}}$ for some $H\in{\mathit{\Omega}}^{\bar1}(X)$, the above condition on ${{A}}$ becomes $dH=0$ and the ${\mathbb{Z}}_2$-graded elliptic complex is the twisted de Rham complex $({\mathit{\Omega}}(X),d+H\wedge\cdot)$. Its analytic torsion $\tau(X,{{\EuScript{F}}},H)$ was studied in [@MW]. Among other properties, the latter is also invariant under the deformation of $H$ by an exact form when $X$ is odd dimensional; the rest of the section will be devoted to generalizing this property to the analytic torsion of flat superconnections.
We return to the general case of a flat superconnection ${\mathbb{A}}$ over a graded vector bundle ${{\EuScript{F}}}$. Suppose $G\in{\mathit{\Omega}}(X,{{\mathrm{End}}}({{\EuScript{F}}}))^{\bar0}$ is point-wisely invertible. Then ${\mathbb{A}}'=G^{-1}{\mathbb{A}}G$ is another flat superconnection on ${{\EuScript{F}}}$; we say that ${\mathbb{A}}'$ is gauge equivalent to ${\mathbb{A}}$. There is an isomorphism of cohomology groups $H^\bullet(X,{{\EuScript{F}}},{\mathbb{A}})\cong H^\bullet(X,{{\EuScript{F}}},{\mathbb{A}}')$, and hence of the corresponding determinant lines, induced by $G$. Now suppose ${\mathbb{A}}$ is deformed to ${\mathbb{A}}_v$ along a path parameterized by $v$ so that each ${\mathbb{A}}_v$ is gauge equivalent to ${\mathbb{A}}$ via $G_v$. Let $$\beta_v=G_v^{-1}\frac{\partial G_v}{\partial v}\in{\mathit{\Omega}}(X,{{\mathrm{End}}}({{\EuScript{F}}}))^{\bar0}.$$
Under deformation of ${\mathbb{A}}$ by gauge equivalence and the natural identification of determinant lines, we have $$\frac{\partial}{\partial v}\log\tau(X,{{\EuScript{F}}},{\mathbb{A}})
={\operatorname{Str}}\big(\beta\,a_{\frac{n}{2}}\big).$$ If $\dim X=n$ is odd, then the above is zero. In this case, the analytic torsion $\tau(X,{{\EuScript{F}}},{\mathbb{A}})$ is invariant under gauge equivalence.
Under the deformation, we have $$\frac{\partial{\mathbb{A}}}{\partial v}=[\beta,{\mathbb{A}}],\qquad
\frac{\partial{\mathbb{A}}^\dagger}{\partial v}=-[\beta^\dagger,{\mathbb{A}}^\dagger].$$ The component of $\beta$ in ${\mathit{\Omega}}^{\bar1}(X,{{\mathrm{End}}}({{\EuScript{F}}})^{\bar1})$ does not contribute to the trace or supertrace, whereas that in ${\mathit{\Omega}}^{\bar0}(X,{{\mathrm{End}}}({{\EuScript{F}}})^{\bar0})$ is even in the degree of differential forms. Following the proofs of Lemmas 3.5 and 3.7 of [@MW], we get the desired variation formula upon a suitable choice of volume elements and identification of determinant lines under the deformation; the rest follows easily.
If $\dim X$ is even, a relative version of analytic torsion (cf. §\[sect:rel\]) is invariant under gauge equivalence.
Analytic torsion of twisted Dolbeault complexes {#sect:dolbeault}
===============================================
Let $X$ be a connected, closed, compact complex manifold and ${{\EuScript{F}}}$, a holomorphic vector bundle over $X$. Denote by ${\mathit{\Omega}}^{p,q}(X,{{\EuScript{F}}})$ the space of smooth differential $(p,q)$-forms on $X$ with values in ${{\EuScript{F}}}$. A holomorphic connection on ${{\EuScript{F}}}$ can act on ${\mathit{\Omega}}^{p,q}(X,{{\EuScript{F}}})$ and splits uniquely as $\partial_{{\EuScript{F}}}+{\bar\partial}_{{\EuScript{F}}}$, where $$\partial_{{\EuScript{F}}}\colon{\mathit{\Omega}}^{p,q}(X,{{\EuScript{F}}})\to{\mathit{\Omega}}^{p+1,q}(X,{{\EuScript{F}}}),\quad
{\bar\partial}_{{\EuScript{F}}}\colon{\mathit{\Omega}}^{p,q}(X,{{\EuScript{F}}})\to{\mathit{\Omega}}^{p,q+1}(X,{{\EuScript{F}}})$$ satisfying ${\bar\partial}_{{\EuScript{F}}}^2=0$. This yields the Dolbeault complex of differential forms with values in ${{\EuScript{F}}}$.
Let ${\mathit{\Omega}}^{p,\bar0}(X,{{\EuScript{F}}})$, ${\mathit{\Omega}}^{p,\bar1}(X,{{\EuScript{F}}})$ be the space of differential forms that is of degree $p$ in the holomorphic part and of even, odd degree, respectively, in the anti-holomorphic part. Consider a differential form $H\in{\mathit{\Omega}}^{0,\bar1}(X)$ that is ${\bar\partial}$-closed, i.e., ${\bar\partial}H=0$. Let ${\bar\partial}_{{{\EuScript{F}}},H}={\bar\partial}_{{\EuScript{F}}}+H\wedge\cdot\;$. We call $H$ a holomorphic flux form and ${\bar\partial}_{{{\EuScript{F}}},H}$, the Dolbeault operator twisted by $H$. Setting ${\bar\partial}_{\bar k}= \bar{\partial}_{{{\EuScript{F}}},H}$ acting on ${\mathit{\Omega}}^{p,\bar k}(X,{{\EuScript{E}}})$ for $k=0,1$, we have ${\bar\partial}_{\bar1}{\bar\partial}_{\bar0}={\bar\partial}_{\bar0}{\bar\partial}_{\bar1}=0$ and a ${\mathbb{Z}}_2$-graded elliptic complex, which we call the [*twisted Dolbeault complex*]{} $$\cdots\to{\mathit{\Omega}}^{p,\bar0}(X,{{\EuScript{F}}})\stackrel{{\bar\partial}_{\bar0}}{\longrightarrow}
{\mathit{\Omega}}^{p,\bar1}(X,{{\EuScript{F}}})\stackrel{{\bar\partial}_{\bar1}}{\longrightarrow}{\mathit{\Omega}}^{p,\bar0}(X,{{\EuScript{F}}})
\stackrel{{\bar\partial}_{\bar0}}{\longrightarrow}{\mathit{\Omega}}^{p,\bar1}(X,{{\EuScript{F}}})\to\cdots.$$ We define the [*twisted Dolbeault cohomology groups*]{} as $$H^{p,\bar0}(X,{{\EuScript{F}}},H)=\ker\,{\bar\partial}_{\bar0}/{\operatorname{im}}\,{\bar\partial}_{\bar1},\quad
H^{p,\bar1}(X,{{\EuScript{F}}},H)=\ker\,{\bar\partial}_{\bar1}/{\operatorname{im}}\,{\bar\partial}_{\bar0}.$$ Like in [@RW; @MW], if the degree of $H$ is $3$ or higher, there is a spectral sequence whose $E_2$-terms are $H^{p,\bullet}(X,{{\EuScript{F}}})$ converging to $H^{p,\bullet}(X,{{\EuScript{F}}},H)$. If $H'$ and $H$ differ by a ${\bar\partial}$-exact form, then there are natural isomorphisms $H^{p,\bullet}(X,{{\EuScript{F}}},H')\cong H^{p,\bullet}(X,{{\EuScript{F}}},H)$.
The above construction is the holomorphic counterpart of the twisted de Rham complex studied in [@RW; @MW]. Holomorphic flux forms arise naturally in a number of prominent situations. Suppose that $X$ is a Calabi-Yau manifold of an odd complex dimension $n$. Then the canonical bundle of $X$ is trivial, i.e., there is a nowhere zero section ${\mathit{\Omega}}$ which satisfies $\partial{\mathit{\Omega}}=0$. Here $H=\overline{\mathit{\Omega}}\in\Omega^{0,n}(X)$ is ${\bar\partial}$-closed. Another example comes from holomorphic gerbes (or holomorphic sheaves of groupoids). The 3-curvature of a holomorphic curving on a holomorphic gerbe on a complex manifold $X$ is a closed holomorphic $3$-form ${\mathit{\Omega}}$ on $X$ (cf. [@Bry], 5.3.17 part (4)). Again, $H=\overline{\mathit{\Omega}}\in{\mathit{\Omega}}^{0,3}(X)$ is ${\bar\partial}$-closed.
The twisted Dolbeault complex is also a special ${\mathbb{Z}}_2$-graded elliptic complex $({{\EuScript{E}}},D)$ with ${{\EuScript{E}}}={\mbox{\fontsize{12pt}{0pt}\selectfont $\wedge$}}^p(T^{1,0}X)^*\otimes{\mbox{\fontsize{12pt}{0pt}\selectfont $\wedge$}}^\bullet(T^{0,1}X)^*\otimes{{\EuScript{F}}}$ and $D={\bar\partial}_{{{\EuScript{F}}},H}$. Suppose $X$ is closed and compact. Given a Riemannian metric on $X$ and an Hermitian form on ${{\EuScript{F}}}$, we have the analytic torsion of the twisted Dolbeault complex (cf. §\[sect:defn\]) $$\tau_p(X,{{\EuScript{F}}},H)=\tau(X,{{\EuScript{E}}},D)\in\det H^{p,\bullet}(X,F,H)$$ with the above choice of $({{\EuScript{E}}},D)$. It is satisfies the functorial properties in §\[sect:prop\]. Since $X$ is always of even (real) dimension, only a relative version of the analytic torsion for the twisted Dolbeault complex is independent of the metric. We conclude from Theorem \[thm:rel\] the following
Let ${{\EuScript{F}}}$ be a holomorphic vector bundle over a compact complex manifold $X$. Suppose $H\in{\mathit{\Omega}}^{0,\bar1}(X)$ is ${\bar\partial}$-closed. For two flat bundles on $X$ given by the representations $\rho_1,\rho_2$ of $\pi_1(X)$ of the same dimension, the relative twisted holomorphic torsion $\tau(X,{{\EuScript{F}}}_{\rho_1},H)\otimes\tau(X,{{\EuScript{F}}}_{\rho_2},H)^{-1}$ is invariant under any deformation of $H$ by an ${\bar\partial}$-exact form, up to natural identification of the determinant lines.
For a non-trivial example of twisted holomorphic torsion, consider the compact Calabi-Yau manifold $T\times M$, where $T$ is a compact complex torus of dimension $1$ and $M$ is a K3 surface. Let ${{\EuScript{L}}}={{\EuScript{L}}}_{u,v}$ be a flat line bundle over $T$ defined by the character $\chi_{u,v}(m,n)=\exp(2\pi{\sqrt{-1}}(mu+nv))$, $0\le u,v\le1$, $m,n\in{\mathbb{Z}}$. If $(m,n)\ne(0,0)$, then the Dolbeault cohomology $H^\bullet(T,{{\EuScript{L}}})$ is trivial. Recall the non-trivial holomorphic torsion of $(T,{{\EuScript{L}}})$ [@RS3] $$\tau_0(T,{{\EuScript{L}}})= \left|e^{\pi{\sqrt{-1}}v^2\tau}
\frac{\theta_1(u-\tau v,\tau)}{\eta(\tau)}\right|,$$ where $\tau$ (with $\mathrm{Im}\,\tau>0$) be the complex moduli of $T$. Here the theta function is defined as $$\theta_1(w,\tau)=-\eta(\tau)e^{\pi{\sqrt{-1}}(w+\tau/6)}\prod_{k=-\infty}^\infty
(1- e^{2\pi{\sqrt{-1}}(|k|\tau-\epsilon_kw)}),$$ where $\epsilon_k={\rm sign}\left(k+\frac{1}{2}\right)$ and $\eta(\tau)$ is the Dedekind eta function. We still denote by ${{\EuScript{L}}}$ the pull-back of ${{\EuScript{L}}}$ to $T\times M$. The Dolbeault cohomology groups of $T\times M$ are trivial and so are the twisted ones. Since $\chi({{\EuScript{O}}}_T({{\EuScript{L}}}))=0$ and $\chi({{\EuScript{O}}}_M)=2$, we have [@RS3] $$\tau_0(T\times M,{{\EuScript{L}}})=\tau_0(T,{{\EuScript{L}}})^{\otimes 2}.$$ Let $H=\bar{\alpha}\wedge\bar{\lambda}$, where ${\alpha}$ is a holomorphic $1$-form on $T$ and ${\lambda}$ a holomorphic $2$-form on $M$. By perturbation theory [@Fa], one has, $$\tau_0(T\times M,{{\EuScript{L}}},H)= e^{o(|H|)}\tau_0(T\times M,{{\EuScript{L}}})
=e^{o(|H|)}\tau_0(T,{{\EuScript{L}}})^{\otimes 2},$$ where $o(H)\to0$ as $H\to0$. Therefore $ \tau_0(T\times M,{{\EuScript{L}}},H)$ non-trivial whenever $|H|$ is sufficiently small.
Relation to topological field theories {#sect:tft}
======================================
In [@Sch], a topological field theory of antisymmetric tensor fields were constructed and the partition function is shown to be equal to the Ray-Singer analytic torsion. The metric independence of the torsion is an evidence that the quantized theory is topological invariant. In this section, we extend the relation to twisted analytic torsion by constructing topological field theories that contain a coupling with the flux form.
Suppose $X$ is a compact, oriented manifold of dimension $n$ and $H$ is a flux form, a closed differential form of odd degree. For $k=0$ or $1$, we define a theory whose action is $$S_{\bar k}[B,C]=\int_X B\wedge d_HC,$$ where $B\in{\mathit{\Omega}}^{\overline{n-k}}(X)$, $C\in{\mathit{\Omega}}^{\bar k}(X)$ are the dynamical fields. Since the operator $d_H=d+H\wedge\cdot\,$ is not compatible with the ${\mathbb{Z}}$-grading, the forms $B,C$ can not be chosen to have fixed degrees. Instead, the degrees of $B,C$ have the same parity when $\dim X$ is odd and opposite parity when $\dim X$ is even. The classical equations of motion are $$d_HC=0,\qquad d_{-H}B=0.$$ The action $S[B,C]$ and the equations of motion are invariant under a set of gauge transformations $$C\mapsto C+d_HC^{(1)},\qquad B\mapsto B+d_{-H}B^{(1)},$$ where $B^{(1)},C^{(1)}$ can be any forms whose degrees have opposite parity with $B,C$, respectively. The phase space is the space of solutions to the equation of motion modulo the gauge transformations. In this case, it is $H^{\overline{n-k}}(X,-H)\oplus H^{\bar k}(X,H)$, expressed in terms of the de Rham cohomology groups twisted by the fluxes $\pm H$.
To quantize the theory, we consider the partition function $$Z_{\bar k}(X,H)=\int{{\EuScript{D}}}B{{\EuScript{D}}}C\;e^{-S_{\bar k}[B,C]}.$$ We need to introduce a Riemannian metric on $X$ which determines the “measures” ${{\EuScript{D}}}B$, ${{\EuScript{D}}}C$. The integration of the transverse parts of $B,C$ yields the determinant ${\operatorname{Det}}'(d_H^\dagger d_H)^{-1/2}$ (defined in §2 of [@MW]); that of the zero modes contributes volume elements on the cohomology groups. The longitudinal modes of $B,C$ are treated by adding Faddeev-Popov ghost fields which contribute to determinant factors in the numerator, and there are secondary and higher ghosts since $B^{(1)},C^{(1)}$ themselves contain redundancies.
We consider a special case when $\dim X=2l+1$ is odd and $H$ is a top-degree form (cf. §5.1 of [@MW]). If $B,C\in{\mathit{\Omega}}^{\bar1}(X)$, then $B\wedge d_HC=BdC$, and the theory is equivalent to an untwisted theory. We now assume that $B,C\in{\mathit{\Omega}}^{\bar0}(X)$. Then the bosonic determinant from the integration of the transverse modes is $${\operatorname{Det}}'{d_0^\dagger d_0+H^\dagger H\quad H^\dagger d_{2l}
\choose\quad\;d_{2l}^\dagger H\qquad\;\;d_{2l}^\dagger d_{2l}}^{\!\!-1/2\,}
\prod_{i=1}^{l-1}({\operatorname{Det}}'d_{2i}^\dagger d_{2i})^{-1/2},$$ where $d_i$ is $d$ on ${\mathit{\Omega}}^i(X)$ for $0\le i\le 2l+1$. The crucial feature in this case is that $H$ does not appear in the gauge transformations $$B\mapsto B+dB^{(1)},\qquad C\mapsto C+dC^{(1)}.$$ Moreover, we can choose $B^{(1)},C^{(1)}\in{\mathit{\Omega}}^{\bar1}(X)$ to be of degree $2l-1$ or less. Further redundancies in $B^{(1)},C^{(1)}$ are described by a hierarchy of gauge transformations $$B^{(i)}\mapsto B^{(i)}+dB^{(i+1)},\qquad
C^{(i)}\mapsto C^{(i)}+dC^{(i+1)},$$ where $B^{(i)},C^{(i)}\in{\mathit{\Omega}}^{\bar i}(X)$ are of degree $2l-i$ or less, for $1\le i\le2l-1$. The Faddeev-Popov procedure yields the determinant factors $$\begin{aligned}
&\prod_{i=0}^{2l}\Big[{\operatorname{Det}}'(d_{2l-i}^\dagger d_{2l-i})
{\operatorname{Det}}'(d_{2l-i-2}^\dagger d_{2l-i-2})\cdots\Big]^{(-1)^{i+1}} \\
&=\prod_{i=0}^l{\operatorname{Det}}'(d_{2i}^\dagger d_{2i})^{-l/2}
\prod_{i=0}^{l-1}{\operatorname{Det}}'(d_{2i+1}^\dagger d_{2i+1})^{(l+1)/2}.\end{aligned}$$ Taking into account the contribution of the zero modes, the partition function is $$Z_{\bar0}(X,H)=\tau(X,H)^{-1}\otimes\tau(X)^{\otimes(-l)}
\in\det H^\bullet(X,H)^{-1}\otimes\det H^\bullet(X)^{\otimes(-l)}.$$ Here $\tau(X)\in\det H^\bullet(X)$ is the classical Ray-Singer torsion [@RS]. The independence of the partition on the metric indicates that the quantum theory is also metric independent although it is necessary to use a metric in the definition.
It would be interesting to construct topological field theories when the flux form $H$ is of an arbitrary degree, when the manifold has a boundary [@Wu], and those related to the analytic torsion of other ${\mathbb{Z}}_2$-graded elliptic complexes such as the twisted Dolbeault complex.
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[^1]: V.M. and S.W. are supported in part by the Australian Research Council. S.W. is supported in part by the Research Grants Council of Hong Kong. We thank the referee for useful comments.
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abstract: 'Let $F=(f_1,..,f_q)$ be a polynomial dominating map from ${\mathbb C ^n}$ to ${\mathbb C ^q}$. In this paper we study the quotient ${{\cal{T}}^1(F)}$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $dR + \sum a_i df_i$, where $R,a_1,..,a_q$ are polynomials. We prove that ${{\cal{T}}^1(F)}$ is always a torsion ${ \mathbb C [t_1,...,t_q]}$-module. Then we determine under which conditions on $F$ we have ${{\cal{T}}^1(F)}=0$. As an application, we study the behaviour of a class of algebraic ${(\mathbb C ^p,+)}$-actions on ${\mathbb C ^n}$, and determine in particular when these actions are trivial.'
author:
- Philippe Bonnet
title: 'Relative exactness modulo a polynomial map and algebraic $\CP$-actions'
---
Section de Mathématiques, Université de Genève,\
2-4, rue du lièvre, 1211 Genève 24, Switzerland\
e-mail: Philippe.bonnet@math.unige.ch
Introduction
============
Let $F=(f_1,..,f_q)$ be a dominating polynomial map from ${\mathbb C ^n}$ to ${\mathbb C ^q}$ with $n>q$. Let $\Omega^k({\mathbb C ^n})$ be the space of polynomial differential $k$-forms on ${\mathbb C ^n}$. For simplicity, we denote by ${\mathbb C}[F]$ the algebra generated by $f_1,..,f_q$, and by ${\mathbb C}(F)$ its fraction field. Our purpose in this paper is to compare two notions of relative exactness modulo $F$ for polynomial 1-forms, and to deduce some consequences on some algebraic groups actions.
The first notion is the [*topological relative exactness*]{}. A polynomial 1-form $\omega$ is topologically relatively exact (in short: TR-exact) if $\omega$ is exact along the generic fibres of $F$. More precisely this means there exists a Zariski open set $U$ in ${\mathbb C ^q}$ such that, for any $y$ in $U$, the fibre $F^{-1}(y)$ is non-critical and non-empty, and $\omega$ has null integral along any loop $\gamma$ contained in $F^{-1}(y)$.
The second notion is the [*algebraic relative exactness*]{}. A polynomial 1-form is algebraically relatively exact (in short: AR-exact) if it is a coboundary of the De Rham relative complex of $F$ ([@Ma2]). Recall this complex is given by the spaces of relative forms: $$\Omega ^k _F= \Omega ^k({\mathbb C ^n})/\sum df_i \wedge
\Omega^{k-1}({\mathbb C ^n})$$ and the morphisms $d_F:\Omega ^k
_F\longrightarrow \Omega ^{k+1} _F$ induced by the exterior derivative.
The module of relative exactness of $F$ is the quotient ${{\cal{T}}^1(F)}$ of TR-exact 1-forms by AR-exact 1-forms. This is a ${\mathbb C}[F]$-module under the multiplication rule $(P(F),\omega)\mapsto P(F)\omega$.
For holomorphic germs, Malgrange implicitly compared these notions of relative exactness in [@Ma2]. He proved that the first relative cohomology group of the germ $F$ is zero if the singular set of $F$ has codimension $\geq 3$; in this case, ${\cal{T}}^1(F)$ is reduced to zero. In [@B-C], Berthier and Cerveau studied the relative exactness of holomorphic foliations, and introduced a similar quotient. For polynomials in two variables, Gavrilov proved that ${\cal{T}}^1(f)=0$ if every fibre of $f$ is connected and reduced ([@Ga]). Concerning polynomial maps, we first prove the following result.
\[torsion\] If $F$ is a dominating map, then ${{\cal{T}}^1(F)}$ is a torsion ${ \mathbb C [F]}$-module.
In other words, every TR-exact 1-form $\omega$ can be written as: $$P(F)\omega = dR +a_1df_1 + ..+ a_qdf_q$$ where $R,a_1,..,a_q$ are all polynomials. In [@B-D], the author in collaboration with Alexandru Dimca studied in a comprehensive way the torsion of this module for any polynomial function $f:{\mathbb C}^2 \rightarrow {\mathbb C}$. We are going to extend these results in any dimension and determine when ${{\cal{T}}^1(F)}$ is zero.
Let $F:X\rightarrow Y$ be a morphism of algebraic varieties, where $Y$ is equidimensionnal and $X$ may be reducible. A property ${\cal{P}}$ on the fibres of $F$ is [*$k$-generic*]{} if the set of points $y$ in $Y$ whose fibre $F^{-1}(y)$ does not satisfy ${\cal{P}}$ has codimension $>k$ in $Y$. A [*blowing-down*]{} is an irreducible hypersurface $V$ in ${\mathbb C ^n}$ such that $F(V)$ has codimension $\geq 2$ in ${\mathbb C ^q}$. If no such hypersurface exists, we say that $F$ has no blowing-downs. Finally $F$ is non-singular in codimension 1 if its singular set has codimension $\geq 2$. It is easy to prove that a non-singular map in codimension 1 has no blowing-downs.
The map $F$ is primitive if its fibres are 0-generically connected and 1-generically non-empty.
Then we show that a polynomial map $F$ is primitive if and only if every polynomial $R$ locally constant along the generic fibres of $F$ can be written as $R=S(F)$, where $S$ is a polynomial. So this definition extends the notion of primitive polynomial ([@D-P]).
\[quasi\] The map $F$ is quasi-fibered if $F$ is non-singular in codimension 1, its fibres are 1-generically connected and 2-generically non-empty. The map $F$ is weakly quasi-fibered if $F$ has no blowing-downs, its fibres are 1-generically connected and 2-generically non-empty.
\[quasi3\] Let $F$ be a primitive mapping. If $F$ is a quasi-fibered mapping, then ${{\cal{T}}^1(F)}=0$. If $F$ is weakly quasi-fibered, then every TR-exact 1-form $\omega$ splits as $\omega=dR + \omega_0$, where $R$ is a polynomial and $\omega_0 \wedge df_1 \wedge ..\wedge df_q=0$.
We apply these results to the study of algebraic ${(\mathbb C ^p,+)}$-actions on ${\mathbb C ^n}$. Such an action is a regular map $\varphi: {\mathbb C}^p \times {\mathbb C ^n}\rightarrow {\mathbb C ^n}$ such that $\varphi(u,\varphi(v,x))=\varphi(u+v,x)$ for all $u,v,x$. Geometrically speaking, $\varphi$ is obtained by integrating a system ${{\cal{D}}}=\{{ \partial}_1,..,{ \partial}_p\}$ of derivations on ${ \mathbb C [x_1,...,x_n]}$ that are pairwise commuting and locally nilpotent ([@Kr]), that is : $$\forall f \in { \mathbb C [x_1,...,x_n]}, \quad \exists k \in \mathbb{N}, \quad { \partial}_i
^k (f)=0$$ The ring of invariants ${ \mathbb C [x_1,...,x_n]}^{\varphi}$ is the set of polynomials $P$ such that $P\circ \varphi=P$. Finally $\varphi$ is [*free at the point $x$*]{} if the orbit of $x$ has dimension $p$, and [*free*]{} if it is free at any point of ${\mathbb C ^n}$. The set of points where $\varphi$ is not free is an algebraic set denoted ${\cal{NL}}(\varphi)$.
An algebraic ${(\mathbb C ^p,+)}$-action on ${\mathbb C ^n}$ satisfies condition $(H)$ if its ring of invariants is isomorphic to a polynomial ring in $(n-p)$ variables.
Under this condition, $\varphi$ is provided with a [*quotient map*]{} $F$ ([@Mu2]) defined as follows: If $f_1,.. ,f_{n-p}$ denote a set of generators of ${ \mathbb C [x_1,...,x_n]}^{\varphi}$, then: $$F: {\mathbb C ^n}\longrightarrow {\mathbb C}^{n-p},\; x\longmapsto (f_1(x),..,f_{n-p}(x))$$ The generic fibres of $F$ are orbits of the action, but this map need not define a topological quotient: For instance, it does not separate all the orbits. The action $\varphi$ is [*trivial*]{} if it is conjugate by a polynomial automorphism of ${\mathbb C ^n}$ to the action: $$\varphi_0(t_1,..,t_p;x_1,..,x_n)= (x_1 + t_1,..,x_p
+t_p,x_{p+1},..,x_n)$$ We are going to search under which conditions the actions satisfying $(H)$ are trivial. According to a result of Rentschler ([@Re]), every fix-point free algebraic $({\mathbb C},+)$-action on ${\mathbb C}^2$ is trivial. We know that $(H)$ is always satisfied for $({\mathbb C},+)$-actions on ${\mathbb C}^3$ ([@Miy]), but we still do not know if fixed-point free $({\mathbb C},+)$-actions on ${\mathbb C}^3$ are trivial ([@Kr]). In dimension $\geq 4$, the works of Nagata and Winkelmann ([@Kr],[@Wi]) prove that $(H)$ need not be satisfied. For $({\mathbb C},+)$-actions satisfying this condition, Deveney and Finston proved that $\varphi$ is trivial if its quotient map defines a locally trivial $({\mathbb C},+)$-fibre bundle on its image (\[D-F\]).
We are going to see how this last result extends via relative exactness. Let $\varphi$ be a ${(\mathbb C ^p,+)}$-action on ${\mathbb C ^n}$ satisfying $(H)$, and consider the following operators: $$\begin{array}{rcl}
[{{\cal{D}}}]: & (R_1,..,R_p) \longmapsto & \det((\partial_i(R_j))) \\ \\
J: & (R_1,..,R_p)\longmapsto & \det(dR_1,..,dR_p,df_1,..,df_{n-p})
\end{array}$$ We say that $[{{\cal{D}}}]$ (resp. $J$) vanishes at the point $x$ if, for any polynomials $R_1,..,R_p$, we have $[{{\cal{D}}}](R_1,..,R_p)(x)=0$ (resp. $J(R_1,..R_p)(x)=0$) . The zeros of $[{{\cal{D}}}]$ correspond to the points of ${\cal{NL}}(\varphi)$, and the zeros of $J$ are the singular points of $F$. We generalise Daigle’s jacobian formula for $({\mathbb C},+)$-actions (\[Da\]).
\[Daigle\] Let $\varphi$ be an algebraic ${(\mathbb C ^p,+)}$-action on ${\mathbb C ^n}$ satisfying condition $(H)$. Then there exists an invariant polynomial $E$ such that $[{{\cal{D}}}]=E \times J$.
From a geometric viewpoint, this means that ${\cal{NL}}(\varphi)$ is the union of an invariant hypersurface and of the singular set of $F$. In particular $E$ is constant if $codim \;
{\cal{NL}}(\varphi) \geq 2$.
\[Triv\] Let $\varphi$ be an algebraic ${(\mathbb C ^p,+)}$-action on ${\mathbb C ^n}$ satisfying condition $(H)$. If $E$ is constant and $F$ is quasi-fibered, then $\varphi$ is trivial.
Therefore the assumption “quasi-fibered” correspond to some regularity in the way that $F$ fibres the orbits. In particular the action is trivial if $F$ defines a topological quotient, i.e. if $F$ is smooth surjective and separates the orbits.
\[Triv2\] Let $\varphi$ be an algebraic $({\mathbb C},+)$-action on ${\mathbb C ^n}$ satisfying condition $(H)$. If $F$ is quasi-fibered, there exists a polynomial $P$ such that $\varphi$ is conjugate to the action $\varphi'(t;x_1,..,x_n)=(x_1 + tP(x_2,..,x_n),x_2,..,x_n)$.
\[Triv3\] Every algebraic $({\mathbb C}^{n-1},+)$-action $\varphi$ on ${\mathbb C ^n}$ such that $codim \; {\cal{NL}}(\varphi)\geq 2$ is trivial. In particular $\varphi$ is free.
We end up with counter-examples illustrating the necessity of the conditions of theorem \[Triv\] and its corollaries.
Proof of Proposition \[torsion\]
================================
In this section, we establish the first proposition announced in the introduction in two steps. First we describe a TR-exact 1-form $\omega$ on every generic fibre of $F$. Second we “glue” all these descriptions by using the uncountability of complex numbers. To that purpose, we use the following definitions. For any ideal $I$, we denote by $I\Omega^1({\mathbb C ^n})$ the space of polynomial 1-forms with coefficients in $I$. We introduce the equivalence relation: $$\omega \simeq \omega'\;[I] \Longleftrightarrow \omega - \omega '
\in d\Omega^0({\mathbb C ^n})+ \sum \Omega^0({\mathbb C ^n})df_i+I\Omega^1({\mathbb C ^n})$$ This equivalence is compatible with the structure of ${\mathbb C}[F]$-module given by the natural multiplication, since $d\Omega^0({\mathbb C ^n})+ \sum
\Omega^0({\mathbb C ^n})df_i$ and $I\Omega^1({\mathbb C ^n})$ are both ${\mathbb C}[F]$-modules.
\[torsion2\] Let $F^{-1}(y)$ be a non-empty non-critical fibre of $F$, where $y=(y_1,...,y_q)$. A polynomial 1-form $\omega$ is exact on $F^{-1}(y)$ if and only if there exists a polynomial $R$ and some polynomial 1-forms $\eta_1,..,\eta_q$ such that $\omega = dR +\sum_i(f_i - y_i)\eta_i$.
[[*Proof:* ]{}]{}Since $\omega$ is exact on $F^{-1}(y)$, it has an holomorphic integral $R$ on this fibre. Since $F^{-1}(y)$ is a smooth affine variety, $R$ is a regular map by Grothendieck’s Theorem ([@Di], p. 182). In other words, $R$ is the restriction to $F^{-1}(y)$ of a polynomial, which will also be denoted by $R$. The $(q+1)$-form $(\omega - dR) \wedge df_1 \wedge ..\wedge df_q$ vanishes on $F^{-1}(y)$. Since $F^{-1}(y)$ is non-critical, $(f_1
- y_1),.., (f_q - y_q)$ define a local system of parametres at any point of $F^{-1}(y)$. So the ideal $((f_1 - y_1),...,(f_q - y_q))$ is reduced and we get: $$(\omega - dR) \wedge df_1 \wedge
..\wedge df_q \equiv 0 \;[f_1-y_1,..,f_q-y_q]$$ The $q$-form $df_1 \wedge ... \wedge df_q$ never vanishes on $F^{-1}(y)$. By de Rham Lemma ([@Sai]), there exist some polynomials $\alpha_i$ and some polynomial 1-forms $\eta_i$ such that: $$\omega - dR=
\sum_{i=1} ^q \alpha_idf_i+ \sum_{i=1} ^q (f_i -y_i)\eta_i$$ which can be rewritten as: $$\omega = d\left(R + \sum_{i=1} ^q
\alpha_i(f_i-y_i) \right) + \sum_{i=1} ^q (f_i
-y_i)(\eta_i-d\alpha_i)$$
$\blacksquare$
[**[Proof of Proposition \[torsion\]]{}:**]{}
Let $\omega$ be a TR-exact 1-form. Let us show there exists a non-zero polynomial $P$ such that $P(F)\omega
\simeq 0\;[(0)]$. By lemma \[torsion2\], there exists a non-empty Zariski open set $U$ in ${\mathbb C ^q}$ such that, for any $y=(y_1,..,y_q)$ in $U$: $$\omega \simeq 0\; [f_1-y_1,..,f_q-y_q]$$ We proceed to an elimination of $f_1-y_1,..,f_q-y_q$. For any point $y=(y_{i+1},..,y_q)$ in ${\mathbb C}^{q-i}$, we denote by $I_i(y)$ the following ideal: $$I_i(y)=(f_{i+1}-y_{i+1},..,f_q-y_q)$$ By convention, ${\mathbb C}^0$ is the space reduced to a point, and $I_q(y)=(0)$. Let us show by induction on $i\leq q$ the following property:\
\
[*There exists a non-empty Zariski open set $U_i$ in ${\mathbb C}^{q-i}$ such that, for any point $y$ in $U_i$, there exists a non-zero polynomial $P$ in ${\mathbb C}[t_1,..,t_i]$ for which $P(f_1,..,f_i)\omega \simeq 0 \;[I_i(y)]$*]{}.\
\
This property is true for $i=0$. Assume it holds to the order $i<q$, and let $U_i$ be such a Zariski open set. We may assume that $U_i$ is a principal open set, i.e $U_i= \{f(y)\not=0 \}$. Write $f=\sum_{k\leq s} f_k(t_{i+2},..,t_q)t_{i+1} ^k$, and set $U_{i+1}=\{f_s(y')\not=0 \}$. Let $y'=(y_{i+2},...,y_q)$ be a point in $U_{i+1}$. For any $z$ such that $f(z,y')\not=0$, the point $y=(z,y')$ belongs to $U_i$. By induction, there exist a non-zero polynomial $P^z$ and a polynomial 1-form $\eta ^z$ such that: $$P^z(f_1,..,f_i)\omega \simeq (f_{i+1}-z)\eta ^z
\;[I_{i+1}(y')]$$ For any such $z$, fix a 1-form $\eta^z$ satisfying this equivalence. The system $\{\eta^z \}$ thus obtained is an uncountable subset of $\Omega^1({\mathbb C ^n})$. Since $\Omega^1({\mathbb C ^n})$ has countable dimension, these forms cannot be linearly independent. There exist some distinct values $z_1,..,z_m$ and some non-zero constants $(\beta_1,..,\beta_m)$ such that: $$\beta_1 \eta ^{z_1}+ ... + \beta_m\eta ^{z_m} = 0$$ Since the equivalence relation is compatible with the structure of ${\mathbb C}[F]$-module, we get with the previous relations: $$\left
(\sum_{j=1} ^m \beta_j P^{z_j}(f_1,..,f_i)\prod _{k\not=j}
(f_{i+1} - z_k) \right ) \omega \simeq 0 \;[I_{i+1}(y')].$$ None of the $\beta_j$ (resp. $P^{z_j}$) is zero by construction. Thus the polynomial $\tilde{P}$: $$\tilde{P}=\sum_{j=1} ^m \beta_j
P^{z_j}(t_1,..,t_i)\prod _{k\not=j}(t_{i+1} - z_k)$$ is non-zero, and satisfies the relation $\tilde{P}(f_1,...,f_{i+1})\omega
\equiv 0 \;[I_{i+1}(y')]$. Since we can perform this process for any point $y'$ in $U_{i+1}$, the induction is proved.
$\blacksquare$
A factorisation lemma
=====================
In this section, we prove an extension of the first Bertini’s theorem and Stein’s factorisation theorem (\[Sh\], p. 139 and \[Ha\], p. 280) to the case of reducible varieties. This result is certainly well-known but I could not find a proper reference for it. So I prefer to give a proof of it, based on Zariski’s Main Theorem.
Let $F:X\rightarrow Y$ be a dominating morphism of complex affine varieties, where $X$ is equidimensional and $Y$ is irreducible. Let $R$ be a regular map on $X$. Assume that:
- [The fibres of $F$ are generically connected,]{}
- [The restriction of $F$ to any irreducible component of $X$ is dominating,]{}
- [The map $G=(F,R)$ is everywhere singular on $X$.]{}
Then $R$ coincides on a dense open set of $X$ with $\alpha(F)$, where $\alpha$ is a rational map on $Y$. In this case, $R$ is said to factor through $F$.
[[*Proof:* ]{}]{}Since the map $G:X \rightarrow Y \times {\mathbb C}$ is everywhere singular, $G$ cannot be dominating. So there exists an element $P$ of ${\mathbb C}[Y][t]$ such that $P(F,R)=0$ on $X$. Note that $P$ has degree $>0$ with respect to $t$, because $F$ is a dominating map. Under the previous assumptions, there exists a Zariski open set $U$ in $Y$ such that:
- [For any irreducible component $X'$ of $X$, $U$ is contained in $F(X')$,]{}
- [For any point $y$ in $U$, $F^{-1}(y)$ is connected,]{}
- [For any point $y$ in $U$, the polynomial $P(y,t)$ is non-zero.]{}
Let $y$ be a point in $U$. Since $P(y,R)=0$ on $F^{-1}(y)$, $R$ is locally constant on $F^{-1}(y)$. Since $R$ is regular and $F^{-1}(y)$ is connected, $R$ is constant on $F^{-1}(y)$. So we can define the correspondence $\alpha: U\rightarrow \mathbb C$ that maps any point $y$ of $U$ to the unique value that takes $R$ on $F^{-1}(y)$. Consider its graph: $$Z=\{(y,\alpha(y)),y \in U
\}$$ If $X'$ is an irreducible component of $X$, then $Z$ coincides with $G(X'\cap F^{-1}(U))$. So $Z$ is constructible for the Zariski topology, and $\overline{Z}$ is irreducible. Therefore $\overline{Z}$ defines in $Y\times\mathbb C$ a rational correspondence from $Y$ to $\mathbb C$ in the sense of Zariski ([@Mu1], pp. 29-51). By Zariski’s Main Theorem, $\alpha$ coincides with a rational map on $Y$. Let $U'$ be an open set contained in $U$ where $\alpha$ is regular. Then $F^{-1}(U')$ is a dense open subset of $X$. Moreover $R$ and $\alpha (F)$ coincide on $F^{-1}(U')$ by construction.
$\blacksquare$
Blowing-downs and primitive mappings
====================================
In this section, we give some properties of blowing-downs and primitive mappings. For this class of maps, we will establish a [*division lemma*]{} (see section 5) that is the key-point for the proof of theorem \[quasi\]. Let $F$ be a polynomial dominating map from ${\mathbb C ^n}$ to ${\mathbb C ^q}$, and let $S(F)$ be its set of singular points. We introduce the following sets: $$\begin{array}{ccl}
B(F)& = & \{y \in \mathbb C ^q,\mbox{\it $F^{-1}(y)$ is non-empty and not
connected}\} \ \\
E(F) & = & \mbox{\it Union of blowing-downs of $F$} \ \\
I(F) & = &\{y \in \mathbb C^q,\mbox{\it $F^{-1}(y)$ is empty}\}
\end{array}$$ Let $H$ be the GCD of all $q$-minors of $dF$, and set: $$\omega _F = \frac{df_1\wedge .. \wedge df_q}{H}$$ Note that for all polynomials $P$ and $R$, we have $P(F) dR \wedge \omega _F =
d(P(F) R) \wedge \omega _F$. Since the sets $B(F),E(F),I(F)$ are all constructible for the Zariski topology, it makes sense to consider their codimensions. Recall that $F$ is primitive if its fibres are 0-generically connected and 1-generically non-empty, i.e. $codim\; B(F) \geq 1$ and $codim\; I(F)\geq 2$.
\[prim\] A polynomial map $F:{\mathbb C ^n}\rightarrow {\mathbb C ^q}$ is primitive if and only if any polynomial $R$ such that $dR \wedge \omega_F = 0$ belongs to ${\mathbb C}[F]$.
[[*Proof:* ]{}]{}Assume that $F$ is primitive. Let $R$ be a polynomial such that $dR \wedge {\omega_F}=0$. Then the map $G=(F,R)$ is everywhere singular. Since the generic fibres of $F$ are connected, $R$ factors through $F$ by the factorisation lemma. Let us set: $$R=b(F)/a(F)$$ where $a,b$ are relatively prime. Let us show by absurd that $a$ is constant. Assume not, and let $a'$ be an irreducible factor of $a$. For any point $y$ in $V(a') - I(F)$, there exists a point $x$ such that $F(x)=y$, which implies that $a(y)R(x)=b(y)=0$. So $b$ vanishes on $V(a') - I(F)$. Since $I(F)$ has codimension $\geq 2$ in ${\mathbb C ^n}$, $V(a') - I(F)$ is dense in $V(a')$ and $b$ vanishes on $V(a')$. By Hilbert’s Nullstellensatz, $a'$ divides $b$, contradicting the fact that $a$ and $b$ are relatively prime. Thus $a$ is constant and $R$ belongs to ${\mathbb C}[F]$.
Assume now that any polynomial $R$ such that $dR\wedge {\omega_F}=0$ belongs to ${\mathbb C}[F]$. The $q$-form ${\omega_F}$ is obviously non-zero, and the polynomials $f_i$ are algebraically independent. So $F$ is a dominating map. Let us prove first that $codim(B(F))\geq 1$. By Bertini first theorem (\[Sh\], p. 139), it suffices to show that ${\mathbb C}(F)$ is algebraically closed in ${\mathbb C}(x_1,...,x_n)$. Let $R$ be a rational fraction that is algebraic over ${\mathbb C}(F)$. Let $P(z,t_1,..,t_q)=\sum_{k\leq s} a_k(t_1,..,t_q)z^k$ be a nonzero polynomial such that $P(R,f_1,..,f_q)=0$. We choose $P$ of minimal degree with respect to $z$. Since $P(R,f_1,..,f_q)=0$, the denominator of $R$ divides $a_s(F)$. By derivation and wedge product, we get: $$\frac{\partial P}{\partial z}(R,f_1,..,f_q)dR \wedge {\omega_F}=0$$ Since $P$ has minimal degree, $dR \wedge {\omega_F}=0$ and $d(a_s(F)R) \wedge {\omega_F}=0$. As $a_s(F)R$ is a polynomial, it belongs to ${\mathbb C}[F]$ and $R$ lies in ${\mathbb C}(F)$.
Let us show by absurd that $codim(I(F))\geq 2$. Assume not, and let $C=V(f)$ be a codimension 1 irreducible component of $\overline{I(F)}$, where $f$ is reduced. Since the intersection $V(f) \cap F({\mathbb C ^n})$ has codimension $\geq 2$, there exists a polynomial $P$ vanishing on $V(f) \cap F({\mathbb C ^n})$ and not divisible by $f$. The function $P(F)$ vanishes on $V(f(F))$. By Hilbert’s Nullstellensatz, there exists an integer $n$ such that $P^n(F)$ is divisible by $f(F)$. The function $P^n/f$ is rational non-polynomial, and $R=P^n(F)/f(F)$ belongs to ${ \mathbb C [x_1,...,x_n]}$. Since $R$ satisfies the equation $dR \wedge {\omega_F}=0$, $R$ belongs to $\mathbb
C[F]$, hence a contradiction.
$\blacksquare$
For $q=1$, a mapping $F$ is primitive if and only if its generic fibres are connected. Indeed any non-constant polynomial map from ${\mathbb C ^n}$ to ${\mathbb C}$ has to be surjective. In this way, the definition of primitive mapping extends the notion of primitive polynomial (\[D-P\]).\
\
[**[Exemple 1:]{}**]{} The polynomial $F(x,y)=x^2$ is not primitive because its generic fibres are not connected. Note that $dx\wedge
d(x^2)=0$, but $x$ does not belong to ${\mathbb C}[x^2]$.\
\
[**[Exemple 2:]{}**]{} Consider the mapping $F: \mathbb C^3 \rightarrow
\mathbb C^2, (x,y,z)\mapsto (x,xy)$. The function $y$ satisfies the relation $dy\wedge dx \wedge d(xy) =0$ but does not belong to $\mathbb C[x,xy]$. So $F$ is not a primitive mapping although its generic fibres are connected. The obstruction lies in the fact that $\overline{I(F)}=\{(y_1,y_2), y_1=0\}$, so $codim(I(F))=1$.\
\
[**[Exemple 3:]{}**]{} Consider the mapping $F: \mathbb C^3
\rightarrow \mathbb C^2, (x,y,z)\mapsto (xy,zy)$. It is easy to see that $F$ is onto and that its generic fibres are isomorphic to $\mathbb C^*$. So $F$ is a primitive mapping.\
\
Recall that a blowing-down is an hypersurface of ${\mathbb C ^n}$ that is mapped by $F$ to a set of codimension $\geq 2$. For instance, the plane $\{y=0\}$ in ${\mathbb C}^3$ is a blowing-down of the map $F(x,y,z)=(xy,zy)$.
\[Blow\] Any blowing-down of $F$ is contained in $S(F)$.
[[*Proof:* ]{}]{}Let $V$ be a blowing-down of $F$, and let $W$ denote the Zariski closure of $F(V)$. Then $W$ is irreducible and there exists a dense open set $W'$ of $W$, consisting only of smooth points of $W$ and containing $F(V)$. So $V'=
F^{-1}(W')\cap V$ is a dense open set of $V$. For any smooth point $x$ in $V'$, the differential of the restriction of $F$ to $V$ has rank $\leq dim W' \leq q-2$. The differential $dF(x)$ maps the hyperplane $T_x V$ to a space of dimension $\leq q-2$. So $dF(x)$ maps ${\mathbb C ^n}$ to a space of dimension $\leq q-1$, and $F$ is singular at $x$. Since any smooth point of $V'$ is a singularity of $F$ and $S(F)$ is closed, we have the inclusion $V\subset S(F)$.
$\blacksquare$
The division lemma
==================
In this section, we are going to establish the essential tool for the proof of theorem \[quasi\]. Let $\omega$ be a TR-exact 1-form $\omega$. By proposition \[torsion\], there exists a non-zero polynomial $P$ in ${ \mathbb C [t_1,...,t_q]}$, and some polynomials $R,a_1,...,a_q$ in ${ \mathbb C [x_1,...,x_n]}$ such that: $$P(F)\omega = dR+ a_1df_1+...+a_qdf_q$$ By using the wedge product with $\omega_F$, we get : $$dR \wedge \omega_F=P(F)\omega \wedge \omega_F\equiv 0\;[P(F)]$$ Assume there exist some polynomials $S,b_1,..,b_q$ such that $\omega=dS + \sum_i b_idf_i$. By an obvious computation, we get $\omega \wedge {\omega_F}= dS \wedge {\omega_F}$ and $d(R-P(F)S)\wedge {\omega_F}=0$. Since $F$ is primitive, there exists a polynomial $A$ such that $R=A(F) + P(F)S$.
More generally, let $R$ be a polynomial satisfying the equation $dR \wedge \omega_F\equiv 0\;[P(F)]$. $R$ is said to be [*${\cal{E}}$-divisible*]{} by $P(F)$ if there exist some polynomials $A$ and $S$ such that $R=A(F) + P(F)S$. In this section we are going to determine under which conditions a polynomial $R$ satisfying this equation is ${\cal{E}}$-divisible by $P(F)$.\
\
[**[Division lemma]{}**]{}
*Let $F$ be a primitive mapping from ${\mathbb C ^n}$ to ${\mathbb C ^q}$. Let $P$ be an element of ${ \mathbb C [t_1,...,t_q]}$, and $R$ a polynomial in ${ \mathbb C [x_1,...,x_n]}$ satisfying the equation $dR \wedge \omega_F \equiv 0\;[P(F)]$. Assume that:*
- [$V(P)\cap B(F)$ has codimension $\geq 2$ in ${\mathbb C ^q}$,]{}
- [$V(P(F)) \cap E(F)$ has codimension $\geq 2$ in ${\mathbb C ^n}$,]{}
- [$V(P)\cap I(F)$ has codimension $\geq 3$ in ${\mathbb C ^q}$.]{}
Then $R$ is ${\cal{E}}$-divisible by $P(F)$.
The weak division lemma
-----------------------
In this subsection, we are going to establish a weak version of the division lemma. A polynomial $R$ is said to be [*weakly ${\cal{E}}$-divisible*]{} by $P(F)$ if there exists a polynomial $B$ coprime to $P$ such that $B(F)R$ is ${\cal{E}}$-divisible by $P(F)$.\
\
[**[Weak division lemma]{}**]{}
*Let $F$ be a primitive mapping from ${\mathbb C ^n}$ to ${\mathbb C ^q}$. Let $P$ be an irreducible polynomial of ${ \mathbb C [t_1,...,t_q]}$. Let $R$ be a polynomial in ${ \mathbb C [x_1,...,x_n]}$ satisfying the equation $dR \wedge \omega_F \equiv 0\;[P(F)]$. Assume that:*
- [$V(P)\cap B(F)$ has codimension $\geq 2$ in ${\mathbb C ^q}$,]{}
- [$V(P(F)) \cap E(F)$ has codimension $\geq 2$ in ${\mathbb C ^n}$,]{}
Then $R$ is weakly ${\cal{E}}$-divisible by $P(F)$.
\
\
The proof splits in two steps. Consider a polynomial $R$ satisfying the equation $dR \wedge \omega_F \equiv 0\;[P(F)]$. First we show that its restriction to $V(P(F))$ factors through $F$. So there exist two polynomials $A,B$, with $B$ coprime to $P$, such that $B(F)R-A(F)$ vanishes on $V(P(F))$. If $h_1 ^{n_1}.. h_r ^{n_r}$ is the irreducible decomposition of $P(F)$ in ${ \mathbb C [x_1,...,x_n]}$, then $h_1 .. h_r $ divides $B(F)R-A(F)$. Second we prove that every factor $h_i$ divides $B(F)R-A(F)$ with multiplicity $\geq n_i$.
Let $P$ be an irreducible polynomial in ${ \mathbb C [t_1,...,t_q]}$. Let $h$ be an irreducible factor of $P(F)$. Let $R$ be a polynomial satisfying the equation $dR \wedge {\omega_F}\equiv 0\;[h]$. Then the map $G:V(h)\rightarrow V(P)\times {\mathbb C}, \;x\mapsto (F(x),R(x))$ is everywhere singular.
[[*Proof:* ]{}]{}It suffices to show that the collection of 1-forms $dR,dh,df_1,..,df_q$ has rank $\leq q$ at any point $x$ of $V(h)$. We are going to check that whenever you choose $q+1$ forms in this collection, their wedge product is divisible by $h$. Consider the first case, when this wedge product contains all the forms $df_1,..,df_q$. Then it is either equal to $dR \wedge df_1 \wedge
..\wedge df_q$ or to $dh \wedge df_1 \wedge ..\wedge df_q$. By assumption $dR \wedge df_1 \wedge ..\wedge df_q$ is divisible by $h$. To see that the second one is divisible by $h$, factor $P(F)=Qh^m$, where $Q$ is coprime to $h$ and $m\geq 1$. By wedge product, we get: $$d P(F) \wedge df_1 \wedge ..\wedge df_q =
mh^{m-1}Q dh \wedge df_1 \wedge ..\wedge df_q + h^{m}dQ \wedge
df_1 \wedge ..\wedge df_q =0$$ This yields $Q dh \wedge df_1
\wedge ..\wedge df_q\equiv 0[h]$. Since $Q$ is coprime to $h$, we find: $$dh \wedge df_1 \wedge ..\wedge df_q\equiv 0[h]$$ Consider now the second case, when $dR$ and $dh$ appear in the wedge product. Assume first that $q>1$. Up to a reordering of the forms $df_i$, we may assume that this wedge product is equal to $dR \wedge dh \wedge df_2 \wedge ..\wedge df_q$. Since $P(F)=
Qh^m$ where $Q$ is coprime to $h$, we get by derivation: $$d\{P(F)\}=\sum_{i=1} ^q \frac{{ \partial}P}{{ \partial}t_i}(F)df_i \equiv 0 \;
[h^{m-1}]$$ By wedge product, we find: $$\frac{{ \partial}P}{{ \partial}t_1}(F)H{\omega_F}= \frac{{ \partial}P}{{ \partial}t_1}(F)df_1 \wedge ..\wedge df_q
= d\{P(F)\}\wedge df_2 \wedge ..\wedge df_q \equiv 0 \; [h^{m-1}]$$ By construction, the coefficients of ${\omega_F}$ have no common factors. Thus $h^{m-1}$ divides ${ \partial}P /{ \partial}t_1 (F)H$. Then write: $$\frac{{ \partial}P}{{ \partial}t_1}(F)H dR \wedge {\omega_F}= dR \wedge
d\{P(F)\}\wedge df_2 \wedge ..\wedge df_q = dR \wedge
d\{Qh^m\}\wedge df_2 \wedge ..\wedge df_q$$ Since $dR \wedge {\omega_F}$ is divisible by $h$, we get: $$dR \wedge d\{Qh^m\}\wedge df_2
\wedge ..\wedge df_q\equiv 0\; [h^m]$$ which leads to: $$mQh^{m-1}dR \wedge dh \wedge df_2 \wedge ..\wedge df_q\equiv 0\;
[h^m]$$ Since $Q$ is coprime to $h$, we deduce: $$dR \wedge dh
\wedge df_2 \wedge ..\wedge df_q\equiv 0\; [h]$$ If $q=1$, we do the same computation and forget the wedge product with $df_2
\wedge ..\wedge df_q$.
$\blacksquare$
\[divv\] Let $P$ be an irreducible polynomial in ${ \mathbb C [t_1,...,t_q]}$. Let $h_1 ^{n_1}.. h_r ^{n_r}$ be the irreducible decomposition of $P(F)$ in ${ \mathbb C [x_1,...,x_n]}$. Let $R$ be a polynomial such that $dR \wedge {\omega_F}\equiv 0\; [h_1.. h_r]$. Assume that:
- [$V(P)\cap B(F)$ has codimension $\geq 2$ in ${\mathbb C ^q}$,]{}
- [$V(P(F)) \cap E(F)$ has codimension $\geq 2$ in ${\mathbb C ^n}$.]{}
Then there exist two polynomials $A,B$, where $B$ is coprime to $P$, such that $B(F)R -A(F)$ is divisible by $h_1.. h_r$.
[[*Proof:* ]{}]{}By the previous lemma applied to all the irreducible components of $V(P(F))$, we can see that the map: $$G:
V(P(F))\rightarrow V(P)\times {\mathbb C}, \; x\mapsto (F(x),R(x))$$ is singular. Since $V(P(F))\cap E(F)$ has codimension $\geq 2$, none of the hypersurfaces $V(h_i)$ is a blowing-down. So $F$ maps every $V(h_i)$ densely on $V(P)$. Since $V(P)\cap B(F)$ has codimension $\geq 2$, the generic fibres of $F:V(P(F))\rightarrow V(P)$ are connected. By the factorisation lemma, there exists a rational map $\alpha$ on $V(P)$ such that $R=\alpha(F)$ on $V(P(F))$. Write $\alpha$ as $A/B$, where $B$ is coprime to $P$. The polynomial $B(F)R -A(F)$ vanishes on $V(P(F))$. By Hilbert’s Nullstellensatz, it is divisible by $h_1.. h_r$.
$\blacksquare$
[**[Proof of the weak division lemma:]{}**]{}
Let $P$ be an irreducible polynomial in ${ \mathbb C [t_1,...,t_q]}$. Let $h_1 ^{n_1}.. h_r ^{n_r}$ be the irreducible decomposition of $P(F)$ in ${ \mathbb C [x_1,...,x_n]}$. Let $R$ be a polynomial such that $dR \wedge {\omega_F}\equiv 0\; [P(F)]$. Then $R$ satisfies the equation: $$dR \wedge
{\omega_F}\equiv 0\; [h_1 .. h_r ]$$ By the previous lemma, there exist some polynomials $A,B$, where $B$ is coprime to $P$, such that $S=B(F)R-A(F)$ is divisible by $h_1..h_r $. Factor $S$ as $S_0h_1
^{k_1}.. h_r ^{k_r}$, where $S_0$ is coprime to each $h_i$. Let us show by absurd that $k_i\geq n_i$ for any $i$.
Assume there exists an index $i$ such that $k_i/n_i <1$. Let $i_0$ be an index for which the ratio $k_i/n_i$ is minimal, and let $u/v$ be its irreducible decomposition. By construction, we have $0<u/v<1$. The function: $$L=S^v /P(F)^u=S_0 ^v h_1 ^{vk_1 -
un_1}.. h_r ^{vk_r - un_r}$$ is polynomial, since $u/v \leq
k_i/n_i \Rightarrow vk_i - un_i\geq 0$. Moreover $L$ satisfies the equation $dL \wedge {\omega_F}\equiv 0\;[h_1..h_r]$. Indeed if $vk_i -
un_i>0$, then $L$ is divisible by $h_i$ and $L=L_i h_i$. We set $P(F)=P_ih_i ^{n_i}$, where $P_i$ is coprime to $h_i$. By an easy computation, we get: $$dP(F) \wedge {\omega_F}= P_i n_i h_i ^{n_i
-1}dh_i \wedge {\omega_F}+ h_i ^{n_i}dP_i \wedge {\omega_F}=0$$ Since $P_i$ is coprime to $h_i$, we deduce $dh_i \wedge {\omega_F}\equiv 0\;[h_i]$, and this implies: $$dL \wedge {\omega_F}=L_idh_i \wedge {\omega_F}+ h_idL_i
\wedge {\omega_F}\equiv 0\;[h_i]$$ If $vk_i - un_i=0$, set $S=S_i h_i
^{k_i}$. By derivation and wedge product, we get: $$SdL \wedge
{\omega_F}= S_i h_i ^{k_i}dL \wedge {\omega_F}= vLdS \wedge {\omega_F}$$ By an easy computation, we obtain: $$dS \wedge {\omega_F}=B(F)dR \wedge {\omega_F}\equiv
0 \;[h_i ^{n_i}]$$ which implies: $$S_i dL \wedge {\omega_F}\equiv 0
\;[h_i ^{n_i -k_i}]$$ Since $n_i -k_i >0$ and $S_i$ is coprime to $h_i$, we deduce $dL \wedge {\omega_F}\equiv 0 \;[h_i]$. Thus $dL \wedge
{\omega_F}$ is divisible by $h_1..h_r$. By lemma \[divv\], there exist two polynomials $A',B'$, where $B'$ is coprime to $P$, such that $B'(F)L - A'(F)\equiv 0\;[h_1..h_r]$.
Let us show by absurd that $vk_i - un_i=0$ for any $i$. Assume that $h_i$ divides $L$. By the previous relation, $h_i$ divides $A'(F)$. Since $V(h_i)$ is not a blowing-down and $P$ is irreducible, $A'$ is divisible by $P$, which implies: $$B'(F)L\equiv 0\;[h_1..h_r]$$ Since none of the $V(h_j)$ are blowing-downs and every $h_j$ divides $P(F)$, every $h_j$ is coprime to $B'(F)$. So $L$ is divisible by $h_1..h_r$, contradicting its construction.
Since $vk_i - un_i=0$, $v$ divides $n_i$ for any $i$. As $0<u/v<1$, $v$ is strictly greater than 1 and $P(F)=T^v$, where $T$ belongs to ${ \mathbb C [x_1,...,x_n]}$. This implies: $$d\{P(F)\} \wedge {\omega_F}= vT^{v-1} dT \wedge {\omega_F}=0$$ Since $F$ is primitive, $T$ belongs to ${\mathbb C}[F]$ by proposition \[prim\]. Therefore $P$ is the $v^{th}$ power of some polynomial, which contradicts the irreducibility of $P$.
$\blacksquare$
Proof of the division lemma
---------------------------
Let $R$ be a polynomial satisfying the equation $dR \wedge {\omega_F}\equiv 0\; [P(F)]$. From an analytic viewpoint, the weak division lemma asserts that $R$ coincides on $V(P(F))$ with $\alpha(F)$, where $\alpha$ is a rational function on $V(P)$. In order to prove the division lemma, we are going to show that $\alpha$ is regular if $V(P)\cap I(F)$ has codimension $ \geq 3$. In other words, we are going to eliminate the “poles” of $\alpha$.
Recall that an ideal $I$ in a local ring $R$ is ${\cal{M}}$-primary if $I$ contains some power of the maximal ideal ${\cal{M}}$ of $R$. We denote by ${\cal{O}}_{{\mathbb C ^q},y}$ the ring of germs of regular functions at the point $y$ in ${\mathbb C ^q}$. For simplicity, we set: $${\mathbb C}[[X]]={\mathbb C}[[x_1,..,x_n]] \quad \mbox{and} \quad {\mathbb C}[[T]]={\mathbb C}[[t_1,..,t_q]]$$
\[div2\] Let $I=(g_1,..,g_n)$ be an ${\cal{M}}$-primary ideal in ${\mathbb C}[[X]]$. If the classes of the formal series $\{e_1,..,e_{\mu}\}$ form a basis of the vector space ${\mathbb C}[[X]]/I$, then $\{e_1,..,e_{\mu}\}$ is a basis of the ${\mathbb C}[[g_1,..,g_n]]$-module ${\mathbb C}[[X]]$.
[[*Proof:* ]{}]{}Since $(g_1,..,g_n)$ is ${\cal{M}}$-primary, ${\mathbb C}[[X]]$ is a finitely generated ${\mathbb C}[[g_1,..,g_n]]$-module (\[Ab\]). By Nakayama lemma (\[Sh\], p. 283), $\{e_1,...,e_{\mu}\}$ forms a minimal set of generators of this module. Let us show by absurd that $e_1,...,e_{\mu}$ are ${\mathbb C}[[g_1,..,g_n]]$-linearly independent.
Assume there exist some formal series $a_i(y_1,..,y_n)$, not all equal to zero, such that $\sum_k a_k(g_1,..,g_n)e_k=0$. Up to a linear change of coordinates on $y_1,..,y_n$, which is equivalent to replacing $g_1,..,g_n$ by another set of formal series generating the same ideal, we may assume there exists an index $i$ for which $a_i(y_1,0,..,0)\not=0$. By setting $a_i(x_1,0,..,0)
=b_i(x_1)$, we find: $$b_1(g_1)e_1 + .. + b_{\mu}(g_1)e_{\mu}\equiv 0 \;[g_2,..,g_n]$$ Let $m$ be the minimum of the orders of all formal series $b_1,...,b_{\mu}$. Then $b_i(x_1) =x_1 ^m c_i(x_1)$ for any $i$, and $c_i(0)\not=0$ for at least one of them. Thus we get: $$g_1 ^m \{ c_1(g_1)e_1 + .. + c_{\mu}(g_1)e_{\mu}
\}\equiv 0 [g_2,..,g_n]$$ Since $(g_1,..,g_n)$ is ${\cal{M}}$-primary, $g_1,..,g_n$ is a regular sequence (\[Sh\], p. 227) and $g_1$ is not a zero-divisor modulo $[g_2,..,g_n]$. We deduce: $$c_1(0)e_1 + .. + c_{\mu}(0)e_{\mu}\equiv 0 [g_1,g_2,..,g_n]$$ So $c_1(0)=..=c_{\mu}(0)=0$, hence contradicting the fact that not all $c_i(0)$ are zero.
$\blacksquare$
\[div3\] Let $y$ be a point in ${\mathbb C ^q}$ such that the fibre $F^{-1}(y)$ is non-empty of dimension $(n-q)$. Let $P,B,A$ be three elements of ${ \mathbb C [t_1,...,t_q]}$ such that $A(F)$ belongs to the ideal $(P(F),B(F)){ \mathbb C [x_1,...,x_n]}$. Then $A$ belongs to $(P,B){\cal{O}}_{{\mathbb C ^q},y}$.
[[*Proof:* ]{}]{}Let $x$ be a point in $F^{-1}(y)$ where the fibre has local dimension $(n-q)$. For simplicity, we may assume $x=0$ and $y=0$. There exists a $q$-dimensional vector space, defined by some linear equations $l_1,...,l_{n-q}$ and intersecting locally $F^{-1}(0)$ only at 0. By Ruckert’s Nullstellensatz (\[Ab\]), the ideal $(f_1,..,f_q,l_1,..,l_{n-q})$ is ${\cal{M}}$-primary in the ring ${\mathbb C}[[X]]$. Let $\{e_1,..,e_{\mu}\}$ be a basis of the vector space ${\mathbb C}[[X]]/(f_1,..,f_q,l_1,..,l_{n-q})$ such that $e_1=1$. By lemma \[div2\], $\{e_1,..,e_{\mu}\}$ is a basis of the ${\mathbb C}[[f_1,..,l_{n-q}]]$-module $\mathbb C[[X]]$. Let $R,S$ be two polynomials in ${ \mathbb C [x_1,...,x_n]}$ such that $A(F)=P(F)R+Q(F)S$. If $R_1(f_1,..,l_{n-q})$ and $S_1(f_1,..,l_{n-q})$ denote their first coordinate in the basis $\{e_1,..,e_{\mu}\}$, we get: $$P(F)R_1(f_1,..,l_{n-q}) + B(F)S_1(f_1,..,l_{n-q})=A(F)$$ After reduction modulo $l_1,..,l_{n-q}$, this implies: $$P(F)R_1(F,0) +
B(F)S_1(F,0)=A(F)$$ Thus $A$ belongs to the ideal $(P,B)\mathbb
C[[T]]$. Since ${\cal{O}}_{{\mathbb C ^q},0}$ is a Zariski ring and $\mathbb
C[[T]]$ is its ${\cal{M}}$-adic completion, we get $(P,B){\mathbb C}[[T]]
\cap {\cal{O}}_{{\mathbb C ^q},0}=(P,B){\cal{O}}_{{\mathbb C ^q},0}$ ([@Ma1], pp. 171-172). So $A$ belongs to $(P,B){\cal{O}}_{{\mathbb C ^q},0}$.
$\blacksquare$
\[div4\] Let $P,B,A$ be three polynomials in ${ \mathbb C [t_1,...,t_q]}$ such that $A(F)$ belongs to $(P(F),B(F)){ \mathbb C [x_1,...,x_n]}$. If $V(P(F),B(F))$ has codimension $\geq 2$ and $V(P(F))\cap I(F)$ has codimension $\geq 3$, then $A$ belongs to $(P,B){ \mathbb C [t_1,...,t_q]}$.
[[*Proof:* ]{}]{}This lemma is obvious if $V(P,B)$ is empty. We assume it is not, and consider the varieties $X=V(P(F),B(F))$ and $Y=V(P,B)$. By assumption, $P(F)$ and $B(F)$ are coprime and $X$ is equidimensionnal of codimension 2 in ${\mathbb C ^n}$. Moreover $P,B$ are coprime and $Y$ is equidimensionnal of codimension 2 in ${\mathbb C ^q}$. As $V(P)\cap I(F)$ has codimension $\geq 3$, the restriction: $$F_R : X\longrightarrow Y, \; x \longmapsto F(x)$$ is a dominating map. We construct a dense open set $U$ in $Y$ such that $F^{-1}(y)$ has dimension $(n-q)$ for any $y$ in $U$. Let $X_i$ be any irreducible component of $X$. If $F(X_i)$ has codimension $\geq 3$, fix a dense open set $U_i$ in $Y$ that does not meet $F(X_i)$. If $F(X_i)$ has codimension 2, we apply the theorem on the dimension of fibres to $F_R: X_i \rightarrow \overline{F(X_i)}$. There exists an open set $V_i$ contained in $F(X_i)$ such that $F^{-1}(y)\cap X_i$ has dimension $(n-q)$ for any $y$ in $V_i$. If $U'$ is the intersection of all $U_i$ and $V'$ is the union of all $V_i$, then $U=U' \cap V'$ is a dense open set in $Y$, and $F^{-1}(y)$ has dimension $(n-q)$ for any $y$ in $U$.
By lemma \[div3\], $A$ belongs to $(P,B){\cal{O}}_{{\mathbb C ^q},y}$ for any $y$ in $U$. This means there exists a polynomial $\beta_y$ such that $\beta_y(y)\not=0$ and $\beta_y A$ belongs to $(P,Q){ \mathbb C [t_1,...,t_q]}$. The zero set of $P,B$ and the $\beta_y$, when $y$ runs through $U$, has codimension $\geq 3$ since it is contained in $Y-U$. The ideal $J$ generated by $P,B$ and the $\beta_y$ has depth $\geq 3$. Since ${ \mathbb C [t_1,...,t_q]}$ is catenary, $J$ contains a polynomial $\beta$ such that $P,B,\beta$ is a regular sequence. By construction $\beta A
\equiv 0\;[P,B]$. As $\beta$ is not a zero divisor modulo $(P,B)$, $A$ belongs to $(P,B){ \mathbb C [t_1,...,t_q]}$.
$\blacksquare$
[**[Proof of the division lemma:]{}**]{}
Let $R$ be a polynomial satisfying the equation $dR \wedge {\omega_F}\equiv 0 \; [P(F)]$. Assume that $V(P)\cap B(F)$ has codimension $\geq 2$, $V(P(F))\cap E(F)$ has codimension $\geq 2$ and $V(P)\cap I(F)$ has codimension $\geq 3$. By the weak division lemma, there exist two polynomials $A,B$, where $B$ is coprime to $P$, and a polynomial $S$ such that: $$B(F)R - A(F)=P(F)S$$ Let us show by absurd that $X=V(P(F),B(F))$ has codimension $\geq 2$. Assume that $X$ contains an hypersurface $V$. Then $F$ maps $V$ to $Y=V(P,B)$, which codimension is $\geq 2$ since $P$ and $B$ are coprime. So $V$ is a blowing-down, and this contradicts the assumption on $V(P(F))\cap E(F)$.
Since $A(F)$ belongs to $(P(F),B(F)){ \mathbb C [x_1,...,x_n]}$ and $V(P)\cap I(F)$ has codimension $\geq 3$, $A$ belongs to $(P,B){ \mathbb C [t_1,...,t_q]}$ by lemma \[div4\]. There exist some polynomials $P_1,B_1$ such that $A=PP_1 + BB_1$. Thus we deduce: $$B(F)\{R-B_1(F)\}=P(F)\{S - P_1(F)\}$$ Since $X=V(P(F),B(F))$ has codimension 2, $P(F)$ and $B(F)$ are coprime. So $P(F)$ divides $R-B_1(F)$ and the division lemma is proved.
$\blacksquare$
Proof of theorem \[quasi3\]
---------------------------
Let $F$ be a primitive mapping that is either quasi-fibered or weakly quasi-fibered. By definition, the following conditions hold:
- [$B(F)$ has codimension $\geq 2$ in ${\mathbb C ^q}$,]{}
- [$E(F)$ is empty,]{}
- [$I(F)$ has codimension $\geq 3$ in ${\mathbb C ^q}$.]{}
Let $\omega$ be a TR-exact 1-form. By proposition \[torsion\], there exists a non-zero polynomial $P$, and some polynomials $R,a_1,..,a_q$ such that: $$P(F)\omega= dR+ a_1df_1+...+a_qdf_q$$ By wedge product with ${\omega_F}$, we can see that $R$ satisfies the equation $dR \wedge {\omega_F}\equiv 0 \;[P(F)]$. According to the conditions given above, $V(P)\cap B(F)$ has codimension $\geq 2$ in ${\mathbb C ^q}$, $V(P(F))\cap E(F)$ is empty and $V(P)\cap I(F)$ has codimension $\geq 3$ in ${\mathbb C ^q}$. By the division lemma, there exist some polynomials $A$ and $S$ such that $R=A(F) + P(F)S$. Therefore a simple calculation yields: $$P(F)\omega= P(F)dS + \sum_{k=1} ^q
\left ( a_k + S \frac{\partial P}{\partial t_k}(F)+ \frac{\partial
A}{\partial t_k}(F) \right )df_k$$ Let $c_k$ denote the coefficient of $df_k$ in this sum. Then $\sum_k c_kdf_k$ is divisible by $P(F)$. If $\omega_0$ is that quotient, we can see: $$\omega_0\wedge df_1 \wedge ..\wedge df_q=0$$ which implies the second part of the theorem. If now $F$ is quasi-fibered, then it is non-singular in codimension 1. By De Rham Lemma ([@Sai]), $\omega_0$ can be written as $\sum_k d_k df_k$, where all $d_k$ are polynomials. Therefore $\omega$ is AR-exact.
$\blacksquare$
Recalls on ${(\mathbb C ^p,+)}$-actions
=======================================
An algebraic ${(\mathbb C ^p,+)}$-action $\varphi$ on an affine variety $X$ consists of a regular map $\varphi: {\mathbb C}^p \times X \rightarrow X$ such that: $$\forall (u,v) \in {\mathbb C}^p \times {\mathbb C}^p, \quad \forall
x \in X, \quad \varphi (u,\varphi (v,x))= \varphi (u+v,x)$$ We denote by ${\mathbb C}[X]^{\varphi}$ its ring of invariants, i.e. the space of regular functions $f$ such that $f\circ \varphi = f$. The action $\varphi$ can be defined as the composition of $p$ pairwise commuting algebraic $({\mathbb C},+)$-actions $\varphi_i$. These latter are the restriction of $\varphi$ to the $i^{th}$ coordinate of ${\mathbb C}^p$. To each $\varphi_i$ corresponds the derivation ${ \partial}_i=\varphi_i ^*(d/dt_i)_{t_i=0}$, which enjoys the remarkable property of being locally nilpotent (see the introduction). Moreover these derivations commute pairwise. Conversely if $\{{ \partial}_1,...,{ \partial}_p\}$ is a system of locally nilpotent pairwise commuting derivations, the exponential map: $$exp(t_1{ \partial}_1+..+t_p{ \partial}_p)(f)= \sum_{k\geq 0}
\frac{(t_1{ \partial}_1+..+t_p{ \partial}_p)^k(f)}{k!}$$ defines a morphism of algebras from ${\mathbb C}[X]$ to ${\mathbb C}[X]\otimes {\mathbb C}[t_1,..,t_p]$. This morphism induces a regular map $\varphi:{\mathbb C}^p \times X
\rightarrow X$ that is an ${(\mathbb C ^p,+)}$-action on $X$. In this case, $\varphi$ is said to be [*generated*]{} by $\{{ \partial}_1,..,{ \partial}_p\}$.
A commutative $p$-distribution ${{\cal{D}}}$ is a system of locally nilpotent pairwise commuting derivations ${ \partial}_1,...,{ \partial}_p$. Its ring of invariants ${\mathbb C}[X] ^{{{\cal{D}}}}$ is the intersection of the kernels of the ${ \partial}_i$ on ${\mathbb C}[X]$.
If $\varphi$ is generated by ${{\cal{D}}}$, then ${\mathbb C}[X] ^{{{\cal{D}}}}$ is the ring of invariants of $\varphi$. Indeed, by definition of $\varphi$ via the exponential map, a regular function $f$ is invariant by $\varphi$ if and only if ${ \partial}_i(f)=0$ for any $i$. Recall that the action $\varphi$ is free at $x$ if the stabilizer of $x$ is reduced to zero, or in other words if the orbit of $x$ has dimension $p$. Let $[{{\cal{D}}}]$ be the operator defined at the introduction. We introduce its evaluation at $x$: $$[{{\cal{D}}}](x): (R_1,..,R_p) \longmapsto det((\partial_i(R_j)))(x)$$
Let $\varphi$ be an algebraic ${(\mathbb C ^p,+)}$-action on $X$, and let ${{\cal{D}}}$ be its commutative $p$-distribution. Then $\varphi$ is not free at $x$ if and only if $[{{\cal{D}}}](x)$ is the null map.
[[*Proof:* ]{}]{}Assume first that $\varphi$ is not free at $x$. Let $(u_1,..,u_p)$ be a non-zero element of the stabilizer of $x$. Let $\varphi ^u$ be the $(\mathbb C , +)$-action defined by $\varphi ^u_t(y)=\varphi _{tu_1,..,tu_p}(y)$. Starting from the relation $\varphi ^u _1(x)=x$, we get by an obvious induction that $\varphi ^u _m(x)=x$ for any integer $m>0$. So $\varphi ^u _t(x)=x$ for any $t$ in ${\mathbb C}$, and $x$ is a fixed point of $\varphi ^u$. For any regular function $R$, we get by derivation: $$\sum u_i \partial_i(R)(x)=0$$ which implies for any $p$-uple $(R_1,..,R_p)$: $$[{{\cal{D}}}](x)(R_1,..,R_p)=det((\partial_i(R_j)))(x)=0$$ Assume now that $[{{\cal{D}}}](x)$ is the null map. Let $({ \partial}_i)_x$ be the evaluation map of ${ \partial}_i$ at $x$, i.e. the map $R\mapsto { \partial}_i(R)(x)$. As ${\mathbb C}$-linear forms on ${\mathbb C}[X]$, the $({ \partial}_i)_x$ are not linearly independent. There exists a non-zero $p$-uple $(u_1,..,u_p)$ such that $\sum_i u_i ({ \partial}_i)_x=0$. Since the $\partial_i$ are locally nilpotent and commute pairwise, the derivation $\delta=u_1 ({ \partial}_1) +..+u_p ({ \partial}_p)$ is itself locally nilpotent. So $\delta$ generates the action $\varphi ^u$ defined by $\varphi ^u_t(y)=\varphi _{tu_1,..,tu_p}(y)$. Since $\sum_i u_i ({ \partial}_i)_x=0$, $x$ is a fixed point of $\varphi ^u$ as can be seen via the exponential map. Therefore the stabilizer of $x$ is not reduced to zero.
$\blacksquare$
Let ${{\cal{D}}}=\{{ \partial}_1,..,{ \partial}_p\}$ be a commutative $p$-distribution on ${\mathbb C}[X]$. Since the exponential map defines a morphism of algebras, the map: $$deg_D : {\mathbb C}[X] \longrightarrow \mathbb{N} \cup \{-\infty \},\;
f \longmapsto deg_{t_1,..,t_p} \left \{ exp(t_1{ \partial}_1 +..+t_p{ \partial}_p)(f) \right
\}$$ satisfies all the axioms of a degree function: This is the [*degree relative to ${{\cal{D}}}$*]{}. By construction, the ring of invariants of ${{\cal{D}}}$ is the set of regular functions of degree $\leq 0$. If $A$ is a domain, we denote by $Fr(A)$ its fraction field. The following lemma is due to Makar-Limanov (\[M-L\]).
\[Makar\] Let $A$ be a domain of characteristic zero. Let ${ \partial}$ be a non-zero locally nilpotent derivation on $A$ and let $A^{{ \partial}}$ be its kernel. Then $Fr(A)$ is isomorphic to $Fr(A^{{ \partial}})(t)$. In particular, for any subfield $k$ of $Fr(A^{{ \partial}})$, the transcendence degrees satisfy the relation: $$deg tr_k \{Fr(A^{{ \partial}})\}= deg tr_k \{Fr(A)\}-1$$
[[*Proof:* ]{}]{}Since ${ \partial}$ is non-zero locally nilpotent, there exists an element $f$ of $A$ such that ${ \partial}(f)\not=0$ and ${ \partial}^2 (f)=0$. So $g={ \partial}(f)$ is invariant. It is then easy to check by induction on $p$ that every element $P$ of $A$, of degree $p$ for ${ \partial}$, can be written in a unique way as $g^p P = a_0 + ..+ a_p f^p$, where all the $a_i$ are invariant.
$\blacksquare$
We end these recalls with the [*factorial closedness*]{} property, which is essential for rings of invariants (\[Da\],\[De\]).
Let $B$ a UFD and let $A$ be a subring of $B$. $A$ is factorially closed in $B$ if every element $P$ of $B$ which divides a non-zero element $Q$ of $A$ belongs to $A$.
Let $X$ be an affine variety such that ${\mathbb C}[X]$ is a UFD. Let ${{\cal{D}}}$ be a commutative $p$-distribution on $X$. Then ${\mathbb C}[X]^{{{\cal{D}}}}$ is factorially closed in ${\mathbb C}[X]$.
[[*Proof:* ]{}]{}Let $Q$ be a non-zero element of ${\mathbb C}[X]^{{{\cal{D}}}}$, and let $P$ divide $Q$ in ${\mathbb C}[X]$. By considering the degree relative to ${{\cal{D}}}$, we get $deg_{{{\cal{D}}}}(Q)=deg_{{{\cal{D}}}}(P) + deg_{{{\cal{D}}}}(Q/P)=0$. This implies $deg_{{{\cal{D}}}}(P)=0$, and $P$ is invariant with respect to ${{\cal{D}}}$.
$\blacksquare$
Jacobian description of $p$-distributions
=========================================
Let $\varphi$ be an algebraic ${(\mathbb C ^p,+)}$-action on ${\mathbb C ^n}$, satisfying the condition $(H)$. Let ${{\cal{D}}}$ be its commutative $p$-distribution, and let $F$ be its quotient map. In this section we are going to prove proposition \[Daigle\]. The main idea is to construct a system of rational coordinates for which calculations will be simple. We obtain this system by adding some polynomials $s_i$ to $f_1,..,f_{n-p}$. By analogy with $(\mathbb C,+)$-actions, we denote them as “rational slices” ([@Da],[@D-F]). With these coordinates, we show there exists an invariant fraction $E$ such that $[{{\cal{D}}}]=E\times J$, and there only remains to show that $E$ is a polynomial.
Let ${{\cal{D}}}$ be a commutative $p$-distribution on ${ \mathbb C [x_1,...,x_n]}$. A diagonal system of rational slices is a collection $\{s_1,..,s_p\}$ of polynomials such that the matrix $({\partial}_i(s_j))$ is diagonal and all its diagonal coefficients are non-zero invariant with respect to ${{\cal{D}}}$.
\[Daigle5\] Every commutative $p$-distribution ${{\cal{D}}}$ satisfying the condition $(H)$ admits a diagonal system of rational slices $\{s_1,...,s_p\}$.
[[*Proof:* ]{}]{}Let ${{\cal{D}}}_k$ be the commutative $(p-1)$-distribution $\{{ \partial}_1,..,{ \partial}_{k-1},{ \partial}_{k+1},..,{ \partial}_p \}$, and let ${ \mathbb C [x_1,...,x_n]}^{{{\cal{D}}}_k}$ be its ring of invariants. By induction on lemma \[Makar\], we get: $$degtr_{{\mathbb C}} Fr({ \mathbb C [x_1,...,x_n]}^{{{\cal{D}}}_k})\geq (n-p+1)$$ Since ${ \mathbb C [x_1,...,x_n]}^{{{\cal{D}}}}$ is isomorphic to a polynomial ring in $(n-p)$ variables, ${ \partial}_k$ cannot be identically zero on ${ \mathbb C [x_1,...,x_n]}^{{{\cal{D}}}_k}$. For any $k$, there exists a polynomial $s_k$ such that ${ \partial}_k(s_k)\not=0$, ${ \partial}_k ^2(s_k)=0$ and ${ \partial}_i(s_k)=0$ if $i\not=k$. The collection $\{s_1,...,s_p\}$ is a diagonal system of rational slices.
$\blacksquare$
Let ${{\cal{D}}}$ be a commutative $p$-distribution satisfying the condition $(H)$. Let $\{s_1,..,s_p\}$ be a diagonal system of rational slices. Then the map $G=(s_1,..,s_p,f_1,..,f_{n-p})$ is dominating.
[[*Proof:* ]{}]{}Let us show by absurd that $G$ is dominating. Assume that $G$ is not, and let $Q$ be an element of ${\mathbb C}[z_1,..,z_p,y_1,..,y_{n-p}]$ such that $Q(G)=0$. We assume $Q$ to have minimal degree with respect to the variables $z_1,..,z_p$. By derivation, we get for all $i$: $$\frac{{ \partial}Q}{{ \partial}z_i}(G) { \partial}_i(s_i)={ \partial}_i(Q(G))=0$$ Since ${ \partial}_i(s_i)\not=0$, this implies ${ \partial}Q /{ \partial}z_i (G)=0$. By minimality of the degree, we deduce that ${ \partial}Q /{ \partial}z_i=0$ for all $i$. So $Q$ belongs to ${\mathbb C}[y_1,..,y_{n-p}]$. Therefore the $f_i$ are not algebraically independent, and we obtain: $$degtr_{{\mathbb C}} {\mathbb C}(F) < n-p$$ But ${\mathbb C}[F]$ is the ring of invariants of ${{\cal{D}}}$. By induction with lemma \[Makar\], we find that $degtr_{{\mathbb C}} {\mathbb C}(F)
\geq n-p$, hence a contradiction.
$\blacksquare$
Let ${{\cal{D}}}$ be a commutative $p$-distribution satisfying $(H)$. Let $\{s_1,..,s_p\}$ be a diagonal system of rational slices. Then ${ \mathbb C [x_1,...,x_n]}\subset {\mathbb C}(f_1,..,f_{n-p})[s_1,..,s_p]$.
[[*Proof:* ]{}]{}Let us show by induction on $r\geq 0$ that every polynomial of degree $r$ with respect to ${{\cal{D}}}$ belongs to ${\mathbb C}(f_1,..,f_{n-p}) [s_1,..,s_p]$. For $r=0$, this is obvious because every polynomial of degree zero is invariant, and belongs to ${\mathbb C}[f_1,..,f_{n-p}]$. Assume the property holds to the order $r$. Let $R$ be a polynomial of degree $r+1$ with respect to ${{\cal{D}}}$. By definition, the polynomials ${ \partial}_i(R)$ have all degree $\leq r$. By induction, there exist some elements $P_i$ of ${\mathbb C}(y_1,..,y_{n-p})[z_1,..,z_p]$ such that ${ \partial}_i(R)=P_i(G)$ for all $i$. Since ${{\cal{D}}}$ is commutative, we get for all $(i,j)$: $$\frac{{ \partial}P_j}{{ \partial}z_i}(G){ \partial}_i(s_i)={ \partial}_i \circ { \partial}_j (R)=
{ \partial}_j \circ { \partial}_i (R)=\frac{{ \partial}P_i}{{ \partial}z_j}(G){ \partial}_j(s_j)$$ By construction, there exists a non-zero polynomial $S_i$ in ${\mathbb C}[y_1,.., y_{n-p}]$ such that ${ \partial}_i(s_i)=S_i(F)$. Since $G$ is dominating, this yields for all $(i,j)$: $$S_i\frac{{ \partial}P_j}{{ \partial}z_i}=S_j\frac{{ \partial}P_i}{{ \partial}z_j}$$ The differential 1-form $\omega=\sum P_i/S_idz_i$ is polynomial in the variables $z_i$. By the above equality, $\omega$ is closed with respect to $z_i$. So $\omega$ is exact and there exists an element $P$ of ${\mathbb C}(y_1,..,y_{n-p})[z_1,...,z_p]$ such that $\omega=dP$. Therefore ${ \partial}_i(R - P\circ G)=0$ for all $i$, and the function $R - P\circ G$ is rational and invariant with respect to ${{\cal{D}}}$. Since the ring of invariants of ${{\cal{D}}}$ is factorially closed, $R -
P\circ G$ belongs to ${\mathbb C}(f_1,..,f_{n-p})$. So $R$ belongs to ${\mathbb C}(f_1,..,f_{n-p})[s_1,..,s_p]$, hence proving the induction.
$\blacksquare$
Following exactly the same argument, we can prove the equality: $${ \mathbb C [x_1,...,x_n]}={\mathbb C}[f_1,..,f_{n-p}][s_1,..,s_p]$$ if the matrix $({ \partial}_i(s_j))$ is the identity. In this case $G$ is an algebraic automorphism. In any case, the previous lemma asserts that $G$ is always a birational automorphism of ${\mathbb C ^n}$.
\[Daigle3\] Let ${{\cal{D}}}$ be a commutative $p$-distribution satisfying $(H)$. Let $\{s_1,...,s_p\}$ be a diagonal system of rational slices. Then $\partial_1(s_1)..\partial_p(s_p)\times J= J(s_1,...,s_p)\times
[{{\cal{D}}}]$.
[[*Proof:* ]{}]{}For any $p$-uple of polynomials $(R_1,...,R_p)$, there exist some rational functions $P_i$ such that $R_i = P_i(G)$. On one hand, we get by the chain rule: $$\begin{array}{ccl}
J(R_1,..,R_p)& = & \det(d(P_1,..,P_p,y_1,..,y_{n-p}))(G) \det(dG) \\ \\
& = & \det(({ \partial}P_i /{ \partial}z_j))(G) J(s_1,..,s_p)
\end{array}$$ On the other hand, we have the following relation: $$[{{\cal{D}}}](R_1,...,R_p)= \det(({ \partial}_i(R_j)))=\det (( \sum_k { \partial}P_j / { \partial}z_k (G)
{ \partial}_i(s_k)))$$ Since the matrix $({ \partial}_i (s_j))$ is diagonal, this yields: $$[{{\cal{D}}}](R_1,...,R_p)= \det (({ \partial}P_i /{ \partial}z_j))(G){ \partial}_1 (s_1)..{ \partial}_p (s_p)$$ which implies the equality $\partial_1(s_1)..\partial_p(s_p) J(R_1,..,R_p)=
J(s_1,..,s_p)\times [{{\cal{D}}}](R_1,..,R_p)$.
$\blacksquare$
\[Daigle4\] Let ${{\cal{D}}}$ be a commutative $p$-distribution satisfying the condition $(H)$. Let $\{s_1,..,s_p\}$ be a diagonal system of rational slices. Then $J(s_1,..,s_p)$ is invariant.
[[*Proof:* ]{}]{}For simplicity, we denote by $J'$ the jacobian of every map from ${\mathbb C ^n}$ to ${\mathbb C ^n}$. Since $\{s_1,..,s_p\}$ is a diagonal system of rational slices, we get via the exponential map the relation $s_i\circ \varphi= s_i + t_i{ \partial}_i(s_i)$, and this yields: $$J'(s_1\circ\varphi,..,s_p\circ \varphi
,f_{1}\circ\varphi,..,f_{n-p}\circ\varphi)=
J'(s_1+t_1{ \partial}_1(s_1),..,s_p+t_p{ \partial}_p(s_p),f_{1},..,f_{n-p})$$ Since every ${ \partial}_i(s_i)$ belongs to ${\mathbb C}[F]$, we deduce: $$J'(s_1\circ\varphi,..,s_p\circ\varphi,f_{1}\circ\varphi,..,f_{n-p}
\circ\varphi)= J'(s_1,..,s_p,f_{1},..,f_{n-p})=J(s_1,...,s_p)$$ Moreover we find by the chain rule: $$J'(s_1\circ\varphi,..,s_p\circ\varphi,f_{1}\circ\varphi,..,f_{n-p}\circ\varphi)=
J'(s_1,..,s_p,f_{1},..,f_{n-p})(\varphi)\times J'(\varphi)$$ Since $\varphi$ is an automorphism of ${\mathbb C ^n}$ for any $(t_1,...,t_p)$, the polynomial $J'(\varphi)$ never vanishes. So it is non-zero constant. As $\varphi_{0,...,0}$ is the identity, $J'(\varphi)\equiv 1$ and that implies: $$J(s_1\circ\varphi,..,s_p\circ\varphi,f_{1}\circ\varphi,..,f_{n-p}\circ\varphi)=
J(s_1,..,s_p,f_{1},..,f_{n-p})(\varphi)$$ which leads to $J(s_1,...,s_p)(\varphi)= J(s_1,...,s_p)$. Thus $J(s_1,...,s_p)$ is invariant.
$\blacksquare$
[**[Proof of proposition \[Daigle\]:]{}**]{}
Let ${{\cal{D}}}$ be a commutative $p$-distribution satisfying the condition $(H)$. By lemmas \[Daigle3\] and \[Daigle4\], there exist two non-zero invariant polynomials $E_1$ and $E_2$ such that: $$E_1
\times [{{\cal{D}}}]=E_2 \times J$$ Since ${\mathbb C}[F]$ is factorially closed in ${ \mathbb C [x_1,...,x_n]}$, we may assume that $E_1$ and $E_2$ have no common factor. Let us show by absurd that $E_1$ is non-zero constant. Assume that $E_1$ is not constant. By definition of $J$, $E_1$ divides all the coefficients of the $(n-p)$-form $df_1\wedge..\wedge df_{n-p}$. So the hypersurface $V(E_1)$ is contained in the singular set of $F$. But that contradicts a result of Daigle ([@Da]), that asserts that $F$ is non-singular in codimension 1.
$\blacksquare$
Trivialisation of algebraic ${(\mathbb C ^p,+)}$-actions
========================================================
In this section, we are going to establish theorem \[Triv\]. The main idea is to refine a diagonal system of rational slices, in order to get the coordinate functions of an algebraic automorphism that conjugates $\varphi$ to the trivial action.\
\
[**[Proof of theorem \[Triv\]:]{}**]{} Let $\varphi$ be an algebraic ${(\mathbb C ^p,+)}$-action on ${\mathbb C ^n}$ satisfying the condition $(H)$. Assume that $E$ is constant and that the quotient map $F$ is quasi-fibered. Let $\{s_1,..,s_p\}$ be a diagonal system of rational slices. Such a system exists by lemma \[Daigle5\]. By proposition \[Daigle\], we have for any $(p-1)$-uple $(R_1,..,R_{i-1},R_{i+1},..,R_p)$: $$J(R_1,..,R_{i-1},s_i,R_{i+1},..,R_p)=[{{\cal{D}}}](R_1,..,R_{i-1},s_i,R_{i+1},..,R_p)/E$$ Let $P_i$ be the polynomial of ${\mathbb C}[t_1,..,t_{n-p}]$ such that ${ \partial}_i(s_i)= P_i(F)$. Since $E$ is constant and ${ \partial}_k(s_i)=0$ if $k\not=i$, the previous equality yields: $$J(R_1,..,R_{i-1},s_i,R_{i+1},..,R_p)\equiv 0 \;[P_i(F)]$$ If we replace $R_k$ by all the polynomials $x_1,..,x_n$, we can see that the coefficients of the differential form $ds_i \wedge df_1 \wedge
..\wedge df_{n-p}$ are all divisible by $P_i(F)$. By Daigle’s result ([@Da]), $F$ is non-singular in codimension 1. So the coefficients of $df_1 \wedge ..\wedge df_{n-p}$ have no common factor. Therefore $s_i$ satisfies the equation: $$ds_i \wedge
\omega_F \equiv 0 \;[P_i(F)]$$ By the division lemma, there exist some polynomials $A_i,S_i$ such that: $$s_i = A_i(F) + P_i(F)S_i$$ By an easy computation, we obtain that $({ \partial}_i(S_j))$ is the identity. By the remark following lemma \[Daigle3\], we have the equality: $${\mathbb C}[x_1,...,x_n]= {\mathbb C}[f_1,..,f_{n-p}][S_1,..,S_p]$$ which implies that $G=(S_1,..,S_p,f_1,..,f_{n-p})$ is an algebraic automorphism of ${\mathbb C ^n}$. Let $\varphi_0$ be the trivial action generated by the commutative $p$-distribution $\{{ \partial}/{ \partial}x_1,..,{ \partial}/{ \partial}x_p\}$. By using the exponential map, we find that $G\circ \varphi = \varphi_0 \circ G$. So $\varphi$ is trivial.
$\blacksquare$
[**[Proof of corollary \[Triv2\]:]{}**]{}
Let $\varphi$ be an algebraic $({\mathbb C},+)$-action on ${\mathbb C ^n}$ satisfying $(H)$, generated by the derivation ${ \partial}$. Assume that the quotient map is quasi-fibered. Since $F$ is nonsingular in codimension 1, the derivation $J$ is locally nilpotent and generates a $({\mathbb C},+)$-action $\varphi'$ such that $\cal{NL}(\varphi')$ has codimension $\geq 2$. By theorem \[Triv\], $\varphi'$ is trivial. Moreover via the automorphism of trivialisation, ${ \partial}$ is conjugate to $P(x_2,..,x_n){ \partial}/{ \partial}x_1$, where $E=P(F)$ is the factor of proposition \[Daigle\].
$\blacksquare$
[**[Proof of corollary \[Triv3\]:]{}**]{}
Let $\varphi$ be an algebraic $({\mathbb C}^{n-1},+)$-action on ${\mathbb C ^n}$, and assume that ${\cal{NL}}(\varphi)$ has codimension $\geq 2$. Then the factor $E$ of proposition \[Daigle\] is constant. Let us prove that $\varphi$ is trivial. By theorem \[Triv\], we only have to show that $\varphi$ satisfies the condition $(H)$ and that its quotient map is quasi-fibered.
Let $f$ be a non-constant invariant polynomial of minimal homogeneous degree on ${ \mathbb C [x_1,...,x_n]}$. Then $f-\lambda$ is irreducible for any $\lambda$. Indeed if $f-\lambda$ were reducible, all its irreducible factors would be invariant by factorial closedness. But that contradicts the minimality of the degree of $f$. Since all the fibres of $f$ are irreducible, they are reduced and connected. So $f$ is quasi-fibered, and there only remains to prove that $f$ generates the ring of invariants of $\varphi$.
Let us show by induction on $r$ that any invariant polynomial $P$ of homogeneous degree $\leq r$ belongs to ${\mathbb C}[F]$. This is obvious for $r=0$. Assume this is true to the order $r$, and let $P$ be an invariant polynomial of degree $\leq r+1$. Let $x$ be a point in ${\mathbb C ^n}$ where $\varphi$ is free, and set $y=f(x)$. Since $P$ is invariant, $P$ is constant on the orbit of $x$. Since this orbit has dimension $(n-1)$ and that $f^{-1}(y)$ is irreducible, this orbit is dense in $f^{-1}(y)$. So $P$ is constant on $f^{-1}(y)$. By Hilbert’s Nullstellensatz, there exists a polynomial $Q$ such that $P=P(x)+ (f-y)Q$. The polynomial $Q$ is invariant by factorial closedness and has degree $\leq r$. By induction, $Q$ belongs to ${\mathbb C}[F]$, and so does $P$, hence giving the result.
$\blacksquare$
A few examples
==============
We can show that the first assertion in theorem \[quasi3\] is an equivalence. More precisely, a primitive mapping $F$ is quasi-fibered if and only if ${{\cal{T}}^1(F)}=0$. We will not prove it here, but we would rather give two examples illustrating the necessity of the conditions given in theorem \[Triv\]. In both cases, the module of relative exactness is not zero. Consider the locally nilpotent derivation on ${\mathbb C}[x,y,z]$: $${ \partial}_1 =x
\frac{\partial}{\partial y} - 2y\frac{\partial}{\partial z}$$ Its ring of invariant is generated by $x$ and $xz+y^2$, and its quotient map is defined by: $$F_1: {\mathbb C}^3 \longrightarrow {\mathbb C}^2,
\quad (x,y,z)\longmapsto (x,xz+y^2)$$ It is easy to check that $F_1$ is surjective and that $\overline{B(F_1)}= \{(u,v) \in {\mathbb C}^2, u=0\}$. So $F_1$ is not quasi-fibered because its fibres are not 1-generically connected, and the action generated by ${ \partial}_1$ is not trivial. Second consider the locally nilpotent derivation on ${\mathbb C}[x,y,u,v]$: $${ \partial}_2 =u\frac{\partial}{\partial x} +
v\frac{\partial}{\partial y}$$ The polynomials $u,v,xv - yu$ are invariant and generate the ring of invariants of ${ \partial}_2$. So the corresponding action $\varphi_2$ satisfies the condition $(H)$, and its quotient map is given by: $$F_2 :{\mathbb C}^4 \longrightarrow
{\mathbb C}^3, \quad (x,y,u,v)\longmapsto (u,v,xv - yu)$$ By an easy computation, we get that $B(F_2)$ is empty, $S(F_2)=V(x,y)$ and $I(F_2)=\{(r,0,0), r \in {\mathbb C}^{*}\}$. So $F_2$ is not quasi-fibered because its fibres are not 2-generically non-empty, and $\varphi_2$ is not trivial.
[M-L]{}
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abstract: 'We have measured the relaxation time, $T_{1}$, of the spin of a single electron confined in a semiconductor quantum dot (a proposed quantum bit). In a magnetic field, applied parallel to the two-dimensional electron gas in which the quantum dot is defined, Zeeman splitting of the orbital states is directly observed by measurements of electron transport through the dot. By applying short voltage pulses, we can populate the excited spin state with one electron and monitor relaxation of the spin. We find a lower bound on $T_{1}$ of 50 $\mu$s at 7.5 T, only limited by our signal-to-noise ratio. A continuous measurement of the charge on the dot has no observable effect on the spin relaxation.'
author:
- 'R. Hanson, B. Witkamp, L.M.K. Vandersypen, L.H. Willems van Beveren, J.M. Elzerman and L.P. Kouwenhoven'
title: 'Zeeman energy and spin relaxation in a one-electron quantum dot'
---
The spin of an electron confined in a semiconductor quantum dot (QD) is a promising candidate for a scalable quantum bit [@loss; @Lieven]. The electron spin states in QDs are expected to be very stable, because the zero-dimensionality of the electron states in QDs leads to a significant suppression of the most effective 2D spin-flip mechanisms [@Khaetskii1]. Recent electrical transport measurements of relaxation between spin triplet and singlet states of two electrons, confined in a pillar etched from a GaAs double-barrier heterostructure (“vertical” QD), support this prediction (relaxation time $>\!$ 200 $\!\mu$s at $T\leq\!$ 0.5 K) [@FujisawaNature]. However, the triplet-to-singlet transition, in which the total spin quantum number *S* is changed from 1 to 0, is forbidden by a selection rule ($\Delta S$=0) that does not hinder relaxation between Zeeman sublevels (which conserves *S*). Therefore, measurements on a single electron spin are needed in order to determine the relaxation time of the proposed qubit.
Relaxation between Zeeman sublevels in closed GaAs QDs is expected to be dominated by hyperfine interaction with the nuclei at magnetic fields below 0.5 T [@Siggi] and by spin-orbit interaction at higher fields [@Khaetskii2]. At 1 T, theory predicts a $T_1$ of 1 ms in GaAs [@Khaetskii2]; at fields above a few Tesla, needed to resolve the Zeeman splitting in transport measurements, no quantitative estimates for $T_1$ exist.
For comparison, in $n$-doped self-assembled InAs QDs containing one resident electron, pump-probe photoluminescence measurements gave a single-electron spin relaxation time of 15 ns (at $B$=0 T, $T$= 10 K) [@Cortez]. In undoped self-assembled InAs QDs, the exciton polarization is frozen throughout the exciton lifetime, giving a relaxation time $>$20 ns [@Paillard].
Electrical measurements of the single-electron spin relaxation time have up to now remained elusive. In vertical QDs, where electrical measurements on a single electron were reported almost a decade ago [@LeoFewEl], it has been difficult to directly resolve the Zeeman splitting of orbitals [@ZeemanVQD]. Recently, the one-electron regime was also reached in single [@Ciorga] and double lateral GaAs QDs [@Jero], which are formed electrostatically within a two-dimensional electron gas (2DEG) by means of surface gates.
In this Letter we study the spin states of a one-electron lateral QD directly, by performing energy spectroscopy and relaxation measurements. We observe a clear Zeeman splitting of the orbital states in electron transport measurements through the QD, and find no signature of spin relaxation in our experimental time window, leading to a lower bound on $T_{1}$ of 50 $\mu$s. This lower bound is two to three orders of magnitude longer than spin relaxation times observed in bulk *n*-type GaAs [@Kikkawa], GaAs quantum wells [@Ohno] and InAs QDs [@Cortez].
![\[Fig 1\] (a) Scanning Electron Micrograph of the metallic surface gates [@HiddenGates]. Gates $M$, $R$ and $T$ are used to form the quantum dot indicated by a white circle. Additionally, gate $Q$ can be used to form a quantum point contact (QPC). To apply high-frequency signals, gate $P$ is connected to a coaxial cable. Currents through the dot, $I_{QD}$, and through the QPC, $I_{QPC}$, are measured as a function of applied bias voltage, $V_{SD}=(\mu_{S}\!-\!\mu_{D})/e$ and $V_{QD}=(\mu_{Q}\!-\!\mu_{D})/e$ respectively. (b)-(d) Differential conductance $dI_{QD}/dV_{SD}$ as a function of $V_{SD}$ and gate voltage near the 0$\leftrightarrow$1 electron transition, at parallel magnetic fields of 6, 10 and 14 T. Darker corresponds to larger $dI_{QD}/dV_{SD}$. The zero-field spin degeneracy of both the ground state (GS) and the first orbital excited state (ES) is lifted by the Zeeman energy as indicated by arrows. (e) Extracted Zeeman splitting $\Delta E_{Z}$ as a function of $B$. At high fields a clear deviation from the bulk GaAs $g$-factor of -0.44 (dashed line) is observed.](ZFig1.eps){width="3.4in"}
The quantum dot is defined in a GaAs/Al$_{0.3}$Ga$_{0.7}$As heterostructure, containing a 2DEG 90 nm below the surface with an electron density ${n_{s}=2.9\times 10^{11}}$ cm${^{-2}}$ (Fig. 1a). A magnetic field (0-14 T) is applied parallel to the 2DEG. All measurements are performed in a dilution refrigerator at base temperature *T* = 20 mK.
We tune the device to the few-electron regime and identify the 0$\leftrightarrow$1 electron transition by the absence of further transitions under applied source-drain voltage up to 10 mV. The electron number is confirmed by using the nearby QPC as a charge detector [@Field; @Jero; @Sprinzak]. We find a charging energy of 2.4 meV and an orbital level spacing of 1.1 meV at $B$ = 0 T.
In a parallel magnetic field, the electron states acquire a Zeeman energy shift, which causes the orbital levels to split by $\Delta E_{Z}\!=\!g \mu_B B$ [@Weis]. Figs. 1b-d show stability diagrams [@LeoFewEl] around the 0$\leftrightarrow$1 electron transition, measured at *B*$\,$=$\,$6 T, 10 T and 14 T. A clear Zeeman splitting of both ground and first orbital excited state is seen directly in this spectroscopy measurement [@Potok]. A least-squares fit of the data to a second-order polynomial, which extrapolates with negligible deviation to the origin, gives $$\left|g\right| =(0.43\pm 0.04)-(0.0077\pm0.0020)\:B\:(T) \;,$$ similar to early measurements on 2DEGs [@Dobers]. If we force the fit to be linear in $B$, we get $\left|g\right|\!=\!0.29\pm0.01$, with a zero-field splitting $\Delta E_{Z,B=0}\!\!=\!\!(34 \pm6) \mu$eV.
Factors which can influence the magnetic field dependence of the $g$-factor include: (1) extension of the electron wavefunction into the Al$_{0.3}$Ga$_{0.7}$As region, where $g\!=\!+0.4$ [@Snelling; @Awschalom], (2) thermal nuclear polarization, which decreases the effective magnetic field through the hyperfine interaction [@BookOptical], (3) dynamic nuclear polarization due to electron-nuclear flip-flop processes in the dot, which enhances the effective magnetic field [@BookOptical], and (4) the nonparabolicity of the GaAs conduction band [@Snelling]. More experiments are needed to separate these effects, which is outside the scope of this Letter.
The two spin states (lowest energy) and can be used as the basis states of a quantum bit. In order to perform quantum operations and to allow sufficient time for read-out of the quantum bit, it is necessary that the spin excited state be stable. We investigate this by measuring the relaxation time from to . By applying short pulses to gate $P$, we can modulate the potential of the dot and thus the position of the energy levels relative to the electrochemical potentials of the leads, $\mu_{S}$ and $\mu_{D}$. This enables us to populate the spin excited state and monitor relaxation to . The applicability of various pulse methods for measuring the spin relaxation time depends on two timescales. If the relaxation rate $W$ ($=\!1/T_{1})$ is at least of the same order as the outgoing tunnel rate $\Gamma_{D}$, i.e. $W\!\geq \!\Gamma_{D}$, we can determine $T_{1}$ by applying single-step pulses. This method has previously been used to measure the relaxation time between orbital levels in a QD ($\sim$10 ns) [@FujisawaNature]. In the other limit, $W\!<\!\Gamma_{D}$, a more elaborate method using double-step pulses is needed [@FujisawaNature]. We proceed as follows. First, we apply single-step pulses to show that $W\!<\!\Gamma_{D}$. Then we apply double-step pulses to measure $T_{1}$. All data shown are taken at $B$ = 7.5 T, and reproduced at 14 T. At fields below 6 T the Zeeman splitting is too small to be resolved in pulse experiments. The bias voltage is always much smaller than the charging energy, thus allowing at most one electron on the dot.
![One-electron spin relaxation studied using single-step pulses at 7.5T. (a) Schematic waveform of the pulse train (rise/fall time of 0.2 ns). (b) Current traces under applied pulses, offset for clarity. (c)-(e) Diagrams showing the position of the energy levels during the two phases of the pulse for threee different gate voltage settings, corresponding to the three peaks in (b). (f) Average number of electrons tunneling per cycle (=$I(t_{l}+t_{h})/e$) through the ground state , as in (c), and through the excited state , as in (d), vs. pulse length $t_{h}$. The shows no decay, as expected for a stable current, whereas saturates. However, this saturation is not due to spin relaxation (see text).[]{data-label="Fig 2"}](ZFig2.eps){width="3.4in"}
The single-step pulses are schematically depicted in Fig. 2a. Fig. 2b shows current traces for different amplitudes of the pulses. Transport of electrons through the ground state takes place when lies in the bias window (i.e. $\mu_S\!>\!E_{\ua}\!>\!\mu_D$). When we apply single-step pulses, this condition is met at two different values of the gate voltage $V_T$ and therefore the Coulomb peak splits in two. Fig. 2c shows the positions of the energy levels during the two phases of the pulse for the left peak in Fig. 2b. Here, electrons flow from source to drain during the “high” phase of the pulse. Similarly, Fig. 2e corresponds to the right peak in Fig. 2b, where ground state transport occurs during the “low” phase of the pulse. When the pulse amplitude exceeds the Zeeman splitting ($\approx$160 $\mu$eV), an extra current peak becomes clearly visible. This peak is due to transient transport via the spin-down state during the “high” phase of the pulse (Fig. 2d). The transient current flows until becomes occupied and Coulomb blockade prohibits other electrons to enter the dot. Occupation of can happen either via tunneling of an electron from the leads into when the dot is empty, or by spin relaxation from to . For both these processes, the probability to have occurred increases with time. Therefore, the number of electrons tunneling via per cycle, , saturates with increasing pulse length $t_{h}$. In particular, if the tunnel rate $\Gamma_{S}$ through the incoming barrier is much larger than the tunnel rate $\Gamma_{D}$ through the outgoing barrier, i.e. $\Gamma_{S}\!\gg\!\Gamma_{D}$ [@Asymmetry], it can be shown that [@FujisawaPRB]
$$<\!n_{\da}\!\!>\: \simeq \:A \Gamma_{D,\da} (1-e^{-Dt_{h}})/D \; ,
\label{naver}$$
where $A \simeq \Gamma_{S,\da}/(\Gamma_{S,\ua}+\Gamma_{S,\da})$ is the injection efficiency into , and $\Gamma_{D,\da}$ is the tunnel rate from to the drain (see Fig. 2c-d). The saturation rate $D$ is the sum of $W$, the spin relaxation rate from to , and $(1\!\!-\!\!A)\Gamma_{D,\da}$, which accounts for direct tunneling into : $$D=W+(1\!\!-\!\!A)\Gamma_{D,\da} \;.
\label{Dtotal}$$ By measuring for different pulse widths $t_{h}$, we can find $D$ and $A \Gamma_{D,\da}$ using Eq. (\[naver\]). Together with the value of $A$, which can be extracted from large-bias measurements without pulses, we can determine the spin-relaxation rate $W$=1/$T_{1}$ via Eq. (\[Dtotal\]).
In Fig. 2f we show the average number of tunneling electrons per cycle for the stable current, , and for the transient current, . Clearly, increases linearly with pulse length, whereas saturates, as expected. From fitting to Eq. (\[naver\]) we find $D$=(1.5 $\pm$ 0.2) MHz and $A \Gamma_{D,\da}$=(0.47 $\pm$ 0.09) MHz. Furthermore, $A$=(0.28 $\pm$ 0.05), leading to $(1\!\!-\!\!A)\Gamma_{D,\da}$=(1.2 $\pm$ 0.3) MHz and $W$=(0.30$\pm$ 0.35) MHz. Averaging over similar measurements, using different tunnel rates and $t_{l}$, leads to $W$=(0.20$\pm$ 0.25) MHz.
We conclude that the spin relaxation rate ($W\!\!<\!$ 0.5 MHz) is much smaller than the tunnel rates ($\Gamma_{S}\!\gg\! \Gamma_{D}\!\approx\!$ 1.6 MHz). This means that the decay of the transient current is dominated by direct injection into , and therefore the single-step pulse method can only provide a weak lower bound on $T_{1}$. To circumvent this, we decouple the read-out stage from the relaxation stage by inserting an extra pulse step. This way, an electron can only tunnel out of the dot *after* the waiting time, enabling us to directly measure the relaxation probabilities as a function of waiting time [@FujisawaNature], as explained below.
![\[Fig 3\] One-electron spin relaxation studied using double-step pulses at 7.5T. (a) Schematic waveform of the pulse train (rise/fall time of 1.5 ns). (b) Typical pulse-excited current trace. The three main peaks correspond to a stable current flowing via when is in the bias window during one of the three stages of the waveform. The small peak is due to transient current via for $V_{P}=V_{m}$ [@HiddenPeak]. (c)-(e) Diagrams depicting the energy levels during the three stages of the pulse for the -peak shown in (b). (c) The dot is emptied during a time $t_{l}$. (d) Both and lie below the electrochemical potentials of the leads and an electron can tunnel into the ; other possible tunnel processes are not indicated since they do not contribute to the current (see text). We allow the electron to relax for a time $t_{h}$. (e) Now lies in the bias window. Only if the electron has spin-down it can tunnel out and contribute to current. (f) Averaged current peaks for $t_{h}$=1, 2.5 and 4 $\mu$s with $t_{m}$=0.4 $\mu$s (for data in (f) and (g) $t_{l}$=$t_{h}$). (g) The probability $P_{\da}(t_{h})/P_{\da}(0)$ that the spin did *not* decay during the waiting time $t_{h}$.](ZFig3.eps){width="3.4in"}
The schematic waveform of the double-step pulses is shown in Fig. 3a. Applying these pulses results in current traces as in Fig. 3b. Figs. 3c-e depict the energy levels for the current peak indicated in Fig. 3b at the three different stages of the pulse cycle. First the dot is emptied (Fig. 3c). In the second stage (Fig. 3d), an electron tunnels into either or . Again, due to the charging energy only one electron can occupy the dot. The probability that it enters , $A$, does not depend on the pulse lengths, which are the only parameters we change. If the electron entered , the probability that it has *not* relaxed to after $t_{h}$ is exp(-$t_{h}$/$T_{1}$) (we assume exponential decay). Finally (Fig. 3e), if the electron is in , it can tunnel out, but only to the drain. In contrast, if the electron is in , it can tunnel out to either the source or the drain when the cycle is restarted (Fig. 3c). Similarly, electrons entering the dot originate from the source or the drain (Fig. 3d). Assuming that $\Gamma_S/\Gamma_D$ is constant throughout the cycle, the average current generated by electrons leaving the dot during the “low” phase of the pulse train (Fig. 3c) is zero. Therefore the current only consists of electrons that entered and have not relaxed during $t_h$:
$$I = e f_{rep} <\!n_{\da}\!\!> \:=\: e f_{rep} C A \:e^{(-t_h/T_{1})},$$
where $f_{rep}$ is the pulse repetition frequency and $C$ a constant accounting for the tunnel probability in the read-out stage. We determine for different $t_{h}$. Normalized to the value for $t_{h}$=0, it is a direct measure of spin relaxation: $$\frac{<\!n_{\da}\!\!>_{t_{h}=t}}{<\!n_{\da}\!\!>_{t_{h}=0}}=\frac{C A\: e^{(-t/T_{1})}}{C A \:e^{(-0/T_{1})}} = \frac{P_{\da}(t)}{P_{\da}(0)} = e^{(-t/T_{1})} \; .
\label{decay}$$ To be able to extract reliable peak heights from the very small currents, we average over many traces. Examples of averaged curves are shown in Fig. 3f for $t_{h}$=1, 2.5 and 4 $\mu$s. In Fig. 3g, data extracted from these and similar curves are plotted as a function of $t_{h}$, up to 7.5 $\mu$s. Longer waiting times result in unmeasurably small currents ($I\propto 1/t_{h}$). The two data sets shown were taken with different gate settings (and thus different tunnel rates) and different $t_{m}$. As a guide to the eye, lines corresponding to an exponential decay with decay times $\tau\!$ = $\!10\, \mu$s, $\tau\!$ = $\!30\, \mu$s and $\tau\!$ = $\!\infty$ are included. There is no clear decay visible. We fit the data in Fig. 3g and similar data, and average the resulting relaxation rates. From an error analysis we find a lower bound of $T_{1}>$ 50 $\mu$s. We emphasize that, since we do not observe a clear signature of relaxation in our experimental time window, $T_{1}$ might actually be much longer.
The lower bound we find for $T_{1}$ is much longer than the time needed for read-out of the quantum bit using proposed spin-to-charge conversion schemes [@Lieven]. In these schemes, spin-dependent tunneling events correlate the charge on the dot to the initial spin state. A subsequent charge measurement thus reveals information on the spin. This can de done in our device using the QPC located next to the QD (see Fig. 1a) [@Jero].
An interesting question is how much the stability of the spin states is affected by such charge measurements. We have studied this by sending a large current through the QPC, set at maximum charge sensitivity, and repeating the $T_{1}$ measurements. The drain lead is shared by the QPC- and the QD-current, which causes some peak broadening and limits the experimental window. However, even for a very large current of $\sim$20 nA through the QPC ($\mu_{Q}\!\!\,-\!\!\,\mu_{D}\!=\!500\,\mu eV$), we still do not find a measurable decay of the spin. For comparison, we can measure the charge on the QD within 50 $\mu$s using a QPC current of only 10 nA [@JeroUnpublished]. Taking these measurements together shows that, by using spin-to-charge conversion, it should be possible to perform single-shot spin readout in this device.
We thank T. Fujisawa, S. Tarucha, T. Hayashi, T. Saku, Y. Hirayama, S.I. Erlingsson, Y.V. Nazarov, O.N. Jouravlev, S. De Franceschi, D. Gammon and R.N. Schouten for discussions and help. This work was supported by the DARPA-QUIST program.
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For clarity, we have hidden the surface gates that are not used in the experiments. The complete gate structure can be found in Ref. [@Jero].
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The condition $\Gamma_{S}\gg\Gamma_{D}$ is easily achieved by tuning the different gate voltages.
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The peak for $V_{h}$ appears at the same voltage on $T$ as the peak belonging to $V_{m}$.
J.M. Elzerman *et al.*, unpublished.
|
---
abstract: |
In this present investigation, we introduce the new class $\mathfrak{R}_{\Sigma ,\gamma }^{\mu ,\rho }\left( \widetilde{\mathfrak{p}}\right) $ of bi-univalent functions defined by using the Tremblay fractional derivative operator. Additionally, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient $\left\vert
a_{n}\right\vert $ of the bi-univalent function class.
Keywords: Bi-univalent functions, subordination, Faber polynomials, Fibonacci numbers, Tremblay fractional derivative operator.
2010, Mathematics Subject Classification: 30C45, 33D15.
---
[**Inclusion properties for bi-univalent functions of complex order defined by combining of Faber polynomial expansions and Fibonacci numbers**]{}\
**Şahsene Altinkaya$^{1,\ast }$, Samaneh G. Hamidi$^{2}$, Jay M. Jahangiri$^{3}$, Sibel Yalçin$^{1}$**\
$^{1}$Department of Mathematics,\
Bursa Uludag University, 16059 Bursa, Turkey\
**E-Mail: sahsenealtinkaya@gmail.com, syalcin@uludag.edu.tr**\
$^{2}$Department of Mathematics, Brigham Young University,\
Provo, UT 84602, USA\
**E-Mail: shamidi@rider.edu**\
$^{3}$Department of Mathematical Sciences, Kent State University,\
Burton, OH 44021-9500, USA\
**E-Mail: jjahangi@kent.edu**\
Introduction, Definitions and Notations
=======================================
Let $\mathbb{C}
$ be the complex plane and $\mathbb{U}=\left\{ z:z\in
\mathbb{C}
\text{ and }\left\vert z\right\vert <1\right\} $ be open unit disc in $\mathbb{C}
$. Further, let $\mathcal{A}$ represent the class of functions analytic in $\mathbb{U}$, satisfying the condition$$f(0)=\ f^{\prime }(0)-1=0.$$Then each function $f$ in $\mathcal{A}$ has the following Taylor series expansion$$f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots =z+\overset{\infty }{\underset{n=2}{\sum
}}a_{n}z^{n}. \label{eq1}$$The class of this kind of functions is represented by $\mathcal{S}$.
With a view to reminding the rule of subordination for analytic functions, let the functions $f,g$ be analytic in $\mathbb{U}$. A function $f$ is *subordinate* to $g,$ indited as $f\prec g,$ if there exists a Schwarz function $$\mathbf{\varpi }(z)=\overset{\infty }{\underset{n=1}{\sum }}\mathfrak{c}_{n}z^{n}\ \ \left( \mathbf{\varpi }\left( 0\right) =0,\text{\ }\left\vert
\mathbf{\varpi }\left( z\right) \right\vert <1\right) ,$$analytic in $\mathbb{U}$ such that $$f\left( z\right) =g\left( \mathbf{\varpi }\left( z\right) \right) \ \ \ \
\left( z\in \mathbb{U}\right) .$$For the Schwarz function $\mathbf{\varpi }\left( z\right) $ we know that $\left\vert \mathfrak{c}_{n}\right\vert <1$ (see [@Duren; @83]).
According to the *Koebe-One Quarter Theorem*, every univalent function $f\in \mathcal{A}$ has an inverse $f^{-1}$ satisfying $f^{-1}\left(
f\left( z\right) \right) =z~~\left( z\in \mathbb{U}\right) $ and $f\left(
f^{-1}\left( w\right) \right) =w~$ $\left( \left\vert w\right\vert
<r_{0}\left( f\right) ;~~r_{0}\left( f\right) \geq \frac{1}{4}\right) ,$ where$$\begin{array}{l}
g(w)=f^{-1}\left( w\right) =w~-a_{2}w^{2}+\left( 2a_{2}^{2}-a_{3}\right)
w^{3} \\
\\
\ \ \ \ \ \ \ \ \ \ \ -\left( 5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right)
w^{4}+\cdots .\end{array}
\label{eq2}$$A function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{U}$ if both $f$ and $f^{-1}$ are univalent in $\mathbb{U}.~$Let $\Sigma $ denote the class of bi-univalent functions in $\mathbb{U}$ given by (\[eq1\]). For a brief historical account and for several notable investigation of functions in the class $\Sigma ,$ see the pioneering work on this subject by Srivastava et al. [@Srivastava; @2010] (see also [@Brannan; @and; @Clunie; @80; @Brannan; @and; @Taha; @86; @Lewin; @67; @Netanyahu; @69]). The interest on estimates for the first two coefficients $\left\vert a_{2}\right\vert $, $\left\vert
a_{3}\right\vert $ of the bi-univalent functions keep on by many researchers (see, for example, [@AA; @Hayami; @2012; @HO; @Seker; @2016; @Srivastava; @2013]). However, in the literature, there are only a few works (by making use of the Faber polynomial expansions) determining the general coefficient bounds $\left\vert a_{n}\right\vert $ for bi-univalent functions ([@AY; @Hamidi; @and; @Jahangiri; @2014; @Hamidi; @and; @Jahangiri; @2016; @S]). The coefficient estimate problem for each of $\left\vert a_{n}\right\vert $ $\left( \ n\in
\mathbb{N}
\backslash \left\{ 1,2\right\} ;\ \
\mathbb{N}
=\left\{ 1,2,3,...\right\} \right) $ is still an open problem.
Now, we recall to a notion of $q$-operators that play a major role in Geometric Function Theory. The application of the $q$-calculus in the context of Geometric Function Theory was actually provided and the basic (or $q$-) hypergeometric functions were first used in Geometric Function Theory in a book chapter by Srivastava [@Srivastava1989]. For the convenience, we provide some basic notation details of $q$-calculus which are used in this paper.
(See [@SO]) For a function $f$ (analytic in a simply-connected region of $\mathbb{C}
$), the fractional derivative of order $\rho $ is stated by$$D_{z}^{\rho }f(z)=\frac{1}{\Gamma (1-\rho )}\frac{d}{dz}\int\limits_{0}^{z}\frac{f(\xi )}{(z-\xi )^{\rho }}d\xi \ \ \ (0\leq \rho <1)$$and the fractional integral of order $\rho $ is stated by$$I_{z}^{\rho }f(z)=\frac{1}{\Gamma (\rho )}\int\limits_{0}^{z}f(\xi )(z-\xi
)^{\rho -1}d\xi \ \ \ (\rho >0).$$
(See [@S]) The Tremblay fractional derivative operator of the function $f $ is defined as$$I_{z}^{\mu ,\rho }f(z)=\frac{\Gamma (\rho )}{\Gamma (\mu )}z^{1-\rho
}D_{z}^{\mu -\rho }z^{\mu -1}f(z)\ \ \ (0<\mu \leq 1,0<\rho \leq 1,\mu \geq
\rho ,0<\mu -\rho <1). \label{eq3}$$
From (\[eq3\]), we deduce that$$I_{z}^{\mu ,\rho }f(z)=\frac{\mu }{\rho }z+\overset{\infty }{\underset{n=2}{\sum }}\frac{\Gamma (\rho )\Gamma (n+\mu )}{\Gamma (\mu )\Gamma (n+\rho )}a_{n}z^{n}.~$$
In this paper, we study the new class $\mathfrak{R}_{\Sigma ,\gamma }^{\mu
,\rho }\left( \widetilde{\mathfrak{p}}\right) $ of bi-univalent functions established by using the Tremblay fractional derivative operator. Further, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient $\left\vert a_{n}\right\vert $ of the bi-univalent function class.
Preliminaries
=============
By utilizing the Faber polynomial expansions for functions $f$ $\in \mathcal{A}$ of the form (\[eq1\]), the coefficients of its inverse map $g=f$ $^{-1}
$ may be stated by [@Airault; @and; @Bouali; @2006; @Airault; @and; @Ren; @2002]:
$$g\left( w\right) =f^{-1}\left( w\right) =w+\overset{\infty }{\underset{n=2}{\sum }}\frac{1}{n}K_{n-1}^{-n}\left( a_{2},a_{3},...\right) w^{n},$$
where
$$\begin{aligned}
K_{n-1}^{-n} &=&\frac{\left( -n\right) !}{\left( -2n+1\right) !\left(
n-1\right) !}a_{2}^{n-1}~+\frac{\left( -n\right) !}{\left[ 2\left(
-n+1\right) \right] !\left( n-3\right) !}a_{2}^{n-3}a_{3}~ \\
&&+~\frac{\left( -n\right) !}{\left( -2n+3\right) !\left( n-4\right) !}a_{2}^{n-4}a_{4}~ \\
&&+\frac{\left( -n\right) !}{\left[ 2\left( -n+2\right) \right] !\left(
n-5\right) !}a_{2}^{n-5}\left[ a_{5}+\left( -n+2\right) a_{3}^{2}\right] \\
&&+\frac{\left( -n\right) !}{\left( -2n+5\right) !\left( n-6\right) !}a_{2}^{n-6}\left[ a_{6}+\left( -2n+5\right) a_{3}a_{4}\right] \\
&&+\overset{}{\underset{j\geq 7}{\sum }}a_{2}^{n-j}V_{j},\end{aligned}$$
such that $V_{j}$ $\left( 7\leq j\leq n\right) $ is a homogeneous polynomial in the variables $a_{2},a_{3},...,a_{n}$. In the following, the first three terms of $K_{n-1}^{-n}$ are stated by
$$\begin{aligned}
\frac{1}{2}K_{1}^{-2} &=&-a_{2}, \\
\frac{1}{3}K_{2}^{-3} &=&2a_{2}^{2}-a_{3}, \\
\frac{1}{4}K_{3}^{-4} &=&-\left( 5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right) .\end{aligned}$$
In general, the expansion of $K_{n}^{p}$ $(p\in
\mathbb{Z}
=\left\{ 0,\pm 1,\pm 2,\ldots \right\} )$ is stated by
$$K_{n}^{p}=pa_{n}+\frac{p\left( p-1\right) }{2}\mathcal{G}_{n}^{2}+\frac{p!}{\left( p-3\right) !3!}\mathcal{G}_{n}^{3}+...+\frac{p!}{\left( p-n\right) !n!}\mathcal{G}_{n}^{n},$$
where $\mathcal{G}_{n}^{p}=$ $\mathcal{G}_{n}^{p}\left(
a_{1},a_{2},...\right) $ and by [@Airault; @2007],
$$\mathcal{G}_{n}^{m}\left( a_{1},a_{2},...,a_{n}\right) =\overset{\infty }{\underset{n=1}{\sum }}\frac{m!\left( a_{1}\right) ^{\delta _{1}}...\left(
a_{n}\right) ^{\delta _{n}}}{\delta _{1}!...\delta _{n}!},$$
while $a_{1}=1$, the sum is taken over all nonnegative integers $\delta
_{1},...,\delta _{n}$ satisfying$$\begin{aligned}
\delta _{1}+\delta _{2}+~...~+\delta _{n} &=&m, \\
\delta _{1}+2\delta _{2}+~...~+n\delta _{n} &=&n.\end{aligned}$$The first and the last polynomials are$$\mathcal{G}_{n}^{1}=a_{n}\ \ \ \ \ \ \ \ \mathcal{G}_{n}^{n}=a_{1}^{n}.$$For two analytic functions $\mathfrak{u}\left( z\right) $, $\mathfrak{v}\left( w\right) $ $\left( \mathfrak{u}\left( 0\right) =\mathfrak{v}\left(
0\right) =0,\ \left\vert \mathfrak{u}\left( z\right) \right\vert <1,\
\left\vert \mathfrak{v}\left( w\right) \right\vert <1\right) ,\ $suppose that$$\begin{array}{l}
\mathfrak{u}\left( z\right) =\sum_{n=1}^{\infty }t_{n}z^{n}\ \ \left(
\left\vert z\right\vert <1,\ z\in \mathbb{U}\right) \ \ \ , \\
\\
\mathfrak{v}\left( w\right) =\sum_{n=1}^{\infty }s_{n}w^{n}\ \ \left(
\left\vert w\right\vert <1,\ w\in \mathbb{U}\right) .\end{array}$$It is well known that
$$\left\vert t_{1}\right\vert \leq 1,\ \ \left\vert t_{2}\right\vert \leq
1-\left\vert t_{1}\right\vert ^{2},\ \ \left\vert s_{1}\right\vert \leq 1,\
\ \left\vert s_{2}\right\vert \leq 1-\left\vert s_{1}\right\vert ^{2}.
\label{eq9}$$
A function $f\in \Sigma $ is said to be in the class$$\mathfrak{R}_{\Sigma ,\gamma }^{\mu ,\rho }\left( \widetilde{\mathfrak{p}}\right) \ \ \ (\gamma \in
\mathbb{C}
\backslash \{0\},\ 0<\mu \leq 1,\ 0<\rho \leq 1,\ z,w\in \mathbb{U})$$if the following subordination relationships are satisfied:$$\left[ 1+\frac{1}{\gamma }\left( \frac{\rho \left( I_{z}^{\mu ,\rho
}f(z)\right) ^{\prime }}{\mu }-1\right) \right] \prec \widetilde{\mathfrak{p}}\left( z\right) =\frac{1+\tau ^{2}z^{2}}{1-\tau z-\tau ^{2}z^{2}}$$and$$\left[ 1+\frac{1}{\gamma }\left( \frac{\rho \left( I_{z}^{\mu ,\rho
}g(w)\right) ^{\prime }}{\mu }-1\right) \right] \prec \widetilde{\mathfrak{p}}\left( w\right) =\frac{1+\tau ^{2}w^{2}}{1-\tau w-\tau ^{2}w^{2}},$$where the function $g$ is given by (\[eq2\]) and $\tau =\frac{1-\sqrt{5}}{2}\approx -0.618.$
The function $\widetilde{\mathfrak{p}}\left( z\right) $ is not univalent in $\mathbb{U}$, but it is univalent in the disc $\left\vert z\right\vert <\frac{3-\sqrt{5}}{2}\approx -0.38$. For example, $\widetilde{\mathfrak{p}}\left(
0\right) =\widetilde{\mathfrak{p}}\left( -\frac{1}{2\tau }\right) $ and $\widetilde{\mathfrak{p}}\left( e^{\pm i\arccos (1/4)}\right) =\frac{\sqrt{5}}{5}$. Also, it can be written as$$\frac{1}{\left\vert \tau \right\vert }=\frac{\left\vert \tau \right\vert }{1-\left\vert \tau \right\vert }$$which indicates that the number $\left\vert \tau \right\vert $ divides $\left[ 0,1\right] $ such that it fulfills the golden section (see for details Dziok et al. [@D]).
Additionally, Dziok et al. [@D] indicate a useful connection between the function $\widetilde{\mathfrak{p}}\left( z\right) $ and the Fibonacci numbers. Let $\left\{ \Lambda _{n}\right\} $ be the sequence of Fibonacci numbers $$\Lambda _{0}=0,\ \Lambda _{1}=1,\ \Lambda _{n+2}=\Lambda _{n}+\Lambda
_{n+1}\ (n\in
\mathbb{N}
_{0}=\left\{ 0,1,2,\ldots \right\} ),$$then$$\Lambda _{n}=\frac{(1-\tau )^{n}-\tau ^{n}}{\sqrt{5}},\ \ \tau =\frac{1-\sqrt{5}}{2}.$$If we set $$\begin{aligned}
\widetilde{\mathfrak{p}}\left( z\right) &=&1+\overset{\infty }{\underset{n=1}{\sum }}\widetilde{\mathfrak{p}}_{n}z^{n}=1+(\Lambda _{0}+\Lambda _{2})\tau
z+(\Lambda _{1}+\Lambda _{3})\tau ^{2}z^{2} \\
&& \\
&&+\overset{\infty }{\underset{n=3}{\sum }}(\Lambda _{n-3}+\Lambda
_{n-2}+\Lambda _{n-1}+\Lambda _{n})\tau ^{n}z^{n},\end{aligned}$$then the coefficients $\widetilde{\mathfrak{p}}_{n}$ satisfy$$\widetilde{\mathfrak{p}}_{n}=\left\{
\begin{array}{ll}
\tau & \left( n=1\right) \\
& \\
3\tau ^{2} & \left( n=2\right) \\
& \\
\tau \widetilde{\mathfrak{p}}_{n-1}+\tau ^{2}\widetilde{\mathfrak{p}}_{n-2}
& \left( n=3,4,\ldots \right)\end{array}\right. . \label{D}$$
Specializing the parameters $\gamma ,\mu $ and $\rho $, we state the following definitions.
For $\mu =\rho =1,$ a function $f\in \Sigma $ is said to be in the class $\mathfrak{R}_{\Sigma ,\gamma }\left( \widetilde{\mathfrak{p}}\right) \left(
\gamma \in
\mathbb{C}
\backslash \{0\}\right) $ if it satisfies the following conditions respectively:$$\left[ 1+\frac{1}{\gamma }\left( f^{\prime }(z)-1\right) \right] \prec
\widetilde{\mathfrak{p}}\left( z\right)$$and$$\left[ 1+\frac{1}{\gamma }\left( g^{\prime }(w)-1\right) \right] \prec
\widetilde{\mathfrak{p}}\left( w\right) ,$$where $g=f^{-1}.$
For $\gamma =\mu =\rho =1,$ a function $f\in \Sigma $ is said to be in the class $\mathfrak{R}_{\Sigma }\left( \widetilde{\mathfrak{p}}\right) $ if it satisfies the following conditions respectively:$$f^{\prime }(z)\prec \widetilde{\mathfrak{p}}\left( z\right)$$and$$g^{\prime }(w)\prec \widetilde{\mathfrak{p}}\left( w\right) ,$$where $g=f^{-1}.$
Main Result and its consequences
================================
For $\gamma \in
\mathbb{C}
\backslash \{0\}$, let $f\in \mathfrak{R}_{\Sigma ,\gamma }^{\mu ,\rho
}\left( \widetilde{\mathfrak{p}}\right) $. If $a_{m}=0~\left( 2\leq m\leq
n-1\right) $, then $$\left\vert a_{n}\right\vert \leq \frac{\left\vert \gamma \right\vert
\left\vert \tau \right\vert \Gamma (\mu +1)\Gamma (n+\rho )}{n\Gamma (\rho
+1)\Gamma (n+\mu )}\ \ \ (n\geq 3).$$
Let $f$ be given by (\[eq1\]). By the definition of subordination yields$$\left[ 1+\frac{1}{\gamma }\left( \frac{\rho \left( I_{z}^{\mu ,\rho
}f(z)\right) ^{\prime }}{\mu }-1\right) \right] =\widetilde{\mathfrak{p}}(\mathfrak{u}(z)) \label{eq16}$$and$$\left[ 1+\frac{1}{\gamma }\left( \frac{\rho \left( I_{z}^{\mu ,\rho
}g(w)\right) ^{\prime }}{\mu }-1\right) \right] =\widetilde{\mathfrak{p}}(\mathfrak{v}(w)). \label{eq17}$$Now, an application of Faber polynomial expansion to the power series $\mathfrak{R}_{\Sigma ,\gamma }^{\mu ,\rho }\left( \widetilde{\mathfrak{p}}\right) $ (e.g. see [@Airault; @and; @Bouali; @2006] or \[[@Airault; @and; @Ren; @2002], equation (1.6)\]) yields $$1+\frac{1}{\gamma }\left( \frac{\rho \left( I_{z}^{\mu ,\rho }f(z)\right)
^{\prime }}{\mu }-1\right) =1+\frac{\Gamma (\rho +1)}{\gamma \Gamma (\mu +1)}\overset{\infty }{\underset{n=2}{\sum }}\mathcal{F}_{n-1}\left(
a_{2},a_{3},...,a_{n}\right) z^{n-1}$$where$$\begin{array}{ll}
\mathcal{F}_{n-1}\left( a_{2},a_{3},...,a_{n}\right) z^{n-1} & =n\frac{\Gamma (n+\mu )}{\Gamma (n+\rho )} \\
& \\
& \times \overset{\infty }{\underset{i_{1}+2i_{2}+\cdots +(n-1)i_{(n-1)}=n-1}{\sum }}\frac{\left( 1-\left( i_{1}+i_{2}+\cdots +i_{n-1}\right) \right) !\left[ \left( a_{2}\right) ^{i_{1}}\left( a_{3}\right) ^{i_{2}}...\left(
a_{n}\right) ^{i_{n-1}}\right] }{\left( i_{1}!\right) \left( i_{2}!\right)
...\left( i_{n-1}!\right) }\end{array}$$$$$$In particular, the first two terms are, $\mathcal{F}_{1}=\frac{2(\mu +1)}{\gamma (\rho +1)}a_{2},\mathcal{F}_{1}=\frac{3(\mu +1)(\mu +2)}{\gamma (\rho
+1)(\rho +2)}a_{3}.$
By the same token, for its inverse map $g=f^{-1}$, it is seen that $$\begin{aligned}
1+\frac{1}{\gamma }\left( \frac{\rho \left( I_{z}^{\mu ,\rho }g(w)\right)
^{\prime }}{\mu }-1\right) &=&1+\overset{\infty }{\underset{n=2}{\sum }}\frac{\Gamma (\rho +1)\Gamma (n+\mu )}{\Gamma (\mu +1)\Gamma (n+\rho )}\frac{n}{\gamma }\times \frac{1}{n}K_{n-1}^{-n}\left( a_{2},a_{3},...\right)
w^{n-1} \\
&& \\
&=&1+\frac{\Gamma (\rho +1)}{\gamma \Gamma (\mu +1)}\overset{\infty }{\underset{n=2}{\sum }}\mathcal{F}_{n-1}\left( b_{2},b_{3},...,b_{n}\right)
w^{n-1}.\end{aligned}$$Next, the equations (\[eq16\]) and (\[eq17\]) lead to$$\begin{aligned}
\widetilde{\mathfrak{p}}\left( \mathfrak{u}\left( z\right) \right) &=&1+\widetilde{\mathfrak{p}}_{1}\mathfrak{u}(z)+\widetilde{\mathfrak{p}}_{2}(\mathfrak{u}(z))^{2}z^{2}+\cdots \\
&& \\
&=&1+\widetilde{\mathfrak{p}}_{1}t_{1}z+\left( \widetilde{\mathfrak{p}}_{1}t_{2}+\widetilde{\mathfrak{p}}_{2}t_{1}^{2}\right) z^{2}+\cdots \\
&& \\
&=&1+\underset{}{\underset{n=1}{\overset{\infty }{\sum }}}\underset{k=1}{\overset{n}{\sum }}\widetilde{\mathfrak{p}}_{k}\mathcal{G}_{n}^{k}\left(
t_{1},t_{2},...,t_{n}\right) z^{n},\end{aligned}$$and $$\begin{aligned}
\widetilde{\mathfrak{p}}\left( \mathfrak{v}\left( w\right) \right) &=&1+\widetilde{\mathfrak{p}}_{1}\mathfrak{v}(w)+\widetilde{\mathfrak{p}}_{2}(\mathfrak{v}(w))^{2}z^{2}+\cdots \\
&& \\
&=&1+\widetilde{\mathfrak{p}}_{1}s_{1}w+\left( \widetilde{\mathfrak{p}}_{1}s_{2}+\widetilde{\mathfrak{p}}_{2}s_{1}^{2}\right) w^{2}+\cdots \\
&& \\
&=&1+\underset{}{\underset{n=1}{\overset{\infty }{\sum }}}\underset{k=1}{\overset{n}{\sum }}\widetilde{\mathfrak{p}}_{k}\mathcal{G}_{n}^{k}\left(
s_{1},s_{2},...,s_{n}\right) w^{n}.\end{aligned}$$Comparing the corresponding coefficients of (\[eq16\]) and (\[eq17\]) yields$$\frac{\Gamma (\rho +1)\Gamma (n+\mu )}{\Gamma (\mu +1)\Gamma (n+\rho )}\frac{n}{\gamma }a_{n}=\widetilde{\mathfrak{p}}_{1}t_{n-1,}$$$$\frac{\Gamma (\rho +1)\Gamma (n+\mu )}{\Gamma (\mu +1)\Gamma (n+\rho )}\frac{n}{\gamma }b_{n}=\widetilde{\mathfrak{p}}_{1}s_{n-1}.$$For $a_{m}=0\ \left( 2\leq m\leq n-1\right) ,$ we get $b_{n}=-a_{n}$ and so$$\frac{\Gamma (\rho +1)\Gamma (n+\mu )}{\Gamma (\mu +1)\Gamma (n+\rho )}\frac{n}{\gamma }a_{n}=\widetilde{\mathfrak{p}}_{1}t_{n-1} \label{eq18}$$and$$-\frac{\Gamma (\rho +1)\Gamma (n+\mu )}{\Gamma (\mu +1)\Gamma (n+\rho )}\frac{n}{\gamma }a_{n}=\widetilde{\mathfrak{p}}_{1}s_{n-1}. \label{eq19}$$Now taking the absolute values of either of the above two equations and from (\[eq9\]), we obtain$$\left\vert a_{n}\right\vert \leq \frac{\left\vert \gamma \right\vert
\left\vert \tau \right\vert \Gamma (\mu +1)\Gamma (n+\rho )}{n\Gamma (\rho
+1)\Gamma (n+\mu )}.$$
For $\gamma \in
\mathbb{C}
\backslash \{0\}$, suppose that $f\in \mathfrak{R}_{\Sigma ,\gamma }\left(
\widetilde{\mathfrak{p}}\right) $. If $a_{m}=0~\left( 2\leq m\leq n-1\right)
$, then $$\left\vert a_{n}\right\vert \leq \frac{\left\vert \gamma \right\vert
\left\vert \tau \right\vert }{n}\ \ \ (n\geq 3).$$
Suppose that $f\in \mathfrak{R}_{\Sigma }\left( \widetilde{\mathfrak{p}}\right) $. If $a_{m}=0~\left( 2\leq m\leq n-1\right) $, then $$\left\vert a_{n}\right\vert \leq \frac{\left\vert \tau \right\vert }{n}\ \ \
(n\geq 3).$$
Let $f\in \mathfrak{R}_{\Sigma ,\gamma }^{\mu ,\rho }\left( \widetilde{\mathfrak{p}}\right) \ (\gamma \in
\mathbb{C}
\backslash \{0\}).$Then$$\begin{aligned}
\left\vert a_{2}\right\vert &\leq &\min \left\{ \dfrac{\left\vert \gamma
\right\vert \left\vert \tau \right\vert }{\sqrt{\left\vert \tfrac{3\gamma
(\mu +1)(\mu +2)}{(\rho +1)(\rho +2)}-\tfrac{12(\mu +1)^{2}}{(\rho +1)^{2}}\right\vert \left\vert \tau \right\vert +\tfrac{4(\mu +1)^{2}}{(\rho +1)^{2}}}},\right. \\
&& \\
&&\left. \left\vert \tau \right\vert \sqrt{\frac{\left\vert \gamma
\right\vert (\rho +1)(\rho +2)}{(\mu +1)(\mu +2)}}\right\}\end{aligned}$$and$$\begin{aligned}
\left\vert a_{3}\right\vert &\leq &\min \left\{ \frac{\left\vert \gamma
\right\vert \tau ^{2}(\rho +1)(\rho +2)}{(\mu +1)(\mu +2)},\right. \\
&& \\
&&\left. \dfrac{\left\vert \tau \right\vert }{\frac{3(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}}\left[ 1+\frac{\left[
\frac{3(\mu +1)(\mu +2)\left\vert \gamma \right\vert \left\vert \tau
\right\vert }{(\rho +1)(\rho +2)}-\frac{4(\mu +1)^{2}}{(\rho +1)^{2}}\right]
}{\left\vert \dfrac{3\gamma (\mu +1)(\mu +2)}{(\rho +1)(\rho +2)}-\dfrac{12(\mu +1)^{2}}{(\rho +1)^{2}}\right\vert \left\vert \tau \right\vert +\dfrac{4(\mu +1)^{2}}{(\rho +1)^{2}}}\right] \right\} .\end{aligned}$$
Substituting $n$ by $2$ and $3$ in (\[eq18\]) and (\[eq19\]), respectively, we find that$$\frac{2(\mu +1)}{\gamma (\rho +1)}a_{2}=\widetilde{\mathfrak{p}}_{1}t_{1},
\label{eq20}$$$$\frac{3(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}a_{3}=\widetilde{\mathfrak{p}}_{1}t_{2}+\widetilde{\mathfrak{p}}_{2}t_{1}^{2}, \label{eq21}$$$$-\frac{2(\mu +1)}{\gamma (\rho +1)}a_{2}=\widetilde{\mathfrak{p}}_{1}s_{1},
\label{eq22}$$$$\frac{3(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}(2a_{2}^{2}-a_{3})=\widetilde{\mathfrak{p}}_{1}s_{2}+\widetilde{\mathfrak{p}}_{2}s_{1}^{2}.
\label{eq23}$$Obviously, we obtain$$t_{1}=-s_{1}. \label{eq24}$$If we add the equation (\[eq23\]) to (\[eq21\]) and use (\[eq24\]), we get $$\frac{6(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}a_{2}^{2}=\widetilde{\mathfrak{p}}_{1}\left( t_{2}+s_{2}\right) +2\widetilde{\mathfrak{p}}_{2}t_{1}^{2}. \label{eq25}$$Using the value of $t_{1}^{2}$ from (\[eq20\]), we get $$\left[ \frac{6(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}\widetilde{\mathfrak{p}}_{1}^{2}-\frac{8(\mu +1)^{2}}{\gamma ^{2}(\rho +1)^{2}}\widetilde{\mathfrak{p}}_{2}\right] a_{2}^{2}=\widetilde{\mathfrak{p}}_{1}^{3}\left( t_{2}+s_{2}\right) . \label{eq26}$$Combining (\[eq26\]) and (\[eq9\]), we obtain $$\begin{aligned}
2\left\vert \frac{3(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}\widetilde{\mathfrak{p}}_{1}^{2}-\frac{4(\mu +1)^{2}}{\gamma ^{2}(\rho +1)^{2}}\widetilde{\mathfrak{p}}_{2}\right\vert \left\vert a_{2}\right\vert ^{2}
&\leq &\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert ^{3}\left(
\left\vert t_{2}\right\vert +\left\vert s_{2}\right\vert \right) \\
&& \\
&\leq &2\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert ^{3}\left(
1-\left\vert t_{1}\right\vert ^{2}\right) \\
&& \\
&=&2\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert ^{3}-2\left\vert
\widetilde{\mathfrak{p}}_{1}\right\vert ^{3}\left\vert t_{1}\right\vert ^{2}.\end{aligned}$$It follows from (\[eq20\]) that$$\left\vert a_{2}\right\vert \leq \dfrac{\left\vert \gamma \right\vert
\left\vert \tau \right\vert }{\sqrt{\left\vert \dfrac{3\gamma (\mu +1)(\mu
+2)}{(\rho +1)(\rho +2)}-\dfrac{12(\mu +1)^{2}}{(\rho +1)^{2}}\right\vert
\left\vert \tau \right\vert +\dfrac{4(\mu +1)^{2}}{(\rho +1)^{2}}}}.
\label{eq28}$$Additionally, by (\[eq9\]) and (\[eq25\]) $$\begin{aligned}
\frac{6(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}\left\vert a_{2}\right\vert ^{2} &\leq &\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert \left( \left\vert t_{2}\right\vert +\left\vert
s_{2}\right\vert \right) +2\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert \left\vert t_{1}\right\vert ^{2} \\
&& \\
&\leq &2\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert \left(
1-\left\vert t_{1}\right\vert ^{2}\right) +2\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert \left\vert t_{1}\right\vert ^{2} \\
&& \\
&=&2\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert +2\left\vert
t_{1}\right\vert ^{2}(\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert
-\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert ).\end{aligned}$$Since $\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert >\left\vert
\widetilde{\mathfrak{p}}_{1}\right\vert $, we get$$\left\vert a_{2}\right\vert \leq \left\vert \tau \right\vert \sqrt{\frac{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}{(\mu +1)(\mu +2)}}.$$Next, in order to derive the bounds on $\left\vert a_{3}\right\vert ,$ by subtracting (\[eq23\]) from (\[eq21\]), we may obtain$$\frac{6(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}a_{3}=\frac{6(\mu +1)(\mu
+2)}{\gamma (\rho +1)(\rho +2)}a_{2}^{2}+\widetilde{\mathfrak{p}}_{1}\left(
t_{2}-s_{2}\right) . \label{eq29}$$Evidently, from (\[eq25\]), we state that$$\begin{aligned}
a_{3} &=&\frac{\widetilde{\mathfrak{p}}_{1}\left( t_{2}+s_{2}\right) +2\widetilde{\mathfrak{p}}_{2}t_{1}^{2}}{\frac{6(\mu +1)(\mu +2)}{\gamma (\rho
+1)(\rho +2)}}+\frac{\widetilde{\mathfrak{p}}_{1}\left( t_{2}-s_{2}\right) }{\frac{6(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}} \\
&& \\
&=&\frac{\widetilde{\mathfrak{p}}_{1}t_{2}+\widetilde{\mathfrak{p}}_{2}t_{1}^{2}}{\frac{3(\mu +1)(\mu +2)}{\gamma (\rho +1)(\rho +2)}}\end{aligned}$$and consequently$$\begin{aligned}
\left\vert a_{3}\right\vert &\leq &\frac{\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert \left\vert t_{2}\right\vert +\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert \left\vert t_{1}\right\vert ^{2}}{\frac{3(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}} \\
&& \\
&\leq &\frac{\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert \left(
1-\left\vert t_{1}\right\vert ^{2}\right) +\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert \left\vert t_{1}\right\vert ^{2}}{\frac{3(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}} \\
&& \\
&=&\frac{\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert +\left\vert
t_{1}\right\vert ^{2}(\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert
-\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert )}{\frac{3(\mu +1)(\mu
+2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}}.\end{aligned}$$Since $\left\vert \widetilde{\mathfrak{p}}_{2}\right\vert >\left\vert
\widetilde{\mathfrak{p}}_{1}\right\vert $, we must write$$\left\vert a_{3}\right\vert \leq \frac{\left\vert \gamma \right\vert \tau
^{2}(\rho +1)(\rho +2)}{(\mu +1)(\mu +2)}.$$On the other hand, by (\[eq9\]) and (\[eq29\]), we have $$\begin{aligned}
\frac{6(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}\left\vert a_{3}\right\vert &\leq &\frac{6(\mu +1)(\mu +2)}{\left\vert
\gamma \right\vert (\rho +1)(\rho +2)}\left\vert a_{2}\right\vert
^{2}+\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert \left( \left\vert
t_{2}\right\vert +\left\vert s_{2}\right\vert \right) \\
&& \\
&\leq &\frac{6(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho
+2)}\left\vert a_{2}\right\vert ^{2}+2\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert \left( 1-\left\vert t_{1}\right\vert ^{2}\right) .\end{aligned}$$Then, with the help of (\[eq20\]), we have$$\frac{3(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}\left\vert a_{3}\right\vert \leq \left[ \frac{3(\mu +1)(\mu +2)}{\left\vert
\gamma \right\vert (\rho +1)(\rho +2)}-\frac{4(\mu +1)^{2}}{\left\vert
\gamma \right\vert ^{2}(\rho +1)^{2}\left\vert \widetilde{\mathfrak{p}}_{1}\right\vert }\right] \left\vert a_{2}\right\vert ^{2}+\left\vert
\widetilde{\mathfrak{p}}_{1}\right\vert .$$By considering (\[eq28\]), we deduce that$$\left\vert a_{3}\right\vert \leq \dfrac{\left\vert \tau \right\vert }{\frac{3(\mu +1)(\mu +2)}{\left\vert \gamma \right\vert (\rho +1)(\rho +2)}}\left\{
1+\frac{\left[ \frac{3(\mu +1)(\mu +2)\left\vert \gamma \right\vert
\left\vert \tau \right\vert }{(\rho +1)(\rho +2)}-\frac{4(\mu +1)^{2}}{(\rho
+1)^{2}}\right] }{\left\vert \dfrac{3\gamma (\mu +1)(\mu +2)}{(\rho +1)(\rho
+2)}-\dfrac{12(\mu +1)^{2}}{(\rho +1)^{2}}\right\vert \left\vert \tau
\right\vert +\dfrac{4(\mu +1)^{2}}{(\rho +1)^{2}}}\right\} .$$
Let $f\in \mathfrak{R}_{\Sigma ,\gamma }\left( \widetilde{\mathfrak{p}}\right) \ (\gamma \in
\mathbb{C}
\backslash \{0\}).$Then$$\left\vert a_{2}\right\vert \leq \min \left\{ \dfrac{\left\vert \gamma
\right\vert \left\vert \tau \right\vert }{\sqrt{3\left\vert \gamma
-4\right\vert \left\vert \tau \right\vert +4}},\left\vert \tau \right\vert
\sqrt{\left\vert \gamma \right\vert }\right\}$$and$$\left\vert a_{3}\right\vert \leq \min \left\{ \left\vert \gamma \right\vert
\left\vert \tau \right\vert ^{2},\dfrac{\left( \left\vert \gamma
-4\right\vert +\left\vert \gamma \right\vert \right) \left\vert \tau
\right\vert ^{2}\left\vert \gamma \right\vert }{3\left\vert \gamma
-4\right\vert \left\vert \tau \right\vert +4}\right\} .$$
Let $f\in \mathfrak{R}_{\Sigma }\left( \widetilde{\mathfrak{p}}\right) .$Then$$\left\vert a_{2}\right\vert \leq \dfrac{\left\vert \tau \right\vert }{\sqrt{9\left\vert \tau \right\vert +4}}$$and$$\left\vert a_{3}\right\vert \leq \frac{4\left\vert \tau \right\vert ^{2}}{9\left\vert \tau \right\vert +4}.$$
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Airault H, Bouali H. Differential calculus on the Faber polynomials. Bulletin des Sciences Mathematiques 2006; 179-222.
Airault H, Ren J. An algebra of differential operators and generating functions on the set of univalent functions. Bulletin des Sciences Mathematiques 2002; 126: 343-367.
Altinkaya Ş, Yalçin S. Coefficient estimates for a subclass of analytic and Bi-univalent functions. Acta Universitatis Apulensis 2014; 40: 347-354.
Altinkaya Ş, Yalçin S. Faber polynomial coefficient bounds for a subclass of bi-univalent functions. C R Acad Sci Paris Ser I 2015; 353: 1075-1080.
Brannan DA, Clunie J. Aspects of contemporary complex analysis. Proceedings of the NATO Advanced Study Instute Held at University of Durham, New York: Academic Press, 1979.
Brannan DA, Taha TS. On some classes of bi-univalent functions. Studia Universitatis Babeş-Bolyai Mathematica 1986; 31: 70-77.
Dziok J, Raina RK, Sokół J. On $\alpha $-convex functions related to shell-like functions connected with Fibonacci numbers. Applied Mathematics and Computation 2011; 218: 996–1002.
Duren PL. Univalent Functions. Grundlehren der Mathematischen Wissenschaften*,* Springer, New York, USA, 259, 1983.
Hamidi SG, Jahangiri JM. Faber polynomial coefficient estimates for analytic bi-close-to-convex functions. C R Acad Sci Paris Ser I 2014; 352: 17–20.
Hamidi SG, Jahangiri JM. Faber polynomial coefficients of bi-subordinate functions, C R Acad Sci Paris Ser I 2016; 354: 365-370.
Hayami T, Owa S. Coefficient bounds for bi-univalent functions. Pan Amer Math J 2012; 22 (4): 15–26.
Özlem Güney H, Murugusundaramoorthy G, Sokol J. Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers. Acta Univ Sapientiae Mathematica 2018; 10 (1): 70-84.
Lewin M. On a coefficient problem for bi-univalent functions.* *Proc Amer Math Soc 1967; 18: 63-68.
Netanyahu E. The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\left\vert z\right\vert <1.$ Archive for Rational Mechanics and Analysis 1969; 32: 100-112.
Şeker B, Mehmetoğlu V. Coefficient bounds for new subclasses of bi-univalent functions. New Trends in Mathematical Sciences 2016; 4 (3): 197-203.
Srivastava HM, Sümer Eker S, Hamidi SG, Jahangiri JM. Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator. Bulletin of the Iranian Mathematical Society 2018; 44 (1): 149–157.
Srivastava HM. Univalent functions, fractional calculus, and associated generalized hypergeometric functions. In: Univalent Functions$,$ Fractional Calculus$,$ and Their Applications (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
Srivastava HM, Owa S. Univalent Functions, Fractional Calculus, and Their Applications. Ellis Horwood Ltd Publ Chichester, 1989.
Srivastava HM, Mishra AK, Gochhayat P. Certain subclasses of analytic and bi-univalent functions. Applied Mathematics Letters 2010;* *23: 1188-1192.
Srivastava HM, Murugusundaramoorthy G, Magesh N. Certain subclasses of bi-univalent functions associated with the Hohlov operator. Global Jour Math Anal 2013; 1 (2): 67–73.
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author:
- Andrei Teleman
date:
title: 'Moduli spaces of $PU(2)$-monopoles'
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[u]{} i[[i]{}]{} \[section\] \[sz\][Theorem]{} \[sz\][Proposition]{} \[sz\][Remark]{} \[sz\][Corollary]{} \[sz\][Definition]{} \[sz\][Lemma]{} \[sz\][Claim]{}
Introduction
============
The most natural way to prove the equivalence between Donaldson theory and Seiberg-Witten theory is to consider a suitable moduli space of “non-abelian monopoles”. In \[OT5\] it was shown that an $S^1$-quotient of a moduli space of quaternionic monopoles should give an homological equivalence between a fibration over a union of Seiberg-Witten moduli spaces and a fibration over certain $Spin^c$-moduli spaces \[PT1\].
By the same method, but using moduli spaces of $PU(2)$-monopoles instead of quaternionic monopoles, one should be able to express any Donaldson invariant in terms of Seiberg-Witten invariants associated with the abelian monopole equations of \[OT6\]. In \[T1\], \[T2\], we have shown that this idea can be further generalized to express Donaldson-type invariants associated with higher symmetry groups in terms of new Seiberg-Witten-type invariants.
The strategy has a very general algebraic-geometric analogon, which we call the “Master Space” strategy. This procedure, developed by Ch. Okonek and the author \[OT7\], \[OST\] reduces the problem of the computation of certain numerical invariants of a GIT moduli space to similar computations on simpler moduli spaces. One “couples” the given GIT problem to a simpler one (having the same symmetry group), and then studies the “Master Space” associated with the coupling as a $\C^*$-space. The fixed point locus of the $\C^*$-action consists of the original moduli space and a union of simpler ones. Then one can use the $S^1$-quotient of the master space to define a homological equivalence between a projective fibration over the initial moduli space and a projective fibration over the other components of the fixed point locus. In the GIT-framework, as in the gauge theoretical one, the technical difficulty is the same: the master space can be singular. The present paper deals with this difficulty in the gauge theoretical situation.\
A program for proving the equivalence between Donaldson theory and Seiberg-Witten theory, which also uses moduli spaces of non-abelian monopoles, is due to Pidstrigach and Tyurin \[PT2\], and was already announced by Pidstrigach in a Conference at the Newton Institute in Cambridge, in December 1994.
There are, however, several important differences between Pidstrigach-Tyurin’s original approach, and the strategy developed by Ch. Okonek in collaboration with the author, which is the strategy we follow in the present paper.
First, our equations have a gauge group of the form $SU(E)$ and hence the moduli spaces which we construct are $S^1$-spaces; in contrast, the Pidstrigach-Tyurin equations \[PT2\] have a gauge group of the form $U(E)$. Whereas we fix the connection in the determinant line bundle, they only fix the curvature of this connection. If $H_1(X,\Z)=0$, their moduli space is the $S^1$-quotient of ours. On the other hand, the $S^1$-operation plays a very important role in our strategy: The description of the ends around the abelian locus at infinity uses in an essential way the $S^1$-equivariance of the local models.
Second, we do not follow Pidstrigach-Tyurin’s program to prove generic regularity results. We show (see section 3.1) that the proofs of the transversality theorems which they use \[PT2\] to get generic regularity are incomplete, by indicating counterexamples to one of the statements on which these proofs are based[^1].
It is interesting to notice that, in fact, [*any*]{} non-abelian solution of the equations in the Kähler case gives a counterexample to their statement. This same statement was also used by the authors in the their definition of the $Spin^c$-polynomial invariants \[PT1\], on which was based their approach to prove the Van de Ven conjecture.
The transversality problem is very complicated, for the $PU(2)$-monopole equations as well as for the non-abelian $Spin^c$-equations. The difficulty is the same in both cases: in the non-abelian points with degenerate spinor component transversality cannot be proved using only perturbations with 0-order operators.
In \[T1\] the author tried to use perturbations with first order operators, and proved that perturbations of this type lead to transversality at least away from the solutions which are abelian on a non-empty open set. However, in order to have a complete transversality result away from the abelian locus, one would need a unique continuation theorem which seems to be difficult to get because of the perturbed symbol.
Another way to achieve transversality is to use an infinite family of “holonomy perturbations” \[FL\].
The present paper solves two fundamental problems concerning the moduli spaces of $PU(2)$-monopoles: [*generic regularity*]{} and [*compactification*]{}.
First we prove an $S^1$-equivariant generic-smoothness theorem: we define perturbations of the equations which lead to $S^1$-spaces which, for generic choices of the perturbing parameters are smooth, at least outside the “Donaldson locus” (the vanishing locus of the projection on the spinor component) and of the abelian locus (Theorem 3.19). The proof of the generic-smoothness theorem is a pure transversality argument; it combines a standard transversality argument with a new method to control the exceptions to transversality.
[*Our result shows that one does get regularity for a generic choice of a system $(g,\sigma,\beta,K)$, consisting of a metric, a compatible $Spin^{U(2)}(4)$-structure $\sigma$ and an order 0- perturbation $(\beta,K)$ of the type considered in \[PT2\].*]{}
We also obtain generic regularity results for the normal bundles of the Donaldson locus and the abelian locus within the moduli space (Theorem 3.21, Proposition 3.22). Similar results, but obtained using quite different lines of reasoning, were obtained by Feehan \[F\] in a preprint distributed around the same time as the first version of the present paper.
Therefore one can go forward towards a proof of the Witten conjecture (see for instance \[OT5\] for a detailed description of the strategy) using relatively simple equations.
Note however that the generic regularity results which we prove for the ASD-$Spin^c$-equations, do automatically solve the transversality problem needed in order to give sense to the $Spin^c$-polynomial invariants, and to use them effectively. For this purpose one would need a pure transversality argument for the ASD-$Spin^c$-equations.[^2] This seems to be difficult. The theory of $Spin^c$-polynomial invariants, and the attempt to prove the Van de Ven conjecture using these invariants, should be therefore revised.
We get our result in two steps. In a first step we prove that, using only the perturbations $(\beta,K)$, one can prove the following partial transversality result: If the Seiberg-Witten map extended to the parameterized moduli space is not a submersion in a point $(A,\Psi,\beta,K)$, then the spinor component $\Psi$ must be degenerate. This is very easy to see.
In the second step we prove that, if we also let the $Spin^{U(2)}(4)$-structure (together with the metric) vary , then the moduli space $\widetilde{{\cal D}{\cal M}}^*_X$ of solutions with non-trivial but degenerate spinor component in the enlarged parameterized moduli space $\widetilde{\cal M}^*_X$ has infinite codimension in every non-abelian point. Using this, we can show (by “weakening” locally the degeneracy equation) that every non-abelian point $[p]$ in $\widetilde{{\cal D}{\cal M}}^*_X$ has a neighbourhood $U_{[p]}$ which is a closed analytic subspace of a manifold $V_{[p]}$ which is Fredholm of negative index over the enlarged parameter space. Taking a countable subcover $(V_{[p]_i})_{i\in\N}$, and using the fact that Fredholm maps are locally proper (\[Sm\]), we prove that the set of parameters for which there exists a non-abelian solution with non-trivial degenerate spinor component is of the first category. The desired set of “generic parameters” is then obtained by intersecting the complement of this set with the set of regular values of the projection of $\widetilde{{\cal M}}^*_X\setminus\widetilde{{\cal D}{\cal M}}^*_X$ on the parameter space.
We believe that this method is in fact a very general one; it can be summarized as follows: Prove first a partial transversality result using perturbations with 0-order differential operators, and show then that the space of solutions which are exceptions to transversality has infinite codimension if one introduces new variable parameters. Such a result is to be expected provided the “exceptional solutions” , the ones which are exceptions to transversality, solve an overdetermined elliptic system.
In particular, the method can be applied to obtain generic regularity along the Donaldson and the abelian locus. More precisely, the moduli space of solutions (with non-vanishing spinor component) of the Dirac-ASD system of \[PT1\] becomes smooth of expected dimension for generic perturbations. The same property has the complement of the zero-section in the fibration of “normal infinitesimal deformations” over the subspace of abelian solutions associated with an abelian reduction of the $Spin^{U(2)}(4)$-bundle.
In this way we obtain perturbed moduli spaces which are smooth except in the abelian points and in the Donaldson-points. These points remain exceptions to transversality, and in general, regularity (smoothness and expected dimension) [*cannot*]{} be achieved in these points by using $S^1$-equivariant perturbations.
The second purpose of the paper, the existence of an “Uhlenbeck compactification” for the perturbed moduli spaces, is achieved in section 4 (see Theorem 4.24). A different proof of the “Uhlenbeck compactification” can be found in \[FL\].
Our arguments follow the same strategy as in the instanton case \[DK\], which can be summarized as follows:\
\
Local estimates – Regularity – Removable Singularities – Compactification.\
\
Some care must be taken, since the monopole equations are only “scale invariant”, not conformal invariant as in the instanton case. On the other hand, many of the results in \[DK\] were obtained by cutting off the solutions and transferring the problem from the 4-ball to the 4-sphere, and then using the conformal invariance of the equations.
Our proof uses the same method, but endows the sphere with a metric with non-negative sectional curvature which is flat in a neighbourhood of the north pole. With this choice, the corresponding first order elliptic operators ($\Dr$, $d^*+d^+$, …) are still injective. For the local computations we work with pairs whose connection component is in Coulomb gauge in the sense of \[DK\], so that all the results in \[DK\] about connections in Coulomb gauge apply automatically. Therefore, we do not use the Coulomb gauge condition for pairs which follows from the elliptic complex of the $PU(2)$-monopole equations (compare with \[FL\]).
A short version of our proof of the Uhlenbeck compactification appeared in \[OT5\], and a very detailed version of it can be found in \[T1\]. The existence of an Uhlenbeck compactification for moduli spaces of non-abelian monopoles was predicted by Pidstrigach and Tyurin in \[PT2\].
Note that in order to prove the equivalence between the Donaldson and the Seiberg-Witten theories, it now remains only to give explicit descriptions of the ends of the moduli space along the abelian locus, and to calculate the corresponding contributions.
My own strategy to study the ends of the moduli spaces of $PU(2)$-monopoles is based on the analytical results in \[T3\]. The $PU(2)$-monopole equations are not conformally invariant, so it is difficult to use the method developed in the case of instantons \[DK\] (which consists of identifying the solutions concentrated in a point with the solutions on the connected sum of $X$ with $S^4$). We use a new strategy \[T4\] which is still based on the gluing method. We obtain concentrated solutions by gluing (non-concentrated) solutions on $X$ corresponding to lower topological data, with concentrated instantons on the [*tangent spaces*]{}, and then we deform the obtained almost-solutions into solutions. This last step makes use of the classical Fredholm $L^p$ theory on $X$, as well as of the Fredholm $L^p$-theory on the tangent spaces (instead of $S^4$) which is developed in the quoted paper.
Progress on this problem, using different methods, was also announced by Feehan and Leness.
I would like to thank professor Ch. Okonek for encouraging me to write this paper, for the careful reading, and for his suggestions. I would also like to thank professor S. T. Yau for suggesting me to submit the paper to AJM. Finally I thank the referee for the very careful checking of the technical arguments and for his valuable observations.
$PU(2)$-monopoles
=================
The $Spin^{U(2)}$ group and $Spin^{U(2)}$-structures
----------------------------------------------------
For a more detailed presentation of the theory of $Spin^{U(2)}$-structures we refer the interested reader to \[T1\], \[T2\]. In these papers we also introduce the concept of $Spin^G$-structures and $G$-monopole equations for quite general compact Lie groups $G$.
The group $Spin^{U(2)}$ is defined by $$Spin^{U(2)}:=Spin\times_{\Z_2} U(2)\ .$$ Using the natural isomorphism $\qmod{U(2)}{\Z_2}\simeq PU(2)\times S^1$, we get the exact sequences $$1\map Spin\map Spin^{U(2)}\textmap{(\bar\delta,\det)}
PU(2)\times S^1\map 1$$ $$1\map U(2)\map Spin^{U(2)}\stackrel{\pi}{\map} SO\map 1
\eqno{(1)}$$ $$1\map \Z_2\map Spin^{U(2)}\textmap{(\pi,\bar\delta,\det)}
SO\times PU(2)\times S^1
\map 1 \ .$$ Let $X$ be a compact manifold and $P^u$ a $Spin^{U(2)}$-bundle over $X$. We consider the following associated bundles $$\pi(P^u):=P^u\times_\pi SO, \ \bar\delta(P^u):=
P^u\times_{\bar\delta} PU(2),\
\det(P^u):=P^u\times_{\det} S^1,$$ $$\G_0:=P^u\times_{Ad\circ\bar\delta} SU(2);\
\gr_0:=P^u\times_{ad\circ\bar\delta} SU(2)
\ ,$$ where $Ad:PU(2)\map Aut(SU(2))$, $ad:PU(2)\map so(su(2))$ are induced by the adjoint morphism $SU(2)\map Aut(SU(2))$, $SU(2)\map so(su(2))$.
The group of sections ${\cal G}_0:=\Gamma(X,\G_0)$ can be identified with the group of automorphisms of $P^u$ over $\pi(P^u)\times_X\det(P^u)$. After suitable Sobolev completions it becomes a Lie group, whose Lie algebra is the corresponding completion of $A^0(\gr_0)$.\
Let $P$ be a $SO$ bundle over $X$. A $Spin^{U(2)}$-structure in $P$ is a morphism $P^u \map P$ of type $\pi$, where $P^u$ is a $Spin^{U(2)}$-bundle. Two $Spin^{U(2)}$-structures $P^u\map P$, $P'^u\map P$ in $P$ are called equivalent if the bundles $P^u$, $P'^u$ are isomorphic over $P$. A $Spin^{U(2)}(n)$-structure in an oriented Riemannian 4-manifold $(X,g)$ is a $Spin^{U(2)}(n)$-structure in the bundle $P_g$ of oriented coframes.\
We refer to \[T1\], \[T2\] for the following classification result:
Let $P$ be a principal $SO$-bundle, $\bar P$ a $PU(2)$-bundle, and $L$ a Hermitian line bundle over $X$.\
i) $P$ admits a $Spin^{U(2)}$-structure $P^u
\rightarrow P$ with $$P^u\times_{\bar \delta}PU(2)\simeq\bar P\ ,\ \ P^u\times_{\det}\C\simeq L$$ if and only if $w_2(P)=w_2(\bar P)+\overline c_1(L)$, where $\overline c_1(L)$ is the mod 2 reduction of $c_1(L)$ .\
ii) If the base $X$ is a compact oriented 4-manifold, then the map $$P^u\longmapsto \left([P^u\times_{\bar\delta} PU(2)],
[P^u\times_{\det}\C]\right)$$ defines a 1-1 correspondence between the set of isomorphism classes of $Spin^{U(2)}$-structures in $P$ and the set of pairs of isomorphism classes $([\bar P],[L])$, where $\bar P$ is a $PU(2)$-bundle and $L$ an $S^1$-bundle with $w_2(
P)=w_2(\bar P)+\overline c_1(L)$. The latter set can be identified with $$\{(p,c)\in H^4(X,\Z)\times H^2(X,\Z) |\ p\equiv (w_2(P)+
\bar c)^2\ {\rm mod}\ 4\}$$
The group $Spin^{U(2)}(4)$ can be written as $$Spin^{U(2)}(4)=\qmod{SU(2)_+\times
SU(2)_-\times U(2)}{\Z_2}\ ,$$ hence it comes with natural orthogonal representations $$\ad_{\pm}: Spin^{U(2)}(4)\map so(su(2)),$$ defined by the adjoint representations of $SU(2)_{\pm}$, and with natural unitary representations $$\sigma_{\pm}:Spin^{U(2)}(4)\map
U(\H_{\pm}\otimes_\C \C^2)$$ obtained by coupling the canonical representations of $SU(2)_{\pm}$ with the canonical representation of $U(2)$. We denote by $\ad_{\pm}(P^u)$, $\Sigma^{\pm}(P^u)$ the corresponding associated vector bundles. The Hermitian 4-bundles $\Sigma^{\pm}(P^u)$ are called the spinor bundles of $P^u$, and the sections in these bundles are called spinors.\
We refer to \[T2\] for the following simple result
Let $P$ be an $SO(4)$-bundle whose second Stiefel-Whitney class admits integral lifts.
There is a 1-1 correspondence between isomorphism classes of $Spin^{U(2)}$-structures in $P$ and equivalence classes of pairs consisting of a $Spin^c(4)$-structure $P^c\map P$ in $P$ and a $U(2)$-bundle $E$. Two pairs are considered equivalent if, after tensoring the first one with a line bundle, they become isomorphic over $P$.
Suppose that $P^u$ is associated with the pair $(P^c,E)$, and let $\Sigma^{\pm}_c$ be the spinor bundles corresponding to $P^c$. Then the associated bundles $P^u\times_\pi\R^4$, $\Sigma^{\pm}(P^u)$, $\bar\delta(P^u)$, $\det(P^u)$, ${\G(P^u)}$, $\G_0(P^u)$ can be expressed in terms of the pair $(P^c,E)$ as follows: $$P^u\times_\pi\R^4=\RSU(\Sigma^+_c,\Sigma^-_c)\ ,$$ $$\Sigma^{\pm}(P^u)=[\Sigma^{\pm}_c]^{\vee}\otimes
E=\Sigma^{\pm}_c\otimes E^{\vee}\otimes[\det(P^u)] ,\
\bar\delta(P^u)\simeq
\qmod{P_E}{S^1},\ \ad_{\pm}(P^u)=su(\Sigma^{\pm}_c)$$ $$\det(P^u)\simeq \det
(P^c)^{-1}\otimes (\det E),\
\G_0(P^u)=SU(E),\ \gr_0(P^u)=su(E)\ .$$ Here we denoted by $\RSU(\Sigma^+_c,\Sigma^-_c)$ the bundle of real multiples of $\C$-linear isometries of determinant 1 from $\Sigma^+_c$ to $\Sigma^-_c$. The euclidean structure and the orientation in this bundle are fibrewise defined by the Pauli matrices associated with a pair of frames $(e_1^{\pm},e_2^{\pm})$ in $\Sigma^{\pm}_c$, satisfying $e_1^+\wedge e_2^+=e_1^-\wedge e_2^-$.
The data of a $Spin^{U(2)}(4)$-structure $P^u\map P$ in an $SO(4)$-bundle $P$ is equivalent to the data of an orientation preserving linear isometry $$\gamma:P\times_{SO(4)}{\R^4}\map
P^u\times_\pi\R^4=\RSU(\Sigma^+_c,\Sigma^-_c)\subset \Hom_{\G_0}
(\Sigma^+(P^u),
\Sigma^-(P^u))$$ which will be called the of the structure.
The Clifford map $\gamma$ induces isomorphisms $$\Gamma_{\pm}:\Lambda^2_{\pm}(P\times_{SO(4)}{\R^4})\map
su(\Sigma^{\pm}_c)=\ad_{\pm}(P^u)\ ,$$ which multiply the norms by 2 (\[DK\] p. 77, \[OT1\]).
The following simple remark will play a fundamental role in this paper:\
[*Suppose that $\Lambda$ is a real oriented 4-bundle, and $\gamma:\Lambda\map
P^u\times_\pi\R^4$ an orientation preserving linear isomorphism. Then $\gamma$ defines an Euclidean structure $g_\gamma$ on $\Lambda$ such that $\gamma$ becomes the Clifford map of a $Spin^{U(2)}(4)$-structure in $(\Lambda,g_\gamma)$.*]{}
The $PU(2)$-monopole equations
------------------------------
Let $\sigma:P^u\map P_g$ be a $Spin^{U(2)}(4)$-structure in the oriented compact Riemannian 4-manifold $(X,g)$. Fix a connection $a\in{\cal A}(\det(P^u))$. Using the third exact sequence in (1), we see that the data of a connection $A\in {\cal A}(\bar\delta(P^u))$ is equivalent to the data of a connection $B_{A,a}$ in $P^u$ which lifts the Levi-Civita connection in $P_g$ and the fixed connection $a$ in $\det(P^u)$ (via the maps $P^u\map P_g$ and $P^u\map \det(P^u)$ respectively). The Dirac operator $\Dr_{A,a}$ associated with the pair $(A,a)$ is the first order elliptic operator $$\Dr_{A,a}:A^0(\Sigma^{\pm}(P^u))\textmap{\nabla_{B_{A,a}}}
A^1(\Sigma^{\pm}(P^u))\stackrel{\gamma}{\map}A^0(\Sigma^{\mp}(P^u))$$ Regarded as operator $\Sigma^+(P^u)\oplus\Sigma^-(P^u)\map\Sigma^+(P^u)\oplus\Sigma^-(P^u)$, the Dirac operator $\Dr_{A,a}$ is also selfadjoint.
We define the quadratic map $\Sigma^{\pm}(P^u)\map \ad_+(P^u)\otimes\gr_0$, $\Psi\longmapsto (\Psi\bar\Psi)_0$ by $$(\Psi\bar\Psi)_0:=pr_{\ad_+(P^u)\otimes\gr_0}(\Psi\otimes\bar\Psi) \ ,$$ where $pr_{\ad_+(P^u)\otimes\gr_0}$ denotes the orthogonal projection $$\Herm(\Sigma^+(P^u))\map{\ad_+(P^u)\otimes\gr_0}\ .$$
We introduce now the $PU(2)$-Seiberg-Witten equations $SW^\sigma_a$ associated to the pair $(\sigma,a)$, which are equations for a pair $(A,\Psi)$ formed by a $PU(2)$-connection $A\in{\cal A}(\bar\delta(P^u))$ and a positive spinor $\Psi\in
A^0(\Sigma^+(P^u))$: $$\left\{\begin{array}{ccc}
\Dr_{A,a}\Psi&=&0\\
\Gamma(F_A^+)&=&(\Psi\bar\Psi)_0
\end{array}\right. \eqno{(SW^\sigma_a)}$$ The natural symmetry group of the equations is the gauge group ${\cal
G}_0:=\Gamma(X,\G_0)$. We denote by ${\cal M}^\sigma_a$ the moduli space $${\cal M}^\sigma_a:=\qmod{\left[{\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+(P^u))\right]^{SW^\sigma_a}}{{\cal G}_0} \ ,$$ where $\left[{\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+(P^u))\right]^{SW^\sigma_a}$ denote the space of solutions of the equations $(SW^\sigma_a)$. Using the well-known Kuranishi method one can endow ${\cal
M}^\sigma_a$ with the structure of a ringed space, which has locally the form $\qmod{Z(\theta)}{G}$, where $G$ is a closed subgroup of $SU(2)$ acting on finite dimensional vector spaces $H^1$, $H^2$, and $Z(\theta)$ is the real analytic space cut-out by a $G$-equivariant real analytic map $H^1\supset U \stackrel{\theta}{\map} H^2\ $ (see \[OT5\], \[T1\], \[T2\] for details).
Smooth moduli spaces
====================
The difficulty
--------------
Equations for pairs $(A,\Psi)$, where $A$ is a unitary connection with fixed determinant connection and $\Psi$ a non-abelian Dirac spinor have been already considered \[PT1\], \[PT2\]. For instance, the definition of $Spin^c$-polynomial invariants starts with the construction of the moduli space of solutions of the $(ASD-Spin^c)$-equations $$\left\{\begin{array}{lll}
\Dr_{A} \Psi&=&0 \ ,\ \ \ \Psi\ne 0\\
F_A^+&=&0\ .
\end{array}\right.$$
The proofs of the corresponding transversality results are incomplete. They are based on the following false statement (\[PT2\], \[PT1\]):\
\
[**(A)**]{} Let $P^c\map P_g$ be a $Spin^c(4)$-structure with spinor bundles $\Sigma^{\pm}(P^c)$ on a Riemannian 4-manifold $(X,g)$, $E$ a Hermitian 2-bundle on $X$, and $A$ a unitary connection in $E$. If the $\Dr_{A}$-harmonic non-vanishing positive spinor $\Psi\in A^0(\Sigma^+(P^c)\otimes E)$ is fibrewise degenerate considered as morphism $E^{\vee}\map \Sigma^+$, then $A$ is reducible.\
In the proof of this assertion (\[PT1\] p. 277) it was used that, in the presence of a $Spin^c(4)$-structure, the Clifford pairing $(\alpha,\sigma)\longmapsto
\gamma(\alpha)\sigma$ between 1-forms and positive spinors has fibrewise no divisors of zero. This is true for real 1-forms, but not for complex ones.
Counterexamples are easy to find:
Every holomorphic section in a holomorphic Hermitian 2-vector bundle ${\cal E}$ on a Kähler surface can be regarded as a degenerate harmonic positive spinor in $\Sigma^+_{\rm can}\otimes {\cal E}$, where $\Sigma^+_{\rm
can}=\Lambda^{00}\oplus\Lambda^{02}$ is the positive spinor bundle of the canonical $Spin^c(4)$-structure in $X$, if we endow ${\cal E}$ with the Chern connection given by the holomorphic structure. Therefore any indecomposable holomorphic 2-bundle ${\cal E}$ with $H^0({\cal E})\ne 0$ gives a counterexample to the assertion [**(A)**]{}.\
Note that these counterexamples occur precisely in the Kähler framework, where all explicit computations of moduli spaces and invariants were carried out.
Partial transversality results
------------------------------
Let $\sigma:P^u\map P_g$ be a $Spin^{U(2)}(4)$-structure on $(X,g)$, denote by $$\gamma:\Lambda^1\map\Hom(\Sigma^+,\Sigma^-)$$ be the associated Clifford map, and let $C_0$ be a fixed $SO(4)$-connection in $P^u\times_\pi SO(4)\simeq P_g$ (not necessarily the Levi-Civita connection). We fix again a connection $a\in{\cal A}(\det(P^u))$. For any connection $A\in {\cal A}(\bar\delta(P^u))$ we have an associated Dirac operator $$\Dr_{a,A}^0=\gamma\cdot \nabla_{C_0,a,A}\ ,$$ where $\nabla_{C_0,a,A}:A^0(\Sigma^+)\map A^1(\Sigma^+)$ is the covariant derivative associated with the connection in $P^u$ which lifts the triple $(C_0,a,A)$.
The role and the properties of these slightly more general Dirac operators will be cleared up in the next section, where $C_0$ will be a ${\cal C}^{\infty}$-connection in the fixed bundle $P^u\times_\pi SO(4)$, but the metric $g$ and the Clifford map $\gamma$ will be variable ${\cal C}^k$-parameters.
Recall that one has a canonical embedding $P^u\times_\pi\C^4\subset\Hom(\Sigma^+,\Sigma^-)$, and that $\sigma$ defines an isomorphism $\Lambda^1_\C\textmap{\simeq}P^u\times_\pi\C^4$. We consider the following equations $$\left\{
\begin{array}{lcc}
\Dr_{a,A}^0(\Psi)+\beta(\Psi)&=&0\\
\Gamma(F_A^+)&=&K(\Psi\bar\Psi)_0
\end{array}\right.\ ,$$ which are equations for a system $$(A,\Psi,\beta,K)\in {\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+)\times A^0(P^u\times_\pi\C^4)
\times \Gamma(X,GL(\ad_+)).$$ Complete the configuration space ${\cal A}:={\cal A}(\bar\delta(P^u))\times
A^0(\Sigma^+)$ with respect to a large Sobolev norm $L^2_l$, and the parameter space $${\cal
Q}:=A^0(P^u\times_\pi\C^4)\times \Gamma(X,GL(\ad_+))$$ with respect to the Banach norm ${\cal C}^k$, $k\gg l$.
The perturbations $(\beta,K)$ were also considered by Pidstrigach and Tyurin in \[PT2\] in their attempt to get transversality for their version of non-abelian monopole equations.
An $SU(2)\times SU(2)\times SU(2)$-reduction of $P^u$ over an open set $U\subset X$ induces isomorphisms $\Sigma^{\pm}(P^u)|_U\simeq S^{\pm}\otimes E$ where $S^{\pm}$, $E$ are $SU(2)$-bundles. A spinor $\Psi\in \Sigma^+(P^u)$ will be called $x\in X$ if, with respect to an $SU(2)\times
SU(2)\times
SU(2)$-reduction around $x$, $\Psi_x\in S_x\otimes E_x=S_x^{\vee}\otimes E_x$ has rank $\leq 1$. $\Psi$ will be called degenerate on $V\subset X$ if it is degenerate in every point of $V$.
A pair $(A,\Psi)\in {\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+)$ will be called if the connection $A$ is reducible, and the spinor $\Psi$ is contained in one of the $A$-invariant summands of $\Sigma^+$.
If $(A,\Psi)\in {\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+)$ is an abelian pair, then $\Psi$ is clearly degenerate on $X$. However, the counterexamples in the previous section show that there exist non-abelian pairs with non-trivial Dirac-harmonic spinor-component which is degenerate on $X$.
Let $sw=sw_{g,\sigma,C_0}:{\cal A}_l\times{\cal Q}^k\map
A^0(\Sigma^-)_{l-1}\times
A^0(\ad_+\otimes\gr_0)_{l-1}$ be the map defined by the left hand side of the equations above, and let $${\cal N}^*:=\qmod{[{\cal A}_l^*\times{\cal Q}^k]\cap
sw^{-1}(0)}{{\cal G}_{l+1}}$$ be the moduli space of solutions with non-trivial spinor-component. ${\cal N}^*$ is the vanishing locus of the induced section $\bar {sw}$ in the Banach bundle $$[{\cal A}_l^*\times{\cal Q}^k]\times_{{\cal G}_{l+1}}[A^0(\Sigma^-)_{l-1}\times
A^0(\ad_+\otimes\gr_0)_{l-1}]$$ over the Banach manifold ${\cal B}^*:=
\qmod{{\cal A}_l\times{\cal Q}^k}{{\cal G}_{l+1}}$ which is defined by $sw$.
The purpose of this section is to prove the following partial transversality theorem
If $sw$ is not a submersion in a solution $p=(A,\Psi,\beta,K)\in
{\cal
A}_l^*\times{\cal Q}^k$, then $\Psi$ must be degenerate on $X$. In particular, ${\cal N}^*$ is smooth away from the closed subspace of solutions with globally degenerate spinor component.
Let $(\Phi,S)\in
A^0(\Sigma^-)_{l-1}\times A^0(\ad_+\otimes\gr_0)_{l-1}$ a pair which is $L^2$-orthogonal to $\im (d_p)$. Using the perturbation $\beta$ we get immediately that $Re(\dot\beta,\Phi\otimes\bar\Psi)$ vanishes for every variation $\dot\beta$ of $\beta$. With respect to any local $SU(2)\times SU(2)\times SU(2)$-reduction $(S^{\pm},E)$ of $P^u|_U$ ($U$ an open set) the contraction of $\Phi\otimes\bar\Psi$ with the Hermitian metric in $E$ must vanish, which shows that pointwise $\Psi(v^+)\bot \Phi(v^-)$ for every $v^{\pm}\in S^{\pm}_u$, $u\in U$. If $\Psi$ has rank 2 in a point $x\in X$, then $\Phi$ must vanish identically on a neighbourhood of $x$.
Also, if $\Psi$ has rank 2 in $x$, then $(\Psi\bar\Psi)_0$ has rank 3 in $x$ as map $\ad_{+,x}^{\vee}\map \gr_{0,x}$, hence the same argument as above shows that $S$ must vanish on a neighbourhood of $x$. Therefore the pair $(\Phi,S)$ must be zero on a neighbourhood of $x$.
We can assume that $A$ is the Coulomb gauge with respect to a smooth connection $A_0$. Therefore, by Agmon-Douglis-Nirenberg’s non-linear-elliptic regularity theorems (see for instance \[B\], p. 467, Theorem 41), it follows that $(A,\Psi)$ is a smooth pair (if the Clifford map $\Lambda^1\map P^u\times\R^4$ had only class ${\cal C}^k$, we would have got a ${\cal C}^{k+1-\varepsilon}$-pair, which is enough to complete the argument). Using now variations $(\dot A,\dot\Psi)$, we see that $(\Phi,S)$ must satisfy an elliptic system of the form $$\tilde D^1_{A,\Psi}[\tilde D^1_{A,\Psi}]^*(\Phi,S)=0 \ .$$ Here $\tilde D^1_{A,\Psi}$ is the first derivative in $(A,\Psi)$ of the map $\tilde sw$ obtained by dividing by 2 the second component of $sw$ such that the symbol of $\tilde
D^1_{A,\Psi}[\tilde D^1_{A,\Psi}]^*$ becomes a scalar, and Aronszajin’s theorem applies. It follows that $(\Phi,S)=0$, because it vanishes on a non-empty open set.
The same result holds if ${\cal Q}^k$ is replaced by any product ${\cal Q}^k\times
{\cal R}$ of ${\cal Q}^k$ with a Banach manifold $R$, and $sw$ by a smooth map $sw':{\cal
A}_l
\times {\cal Q}^k\times {\cal R}$ whose restriction to any fibre ${\cal A}_l \times
{\cal Q}^k\times \{r\}$ has the form $sw_{g,\sigma,C_0}$ for a metric $g$, a $Spin^{U(2)}$-structure $\sigma$ in $(X,g)$, and an $SO(4)$-connection $C_0$.
An easy way to parameterize the space of pairs consisting of a metric and a $Spin^{U(2)}(4)$-structure will be given in the next section.
$PU(2)$-monopoles with degenerate spinor component. Generic regularity
----------------------------------------------------------------------
Let $P^u$ be a $Spin^{U(2)}$-bundle. Suppose that the spinor $\Psi\in A^0(\Sigma^+)$ is degenerate on a whole neighbourhood of a point $x\in X$ but $\Psi_x\ne 0$, and let $A\in\bar\delta(P^u))$ be a $PU(2)$-connection. The pair $(A,\Psi)$ will be called if the second fundamental form of the line subbundle $L\subset E$ generated by $\Psi$ around $x$ is non-zero in $x$.
We recall that if $P^u$ is associated with a pair $(P^c,E)$, where $P^c$ is a $Spin^c(4)$ bundle $P^c$ of spinor bundles $\Sigma^{\pm}_c$ and $E$ is a $U(2)$-bundle, then $\Sigma^{\pm}=[\Sigma^{\pm}_c]^{\vee}\otimes E=\Sigma^{\pm}_c\otimes
E^{\vee}\otimes\det(P^u)$ and $P^u\times_\pi\R^4=\RSU(\Sigma^+_c,\Sigma^-_c)\subset
\Hom(\Sigma^+_c,\Sigma^-_c)\subset\Hom(\Sigma^+,\Sigma^-)$. The euclidean structure and the orientation in the real 4-bundle $\RSU(\Sigma^+_c,\Sigma^-_c)$ are fibrewise defined by the Pauli matrices associated with frames $(e_1^{\pm},e_2^{\pm})$ of $\Sigma^{\pm}_x$ satisfying $e_1^+\wedge e_2^+=e_1^-\wedge e_2^-$.
Let $P^u$ be a $Spin^{U(2)}$-bundle with $P^u\times_\pi\R^4
\simeq \Lambda^1$. A Clifford map is an orientation preserving linear isomorphism $$\gamma:\Lambda^1\map
P^u\times_\pi\R^4=\RSU(\Sigma^+_c,\Sigma^-_c)\subset
\Hom(\Sigma^+,\Sigma^-)\ .$$
Every ${\cal C}^k$ Clifford map $\gamma:\Lambda^1 \map P^u\times_\pi \R^4$ defines a ${\cal C}^k$ metric $g_\gamma$ on $X$ which makes $\gamma$ an isometry, so that $\gamma:\Lambda^1 \map P^u\times_\pi
\R^4\subset
\Hom(\Sigma^+,\Sigma^-)$ becomes the Clifford map of a $Spin^{U(2)}$-structure $\sigma_\gamma$ in $(X,g_\gamma)$.
This formalism will play an important role in this paper. The space $$Clif:=\Gamma(X,{\rm
Iso}_+(\Lambda^1,P^u\times_\pi\R^4))$$ of Clifford maps parameterizes the set of pairs consisting of a metric and a $Spin^{U(2)}(4)$-structure for that metric. Note that the metric determines a $Spin^{U(2)}$-structure with a given bundle $P^u$ only up to an $SO(4)$-gauge transformation of the cotangent bundle.
As in the previous section fix a ${\cal C}^{\infty}$ $SO(4)$-connection $C_0$ in $P^u\times_\pi SO(4)$. To any pair of connections $(a,A)\in
{\cal A}(\det(P^u))\times{\cal
A}(\bar\delta(P^u))$ we associate a Dirac operator $\Dr_{\gamma,a,A}^0$ using the Clifford map $\gamma$ and the lift $\nabla_{C_0,a,A}:A^0(\Sigma^+)\map A^1(\Sigma^+)$ of $(C_0,a,A)$: $$\Dr_{\gamma,a,A}^0=\gamma\cdot \nabla_{C_0,a,A}\ .$$ This Dirac operator does not coincide with the standard Dirac operator $\Dr_{\gamma,a,A}$ associated with $(A,a)$ and the $Spin^{U(2)}$-structure on $(X,g_\gamma)$ defined by $\gamma$, because $\gamma^{-1}(C_0)$ may be different from the Levi-Civita connection in $(\Lambda^1,g_\gamma)$; however, it has the same symbol as the standard one. The advantage of using these Dirac operators, is that they depend in a very simple way on $\gamma$ and that they are operators with ${\cal C}^k$-coefficients if $\gamma$ is of class ${\cal C}^k$. The coefficients of the Levi-Civita connection in $(\Lambda^1,g_\gamma)$ are in general only of class ${\cal C}^{k-1}$, and the coefficients of the induced Levi-Civita connection in $P^u\times_\pi \R^4$ are also of class ${\cal C}^{k-1}$, so that the coefficients of the standard Dirac operator $\Dr_{\gamma,a,A}$ have a regularity-class smaller by 1 than the regularity class of $\gamma$.
The use of these Dirac operators, whose coefficients do not contain the derivatives of the Clifford map, is essential in our proofs.
There exists a section $\beta=\beta(\gamma,C_0)\in {\cal
C}^{k-1}(P^u\times_\pi\C^4)$ such that $\Dr_{\gamma,a,A}^0=\Dr_{\gamma,a,A}+ \beta $.
To see this, let $C_\gamma$ be the $SO(4)$-connection in $P^u\times_\pi\R^4$ induced via $\gamma$ by the Levi Civita connection in $(\Lambda^1,g_\gamma)$. The difference $\alpha:=\nabla_{C_\gamma,a,A}-\nabla_{C_0,a,A}$ is an $\ad_+$-valued 1-form of class ${\cal C}^{k-1}$, hence an element in $${\cal C}^{k-1}(\Lambda^1(\ad_+))=
{\cal C}^{k-1}(\Lambda^1(su(\Sigma^+_c)))\subset {\cal
C}^{k-1}(\Lambda^1(\End(\Sigma^+)))$$ which does not depend on $(A,a)$. In local coordinates, $\alpha$ has the form $\alpha=\sum u^i\otimes \alpha_i $, with local sections $\alpha^i$ in $su(\Sigma^+_c)$. Its contraction with $\gamma$ has locally the form $\sum\gamma(u^i)\circ
\alpha_i$, and defines a ${\cal C}^{k-1}$-section $\beta$ in $\Hom(\Sigma^+_c,\Sigma^-_c)=P^u\times_\pi\C^4 $.
Consider the following $PU(2)$-monopole equations $$\left\{\begin{array}{lcl}
\Dr_{\gamma,a,A}^0\Psi&=&0\\
\Gamma_\gamma(F_A)&=&(\Psi\bar\Psi)_0 \ .
\
\end{array}\right.\eqno{(SW_a)}$$ for a triple $(A,\Psi,\gamma)\in{\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+)\times Clif$. The map $$\Gamma_\gamma:\Lambda^2\map
\End(\Sigma^+_c)\subset\End(\Sigma^+)$$ is determined by $\gamma$ via the formula $$\Gamma_\gamma(u\wedge
v)=\frac{1}{2}\left(-\gamma(u)^*\gamma(v)+\gamma(v)^*\gamma(u)\right)$$ and vanishes identically on $\Lambda^2_{-,g_\gamma}$, so that we could have written $F_A^{+_{g_{\gamma}}}$ instead of $F_A$ in the second equation. In the form above it will be easier to compute the derivative with respect to $\gamma$.
Complete the configuration space ${\cal A}:={\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+)$ with respect to a large Sobolev norm $L^2_l$ and the space of Clifford maps $Clif$ with respect to the Banach norm ${\cal C}^k$, $k\gg l$.
Before stating the main result of this section, we begin with two simple remarks
Let $A$, $F$ be subspaces of a normed space $H$ with $F$ finite dimensional. Then $$\overline{A+F}=\bar A +F\ .$$
Indeed, $\overline{A+F}\supset
\bar A$, and $\overline{A+F}\supset F$, hence $\overline{A+F}\supset \bar A +F$. To prove the opposite inclusion, it is enough to notice that $\bar A+ F\supset A+F$ and to prove that $\bar A+ F$ is closed. Let $q: H\map \qmod{H}{\bar A}$ be the canonical projection. The right hand space is also normed, hence $q(F)\subset
\qmod{H}{\bar A}$ is closed (being finite dimensional), and therefore $q^{-1}(q(F))=\bar A+ F$ is closed in $H$, since $q$ is continuous. This proves the remark
Let $f:H_1\map H_2$ be a continuous operator with closed image and finite dimensional kernel between Banach spaces , and let $A\subset H_1$ be a closed subspace. Then $f(A)$ is closed.
$f$ factorizes as $H_1\stackrel{p}{\map} \qmod{H_1}{\ker f}
\stackrel{\simeq}{\map}
f(H_1)\hookrightarrow H_2$, where the middle arrow is an isomorphism by the Banach Theorem. Therefore it is enough to show that $p(A)$ is closed, or equivalently that $p^{-1}(p(A))=A+\ker f$ is closed. But this follows by the remark above. Let $[{\cal A}_l\times Clif^k]^{SW}$ be the space of solutions $(A,\Psi,\gamma)$ of the equations above, and let $[{\cal A}_l\times Clif^k]^{SW}_U$ be the subspace of solutions whose spinor component is degenerate on the open set $U$.
The space $[{\cal A}_l\times Clif^k]^{SW}_U$ is a closed real analytic subspace of the space $[{\cal A}_l\times Clif^k]^{SW}$, since it is the vanishing locus of the (real analytic) map $${\cal A}_l\map A^0(\Sigma^+)_l\stackrel{\det}{\map}
A^0(\det(P^u))_l\stackrel{res_U}{\map}A^0(\det(P^u)|_U)_l \ .$$ We can now state the main result of this section.
Let $\theta=(A,\Psi,\gamma)\in [{\cal A}_l\times
Clif^k]^{SW}_U$, and suppose that for a point $u\in U$, one has $\Psi_u\ne 0$, and the pair $(A,\Psi)$ is non-abelian in $u$. Then the image of the Zariski tangent space $T_{\theta}[{\cal A}_l\times
Clif^k]^{SW}_U$ under the projection $$T_{\theta}[{\cal A}_l\times
Clif^k]^{SW} \map T_\gamma(Clif^k)={\cal C}^k(\Hom(\Lambda^1,
P^u\times_\pi\R^4))$$ has infinite codimension.
For the proof of the theorem, we need some preparations:
Note first (using \[DK\], p. 135) that we may assume that the Sobolev connection $A$ is in Coulomb gauge with respect to a smooth connection $A_0$ and a fixed smooth metric $g_0$, i.e. $$d_{A_0}^{*_{g_0}}(A-A_0)=0 \ .$$ Put $\alpha:=A-A_0$, hence $F_A=d_{A_0}\alpha+\alpha\wedge\alpha+F_{A_0}$. The differential operator $\Gamma_\gamma\circ d_{A_0}+d_{A_0}^{*_{g_0}}$ is [*elliptic*]{} although the metrics $g_0$ and $g_\gamma$ may be different, and it has coefficients of class ${\cal C}^k$. Note also that $\Gamma_\gamma\circ
d_{A_0}+d_{A_0}^{*_{g_0}}$ is an operator between ${\cal C}^{\infty}$-bundles.
The Dirac operator $\Dr_{\gamma,a,A_0}^0=\Dr_{\gamma,a,A}^0-\gamma(\alpha)$ has coefficients of class ${\cal C}^k$. Therefore, the pair $(\alpha,\Psi)$ is a solution of the non-linear elliptic system $$\left\{\begin{array}{lcl}
\Dr_{\gamma,a,A_0}^0\Psi+\gamma(\alpha)\Psi&=&0\\
\Gamma_\gamma(d_{A_0}\alpha+\alpha\wedge\alpha+F_{A_0})&=&(\Psi\bar\Psi)_0\\
d^{*_{g_0}}_{A_0}\alpha&=&0 \ \ \ \ .
\end{array}\right.$$ Writing the left hand side as a function of $x^j$, $\alpha^k$, $\Psi^l$, $\partial_j \alpha^k$, $\partial_j \Psi^l$ (with respect to a smooth chart and bundle trivializations), we see that this function has class ${\cal C}^k$ in this system of variables (in fact it is polynomial of degree 2 in the last four group of variables). It follows, by Agmon-Douglis-Nirenberg’s non-linear-elliptic regularity theorems (\[B\], p. 467, Theorem 41) that $\alpha$, $\Psi$, hence also the pair $(A,\Psi)$, have class ${\cal C}^{k+1-\varepsilon}$. It would have class ${\cal C}^{k+1}$ if we had chosen a non-integer index $k=[k]+\varepsilon$, i.e. if we had worked with the Hölder space ${\cal
C}^{[k],\varepsilon}$.\
Let $sw:{\cal A}_l\times Clif^k\map
A^0(\Sigma^-)_{l-1}\times A^0(\ad_+\otimes\gr_0)_{l-1}$ be the map given by the left hand side of the equations $(SW_a)$, and put $\det_U:=res_U\circ \det$.
The tangent space $T_\theta[{\cal A}_l\times Clif^k]^{SW}_U$ is the space of solutions $(\dot
A,\dot\Psi,\dot\gamma)$ of the linear system $$\left\{
\begin{array}{lcl}
\frac{\partial sw}{\partial(A,\Psi)}|_\theta(\dot A,\dot\Psi)+\frac{\partial
sw}{\partial\gamma}|_\theta(\dot\gamma)&=&0\\
d_\Psi(\det_U)(\dot \Psi)&=&0 \ .
\end{array}
\right.$$ Denote by $$D_l^U=:\ker[d_\Psi({\det}_U)]\subset
A^0(\Sigma^+)_l$$ the Zariski tangent space at $\Psi$ to the space ${\cal D}_l^U:=\det_U^{-1}(0)$ of $L^2_l$ positive spinors which are degenerate on $U$.\
Theorem 3.7 can now be reformulated as follows
The subspace $$\left(\frac{\partial sw}{\partial\gamma}|_\theta\right)^{-1}
\left(\frac{\partial sw}{\partial(A,\Psi)}|_\theta \ (A^1(\gr_0)_l\times
D_l^U)\right)$$ has infinite codimension in ${\cal C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4))$.
In order to prove Proposition 3.8 we start by giving explicit formulas for the partial derivatives above.
The derivative with respect to $\gamma$, $$\left(\frac{\partial sw}{\partial\gamma}|_\theta\right):{\cal
C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4))\map A^0(\Sigma^-)_{l-1}\times
A^0(\ad_+\otimes\gr_0)_{l-1}\ ,$$ is given by $$\left(\frac{\partial
sw}{\partial\gamma}|_\theta\right)(\dot\gamma)=
\left(\matrix{\dot\gamma(\nabla_{C_0,a,A}\Psi)\cr \cr
\frac{d}{d\gamma}(\Gamma_\gamma(F_A))(\dot\gamma)}\right)\ .
\eqno{(1)}$$ The derivative with respect to the pair $(A,\Psi)$, $$\left(\frac{\partial sw}{\partial(A,\Psi)}|_\theta\right): A^1(\gr_0)_l\times
A^0(\Sigma^+)_l
\map A^0(\Sigma^-)_{l-1}\times
A^0(\ad_+\otimes\gr_0)_{l-1}\ ,$$ is $$\left(\frac{\partial sw}{\partial(A,\Psi)}|_\theta\right)(\dot A,\dot \Psi)=
\left(\matrix{\Dr_{\gamma,a,A}^0\dot \Psi+\gamma(\dot
A)\Psi\cr \cr
\Gamma_\gamma(d_A\dot A)-[(\dot\Psi\bar
\Psi)_0+(\Psi\bar{\dot\Psi})_0]}\right) \ .
\eqno{(2)}$$
The next two lemmata will translate the problem into a similar one which involves only Sobolev completions.
Let $j^k_{l-1}$ be the compact embedding $$j^k_{l-1}:{\cal
C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4)) \map
A^0(\Hom(\Lambda^1,P^u\times_\pi \R^4))_{l-1}\ .$$
1\. The linear operator $\left(\frac{\partial
sw}{\partial\gamma}|_\theta\right)$ has a continuous extension to the Sobolev completion $A^0(\Hom(\Lambda^1,P^u\times_\pi \R^4))_{l-1}$. More precisely, formula (1) defines a linear continuous map $$a_{l-1}: A^0(\Hom(\Lambda^1,P^u\times_\pi \R^4))_{l-1}\map
A^0(\Sigma^-)_{l-1}\times A^0(\ad_+\otimes\gr_0)_{l-1}$$ such that $$\left(\frac{\partial
sw}{\partial\gamma}|_\theta\right)=a_{l-1}\circ j^k_{l-1}\ .$$ 2. The space $ \frac{\partial
sw}{\partial(A,\Psi)}|_\theta \ (A^1(\gr_0)_l\times D_l^U) $ is closed in $$A^0(\Sigma^-)_{l-1}\times A^0(\ad_+\otimes\gr_0)_{l-1} .$$
1\. The first assertion follows easily, since $\nabla_{C_0,a,A}\Psi$ and $F_A$ have regularity class ${\cal C}^{k-\varepsilon}$, and $\gamma$ has regularity class ${\cal C}^k$. Therefore, working in local ${\cal C}^\infty$-coordinates, the expression $$(\frac{d}{d\gamma}\Gamma) (\dot\gamma)(F_A)=
\frac{d}{d\gamma}
\left( \frac{1}{2}[-{\gamma(u^i)}^*\gamma(u^j)+
(\gamma(u^j)^*\gamma(u^i)]\otimes
F_{A,ij}\right)(\dot \gamma)$$ is a linear operator of order 0 with ${\cal C}^{k-\varepsilon}$ coefficients in the variable $\dot\gamma$.\
2. Decompose $A^1(\gr_0)_l\times A^0(\Sigma^+)_{l}$ as $$A^1(\gr_0)_l\times A^0(\Sigma^+)_{l}=
D^0_{(A,\Psi)} [A^0(\gr_0)_{l+1}]\oplus\ker [D^0_{(A,\Psi)}]^*=
\im D^0_{(A,\Psi)}\oplus\ker [D^0_{(A,\Psi)}]^*$$ where $D^i_{(A,\Psi)}$ are the differential operators in the fundamental elliptic complex associated with the pair $(A,\Psi)$ and the metric $g_\gamma$. The decomposition is $L^2_{g_\gamma}$-orthogonal.
The subspace $A^1(\gr_0)_l\times D_l^U\subset A^1(\gr_0)_l\times
A^0(\Sigma^+)_{l} $ is closed, and contains the first summand $\im D^0_{(A,\Psi)}$ by the gauge-invariance property of the degeneracy-condition. Using the fact that $D^1_{(A,\Psi)}\circ D^0_{(A,\Psi)}=0$, we get $$\frac{\partial
sw}{\partial(A,\Psi)}|_\theta \ (A^1(\gr_0)_l\times
D_l^U)= D^1_{(A,\Psi)}
\left[(A^1(\gr_0)_l\times D_l^U)\cap\ker (D^0_{(A,\Psi)})^*\right]=$$ $$=D^1_{(A,\Psi)}|_{_{\ker (D^0_{(A,\Psi)})^*}}
\left[(A^1(\gr_0)_l\times
D_l^U)\cap\ker (D^0_{(A,\Psi)})^*\right]\ .$$ But $D^1_{(A,\Psi)}|_{\ker (D^0_{(A,\Psi)})^*}:
{\ker (D^0_{(A,\Psi)})^*}\map
A^0(\Sigma^-)_{l-1}\times A^0(\ad_+\otimes\gr_0)_{l-1}$ is Fredholm and the subspace $\left[(A^1(\gr_0)_l\times
D_l^U)\cap\ker (D^0_{(A,\Psi)})^*\right]$ of $\ker (D^0_{(A,\Psi)})^*$ is closed, so that the assertion follows from Remark 3.6.
If $$V:=\left(\frac{\partial sw}{\partial\gamma}|_\theta\right)^{-1}
\left(\frac{\partial sw}{\partial(A,\Psi)}|_\theta \ (A^1(\gr_0)_l\times
D_l^U)\right)$$ had finite codimension in ${\cal C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4))$, then $$V_{l-1}:=a_{l-1}^{-1}
\left(\frac{\partial sw}{\partial(A,\Psi)}|_\theta \ (A^1(\gr_0)_l\times
D_l^U)\right)$$ would have finite codimension in $A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}$.
Suppose there exists a finite dimensional subspace $F$ of the space ${\cal
C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4))$, such that $$V+F={\cal
C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4))\ .$$ Then we have $$j^k_{l-1}(V)+j^k_{l-1}(F)=j^k_{l-1}({\cal
C}^k(\Hom(\Lambda^1,P^u\times_\pi \R^4))\subset
A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}\ ,$$ hence $$\overline{j^k_{l-1}(V)+j^k_{l-1}(F)}=A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}\eqno{(4)}$$ by the density property of smooth sections in any Sobolev completion.
Therefore, under the hypothesis of the lemma, and using (4) and Remark 3.5, one gets $$\overline{j^k_{l-1}(V)}+j^k_{l-1}(F)=
A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}\ .\eqno{(5)}$$ On the other hand, we know that $\frac{\partial
sw}{\partial\gamma}|_\theta=a_{l-1}\circ j^k_{l-1}$. Therefore $$V=[j^k_{l-1}]^{-1}(V_{l-1})\ ,$$ which shows that $j^k_{l-1}(V)\subset V_{l-1}$. But $V_{l-1}$ is closed by Lemma 3.9., hence $\overline{j^k_{l-1}(V)}\subset V_{l-1}$. From (5) it follows that $$V_{l-1}+j^k_{l-1}(F)=A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}$$ which proves Lemma 3.10.
The proof of Proposition 3.8 is now reduced to showing that $V_{l-1}$ cannot have finite codimension in $A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}$. To prove this, we show that the sections in $V_{l-1}$ must fulfill a very restrictive condition, which is not of finite codimension.
Let $v\in V_{l-1}$. Then, by definition $$a_{l-1}(v)\in \frac{\partial sw}{\partial
(A,\Psi)}|_{\theta}(A^1(\gr_0)_{l}\times D_l^U)\ ,$$ hence there exists a pair $(\dot A,\dot \Psi)\in A^1(\gr_0)_{l}\times D_l^U$ such that
$$\left\{\begin{array}{lcl}
\Dr_{\gamma,a,A}^0\dot \Psi+\gamma(\dot
A)\Psi&=& v(\nabla_{C_0,a,A}\Psi)\\ \\
\Gamma_\gamma(d_A\dot A)-[(\dot\Psi\bar
\Psi)_0+(\Psi\bar{\dot\Psi})_0]&=&
\frac{d}{d\gamma}(\Gamma_\gamma(F_A))(v)
\ .
\
\end{array}\right.$$\
Consider now small balls $U_1$, $U_2$ centered in $u$ such that $U_1\subsetint U_2\subset U$, and such that the following two conditions hold:\
1. $\Psi$ is nowhere vanishing on $U_2$.\
Let $S^{\pm}$, $E$ be the trivial $SU(2)$-bundles associated with a $SU(2)\times SU(2)\times SU(2)$-reduction of $P^u|_{U_2}$. The connection $C_0$ induces ${\cal C}^{\infty}$-connections in $S^{\pm}$, and the pair $(A,a)$ induces a connection $B_A$ (with ${\cal C}^{k+1-\varepsilon}$-coefficients) in $E$ which lifts the connection $A|_{U_2}$ in $ \bar\delta(P^u)|_{U_2}=
\qmod{P_E}{S^1}$ and the connection $a|_{U_2}$ in $\det(P^u)|_{U_2}=\det(E)$. Since $\Psi$ has rank 1 in every point of $U_2\subset U$, it defines a ${\cal C}^{k+1-\varepsilon}$-splitting $E=L\oplus M$ with $\Psi|_{U_2}\in A^0(S^+\otimes L)$.\
2. The second fundamental form $b\in {\cal C}^{k+1-\varepsilon}(\Lambda^1_\C)$ of $L$ with respect to the unitary connection $B_A$ (or, equivalently, with respect to $A$) is nowhere vanishing on $U_2$.\
Let $l,\ m$ be ${\cal C}^{k+1-\varepsilon}$ sections of $E$ giving unitary frames in $L$ and $M$. Then we can write $\Psi|_{U_2}=s_0^+\otimes l$, where $s_0^+$ is a nowhere vanishing ${\cal C}^{k+1-\varepsilon}$-section of $S^+$. Once we have fixed this trivialization of $E$, we can identify the connections with the associated connection matrices, and write $B_A=A+\frac{1}{2}a\id$\
Recall that $b$ is defined by $b:=(\nabla_{B_A}l,m)$, and for any section $\varphi l$ of $L$ one has $\nabla_{B_A}(\varphi l)=
\nabla_{B_L}(\varphi l)+ \varphi b\otimes m$, where $B_L$ (resp. $B_M$) are the connections induced by $B_A$ in $L$ (resp. $M$).
By the Dirac harmonicity condition, one has, taking the component of $\Dr_{\gamma,a,A}^0\Psi$ in $S^-\otimes M$, $$\gamma(b)(s_0^+)=0\ .$$ Denote by $S_0$ the rank 1 subbundle of $S^+$ generated by the section $s_0^+$, and by $S^{\bot}_0$ its orthogonal complement. Let $\Psi_t$ be a path of spinors with $\Psi_0=\Psi$ and $\det(\Psi_t)=0$. Derivating it in 0, we get that the component of $\dot\Psi_0$ in $S^\bot_0\otimes M$ must vanish. Therefore, the restriction $\dot\Psi|_{U_1}$ of an element $\dot\Psi\in D_l^U=T_\Psi({\cal D}^U_l)$ must have the form $$\dot\Psi|_{U_1}=\dot
\sigma^+\otimes l+
\dot\zeta s_0^+\otimes m\ ,\ \ \dot \sigma^+\in
L^2_l(S^+|_{U_1}) \ ,\ \ \dot\zeta\in L^2_l(U_1,\C) \ .$$
Take now the component in $(S^-\otimes M)|_{U_1}$ of the restriction of the first equation to $U_1$. Put $\nabla_{B_M}(m)=\lambda\otimes m$, where $\lambda$ is a ${\cal
C}^{k-\varepsilon}$ pure imaginary 1-form.
One gets the following equation on $U_1$: $$\Dr_{\gamma}^0(\dot\zeta s_0^+) +\dot\zeta\gamma(\lambda)(s_0^+)
+\gamma(b)(\dot
\sigma^+) +\gamma(\dot A^2_1)(s_0^+)=v(b)(s_0^+) \ .
\eqno{(6)}$$ Here $\Dr_{\gamma}^0:A^0(S^+)_{s.}\map A^0(S^-)_{s-1}$ , $s\leq k$, stands for the Dirac operator associated with the Spin(4) structure on $(U_2, g_{\gamma})$ defined by $\gamma$ and the $SO(4)$-connection $C_0|_{U_2}$ in $\RSU(S^+,S^-)$. $\Dr_{\gamma}^0$ is a first order elliptic operator with ${\cal C}^k$-coefficients. The complex 1-form $\dot A^2_1$ is the component of $\dot A$ written in the matricial form with respect to the decomposition $E=L\oplus M$.
The idea to prove Proposition 3.8 is the following:
By the properties 1., 2. above it follows that, varying $v$ in the equation (6), one can get the $L^2_{l-1}$-sections of the bundle $(S^-\otimes M)|_{U_1}$. But on the left of the same equation one has a differential operator of order 1 with ${\cal C}^{k-\varepsilon}$ coefficients in $(\dot
\zeta,
\dot\sigma^+,\dot A^2_1)$ which has a - symbol: only the complex valued function $\dot \zeta$, which is a section in a bundle on $U_1$, is derivated on the left.
The problem comes down to showing that the map $L^2_l\map L^2_{l-1}$ associated with such an operator, cannot have a range of finite codimension.\
We define the following operators:
$$res_{U_1}:A^0(\Hom(\Lambda^1,P^u\times_\pi \R^4))_{l-1}\map
A^0(\Hom(\Lambda^1,P^u\times_\pi \R^4)_{U_1})_{l-1}\ ,$$ $$ev_{b,s_0^+}:A^0(\Hom(\Lambda^1,
P^u\times_\pi \R^4)_{U_1})_{l-1} \map
A^0(S^-|_{U_1})_{l-1}\ ,\ v'\longmapsto v'(b)(s_0^+)\ ,$$ $$[\Dr_{\gamma}^0]^-:A^0(S^-|_{U_1})_{l-1}
\map A^0(S^+|_{U_1})_{l-2} \ ,$$ $$pr^\bot:A^0(S^+|_{U_1})_{l-2}\map
A^0(S_0^\bot|_{U_1})_{l-2}\ .$$ Here $[\Dr_{\gamma}^0]^-$ is the Dirac operator associated with the connection $C_0$ and the Clifford map $\gamma^-:\Lambda^1\map \RSU(S^-,S^+)$ given by $$\gamma^-(u)=-\gamma(u)^*\ .$$
In general, the operator $[\Dr_{\gamma}^0]^-$ is not the formal adjoint of $\Dr_\gamma^0$, because $\gamma^{-1}(C_0)$ can have non-vanishing torsion, but it has the same symbol as $[\Dr_\gamma^0]^*$ and it is an operator with ${\cal C}^k$-coefficients. The associated Laplacian $[\Dr_{\gamma}^0]^-\circ \Dr_\gamma^0$ has scalar symbol given by $\xi\mapsto
-g_\gamma(\xi,\xi)\id_{S^+}$.
1\. The operators $res_{U_1}$, $pr^\bot$ are surjective.\
2. The image of the operator $[\Dr_{\gamma}^0]^-:A^0(S^-|_{U_1})_{l-1}\map
A^0(S^+|_{U_1})_{l-2}$ has finite codimension.\
3. The operator $ev_{b,s_0^+}$ is surjective.\
1\. The surjectivity of $res_{U_1}$ follows from the extension theorems for Sobolev spaces (\[Ad\], p. 83); the surjectivity of $pr^{\bot}$ is obvious.
2\. The fact that the image of $[\Dr_{\gamma}^0]^-:A^0(S^-|_{U_1})_{l-1}\map A^0(S^+|_{U_1})_{l-2}$ has finite codimension follows from the general theory of elliptic operators (see for instance \[BB\]); It can also be directly verified as follows: We may suppose that $X$ is the 4-sphere $S^4$ and that $S^\pm|_{U_1}$ are the restrictions to $U_1$ of the spinor bundles $S'^{\pm}$ associated with a $Spin(4)$-structure on $S^4$ whose Clifford map $\gamma'$ extends $\gamma|_{U_1}$. We can also find a connection $C_0'$ in the associated $SO(4)$-bundle extending $C_0|_{\bar U_1}$.
The image of $[\Dr_{\gamma}^0]^-$ contains the image of the composition $res_{U_1}\circ[\Dr_{\gamma'}^0]^-$, where $[\Dr_{\gamma'}^0]^-:A^0(S'^-)_{l-1}\map A^0(S'^+)_{l-2}$ is the Dirac operator on the sphere associated with $(\gamma')^-$ and $C_0'$. But $res_{U_1}$ is surjective and $[\Dr_{\gamma'}^0]^-$ is Fredholm.
Note that $[\Dr_{\gamma}^0]^-$ is in fact , if $U_1$ is sufficiently small.
3\. The surjectivity of $ev_{b,s_0^+}$ is the crucial point in which the fact that $s^+_0$ and $b$ are nowhere vanishing on $U_2$ is used in an essential way.
We begin by choosing a Clifford map $$\gamma_0:\Lambda^1_{U_2}
\map P^u|_{U_2}\times_\pi\R^4$$ such that $\gamma_0(b): S^+\map S^-$ is an isomorphism in every point $u\in U_2$.
This can be achieved as follows: We know that $\gamma(b)(s_0^+)=0$, so the determinant $\det(\gamma(b))$ of the induced morphism $\gamma(b):S^+\map S^-$ must vanish. Therefore $g_\gamma^\C(b)=\det(\gamma(b))=0$, hence the real forms ${\rm Re}(b)$, ${\rm Im}(b)$ have pointwise in $U_2$ the same (non-zero !) $g_\gamma$-norm and are pointwise $g_\gamma$-orthogonal. It suffices to choose $\gamma_0$ such that ${\rm Re}(b)$, ${\rm Im}(b)$ are nowhere $g_{\gamma_0}$-orthogonal on $U_2$. With this choice $\gamma_0(b)(s_0^+)$ will be a nowhere vanishing section of $S^-$ on $U_2$.
Let now $s'\in A^0(S^-|_{U_1})_{l-1}$ be an arbitrary $L^2_{l-1}$-negative spinor.
One can find a unique $L^2_{l-1}$ section $\delta\in A^0(\RSU(S^-,S^-)|_{U_1}))_{l-1}$, such that $\delta(\gamma_0(b)(s_0^+))=s'$: To see this, one uses the bilinear bundle map $$\RSU(S^-,S^-)\times S^-\map S^-\ .$$ The section $\delta$ is obtained by fibrewise dividing (in the quaternionic sense) $s'$ by the smooth nowhere vanishing spinor $\gamma_0(b)(s_0^+)$ which is a ${\cal
C}^{k-\varepsilon}$-section on $U_2\supset \bar U_1$.
One also has a bilinear bundle map $$\RSU(S^+,S^-)\times \RSU(S^-,S^-)\map \RSU(S^+,S^-)$$ which in local coordinates looks like quaternionic multiplication.
Now define the $L^2_{l-1}$-morphism $v':\Lambda^1_{U_1}\map
\RSU(S^+|_{U_1},S^-|_{U_1})$ by $$v'(\alpha):=
\delta\cdot [\gamma_0(\alpha)]\ ,\ \forall\ \alpha\in \Lambda^1_{U_1}\ .$$ This morphism defines a section in $$A^0(\Hom(\Lambda^1_{U_1},P^u|_{U_1}\times_\pi\R^4)_{l-1}
=A^0(\Hom(\Lambda^1_{U_1},\RSU(S^+,S^-)|_{U_1})_{l-1}$$ which acts on complex 1-forms $\alpha$ by $$v'(\alpha)(\cdot)=\delta[\gamma_0(\alpha)(\cdot)] \ .$$ In particular, $v'(b)(s_0^+)=\delta[\gamma_0(b)(s_0^+)]=s'$.
After these preparations we can finally prove Proposition 3.8.
We have to show that $V_{l-1}$ has infinite codimension in $$A^0(\Hom(\Lambda^1,P^u\times_\pi \R^4))_{l-1}\ .$$ Take $v\in V_{l-1}$ and apply $[pr^\bot\circ [\Dr_{\gamma}^0]^-]$ to both sides of (6).
On the left, the only term containing second order derivatives of the sections $(\dot\zeta,\dot \sigma^+,\dot A^2_1)$ is $$[pr^\bot\circ
[\Dr_{\gamma}^0]^-](\Dr_{\gamma}^0(\dot\zeta s_0^+))\ .$$ But, denoting by $i_0$ the bundle inclusion $U_1\times\C \map S^+|_{U_1}$, $z\longmapsto z s_0^+$, one sees that the 2-symbol of the composition $$pr^\bot\circ \left[[\Dr_{\gamma}^0]^-
\circ \Dr_{\gamma}^0\right]\circ i_{0}$$ vanishes, since the symbol of the Laplacian $ [\Dr_{\gamma}^0]^-\circ \Dr_{\gamma}^0$ is scalar.
Therefore, applying $[pr^\bot\circ [\Dr_{\gamma}^0]^-]$ on the left, one gets an expression containing only first order derivatives of the Sobolev $L^2_l$ sections $(\dot \zeta,
\dot\sigma^+,\dot A^2_1)$, hence an $L^2_{l-1}$-section of $S_0^{\bot}$.
On the other hand applying $[pr^\bot\circ [\Dr_{\gamma}^0]^-]$ on the right of (6), one gets precisely $$\left[pr^\bot\circ [\Dr_{\gamma}^0]^-\circ ev_{b,s_0^+}\circ
res_{U_1}\right](v)\ .$$
Now consider the operator $$P:=\left[pr^\bot\circ
[\Dr_{\gamma}^0]^-\circ ev_{b,s_0^+}\circ
res_{U_1}\right]:A^0(\Hom(\Lambda^1,
P^u\times_\pi \R^4))_{l-1}\map
A^0(S^\bot_0 |_{U_1})_{l-2}$$ and the following exact sequence $$0\rightarrow\qmod{\im(P)}{P(V_{l-1})}\rightarrow
\qmod{A^0(S^\bot_0)_{l-2}}{P(V_{l-1})}\rightarrow
\coker(P)\rightarrow 0\ .$$ We have seen that $P(V_{l-1})$ is contained in $A^0(S^\bot_0)_{l-1}$, which has infinite codimension in $A^0(S^\bot_0)_{l-2}$. [^3]
Therefore $\qmod{A^0(S^\bot_0)_{l-2}}{P(V_{l-1})}$ has infinite dimension. By Lemma 3.11 $\coker(P)$ has finite dimension, so that $\qmod{\im(P)}{P(V_{l-1})}$ must have infinite dimension. But $\qmod{\im(P)}{P(V_{l-1})}$ is a quotient of $$\qmod{A^0(\Hom(\Lambda^1,P^u\times_\pi
\R^4))_{l-1}}{V_{l-1}}\ ,$$ so that the latter must also have infinite dimension.\
Let ${\cal M}^*$, ${\cal D}{\cal M}^*_U$ be the moduli spaces $${\cal M}^*:=\qmod{[{\cal A}_l^*\times
Clif^k]^{SW_a}}{{\cal G}_{l+1}}\ ,\
{\cal D}{\cal M}^*_U:=\qmod{[{\cal A}_l^*
\times Clif^k]^{SW_a}_U}{{\cal G}_{l+1}}\ ,$$ where the upper script $(\ )^*$ denotes the subspace with non-zero spinor component.
Let $p=(A,\Psi,\gamma)\in [{\cal A}_l^* \times Clif^k]^{SW_a}_U$ such that for some $u\in U$, $\Psi_u\ne 0$ and $(A,\Psi)$ is non-abelian in $u$. Then the Zariski tangent space $T_{[p]}{\cal D}{\cal M}^*_U$ has infinite codimension in $T_{[p]}{\cal M}^*$. In particular, $T_{[p]}{\cal D}{\cal M}^*_X$ has infinite codimension in $T_{[p]}{\cal M}^*$ for every solution $p$ with non-abelian $(A,\Psi)$-component.
We have $$pr_{T_\gamma(Clif^k)}(T_{[p]}{\cal M}^*)=
\frac{\partial sw}{\partial\gamma}^{-1}
\left[D^1_{(A,\Psi)}(A^1(\gr_0)_l\times
A^0(\Sigma^+)_l)\right]\ ,$$ and the vector space $D^1_{(A,\Psi)}(A^1(\gr_0)_l
\times A^0(\Sigma^+)_l)$ has finite codimension in $A^0(\Sigma^-)_{l-1}
\times A^0(\ad_+\otimes \gr_0)_{l-1}$.
Therefore, also the image of $T_{[p]}{\cal M}^*$ under the projection to $T_\gamma(Clif^k)$ has finite codimension.
But, by Theorem 3.7, the image of $T_{[p]}{\cal D}{\cal M}^*_X$ under the same projection has infinite codimension. This proves the first assertion.
The second assertion follows from Aronszajin’s unique continuation theorem and the fact that the vanishing locus of an harmonic spinor cannot separate domains \[FU\]. Alternatively, one can use the Unique Continuation Theorem for monopoles \[FL\] to see that a mnopole with non-vanishing spinor component, and which is abelian on a non-empty open set, must be globally abelian.
Therefore in the condition of the proposition we can find a point $x\in X$ with $\Psi_x\ne 0$ such that $(A,\Psi)$ is non-abelian in $x$.
Using this result we can prove that for a generic Clifford map $\gamma$, the only degenerate solutions in the moduli space ${\cal M}^*\cap p_{Clif^k}^{-1}(\gamma)$ are the abelian ones. The idea is the following:
Let ${\cal D}{\cal M}^\circ_X\subset {\cal D}{\cal M}^*_X$ be the subspace of ${\cal D}{\cal
M}^*_X$ consisting of solutions with non-abelian $(A,\Psi)$-component. We have proven that ${\cal D}{\cal M}^\circ_X$ has infinite codimension in ${\cal M}^*$. Since the projection ${\cal D}{\cal M}^\circ_X\map Clif^k$ has “index $-\infty$”, the generic fibre should be empty. There are of course two serious problems with this argument:\
1. ${\cal D}{\cal M}^\circ_X$ is not smooth.\
2. The restriction of the projection ${\cal D}{\cal M}^\circ_X\map Clif^k$ to the smooth part is not Fredholm.
The idea to proceed is to weaken locally the equation defining ${\cal D}{\cal
M}^\circ_X$, such that the resulting spaces of solutions become smooth manifolds which are Fredholm of negative index over $Clif^k$. This can be achieved, since ${\cal D}{\cal
M}^\circ_X$ is embedded in the space ${\cal M}^*$, which, though possibly singular, is Fredholm over $Clif^k$.\
In order to carry out this idea, we will need the following two general lemmata.
Let $f$ be a smooth map taking values in a Banach space, and denote by $Z(f)$ its vanishing locus. For a point $p\in Z(f)$ define the Zariski tangent space to $Z(f)$ in $p$ by $$T_p(Z(f)):=\ker(d_p f )\ .$$
Let $\Sigma$ be a Banach manifold, $p\in\Sigma$, $E$ a Banach space, and $s:\Sigma\map E$ a smooth map such that $s(p)=0$. Suppose\
i) $\ker d_p s$ has a topological complement.\
ii) $\im d_p s$ is closed and has a topological complement.
Then there exists an open neighbourhood $\Sigma'$ of $p$ in $\Sigma$ and a submanifold $W$ of $\Sigma$ containing $p$, such that\
1. $\Sigma'\cap Z(s)$ is a closed subset of $W$.\
2. $T_p(Z(s))=T_p(W)$.
Put $T:=\im d_p s$, and denote by $pr_T$ the projection on $T$ associated with a topological complement of $T$.
The composition $pr_T\circ s$ is a submersion in $p$, since its derivative in $p$ is surjective and $\ker(d_p(pr_T\circ s))=\ker(d_p s)$ has a topological complement by assumption. Let $\Sigma'$ be an open neighbourhood of $p$ such that $pr_T\circ s$ is a submersion in every point of $\Sigma'$.
Then $$\Sigma'\cap Z(s)=\Sigma'\cap Z(pr_T\circ s)\cap Z(s)= Z(pr_T\circ
s|_{\Sigma'}))\cap Z(s)\ .$$ Therefore, taking $W:=Z(pr_T\circ s|_{\Sigma'}))$, claim 1. follows. Clearly $$T_p(W)=\ker(d_p(pr_T\circ s))=\ker(d_p s)=T_p(Z(s))\ .$$
Let $W$ be a Banach manifold, $E$ a Banach space, $p\in W$, and $\delta:W\map E$ a smooth map such that $\ker (d_p \delta)$ has infinite codimension in $T_p(W)$.
Then, for every $n\in \N$ there exists an open neighbourhood $W'_n$ of $p$ in $W$ and a codimension $n$ submanifold $V_n$ of $W$ such that $W'_n\cap Z(\delta)$ is a closed subset of $V_n$.
Since $\ker (\delta_p d)$ has infinite codimension in $T_p(W)$, it follows that $\im(d_p \delta)$ has infinite dimension. Let $F_n\subset
\im(d_p \delta)$ be a subspace of dimension $n$, and $pr_{F_n}$ the projection associated with a topological complement of $F_n$ in $E$. The composition $pr_{F_n}\circ \delta$ is a submersion in $p$. Indeed, the derivative in $p$ is surjective and the kernel of the derivative is closed of finite codimension, hence it has a topological complement. Let $W'_n$ be an open neighbourhood of $p$ such that $pr_{F_n}\circ \delta$ is a submersion in every point of $W'_n$. Then $$W'_n\cap Z(\delta)=W'_n\cap Z(pr_{F_n}\circ \delta)
\cap Z(\delta)=Z(pr_{F_n}\circ
\delta|_{W'_n})
\cap Z(\delta)\ .$$ Take $V_n:=Z(pr_{F_n}\circ \delta|_{W'_n})$.
Every non-abelian point $[p]\in {\cal D}{\cal M}^*_X$ has a neighbourhood $U_{[p]}$ which is a closed analytic subspace of a submanifold $V_{[p]}\subset \qmod{[{\cal
A}^*_l\times Clif^k]}{{\cal G}_{l+1}}$ such that the projection $V_{[p]}\map
Clif^k$ is Fredholm of negative index.
Put $p=(\theta,\gamma)$ with $\theta\in{\cal A}_l^*$ and $\gamma\in Clif^k$. Consider a slice $S_\theta\subset \theta+\ker(D^0_\theta)^*\subset
{\cal A}^*_l$ through $\theta$ to the orbits of the ${\cal G}_{l+1}$-action, such that the restriction of the canonical projection to $S_\theta$ defines a parameterization of the quotient $\qmod{{\cal A}^*_l}{{\cal G}_{l+1}}$ around $[\theta]$.
Note first, that the image $T$ of the differential $d_p({sw|_{S_\theta\times Clif^k}})$ is closed and has finite codimension in the Hilbert space $A^0(\Sigma^-)_{l-1}\times
A^0(\ad_+\otimes\gr_0)_{l-1}$.
Indeed, $T$ contains the image of $\frac{\partial sw}{\partial
(A,\Psi)}|_p$ , which is the operator $D^1_{\theta}$ associated with the deformation elliptic complex of the solution $\theta=(A,\Psi)$, and the image of $D^1_{\theta}$ is already closed of finite codimension.
Now put $\Sigma:=S_\theta\times Clif^k$, and note that the restriction $$q:\Sigma\map\qmod{[{\cal
A}^*_l\times Clif^k]}{{\cal G}_{l+1}}$$ of the canonical projection is a parametrisation of the Banach manifold $\qmod{[{\cal
A}^*_l\times Clif^k]}{{\cal G}_{l+1}}$ around $[p]$.\
\
Put $s:=sw|_\Sigma$. Then the projection $$T_\theta S_\theta\times T_\gamma(Clif^k)\supset
\ker(d_p s)\map T_\gamma(Clif^k)$$ is Fredholm. In particular $\ker(d_p s)$ has a topological complement in the tangent space $T_p(\Sigma)=T_\theta S_\theta\times T_\gamma(Clif^k)$.\
Indeed, the kernel of this map is $\H^1_\theta$ and its image can be identified with the subspace $\left(\frac{\partial {sw}}{\partial \gamma}\right)^{-1}[\im
D^1_{\theta}]$, whose codimension is at most $\dim \H^2_\theta$. If $\Lambda$ is a topological complement of $\H^1_\theta$ in $T_\theta S_\theta=
\ker(D^0_\theta)^*$ and $F$ is a topological complement of $\left(\frac{\partial {sw}}{\partial
\gamma}\right)^{-1}[\im D^1_{\theta}]$ in $T_\gamma(Clif^k)$, then $(\Lambda\times\{0\})\oplus(\{0\}\times F)$ is a topological complement of $\ker(d_p s)$ in $T_\theta S_\theta\times T_\gamma(Clif^k)$.\
Applying Lemma 3.13 to the Banach manifold $\Sigma$ and the map $s$, we get a neighbourhood $\Sigma'$ of $p$ and a submanifold $W$ such that $\Sigma'\cap Z(s)$ is a closed subset of $W$ and $$T_p(W)=T_p(Z(s))\simeq T_{[p]}({\cal M}^*)\ .$$ The restriction $\det|_W$ of the determinant map $\det:\Sigma\map
A^0(\det(P^u))_{l}$ satisfies the hypothesis of Lemma 3.14.
Indeed, $$\ker d_p(\det|_W)=\ker(d_p(\det|_\Sigma))\cap T_p(W)=
\ker(d_p(\det|_\Sigma))\cap\ker
d_p(sw|_\Sigma)\simeq$$ $$\simeq T{[p]}({\cal D}{\cal M}^{*}_X)\ ,$$ which has infinite codimension in $T_{[p]}({\cal M}^*)\simeq T_p(W)$ by Corollary 3.12.
Using now Lemma 3.14 we get, for any $n\in \N$, an open neighbourhood $W'_n$ of $p$ in $W$ and a codimension $n$ submanifold $V_n$ of $W$ such that $W'_n\cap Z(\det|_W)$ is a closed subspace of $V_n$.
Let $\Sigma'_n\subset\Sigma'$ be an open neighbourhood of $p$ in $\Sigma$ such that $$W'_n=\Sigma'_n\cap W\ .$$ Then we have $$\Sigma'_n \cap q^{-1}(({\cal D}{\cal M}^{*}_X)= Z(sw|_{\Sigma'_n})\cap
Z(\det|_{\Sigma'_n})=$$ $$=Z(pr_T\circ sw|_{\Sigma'_n})\cap Z(sw|_{\Sigma'_n})\cap
Z(\det|_{\Sigma'_n})=W'_n\cap Z(sw|_{\Sigma'_n})\cap
Z(\det|_{\Sigma'_n})=$$ $$= [W'_n\cap Z(\det|_{W'_n})]\cap Z(sw|_{\Sigma'_n})\ .$$ Therefore $\Sigma'_n \cap q^{-1}(({\cal D}{\cal M}^{*}_X)$ is a closed subspace of $
[W'_n\cap Z(\det|_{W'_n})]$, which is closed in $V_n$.
On the other hand we know that the projection $$T_p(W)=\ker(d_p s )\map
T_{\gamma} Clif^k$$ is Fredholm. Since being Fredholm is an open property, we may assume (taking $\Sigma'$ small) that the projection of $W$ on $Clif^k$ is Fredholm of constant index.
Now choose $n$ larger than the index of this projection, and put $$V_{[p]}:=q(V_n)\ , U_{[p]}:=q(\Sigma'_n \cap q^{-1}
(({\cal D}{\cal M}^{*}_X))=
q(\Sigma'_n)\cap {\cal D}{\cal M}^{*}_X\ .$$
The set $$\{\gamma\in Clif^k|\ {\cal D}{\cal M}^*_X
\cap{\rm pr}_{Clif^k}^{-1}(\gamma)\ {\rm contains\
a\ non-abelian\ pair}\}$$ is a set of the first category in $Clif^k$.
Indeed, let again ${\cal D}{\cal M}^\circ_X$ be the open subspace of ${\cal D}{\cal M}^*_X$ consisting of solutions with non-abelian $(A,\Psi)$-component. By Lemma 3.15 and the Lindelöf Theorem (\[Ke\], p. 49) we can find a countable cover $(U_i)_i$ of ${\cal D}{\cal M}^\circ_X$ such that every $U_i$ is a closed analytic subspace of a smooth submanifold $V_i\subset
\qmod{ [{\cal A}^*_l\times Clif^k]}{{\cal G}_{l+1}}$ which projects on the parameter space $Clif^k$ via a Fredholm map of negative index. Since Fredholm maps are locally proper \[Sm\], it follows that ${\rm pr}_{Clif^k}({\cal D}{\cal M}^\circ_X)$ is a countable union of closed sets; each of these closed sets is contained in a set of the form ${\rm pr}_{Clif^k}(V_i)$, which is of the first category, by the Sard-Smale theorem.
Corollary 3.12, Lemma 3.15, Corollary 3.16 hold for family of order 0-perturbations of the equations which contains the perturbations of the Clifford map which we have studied above. We need the following particular case:
Define the space of parameters ${\cal P}^k$ by $${\cal P}^k:={\cal C}^k(P^u\times_\pi\C^4)
\times {\cal C}^k(GL(\ad_+))\times
Clif^k\ .$$ Recall that a section $\beta$ in the bundle $$P^u\times_\pi\C^4=\Hom(\Sigma^+_c,\Sigma^-_c)\subset
\Hom(\Sigma^+,\Sigma^-)$$ defines an order 0-operator $A^0(\Sigma^+)\map A^0(\Sigma^-)$, commuting with the gauge action.
Consider now the equations $$\left\{\begin{array}{lcc}
\Dr_{\gamma,a,A}^0\Psi+ \beta (\Psi)&=&0\\
\Gamma_\gamma(F_A)&=&K (\Psi\bar\Psi)_0
\end{array}\right. \eqno{(\tilde {SW_a})}$$ for a system $$(A,\Psi,\beta,K,\gamma)\in \tilde {\cal A}:=
{\cal A}(\bar\delta(P^u))_l\times
A^0(\Sigma^+)_l\times {\cal P}^k\ .$$
Let $[{\cal A}_l\times{\cal P}^k]^{\tilde {SW}_a}$ ($[{\cal A}_l\times{\cal P}^k]^{\tilde
{SW}_a}_U$) be the space of solutions of the equations $(\tilde {SW}_a)$ (whose spinor component is degenerate on $U$), and denote also by $\widetilde {\cal M}^*$ ($\widetilde{{\cal D}{\cal M}}^*_U$) the moduli space of solutions (whose spinor component is degenerate on $U$) with non-vanishing spinor component.
Let $p=(A,\Psi,\beta,K,\gamma)\in
[{\cal A}_l\times{\cal P}^k]^{\tilde
{SW}_a}_U$ such that for some $u\in U$, $\Psi\ne 0$ and $(A,\Psi)$ non-abelian in $u$. Then the Zariski tangent space $T_{[p]}\widetilde{{\cal D}{\cal M}}^*_U$ has infinite codimension in $T_{[p]}\widetilde {\cal M}^*$.
Consider the image of $T_p([{\cal A}_l\times{\cal P}^k]^{\tilde {SW}_a}_U)$ under the projection to the tangent space $T_{(\beta,K,\gamma)}{\cal P}^k\ .$ This image has again infinite codimension. To see this it is enough to notice that the intersection of this image with the subspace $\{0\}\times\{0\}\times T_\gamma Clif^k$ has infinite codimension in $\{0\}\times\{0\}\times T_\gamma Clif^k$. But this follows by precisely the same arguments as in Theorem 3.7; one just has to replace the equations $(SW_a)$ by their $(\beta,K)$-perturbations. The left hand side in the crucial identity (6) will only be modified by the 0-order term $\dot\zeta\beta(s_0^+)$. Using this result and the same arguments as above, we get
The set $$\{\pg\in{\cal P}^k|\ \widetilde{{\cal D}{\cal M}}^*_X
\cap{\rm pr}_{{\cal P}^k}^{-1}(\pg)\ {\rm
contains\ a\ non-abelian\ pair}\}$$ is a set of the first category in ${\cal P}^k$ .
We can state now our generic regularity result:
There is a dense second category subset ${\cal P}^k_0$ of ${\cal P}^k$ such that for every $\pg\in{\cal P}^k_0$ the moduli space ${\cal M}^*_\pg:=
\widetilde{\cal M}^*\cap{\rm pr}_{{\cal
P}^k}^{-1}(\pg)$ is smooth away from the abelian locus.
We know by Theorem 3.1 and Remark 3.2 that $\tilde{\cal M}^*\setminus \widetilde
{{\cal D}{\cal M}}^*_X$ is a smooth manifold. Applying the Sard-Smale theorem to the Fredholm map $$\widetilde {\cal M}^*\setminus \widetilde{{\cal D}{\cal M}}^*_X\map{\cal P}^k$$ it follows that there exists a first category subset ${\cal P}^k_1\subset{\cal P}^k$ such that the moduli space $[\widetilde {\cal M}^*\setminus
\widetilde{{\cal D} {\cal M}}^*_X]\cap{\rm pr}_{{\cal P}^k}^{-1}(\pg)$ is smooth for every $\pg\in{\cal P}^k\setminus{\cal P}^k_1$. Let ${\cal P}^2_k$ be the first category set given by Corollary 3.18, and take ${\cal P}^k_0:={\cal P}^k
\setminus({\cal P}_k^1\union {\cal
P}_k^2)$.
Finally consider the following parameterized ASD-$Spin^c$- equations $$\left\{\begin{array}{lcc}
\Dr^0_{\gamma,a,A}\Psi+ \beta(\Psi)&=&0\\
\Gamma_\gamma(F_A)&=&0\end{array}\right.$$ for a system $(A,\Psi,\beta,\gamma)\in {\cal A}(\bar\delta(P^u))_l\times
A^0(\Sigma^+)_l\times{\cal C}^k(P^u\times_\pi\C^4)\times Clif^k$.
Let ${\cal M}'^*$ be the moduli space of solutions with non-trivial spinor component, and let ${\cal P}'^k$ be the parameter space ${\cal
P}'^k:={\cal C}^k(P^u\times_\pi\C^4)\times Clif^k$. Denote also by ${\cal D}{\cal M}'^*_X$ the subspace of solutions with degenerate spinor component, and by ${\cal
M}'^*_{\rm red}$ the subspace of solution with reducible connection-component.
Using the methods of section 3.2, one can prove the following partial transversality result
Suppose that the base manifold is simply connected. Then the moduli space ${\cal
M}'^*$ is smooth away from the union ${\cal D}{\cal M}'^*_X\union{\cal
M}'^*_{\rm red}$.
Indeed, let $ p=(A,\Psi,\beta,\gamma)$ be a solution with non-degenerate spinor component and non-reducible connection component, and suppose as in the proof of Theorem 3.1 that $(\Phi,S)$ is $L^2_{g_\gamma}$-orthogonal on the image of the differential in $p$ of the map cutting out the space of solutions. Using variations $\dot \beta$ of $\beta$ one sees that $\Phi$ must vanish on a non-empty open set. But using variations of $\Psi$, it follows that $\Phi$ must solve a Dirac equation, hence by Aronszajin’s unique continuation theorem, it must vanish on $X$. Then using variations $\dot \gamma$ of $\gamma$ we get as in \[DK\], p. 154 that $S=0$. It is enough to notice that $A$ is $g_\gamma$-ASD, and that any variation of the metric $g_\gamma$ is induced by a variation of the Clifford map $\gamma$.
In the proof of Theorem 3.7 we have only used the Dirac equation and the ellipticity (modulo the gauge group) of the system . Therefore the same arguments as above give the following important
1\. There exists a first category subset ${\cal P}'^k_2\subset{\cal
P}'^k$ such that for every $\pg\in {\cal P}'^k\setminus
{\cal P}'^k_2$ the only solutions with degenerate spinor component in the moduli space ${\cal M}'^*\cap p_{{\cal P}'^k}^{-1}(\pg)$ are the abelian ones.\
2. If the base manifold $X$ is simply connected, there exists a dense second category subset ${\cal P}'^k_0\subset{\cal P}'^k$ such that for every $\pg\in{\cal P}'^k_0$ the $Spin^c$-moduli space ${\cal M}'^*\cap p_{{\cal P}'^k}^{-1}(\pg)$ is smooth away from ${\cal M}'^*_{\rm red}\cap
p_{{\cal P}'^k}^{-1}(\pg)$.
The results above are sufficient to go forward towards a complete proof of the Witten conjecture.
Moreover, one can use the same method to prove a generic regularity theorem along the abelian part of the moduli space.
More precisely, let ${\cal M}^{\rm ab}_\pg\subset {\cal M}^*_\pg$
be the abelian part of the moduli space ${\cal M}^*_\pg$ of solutions of the monopole equations associated with the perturbation parameter $\pg$. The space ${\cal M}^{\rm
ab}_\pg$ can be identified with the disjoint union of the $Spin^c$-Seiberg-Witten moduli spaces associated with the abelian reductions of $P^u$ (\[OT5\], \[OT7\], \[T1\]).
Let $[p]\in {\cal M}^{\rm ab}_\pg$ be an abelian solution. The elliptic deformation complex ${\cal C}_p$ of $p$ splits as the sum $${\cal C}_p={\cal C}_p^{\rm ab}\oplus {\cal N}_p$$ where the first summand ${\cal C}_p^{\rm ab}$ can be identified with the elliptic deformation complex of $p$ regarded as solution of the abelian monopole equations, and ${\cal N}_p$ is the so called normal elliptic complex of $p$.
The union ${\cal H}^1_\pg:=\union\limits_{[p]
\in {\cal M}^{\rm ab}_\pg}\H^1({\cal N}_p)$ is a real analytic space which fibres over ${\cal M}^{\rm ab}_\pg$, but in general is not locally trivial over ${\cal M}^{\rm ab}_\pg$, and local triviality cannot be achieved in the class of $S^1$-equivariant perturbations.
Using the method from above one can prove
For a generic parameter $\pg\in{\cal P}^k$, the complement of the zero section in ${\cal H}^1_\pg$ is smooth of the expected dimension in every point.
The Uhlenbeck Compactification
==============================
Local estimates
----------------
The essential difference between the anti-self-dual and the monopole equations is that the latter are not conformal invariant. Under a conformal rescaling of a metric $g\mapsto\tilde g=\rho^2 g$ on a 4-manifold $X$, the associated objects change as follows $$\begin{array}{c}
\tilde{g^*}=\rho^{-2}g^*\ {\rm on\ 1-forms} ; \ \
vol_{\tilde g}=\rho^4 vol_g \ ;\ \ s_{\tilde g} =\rho^{-2}s_g+
2\rho^{-2}\Delta \rho\\
\Sigma^{\pm}_{\tilde g}=\Sigma^{\pm}_{g}\ ({\rm as\ Hermitian\ bundles}) , \
\tilde\gamma=\rho^{-1}\gamma\ ;\
\tilde\Gamma=\rho^{-2}\Gamma\ ;
\
\
\Dr_{\tilde g}=\rho^{-\frac{5}{2}}\Dr_g\rho^{\frac{3}{2}} \ .
\end{array}$$
A standard procedure used in proving regularity and compactness theorems for instantons is the following: restrict the equations on small balls in the base manifold, and then rescale the metric. In this way, using the conformal invariance of the equations, one can reduce the local computations to the unit ball endowed with a metric close to the euclidean one.
A similar procedure will be used in the case of $PU(2)$-monopoles. The problem here is that the perturbed equations depend on a much larger system of parameters (data). Using rescalings of the Clifford map (and hence of the metric), we show first that one can reduce the local computations to computations on the unit ball endowed with a system close to a system of “standard data” (see Definition 4.4).
First of all notice that if $(A,\Psi)\in {\cal A}(\bar\delta(P^u))
\times A^0(\Sigma^+)$ is a solution of the non-perturbed $PU(2)$-monopole equations $SW^\sigma_a$ for the metric $g$ with respect to the $Spin^{U(2)}(4)$-structure $\sigma$, and if $\rho$ is a , then $(A,\rho^{-1}\Psi)$ is a solution of the monopole equations $SW^{\tilde\sigma}_a$ for $\tilde g=\rho^2 g$ with respect to the $Spin^{U(2)}(4)$-structure $\tilde\sigma$ defined by the correspondingly rescaled Clifford map $\tilde\gamma=\rho^{-1}\gamma$.
The case of the perturbed equations is more delicate. Fix a $Spin^{U(2)}(4)$-bundle $P^u$. To write down the general perturbed $PU(2)$-monopole equations we considered, one also needs [*a system of data*]{} of the form $\pg=(\gamma,C,a,\beta,K)$, where $\gamma$ is a Clifford map (see Definition 3.3), $C$ is an $SO(4)$-connection in $P^u\times_\pi\R^4$, $a$ is a connection in the line bundle $\det(P^u)$, $\beta$ is a section in $P^u\times_\pi\C^4$, and $K$ is a section in $\End(\ad_+)$.
The rescaling rule is:
If $(A,\Psi)\in{\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+)$ solves the perturbed $PU(2)$-monopole equations associated with the data $(\gamma,C,a,\beta,K)$. Then $(A,\rho^{-1}\Psi)$ solves the perturbed $PU(2)$-monopole equations associated with the data $$(\rho^{-1}\gamma,C,a,\rho^{-1}\beta,K).$$
Let $\bar B$ be the standard closed 4-ball with interior $B$. Fix two copies $\H_{\pm}$ of the quaternionic skew-field $\H$ regarded as right complex and quaternionic vector spaces and consider the two trivial $SU(2)$-bundles $S_0^{\pm}:=\bar B\times\H_{\pm}$. Let also $E_0=\bar B\times\C^2$ be the trivial Hermitian rank 2-vector bundle on $\bar B$.
Let $P^u_0$ be the trivial $Spin^{U(2)}(4)$-bundle associated with $S_0^{\pm}$, $E_0$ via the morphism $SU(2)\times SU(2)\times U(2)\rightarrow
Spin^{U(2)}(4)$ (section 2.1, Prop. 2.2).
A Clifford map for $P^u$ is an orientation preserving linear isomorphism $\gamma:\Lambda^1_{\bar B}\map \Hom_\H(S^+_0,S^-_0)=
\bar B\times\H$. To every such a Clifford map $\gamma$, we can associate the constant Clifford $\gamma^c$ given by the composition $$\Lambda^1_{\bar B}\map\bar B\times
\Lambda^1_0\textmap{\id\times{\gamma|_{\Lambda^1_0}} } B\times\H$$ Note that the corresponding metric $g_{\gamma^c}$ is flat.
Denote by $ h_r:\bar B\map \bar B_r\subset\bar B$ the homothety of slope $r<1$.
The Clifford maps $\gamma_r:=r h_r^*(\gamma|_{B_r})$ converge in the ${\cal
C}^{\infty}$-topology to $\gamma_0$, which is a Clifford map for the flat metric $g_{\gamma^c}$. In particular the metrics $g_r:=r^{-2}h_r^*(g)$ converge to the flat metric $g_{\gamma^c}$.
Indeed, one has $$\gamma_r(x,\lambda)=r \gamma((h_r)_*(x,\lambda))=r
\gamma(rx,r^{-1}\lambda)=\gamma(rx,\lambda)$$
The data of a $PU(2)$-connection $A\in{\cal A}(\bar\delta(P^u_0))$ is equivalent to the data of a connection matrix, i.e. an element in $A^1(\bar B,su(2))$. Similarly, the data of a $U(1)$-connection in $\det(P^u_0)$ is equivalent to the data of a 1-form in $A^1(\bar
B,u(1))$.
Let $(A,\Psi)\in{\cal A}(\bar\delta(P^u_0))\times
A^0(\Sigma^+(P^u_0))$ be a pair which solves the monopole equations for the data $(\gamma,C,a,\beta,K)$. Then $(h_r^*(A), r h_r^*(\Psi))$ solves the $PU(2)$-monopole equations for the data $(\gamma_r,h_r^*(C),h_r^*(a),rh_r^*(\beta),h_r^*(K))$.
Note that, as $r\rightarrow 0$ ,\
$\gamma_r\rightarrow \gamma^c$ (which is a Clifford map for the flat metric $g_{\gamma^c}$),\
$rh_r^*(\beta)\rightarrow 0$, $h_r^*(K)\rightarrow K(0)$,\
$h_r^*(a)$ converges to the flat connection in $B\times\C=\det(P^u_0)$,\
$h_r^*(C)$ converges to the flat connection in $B\times\H$, and\
$\gamma^{-1}_r(h_r^*(C))$ converges to the flat connection in $(\Lambda^1_B=B\times\R^4,g_{\gamma^c})$, which is precisely the Levi-Civita connection for $g_{\gamma^c}$).
A system of data for the bundle $P^u_0$ will be called a standard system, if it has the form $(\gamma_0, C_0,0,0,K_0)$, where:\
$\gamma_0$ is the standard identification $\Lambda^1_{B}
=B\times\R^4\map
B\times\H$,\
$C_0$ the flat $SO(4)$-connection in $B\times\H$, and\
$K_0$ is a automorphism of the trivial bundle $su(S^+_0)=B\times
su(2)_+$.
The metric associated with the standard identification $\Lambda^1_{B}=B\times\R^4\rightarrow B\times\H$ is the standard Euclidean metric $g_0$ on the ball.
For any $K_0\in \End(su(2))$, let $\pg_{K_0}$ be the standard system of data on $\bar B$ defined by $K_0$.\
Let $X$ now be 4-manifold, and $P^u$ a $Spin^{U(2)}(4)$-bundle on it. Let $x_0$ be a point in $X$ and $U$ an open neighbourhood of $x_0$. Fix an identification of $P^u|_U$ with the the trivial $Spin^{U(2)}(4)$-bundle on $U$, i. e. with the $Spin^{U(2)}(4)$-bundle associated with the triple $U\times\H_\pm$, $U\times \C^2$ (see section 2.1).
Given a system of data $(\gamma,C,a,\beta,K)$ for $P^u$, we consider a parameterization $B_{r_0}\stackrel{f}{\map} U\subset X$ around $x_0$ such that $f(0)=x_0$ and $\gamma|_{\Lambda^1_{x_0}}\circ
[f_*]_{\Lambda^1_0}$ is the standard identification $\Lambda^1_0=\R^4\map \H$.
For any pair $(A,\Psi)$ solving the monopole equations for the data $(\gamma,C,a,\beta,K)$, the pair $\left((f\circ h_r)^*(A),
r(f\circ h_r)^*(\Psi)\right)$ solve the monopole equations associated with the system $$\left(f^*(\gamma)_r,(f\circ h_r)^*(C),(f\circ h_r)^*(a),
r(f\circ h_r)^*(\beta),(f\circ h_r)^*(K)\right)\ .$$ This system converges to a system of standard data on the ball, as $r\rightarrow 0$.
Therefore, as long as we are interested only in local computations, we can work on the standard ball and assume (via the transformation defined in Remark 4.5) that our system of data belongs to a small neighbourhood of a standard system.\
We recall now the following important “gauge fixing” theorem (see Theorem 2.3.7 in \[DK\]).
(Gauge-fixing) There are constants $\varepsilon_1,\ M>0$ such that the following holds:
Any connection $A$ on the trivial bundle $E_0$ over $\bar B$ with $\parallel F_A\parallel_{L^2}<\varepsilon_1$ is gauge equivalent to a connection $\tilde A$ over $B$ with\
(i) $d^*_0\tilde A=0$, where $d^*_0$ is the normal adjoint of $d$ with respect to the standard flat metric $g_0$.\
(ii) $\lim_{r\rightarrow 1} A_r=0$ on $S^{3}$,\
(iii) $\parallel\tilde A\parallel_{L^2_1}\leq M\parallel F_{\tilde
A}\parallel_{L^2}$.\
The corresponding gauge transformation is unique up to a constant matrix.
Using this result we can prove the following
(Local estimates for data close to the standard data) There is a positive constant $\varepsilon_2=\varepsilon_2(K_0)>0$ such that for any system of data $\pg'$ on $\bar B$ which is sufficiently ${\cal C}^2$-close to the standard system $\pg_{K_0}$, the following holds:
For any solution $(A,\Psi)$ of the $PU(2)$-monopole equation for the monopole equations associated with $\pg$ over the open ball $B$ satisfying the conditions $d^*_0 A=0$, $\nr(A,\Psi)\nr_{L^4}\leq\varepsilon_2$, and any interior domain $D\subsetint B$, one has estimates of the form : $$\nr (A,\Psi)\nr _{L^2_l(D)}\leq
C_{D,l,\pg'} \nr (A,\Psi)\nr_{L^4}\ ,\\$$ with positive constants $C_{D,l,\pg'}$, for all $l\geq 1$.
First of all we identify the ball with the upper semi-sphere of $S:=S^4$ and we endow the sphere with a metric $g_s$ which extends the standard flat metric $g_0$ on the ball, and which has non-negative sectional curvature[^4].
We fix a $Spin(4)$-structure on the sphere with spinor bundles $S^{\pm}_s$ given by a Clifford map $\gamma_s:\Lambda^1_S\map \Hom_\H(S_s^+ , S^-_s)$, which, with respect to fixed trivializations $S^{\pm}|_{\bar B}=\bar B\times \H_{\pm}$, extends the standard Clifford map $\gamma_0$ on the ball. Let also $C_s$ be the Levi-Civita connection induced by $\gamma_s$ in $\Hom_\H(S_s^+ , S^-_s)$. Its restriction to the ball is the standard flat connection $C_0$ in $\bar B\times \H$. Let finally $K_s$ be an extension of $K_0$ to an endomorphism $K_s\in A^0(\End(su(S_s^+)))$.
We denote by $E_s$ the trivial $U(2)$-bundle over $S$, and by $P^u_s$ the $Spin^{U(2}(4)$-associated with the triple $(S^{\pm}_s,E_s)$. $P^u_s$ comes with an identification $P^u_s|_{\bar B}=P^u_0$, induced by the fixed trivializations of $S^{\pm}_s$.
The system $(\gamma_s,C_s,0,0,K_s)$ is an extension on the sphere of the standard system $\pg_s:=(\gamma_0,C_0,0,0,K_0)$. The point is now that any system $\pg'$ of data which is close to $\pg_{K_0}$ has an extension $\pg$ which is close to $\pg_s$.
Put $\pg=(\gamma,C,a,\beta,K)=(\qg,K)$. The system $\qg$ defines two first order elliptic operators on the sphere $$\begin{array}{ccccc}
\Dr_{\qg}&:& A^0(S^+_s\otimes E_s)& \map&
A^0(S^-_s\otimes E_s)\\
\delta_\gamma:=d_s^*+ \Gamma_\gamma\circ d &: &A^1(su(2))&
\map& A^0(su(2))^\bot
\oplus A^0(su(S^+_s) \otimes su(E_s))
\end{array}$$ The symbol $d^*_s$ means the adjoint of $d:A^0(su(2))\map
A^1(su(2))$ with respect to the fixed metric $g_s$, and $\Dr_{\qg}:=\Dr_\gamma^C+\beta+\gamma(\frac{a}{2})$. $A^0(su(2))^\bot$ denotes the $L^2_{g_s}$-orthogonal complement of the 3-dimensional space of constant sections.
These operators are injective in the special case $\qg=\qg_s:=
(\gamma_s,C_s,0,0)$, by the Weitzenböck formula for the Dirac operator and because the cohomology group $H^1_{\rm DR}(S)$ vanishes. Since the coefficients of both operators in local coordinates are algebraic expressions in the components of $\qg$, it follows by elliptic semicontinuity that the two operators remain injective if $\qg$ is sufficiently ${\cal C}^0$-close to $\qg_s$. Denote by $D_\qg$ the direct sum of these operators. We get operator valued maps $$\qg\mapsto D_{\qg}\in$$ $$\Iso\left[A^0(S^+_s\otimes
E_s\oplus \Lambda^1(su(2)))_{k+1}, A^0(S^-_s\otimes E_s
\oplus su(S^+_s)\otimes
su(E_s)_k\oplus A^0(su(2))_k^\bot\right]$$ which are continuous with respect the ${\cal C}^k$-topology on the space of data $\qg$ on the sphere.
Therefore one has elliptic estimates $$\nr u\nr_{L^2_{k+1}}\leq const(\qg)\nr D_\qg u\nr_{L^2_{k}}\\
\eqno{(el_k)}$$ where $const(\qg)$ depends continuously on $\qg$ w. r. t. the ${\cal C}^k$-topology. In a sufficiently small ${\cal C}^2$-neighbourhood of $\qg_s$ one has the following estimates with $\qg$- $$\nr u\nr_{L^2_{k+1}}\leq const\nr D_{\qg}
u\nr_{L^2_{k}}\\
\eqno{(el)}$$
Since $D_\qg$ is a first order operator, we have an identity of the form: $$D_\qg(\varphi v)=\varphi D_\qg (v)+ A_{\qg,\partial\varphi}(v) \eqno{(*)}$$ where $A_{\qg,\partial\varphi}$ is an operator of order 0 depending on $\qg$ and depending linearly on the first order derivatives of $\varphi$.
The first step is an input-estimate for the ${L^2_1(D)}$-norms:
Denote by $u$ the pair $(A,\Psi)$. Let $\varphi_1$ be a cut-off function supported in the open ball $B$ which is identically 1 in a neighbourhood of $\bar D$. Then $u_1:=
\varphi_1 u$ extends as section in the bundle $\Lambda^1(su(2))\oplus S^+_s\otimes E_s$ on the sphere. Taking into account that $u$ solves the monopole equations associated with the data $\pg'$, its connection component is in Coulomb gauge, and that $\pg=(\qg,K)$ extends $\pg'$ one gets by $(*)$ $$D_\qg(u_1)=A_{q,\partial\varphi_1}(u)+\varphi_1
\left[ \matrix{ -\gamma(A)\Psi\cr
-\Gamma_\gamma(A\wedge A)+K(\Psi\bar\Psi)_0}\right]=$$ $$=A_{q,\partial\varphi_1}(u)+\varphi_1 B_{\gamma,K}(u) \eqno{(1)}$$ where $B_{\gamma,K}$ is a quadratic map.
Then by $(el)$ we obtain an elliptic estimate of the form $$\
\nr u_1\nr_{L^2_1}\leq
c\nr D_{\qg} u_1\nr_{L^2}
\leq c'(\nr u\nr_{L^4}^2+\nr
d\varphi\nr_{L^4}\nr u\nr_{L^4})\leq$$ $$\leq c''(\nr u\nr_{L^4} \nr u\nr_{L^2_1}+\nr
d\varphi\nr_{L^4}\nr u\nr_{L^4})$$ where, for the second inequality we have used on the right the bounded Sobolev embedding $L^2_1\subset L^4$. The constants $c$, $c'$ can be chosen to depend continuously on $\pg$, so that we can assume that they are independent of $\pg$ on a small neighbourhood of $\pg_s$. We use now the standard rearrangement procedure described in \[DK\], p. 60, 62. For a sufficiently small (independent of $D$) apriori bound $\varepsilon(K_0)$ of the norm $\nr u\nr_{L^4}$, we get an estimate of the type $$\nr\ u_1\nr_{L^2_1}\leq const_D\nr u\nr_{L^4} \ .$$ The constant $const_D$ in this estimate is independent of $\pg$ in a sufficiently small neighbourhood of $\pg_s$, but it depends on $D$ via $\nr d\varphi_1\nr_{L^4}$.\
In a next step we estimate the $L^2_2$-norms:
Put $u_2=\varphi_2 u$, where $\varphi_2$ is identically 1 on $D$, but the support ${\rm supp}\varphi_2$ is contained in the interior of $\varphi_1^{-1}(1)$. Then we can also write $u_2=\varphi_2 u_1$, and we have $A_{\qg,\partial\varphi_2}(u)=A_{\qg,\partial\varphi_2}(u_1)$.
We estimate first the $L^2_1$-norm of the right hand side of the formula obtained by replacing $\varphi_1$ with $\varphi_2$ in (1) . We find $$\nr D_\qg(u_2)\nr_{L^2_1}\leq const \nr\varphi_2
B_{\gamma,K}(u_1)\nr_{L^2_1}
+const_{D}\nr u_1\nr_{L^2_1}\ ,\eqno{(2)}$$ and again we can assume that the constants do not depend on $\qg$. The term $\varphi_2B_{\gamma,K}(u_1)$ can be written as $\tilde B_{\gamma,K}(\varphi_2
u_1\otimes u_1 )$, where $\tilde B_{\gamma,K}$ is the linear map defined on the tensor product $\left(\Lambda^1(su(2))\oplus S^+_s\otimes E_s\right)^{\otimes 2}$ associated with the quadratic map $B_{\gamma,K}$.
In local coordinates we can write: $$\partial_i [\tilde B_{\gamma,K}(\varphi_2 u_1\otimes
u_1) ]=\partial_i(\tilde B_{\gamma,K})
(\varphi_2\otimes
u_1)\otimes u_1 + \tilde B_{\gamma,K}
\left[\partial_i(\varphi_2 u_1)\otimes u_1+
u_1\otimes(\varphi_2\partial_i u_1)\right]$$ $$=\partial_i(\tilde B_{\gamma,K}) (\varphi_2\otimes
u_1)\otimes u_1+\tilde B_{\gamma,K}
\left[\partial_i(\varphi_2 u_1)\otimes
u_1+u_1\otimes\partial_i (\varphi_2 u_1)-
\partial_i(\varphi_2) u_1\otimes u_1\right]$$ This gives an estimate of the form $$\nr \tilde B_{\gamma,K}(\varphi_2 u_1\otimes
u_1) \nr_{L^2_1} \leq const \nr u_2\nr_{L^4_1}
\nr u_1\nr_{L^4} +const_D \nr
u_1\nr_{L^4}^4\ ,$$ which together with (2) and $(el)$ gives $$\nr u_2\nr_{L^2_2}\leq const \nr u_2\nr_{L^4_1}
\nr u_1\nr_{L^4} +const_D(\nr
u_1\nr_{L^2_1}+\nr u_1\nr_{L^4})\ .$$ By the same rearrangement argument and using the existence of a bounded inclusion $L^2_2\subset L^4_1$, we get, for a sufficiently small, independent of $D$, apriori bound of $\nr u\nr_{L^4}$, an estimate of the form $$\nr u_2\nr_{L^2_2}\leq const_D \nr u\nr_{L^4}\ .$$
The estimates for the third step can be proved by the same algorithm, using the existence of a bounded inclusion $L^2_3\subset L^4_2$.
Since $L^2_3$ is already a Banach algebra, the estimates for the higher Sobolev norms follow by the usual bootstrapping procedure using the estimates $(el_k)$. Note in particular that we no longer need to use the rearrangement argument, so we do have to take smaller bounds for $\nr u\nr_{L^4}$ to get estimates of the higher Sobolev norms, so that a positive number $\varepsilon_2=\varepsilon(K_0)$ (independent of $l$ and $D$ !) with the required property does exist. Let $V_+$, $F$ Hermitian vector spaces of rang 2. One can easily check that there exists a universal constants $\eg>0$, $C$, $C_1>0$, $C_2>0$ such that for every $K\in\End(su(V_+))$ with $|K-\id|<\eg$, and every $\Psi\in V_+\otimes F$ the following inequalities hold $$C_1|\Psi|^2\leq |K(\Psi\bar\Psi)_0|\leq C_2|\Psi|^2 \eqno{(3)}$$ $$C|\Psi|^4\leq \left(K(\Psi\bar\Psi)_0,
(\Psi\bar\Psi)_0\right)=
\left(K(\Psi\bar\Psi)_0(\Psi),\Psi\right)
\eqno{(4)}$$ From now on we’ll assume the last component $K$ of a system of data $(\gamma,C_0,a,\beta,K)$ satisfies in every point $x$ the inequality $|K(x)-\id_{\ad_+}|<\eg$.
(Estimates in terms of the curvature) There exists a constant $\varepsilon>0$, such that for any system $\pg'$ of data on the closed ball which is sufficiently ${\cal
C}^2$-close to a system of standard data $(\gamma_0,C_0,0,0,K_0)$ with $|K_0-\id|<\eg$ the following holds:
For any interior ball $D\subsetint B$ and any $l\geq 1$ there exist a positive constants $C_{D,l,\pg'}$, $C'_{D,l,\pg'}$ such that every solution $(A,\Psi)$ of the $PU(2)$-monopole equations on $\bar B$ associated with $\pg'$ satisfying $\nr F_A\nr_{L^2}\leq\varepsilon$, is gauge equivalent on $B$ to a pair $(\tilde
A,\tilde\Psi)$ satisfying the estimates $$\nr\tilde A\nr_{L^2_l(D)}\leq C_{D,l,\pg'}\nr
F_A\nr_{L^2}\ ,\ \ \nr\tilde
\Psi\nr_{L^2_l(D)}\leq C'_{D,l,\pg'}\nr F_A
\nr_{L^2}^{\frac{1}{2}} \ .$$
Note first that all the pairs $(A,\Psi)$ with $\nr F_A\nr_{L^2}\leq
\varepsilon_1$ are gauge equivalent to pair $(\tilde A,\tilde\Psi)$ whose connection component is in the Coulomb gauge with respect to the trivial connection and such that $$\nr\tilde A \nr_{L^2_1}\leq M\nr
F_{\tilde A}\nr_{L^2}\eqno{(5)}$$ Since now the constant $K_0$ is supposed to belong to the bounded set $B(\id,\eg)$ the conclusion of Theorem 4.7 holds for a constant $\varepsilon_2$ which can be chosen [*independently*]{} of $K_0$.
On the other hand, by the estimate (3) and the second monopole equation, one has $$\nr \tilde\Psi\nr_{L^4_{g_{\gamma'}}}\leq \frac{1}
{C_1^{\frac{1}{2}}} \nr
\Gamma_{\gamma'}(F_{\tilde A})
\nr_{L^2_{g_{\gamma'}}}^{\frac{1}{2}}
= \frac{\sqrt 2}{C_1^{\frac{1}{2}}} \nr F_{\tilde
A} ^{+_{g_{\gamma'}}}\nr_{L^2_{g_{\gamma'}}}^{\frac{1}{2}}
\leq\frac{\sqrt 2}{C_1^{\frac{1}{2}}} \nr F_{\tilde
A} \nr_{L^2_{g_{\gamma'}}}^{\frac{1}{2}}
\eqno{(6)}$$ Since $\gamma'$ is supposed to belong to a small neighbourhood of $\gamma_0$ this gives an uniform estimate of $\nr \tilde\Psi\nr_{L^4}$ in terms of $\nr
F_{\tilde
A} \nr_{L^2}^{\frac{1}{2}}$. Using the bounded inclusion $L^2_1\subset L^4$, and the estimates (5), (6) we see now that the $L^4$ norm of the pair $(\tilde A,\tilde\Psi)$ can be made as small as we please by choosing $\varepsilon$ small, in particular smaller than the constant $\varepsilon_2$. With this choice the conclusion of Theorem 4.7 holds, and we get estimates of the Sobolev norms of the restrictions on smaller disks $D\subsetint B$ in terms of $\nr(\tilde
A,\tilde\Psi)\nr_{L^4}$, hence in terms of $\nr
F_{\tilde A} \nr_{L^2}^{\frac{1}{2}}$.
On the other hand, the same cutting off procedure as in the proof of Theorem 4.7, gives on the sphere an identity of the form $$(d_s^*+\Gamma_\gamma d)(\varphi_1\tilde
A)=A'_{\qg,\partial\varphi }(\tilde A)+\varphi [-\Gamma_\gamma
(\tilde A\wedge
\tilde A)+(\tilde \Psi\bar{\tilde \Psi})_0]\ ,$$ which is similar to the identity (1) in the proof of the theorem. Using Theorem 4.7 to estimate the quadratic term on the right, it follows that the $L^2_l$-norm of $\tilde A|_D$ can be estimated in terms of the $L^2_{l-1}$-norm of the restriction of $\tilde A$ to a slightly larger disk $D_l \subsetint B$ and $\nr(\tilde A,\tilde \Psi)\nr_{L^4}^2$. Inductively we get an estimate of the $L^2_l$-norm of $\tilde A|_D$ in terms of the $L^2_1$-norm of $\tilde
A$ and of $\nr(\tilde A,\tilde \Psi)\nr_{L^4}^2$. But both terms can be estimated now in terms of $\nr F_{\tilde A}\nr_{L^2}$.
Note that the estimate in terms of $\nr F_{\tilde A}
\nr_{L^2}^{\frac{1}{2}}$ which we obtained by applying directly Theorem 4.7, is in fact fully sufficient for our purposes. However it is interesting to notice that the Sobolev norms of the connection component $\tilde A$ can be estimated as in the instanton case in terms of $\nr F_{\tilde
A}\nr_{L^2}$.
(Local compactness) There exists a constant $\varepsilon>0$ such that the following holds:
For any pair system of data $\pg$ which is sufficiently close to a system of standard data $\pg_{K_0}$ on the ball with $|K_0-\id|<\eg$ , and any sequence $(A_n,\Psi_n)$ of solutions of the $PU(2)$-monopole equations for $\pg$ with $\nr
F_{A_n}\nr_{L^2}\leq
\varepsilon$, there is a subsequence $m_n$ of $\N$ and gauge equivalent solutions $(\tilde A_{m_n},
\tilde \Psi_{m_n})$ converging in the ${\cal C}^{\infty}$-topology on the open ball $B$.
We can prove now the following result, which is the analogon of Proposition 4.4.9 p. 161 \[DK\].
(Global compactness) Let $\Omega$ be a 4-manifold and let $P^u$ be a $Spin^{U(2)}(4)$-bundle on $\Omega$ such that $\Lambda^1_\Omega\simeq
P^u\times_\pi\R^4$ as oriented 4-bundles. Let $\pg=(\gamma,C,a,\beta,K)$ be an arbitrary system of data for $(\Omega,P^u)$ satisfying the condition $|K(x)-\id_{\ad_{+}}|<\eg$ in every point $x\in\Omega$.
Let $(A_n,\Psi_n)$ be a sequence of solutions of the $PU(2)$-monopole equations associated with $\pg$ such that every point $x\in
\Omega$ has a geodesic ball neighbourhood $D_x$ such that for all large enough $n$, $$\int_{D_x}| F_{A_n}|_{g_\gamma}^2
vol_{g_\gamma}< \varepsilon^2$$ where $\varepsilon$ is the constant in Corollary 4.9. Then there is a subsequence $(m_n)\subset\N$ and gauge transformations $u_{n}\in{\cal G}_0$ such that $ u_{n} (A_{m_n},\Psi_{m_n})$ converges in the ${\cal C}^{\infty}$-topology on $\Omega$.
First of all note that every point has a geodesic ball neighbourhood $D'_x\subset D_x$ such that for a suitable subsequence $(m^x_n)_n\subset\N$ and suitable gauge transformations $u^x_{n}$ over $D'_x$ the sequence $(u^x_n(A_{m^x_n}|_{D'_x},
\Psi_{m^x_n}|_{D'_x})_n$ converges in the ${\cal C}^{\infty}$ topology on $D'_x$. This follows from Remark 4.5, Corollary 4.9 and the conformal invariance of the $L^2$-norm of 2-forms.
Using now Corollary 4.4.8 p. 160 \[DK\] we get a subsequence $(m_n)_n$ of $\N$ and gauge transformations $u_n$ such that $u_n(A_{m_n})$ converges in the ${\cal
C}^{\infty}$-topology on $\Omega$ to a connection $A$. But using the first monopole equation we see that the convergence of the connection component together with the local $L^4$-bound of the spinor component (provided by the local $L^2$-boundedness of the curvature) implies the local boundedness of the spinor component in any $L^2_l$-norm.
(Apriori ${\cal C}^0$-boundedness of the spinor) Let $X$ be a compact oriented 4-manifold, $ P^u$ a $Spin^{U(2)}(4)$-bundle on $X$ with $P^u\times_\pi\R^4\simeq
\Lambda^1_X$ as oriented 4-bundles, and $\pg=(\gamma,C,a,\beta,K)$ a system of data for the pair $(X,P^u)$ satisfying the condition $|K(x)-\id_{\ad_{+}}|<\eg$ in every point $x\in X$.\
1. If $\beta=0$, and $C$ is induced via $\gamma$ by the Levi-Civita connection in $(\Lambda^1,g_\gamma)$, then for any solution $(A,\Psi)\in{\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+(P^u))$ of the $PU(2)$-monopole equations associated with $\pg$, the following apriori estimate holds: $$\sup\limits_X|\Psi|^2_{g_\gamma}\leq \max\left(0,
C^{-1}\sup\limits_X(-\frac{s}{4}+c|F_a^+|_{g_\gamma})\right)$$ Here $s$ stands for the scalar curvature of $g_\gamma$, $c$ is a universal positive constant, and $C$ is the universal positive constant in (4) .\
\
2. In the general case one has an apriori estimate of the form $$\sup\limits_X|\Psi|^2_{g_\gamma}\leq \max\left(0,
C^{-1}\left[\sup\limits_X(-\frac{s}{4}+c|F_a^+|_{g_\gamma})+
\sigma(\gamma,C,\beta)\right]
\right)
\ ,$$ where $\sigma(C,\beta,\gamma)$ depends continuously on the coefficients of $\gamma,C,\beta$ with respect to the ${\cal C}^2\times{\cal
C}^1\times{\cal C}^1$-topology.
We prove the second assertion. Using Remark 3.4, it follows that, modifying $\beta$ if necessary, we may assume that $C$ is induced via $\gamma$ by the Levi-Civita connection in $(\Lambda^1,g_\gamma)$, so that the Dirac operator $\Dr^C_{\gamma,a,A}$ associated with $C$ coincides with the standard Dirac operator $\Dr_{\gamma,a,A}$.
The Weitzenböck formula for coupled Dirac operators gives for any triple $(A,a,\Psi)\in {\cal A}(\bar\delta(P^u))\times
{\cal A}(\det(P^u))\times
A^0(\Sigma^+(P^u))$ $$\Dr_{\gamma,A,a}^2\Psi=\nabla_{A,a}^*\nabla_{A,a}\Psi+
\Gamma_\gamma[(F_A+\frac{1}{2}F_a)^{+_{g_\gamma}}]\Psi+
\frac{s}{4}\ \Psi \cdot$$ On the other hand $$\Dr_{\gamma,a,A}(\Dr_{\gamma,a,A}+\beta)=
\Dr_{\gamma,a,A}^2+
\gamma\cdot\nabla_{a,A}\circ\beta$$ If $(A,\Psi)$ solves the $PU(2)$-monopole equations for the system of data $\pg$, it most hold pointwise $$\begin{array}{c}(\nabla_{A,a}^*\nabla_{A,a}\Psi,\Psi)+
(K (\Psi\bar\Psi)_0(\Psi),\Psi)+
\frac{1}{2}(\Gamma_\gamma(F_a)(\Psi),\Psi)+\\ \\
+\frac{s}{4}|\Psi|^2+
(\gamma\cdot \nabla_{a,A}\circ\beta(\Psi),\Psi)=0 \ .
\end{array}$$ Using the inequality (4), we get $$\begin{array}{l}
\frac{1}{2}\Delta|\Psi|^2= (\Delta_{A,a}\Psi,\Psi)-
|\nabla_{A,a}\Psi|^2\leq \\ \\
\ \ \ \ \ \leq -C|\Psi|^4+ (c
|F_a^+|-\frac{s}{4})|\Psi|^2
+|(\gamma\cdot\nabla_{a,A}\circ\beta(\Psi),\Psi)|-
|\nabla_{A,a}\Psi|^2\ .
\end{array}\eqno{(7)}$$ On the other hand $$\gamma \cdot \nabla_{a,A}\circ\beta(\Psi)=\gamma\cdot
[(\nabla_{C}\beta)(\Psi)+\beta\nabla_{A,a}\Psi]\ .$$ Therefore the term $(\gamma\cdot\nabla_{a,A}\circ\beta(\Psi),\Psi)$ can be estimated as follows $$|(\gamma \nabla_{a,A}\circ\beta(\Psi),\Psi)|
\leq c'\left(|\nabla_{C}(\beta)| |\Psi|^2 +
|\beta||\nabla_{A,a}\Psi| |\Psi|\right)\leq$$ $$\leq c'\left[|\nabla_{C}(\beta)| |\Psi|^2 +
|\beta| \left(\varepsilon\ |\nabla_{A,a}\Psi|^2 +
\frac{1}{\varepsilon}\ |\Psi|^2\right)\right] \ ,
\eqno{(8)}$$ where $c'$ is a universal constant and $\varepsilon$ is any positive number. Choose now $\varepsilon:=\frac{1}{2(c'\sup|\beta|+1)}$, so that the total coefficient of $|\nabla_{A,a}\Psi|^2$ in the expression obtained by replacing $|(\gamma\cdot \nabla_{a,A}\circ\beta(\Psi),\Psi)|$ in (7) with the right hand term of (8) becomes negative. Then we get an inequality of the form $$\frac{1}{2}\Delta|\Psi|^2\leq -C|\Psi|^4+ \sup\left(c|
F_a^+|+c'|\nabla_{C}(\beta)| -\frac{s}{4}\right)|\Psi|^2
+\frac{c'\sup|\beta|}{\varepsilon}|\Psi|^2\ ,$$ and the assertion follows easily by the maximum principle.
If $\Omega$ is compact, the condition “$\int\limits_{D_x}|
F_{A_n}|_{g_\gamma}^2 vol_{g_\gamma}<\varepsilon^2$ for all sufficiently large $n$” in Corollary 4.10 can be replaced by the condition $$"\int_{D_x}| F_{A_n}^{-_{g_\gamma}}|_{g_\gamma}^2
vol_{g_\gamma}<\frac{\varepsilon^2}{2}
\ for\ all\ sufficiently\ large\ n\ ".$$
By Proposition 4.11 and the inequality (3), the pointwise norm $|F_{A_n}^{+_{g_\gamma}}|$ of the $g_\gamma$-self-dual component of the curvature is apriori bounded by a constant (depending on $s_{g_\gamma}$ and $\pg$) hence $\int_{D_x}| F_{A_n}^{+_{g_\gamma}}|_{g_\gamma}^2$ can be made arbitrarily small, by replacing eventually $D_x$ with a smaller ball.
Regularity
----------
We begin with the following simple
Let $X$ be a 4-manifold and $g$, $g'$ two metrics on $X$. Then the operator $d_{g}^*+d^{+_{g'}}:A^1\map A^0 \oplus A^2_{+_{g'}}$ is elliptic. If $X$ is compact then the kernel of this operator is the harmonic space $\H^1_{g}$. The image of its extension $L^2_{k+1}\map L^2_{k}$ is $(A^0)_{k}^{\bot}\oplus
(A^2_{+_{g'}})^{\bot}_{k}$, where $(A^0)_{k}^{\bot}$ is the $L^2_{g}$-orthogonal complement of $\R\subset (A^0)_{k}$, and $(A^2_{+_{g'}})^{\bot}_{k}$ is the $L^2_{g'}$-orthogonal complement of $\H^2_{+_g'}\subset
(A^2_{+_{g'}})_{k}$.
Indeed, one checks easily that the symbol $\sigma$ of $d_{g}^*+d^{+_{g'}}$ is injective for non-vanishing cotangent vectors $\xi$. Indeed, if $\sigma_\xi(\alpha)=0$, then $(\xi\wedge\alpha)_{+_{g'}}=0$, hence $ \xi\wedge\alpha=0$. Therefore $\alpha$ has the form $\alpha=c\ \xi$, $c\in\R$. Using now the $\Lambda^0$-component of the equation $\sigma_\xi(\alpha)=0$, we get $c\ |\xi|_{g}^2=0$, i. e. $c=0$. But $\Lambda^1$, $\Lambda^0\oplus \Lambda^2_{+_{g'}}$ have both rang 4, so $\sigma_\xi$ must be isomorphism.
On compact 4-manifolds one has $\ker d^{+_{g'}}=\ker d$. Therefore $\ker(d_{g}^*+d^{+_{g'}})=\ker(d_{g}^*+d)=\H^1_{g}(X)$. The image of the $L^2_{k+1}\map L^2_{k}$ extension of $d_{g}^*+d^{+_{g'}}$ is obviously contained in $(A^0)_{k}^{\bot}\oplus (A^2_{+,{g'}})^{\bot}_{k}$. Therefore it must coincide with this space, because ${\rm index}(d_{g}^*+d^{+_{g'}})=
{\rm index}(d_{g}^*+d^{+_{g}})=b_1-b_+-1$.
As in the section above we fix $SU(2)$-bundles $S^{\pm}_s$ on the 4-sphere $S$ such that $\Lambda^1_S\simeq\RSU(S^+_s,S^-_s)=\Hom_\H(S^+_s,S^-_s)$ as oriented 4-bundles. The pairs consisting of a metric on the sphere and a $Spin(4)$-structure for that metric are parameterized by linear isomorphic Clifford maps $$\gamma:\Lambda^1_S\map \Hom_\H(S^+_s,S^-_s)\ .$$
We denote by $Clif(S)$ the space of Clifford maps on the sphere. Let again $E_s$ be the trivial $U(2)$ bundle on $S$.
We fix a Clifford map $\gamma_s:\Lambda^1_S\map \Hom_\H(S^+_s,S^-_s)$ such that $g_s:=g_{\gamma_s}$ has non-negative scalar curvature, strictly positive in the south pole $\infty$. Therefore the associated selfadjoint Dirac operator $\Dr_{\gamma_s}$ is injective, by the Weitzenböck formula. Denote by $C_s$ the Levi-Civita connection induced by $\gamma_s$ in the $SO(4)$-bundle $P^u_s\times_\pi\R^4=\Hom_\H(S^+_s,S^-_s)$ and denote by $\qg_s$ the system of data $$\qg_s:=(\gamma_s,C_s,0,0)\in Clif(S)\times{\cal
A}(P^u_s\times_\pi\R^4) \times{\cal A}(\det(P^u_s))\times
A^0(P^u_s\times_\pi\R^4)\ ,$$ where we used as usually the identification ${\cal A}(\det(P^u_0))=A^1(u(1))$.
Denote by $$sw_\pg:{\cal A}(\bar\delta(P^ u_s))\times
A^0(S^+_s\otimes E_s)\map
A^0(S^-_s\otimes E_s)\times A^0(su(S^+_s)\otimes su(2))$$ the Seiberg-Witten map associated with a system of data $\pg$ for the pair $(S,P^u_s)$.
(Regularity of $L^4$-small $L^2_1$-almost solutions with connection component in Coulomb gauge) Let $g$ be an arbitrary fixed metric on the sphere. There are positive constants $\alpha$, $\mu$, $c$ (depending on $g$ and $\gamma_s$) such that for any system of data $\pg=(\qg,K)$ with $\qg$ sufficiently close to $\qg_s$ and $|K-\id_{su(S^+_s)}|<\eg$ the following holds.
Any pair $u=(A,\Psi)\in L^2_1(\Lambda^1 (su(2)))\times
L^2_1(S^+_S\otimes E_s )$ satisfying:\
(i) $d^*_g(A)=0$,\
(ii) $\nr u\nr_{L^4}< \alpha$,\
satisfies the inequality $$\nr u\nr_{L^2_1}\leq c \ \nr sw_{\pg}(u)\nr_{L^2}\ .$$ If, moreover\
(iii) $\nr sw_{\pg}(u)\nr_{L^2}<\mu$,\
(iv) $sw_{\pg}(u)$ is smooth,\
then $u$ is also smooth.
We use the method of continuity as in the proof of 4.4.13 \[DK\]. The essential fact used in the proof of that theorem is that the map $$B\longmapsto (d^* B, F_B^+)$$ can be written as the sum of an elliptic first order operator and a quadratic map. By Remark 4.13, the map $(d_g^*,sw_\pg)$ has the same property. Note that we do require the metric $g$ to be close to $g_{\gamma_s}$.
As in the proof of Theorem 4.7, the system $\qg=(\gamma,C,a,\beta)$ defines an elliptic first order operator on the sphere $$D_{\qg}:=
\begin{array}{ccccc}
\Dr_{\qg}&:&A^0(S^+_s\otimes E_s)&\map&
A^0(S^-_s\otimes E_s)\\
\oplus&& \oplus&&\oplus\\
d_g^*+\Gamma_\gamma\circ d &:&A^1(su(2))&\map& A^0(su(2))^\bot
\oplus A^0(su(S^+_s)\otimes su(E_s))
\end{array}$$ Here $\Dr_{\qg}$ stands for the Dirac operator $\Dr_\gamma^C+\beta+
\gamma(\frac{a}{2})$, and $A^0(su(2))^\bot$ for the $L^2_{g}$-orthogonal complement of the 3-dimensional space of constant $su(2)$-valued functions.
By Remark 4.13 and elliptic semicontinuity, it follows that $D_{\qg}$ is injective if $\qg$ is sufficiently ${\cal C}^0$-close to $\qg_s$. Moreover, the $L^2_{k+1}\map L^2_{k}$ extension of $D_{\qg}$ is an isomorphisms depending continuously on $\qg$ with respect to the ${\cal
C}^k$-topology. We extend the operator $d_g^*$ on pairs by putting $d_g^*(B,\Phi):=d_g^*(B)$. With this convention note that the map $d_g^*+sw_\pg$ can be written as $$(d_g^*+sw_\pg)(B,\Phi)=D_\qg(B,\Phi)+\left[ \matrix{\gamma(B)\Phi\cr
\Gamma_\gamma(B\wedge
B)-K(\Phi\bar\Phi)_0}\right]=D_\qg(B,\Phi)+B_{\gamma,K}(B,\Phi)\ ,$$ where $B_{\gamma,K}$ is the quadratic map defined by the square bracket.\
If $\alpha$ is sufficiently small, there exists a constant $c=c(g,\gamma_s)$ such that for any $L^2_1$-pair $v$ with $d_g^*v=0$, $\nr v\nr_{L^4}<\alpha$, one has the estimate $$\nr v\nr_{L^2_1}< c \nr sw(v)\nr_{L^2}\ . \eqno{(1)}$$
Indeed, the Coulomb gauge condition $d_g^*(v)=0$ implies $$D_\qg(v)=-B_{\gamma,K}(v)+sw(v)\ . \eqno{(2)}$$ This gives an estimate of the form $$\nr v\nr_{L^2_1}\leq C_\qg\nr D_\qg(v)\nr_{L^2}\leq C_\qg C_{\gamma,K}\nr
v\nr_{L^4}^2+\nr sw(v)\nr_{L^2}\leq$$ $$\leq C C_\qg C_{\gamma,K}\nr v\nr_{L^4}\nr
v\nr_{L^2_1}+\nr sw(v)\nr_{L^2}\ ,$$ Since $\qg$ is assumed to be close to $\qg_s$ and $K$ belongs to a bounded family, it follows that the constants $C_\qg$, $C_{\gamma,K}$ can be chosen independently of $\pg$. The claim follows by the same rearrangement argument used in the proof of Theorem 4.7, taking $\alpha\leq \frac{1}{2C C_\qg C_{\gamma,K}}$. This proves the claim and the first part of the theorem.\
If $\alpha$ is sufficiently small, then for any two $L^2_1$-sections $v_1$, $v_2$ with $d_g^*(v_i)=0$, $\nr v_1\nr_{L^4}<\alpha$, $\nr v_2\nr_{L^4}<\alpha$ and $sw(v_1)=sw(v_2)$ it follows $v_1=v_2$.\
Indeed, let $b_{\gamma,K}$ be the $\R$-bilinear map associated with $B_{\gamma,K}$. One has $$D_\qg(v_1-v_2)=b_{\gamma,K}((v_2-v_1),v_1)+
b_{\gamma,K}(v_1,(v_2-v_1))\ ,$$ hence, by the injectivity of $D_\qg$, we get an estimate of the form $$\nr v_1-v_2\nr_{L^4}\leq C \nr v_1- v_2\nr_{L^2_1}\leq C C_\qg
\nr b_{\gamma,K}((v_2-v_1),v_1)+
b_{\gamma,K}(v_1,(v_2-v_1))\nr_{L^2}$$ $$\leq C C_\pg (\nr v_1\nr_{L^4}+\nr v_2\nr_{L^4})\nr v_2-
v_1\nr_{L^4}$$ where $C_\pg$ is a constant depending continuously of $\pg$ with respect to the ${\cal
C}^0$-topology. Therefore, we may suppose as above that $C_\pg=C_1$ is independent of $\pg$. Take $\alpha\leq\frac{1}{4C C_1}$.\
If $\alpha$ is sufficiently small, then for any pair $v$ with $d_g^*(v)=0$, $\nr v\nr_{L^4}<\alpha$ one has estimates of the form $$\nr v \nr_{L^2_{k+1}}\leq C_{\pg,k} \nr sw(v) \nr_{L^2_k} +
P_{\pg,k}(\nr sw(v)
\nr_{L^2_{k-1}})\ ,$$ where $C_{\pg,k}$ is a positive constant and $P_{\pg,k}$ is a polynomial with positive coefficients and without constant term.\
To see this use again the rearrangement argument above to estimate the $L^2_2$ and the $L^2_3$ norms of $v$ (compare with the proof of Theorem 3.7). For the higher Sobolev norms apply the usual bootstrapping procedure to the elliptic equation $(2)$.\
If $\alpha$ is sufficiently small, there exists a positive number $\mu$ such that for every smooth section $f\in A^0(S^-_s\otimes E_s \oplus su(S^+_s)
\otimes su(2))$ with $\nr f\nr_{L^2}<\mu$, the equation $$sw(v)=f\ ,\ d_g^*(v)=0$$ has a smooth solution $v$ satisfying $\nr v\nr_{L^4}<\alpha$.\
Indeed, choose first $\alpha$ such that the conclusions of Claims 1-3 hold. We use the continuity method to find a smooth solution of the equations $sw(v)=f$, $d_g^*(v)=0$ . Let $(SW^t)$ be the equation $$(d_g^*+sw_\pg)(v)=t\ f \ .\eqno{(SW^t)}$$ We have to find a smooth solution of $(SW^1)$ whose $L^4$-norm is bounded by $\alpha$. Let $N$ be the set $$N:=\{t\in [0,1]| \ (SW^t)\ {\rm has\ a\ smooth\ solution}\
v\ {\rm with}\ \nr
v\nr_{L^4}<\alpha\}$$
The set $N$ contains $0$. We assert that, taking a smaller bound $\alpha$ if necessary, $N$ becomes an open set. We use the implicit function theorem. Let $v_0$ be a solution of $(SW^{t_0})$ satisfying $d_g^*(v_0)=0$, $\nr
v_0\nr_{L^4}<\alpha$. We have $$\frac{\partial}{\partial v}(d_g^*+sw_\pg)(\dot v)=D_\qg(\dot v)+
b_{\gamma,K}(\dot v,v)+b_{\gamma,K}(v,\dot v)$$ This shows that, for $v=0$, the operator $ \frac{\partial}{\partial v}|_{_0}
(d_g^*+sw_\pg)$ defines an isomorphism: $$\begin{array}{ccc}
L^2_1(S^+_s\otimes E_s)&\map&
L^2(S^-_s\otimes E_s)\\
\oplus&&\oplus\\
L^2_1(\Lambda^1(su(2)))&\map& L^2(su(2))^\bot
\oplus L^2(su(S^+_s)\otimes su(E_s))
\end{array}$$ If $\nr v\nr _{L^4}$ is sufficiently small, then the $L^2_1 \map L^2$ extension of $\frac{\partial}{\partial v}(d_g^*+sw_\pg) $ is still an isomorphism. By the Fredholm alternative it follows that the $L^2_3 \map L^2_2$ extension is an isomorphism, too. Therefore, there exists $\varepsilon>0$ and an $L^2_3$ solution $v_t$ of $(SW^t)$ for any $t\in (t_0-\varepsilon,t_0+\varepsilon)$ such that $v_{t_0}=v_0$. Using the usual bootstrapping procedure, it follows that $v_t$ must be smooth.
We claim that $N$ is closed, if the bound $\mu$ of $\nr f\nr_{L^2}$ is sufficiently small. Indeed, if $t_n\rightarrow t_0$, and if $v_n$ is a smooth solution of $(SW^{t_n})$ with $\nr v_{n}\nr_{L^4}<\alpha$, then Claim 3. shows that there is a subsequence $(v_{n_m})_{m\in \N}$ converging in the ${\cal
C}^{\infty}$-topology to a smooth section $v_0$, which must solve the equation $(SW^{t_0})$. Of course, it is not clear that the strict inequality $\nr v_{n_m}\nr_{L^4}<\alpha$ is preserved at the limit. On the other hand, using the estimate (1) proved in Claim 1. and the boundedness of the inclusion $L^2_1\subset L^4$, we see that, choosing $\mu$ sufficiently small, we can assure that $$\nr v_n\nr_{L^4}\leq \frac{\alpha}{2}\ .$$ Therefore $v_0$ satisfies the stronger inequality $\nr v_0\nr_{L^4}\leq\frac{\alpha}{2}$. Now the second assertion in the theorem follows immediately: If $\nr u\nr_{L^4}<
\alpha$, $d_g^*(u)=0$, $\nr sw_{\pg}(u)\nr_{L^2}<\mu$, and $sw_{\pg}(u)$ is smooth, we can find a smooth solution $v$ of the equations $d_g^* v=0$, $sw_{\pg}(v)=sw_{\pg}(u)$ with $\nr
v\nr_{L^4}<\alpha$. But, by Claim 2., this solution must coincide with $u$.
With the notations and assumptions of the theorem, the following holds: There exists a positive constant $\alpha_1$ (depending on $(g,\gamma_s)$) such that any $L^2_1$-pair $u=(A,\Psi)$ with $d_g^*(A)=0$, $\nr u\nr_{L^2_1}\leq \alpha_1$ and $sw_\pg(u)$ smooth, is also smooth.
Removable singularities
-----------------------
We notice first that Corollary 4.8 (Estimates in terms of the curvature) can be easily generalized to an [*arbitrary*]{} system of data $\pg'=(\qg',K')$ for the pair $(\bar B, P^u_0)$, not necessarily close to a standard system. The only difference is that the constant $\varepsilon$ in the conclusion of the theorem will depend on $\pg'$. To see this it is enough to notice that the operator $D_\qg$ constructed in the proof of Theorem 4.7 is always elliptic by Remark 4.13 (even if the metric $g_\gamma$ is not close to the metric $g_s$). We can use in fact the standard constant curvature metric on the sphere for the Coulomb condition, as in \[DK\]. $D_\qg$ will be in general non-injective, but the injectivity of this operator is not essential in the proof of 4.7: the corresponding elliptic estimates $(el)$, $(el)_k$ will contain on the right the additional term $\nr u\nr_{L^2}$, which can be estimated in terms of $\nr u\nr_{L^4}$ using the volume of the sphere endowed with the metric $g_\gamma$.
An alternative argument uses a division of the unit ball in small balls, the scale invariance of the equations (Remark 4.5), the original Theorem 4.7, and the patching arguments explained on p. 162 \[DK\] in the instanton case.
Using this generalization of Corollary 4.8, we get the following analogon of Proposition 4.4.10 \[DK\]:
Let $\Omega$ be a strongly simply connected 4-manifold endowed with a $Spin^{U(2)}(4)$-bundle $P^u$ with $P^u\times_\pi\R^4\simeq
\Lambda^1_\Omega$, $\bar\delta(P^u)\simeq \Omega\times PU(2)$. Fix a trivialization of the $PU(2)$-bundle $\bar\delta(P^u)$.
Let $\pg=(\gamma,C,a,\beta,K)$ be a system of data for the bundle $P^u$ such that pointwise $|K-\id|<\eg$.
There exists a positive constant $\varepsilon_\pg$, and for every precompact interior domain $\Omega'\subsetint\Omega$ there exists a positive constant $M_{\pg,\Omega'}$ such that any solution $(A,\Psi)$ of the $PU(2)$-monopole equations for $\pg$ with $\nr
F_A\nr_{L^2_{g_\gamma}}<\varepsilon_\pg$ is gauge equivalent over $\Omega'$ to a pair $(A',\Psi')$ satisfying $$\nr A'\nr_{L^4_{g_\gamma}(\Omega')}<
M_{\pg,\Omega'}\nr F_A\nr_{L^2_{g_\gamma}}\ .$$
Given a fixed system of data $\pg_0$, we can find constants $\varepsilon_0$, $M_{0,\Omega'}$ (independent of $\pg$) such that the conclusion of the theorem holds with these constants, for every $\pg$ sufficiently close to $\pg_0$. Moreover, the statement is true if we use the fixed metric $g_{\gamma_0}$ to compute the Sobolev norms.
We will need these results in the following particular case:
Let ${\cal N}$, ${\cal N}'$ be the annuli $${\cal N}:=\{x\in B|\ \frac{1}{2}<|x|<1\}\ ,\ \ {\cal N}':=\{x\in B|\
\frac{4}{6}<|x|<\frac{5}{6}\} \ .$$ Denote by ${\cal N}_r$, ${\cal N}'_r$ the images of ${\cal N}$, ${\cal N}'$ under the homothety $h_r$. We recall that we denoted by $P^u_0$ the trivial $Spin^{U(2)}(4)$-bundle on $\bar B$, which is associated with the triple of $SU(2)$-bundles $S^{\pm}_0:=\bar B\times\H_{\pm}$, $E_0:=\bar
B\times\C^2$.
Let $\pg=(\gamma,C,a,\beta,K)$ be a system of data for the trivial $Spin^{U(2)}(4)$-bundle $P^u_0$ on the ball $\bar B$, such that pointwise $|K-\id|<\eg$, and such that $\gamma|_{\Lambda^1_0}:\R^4\map\H=(P^u_0\times_\pi\R^4)_0$ is the standard identification.. Then there exists constants $\varepsilon(K_0)>0$, $M(K_0)$ such that for any sufficiently small $r>0$ the following holds:\
Any solution $(A,\Psi)$ of the $PU(2)$-monopole equations for $\pg|_{{\cal N}_r}$ with $\nr F_A\nr_{L^2({\cal N}_r)}<\varepsilon(K_0)$ is gauge equivalent over ${\cal N}'_r$ to a pair $(A',\Psi')$ satisfying $$\nr A'\nr_{{L^4}({\cal N}_r')}<
M(K_0)\nr F_A\nr_{{L^2}({\cal N}_r)}\ .$$
The constants $\varepsilon(K_0)>0$, $M(K_0)$ are independent of $r$, and the Sobolev norms are computed with respect to the standard euclidean metric.
We use the same argument as in Remark 4.5. Let $$h_r:({\cal N}',{\cal N})\map ({\cal N}'_r,{\cal N}_r)$$ the homothety of slope $r$.
The pair $(h_r^*(A),rh_r^*(\Psi))$ solves the monopole equations associated with the system of data $(\gamma_r:=rh_r^*(\gamma),h_r^*(C),h_r^*(a),
rh_r^*(\beta),h_r^*(K))$, which converges to the standard system $\pg_{K_0}$ restricted to ${\cal N}$, as $r\rightarrow 0$.
The result follows now from 4.16, 4.17 and the conformal invariance of the $L^4$-norm on 1-forms and of the $L^2$-norm on 2-forms.
We shall use the following notations $$\Omega_r:=B\setminus \bar B(r)\ ,\ \ B^{\bullet}=B\setminus\{0\}\ , \ \
B^{\bullet}(R)=B(R)\setminus\{0\}\ \ , S^{\bullet}=S\setminus\{0\}\ .$$
Let $\pg=(\gamma,C,a,\beta,K)=(\qg,K)$ be a system of data for the trivial bundle $P^u_0$ on the ball $B$, and let $(A,\Psi)$ be a pair on $B^{\bullet}$ solving the monopole equations for $\pg|_{B^{\bullet}}$ such that $$\int\limits_{B^\bullet}|F_A|^2<\infty$$ Then for any sufficiently small $r>0$, there exist an $SU(2)$-bundle $E_r$ over $B$, a pair $(A_r,\Psi_r)\in{\cal A}(E_r)\times A^0(S^+_0\times E_r)$ and an $SU(2)$-isomorphism $$\rho_r:E_r|_{\Omega(r)}\map E|_{\Omega(r)}$$ such that:\
i) $\rho_r^*(A,\Psi)=(A_r,\Psi_r)$,\
ii) $\nr sw_\pg(A_r,\Psi_r)\nr_{L^2(B)}\rightarrow 0$ as $r\rightarrow 0$.
Let $\varphi$ be a cut-off map $\varphi: B\map [0,1]$ which is identically 1 on $B\setminus B(\frac{5}{6}-\varepsilon)$ and identically 0 on $B(\frac{4}{6}+\varepsilon)$.
Put $\varphi_r:=\varphi\circ h_r^{-1}$. Note first that, by the conformal invariance of the $L^4$-norm on 1-forms, the norm $\nr d\varphi_r\nr_{L^4}$ (computed with the euclidean metric) does not depend on $r$.
Consider now the restriction of the pair $(A,\Psi)$ to ${\cal N}_r$. Since the total integral of $|F_A|^2$ on the ball is finite, it follows that for any sufficiently small $r>0$ we have $$\nr F_A\nr_{L^2({\cal N}_r)}<\varepsilon(K_0)\ ,$$ so that Lemma 4.18 applies. The conclusion of this Lemma can be reformulated as follows: There exists an $SU(2)$-trivialization ${\cal N}'_r\times\C^2\textmap{\tau_r} E_0|_{{\cal N}'_r}$ such that the connection matrix of $\tau_r^*(A)$ (which we also denote by $\tau_r^*(A)$) satisfies the estimate $$\nr \tau_r^*(A)\nr_{L^4({\cal N}'_r)}\leq M(K_0)\nr F_A\nr_{L^2({\cal N}_r)}
\eqno{(1)}$$ We define the $SU(2)$-bundle $E_r$ by gluing (over the annulus ${\cal N}'_r$) the trivial bundles $B(0,\frac{5r}{6})\times\C^2$, $E_0|_{\Omega(\frac{4r}{6})}$ via the isomorphism $\tau_r$.
Let $P^u_r$ be the $Spin^{U(2)}(4)$-bundle associated with the triple $(S^{\pm}_0,E_r)$. The system $\pg$ can be also regarded as a system of data for the bundle $P^u_r$.
Now denote by $u$ the initial pair $u:=(A,\Psi)$, and by $u_r$ the pair $$u_r\in {\cal A}(E_r)\times A^0(S^+_0\otimes E_r)\ ,$$ which coincides with $u$ on $\Omega((\frac{5}{6}-\varepsilon)r)$ and with the cut-off $\varphi_r \tau_r^*(u)$ of $\tau_r^*(u)$ on $B(0,\frac{5r}{6})$.
The section $sw_\pg(u_r)$ vanishes identically on ${\Omega(\frac{5r}{6})}$, where $u_r$ coincides with $u$. Therefore, in order to prove $ii)$ we only have to estimate the $L^2$ norm of $sw_\pg(\varphi_r \tau_r^*(u))$ on $B(0,\frac{5r}{6})$, where $E_r$ coincides with the trivial bundle $B(0,\frac{5r}{6})\times\C^2$.
On $B(0,\frac{5r}{6})$ the Seiberg-Witten map $sw_\pg$ can be written as a sum between a first order differential operator and a quadratic map: $$sw_{\pg}(B,\Phi)=\left[\matrix{\Dr_\qg \Phi\cr
\Gamma_\gamma(d B) }\right]+
\left[\matrix{\gamma(B)(\Phi)\cr \Gamma_\gamma(B\wedge
B)-K(\Phi\bar\Phi)_0}\right]=T_\qg(B,\Phi)+B_{\gamma,K}(B,\Phi)$$ Since $T_\qg$ is a first order operator, we have an identity of the form $$T_\qg(f v)=A_\qg(df)(v)+fT_\qg(v)\ ,$$ where $A_\qg(df)$ is a 0-order operator whose coefficients depend linearly on the first order derivatives of the real function $f$.
Therefore $$sw_\pg(\varphi_r \tau_r^*(u))=A_\qg(d\varphi_r)(\tau_r^*(u))+\varphi_r
T_\qg(\tau_r^*(u))+\varphi_r^2 B_{\gamma,K}(\tau_r^*(u))=$$ $$=\varphi_r
sw_\pg(\tau_r^*(u))+A_\qg(d\varphi_r)(\tau_r^*(u))+
(\varphi_r^2-\varphi_r)B_{\gamma,K}(\tau_r^*(u))=$$ $$A_\qg(d\varphi_r)(\tau_r^*(u))+
(\varphi_r^2-\varphi_r)B_{\gamma,K}(\tau_r^*(u))\ .$$ Therefore, taking into account that $d\varphi_r$ and $(\varphi_r^2-\varphi_r)$ vanish outside ${\cal N}'_r$, we get $$\nr sw_\pg(u_r)\nr_{L^2(B)}=
\nr sw_\pg(\tau_r^*(u))\nr_{L^2(B(\frac{5r}{6}))}\leq$$ $$\leq C_\qg\nr d\varphi_r\nr_{L^4}
\nr\tau_r^*(u)\nr_{L^4({\cal N}'_r)}+
C'_\pg \nr\tau_r^*(u)\nr_{L^4({\cal N}'_r)}^2$$ Since $\nr d\varphi_r\nr_{L^4}$ does not depend on $r$ we have only to prove that $\nr\tau_r^*(u)\nr_{L^4({\cal N}'_r)}$ converges to 0 as $r\rightarrow 0$. But the estimate $(1)$ shows that the $L^4$-norm of the connection component of $\tau_r^*(u)$ converges to 0 as $r\rightarrow 0$.
On the other hand, by the inequality (3) Section 4.1 and the second monopole equation, one has pointwise in ${\cal N}'_r$. $$|\tau_r^*(\Psi)|^4=|\Psi|^4\leq
\left[C_1^{-1}|\Gamma_\gamma(F_A)|\right]^2 \ .
$$ This gives an estimate of $\nr\tau_r^*(\Psi)\nr_{L^4({\cal N}'_r)}$ in terms of $\nr F_A\nr_{L^2({\cal N}'_r)}^{\frac{1}{2}}$, which obviously converges to 0 as $r\rightarrow 0$.
We recall from \[DK\] the following important
(Gauge fixing on the sphere [^5]) Let $g_c$ be the standard constant curvature metric on the sphere $S^4$. Then there are constants $\varepsilon_c$, $M_c$ such that any connection $A$ in the trivial $SU(2)$-bundle $E_s$ with $\nr F_A\nr_{L^2}<\varepsilon_c$ is gauge equivalent to a connection $\tilde A$ satisfying $$d_{g_c}^*(\tilde A)=0\ ,\ \ \nr \tilde A\nr_{L^2_1}<M_c\nr F_A\nr_{L^2} \ .$$
We can prove now
(Removable singularities) Let $\pg=(\qg,K)$ a system of data for the trivial $Spin^{U(2)}(4)$-bundle $P^u_0$ on $\bar B$ and let $u=(A,\Psi)$ be a pair on the punctured ball solving the monopole equations for $\pg|_{B^{\bullet}}$ such that $$\int_{B^{\bullet}}|F_{A}|^2<\infty\ .$$ There exists an $SU(2)$-bundle $F$ on the ball, and an $SU(2)$-isomorphism $\rho:F|_{B^{\bullet}}\map E_0|_{B^{\bullet}}$ such that $\rho^*(A,\Psi)$ extends to a global smooth solution of the monopole equations associated with $\pg$ and the $Spin^{U(2)}(4)$-bundle defined by $(S^{\pm}_0,F)$.
We use similar arguments as in the proof of the “Removable singularities” theorem for the instanton equation (Theorem 4.4.12 \[DK\]). The only difference is that the $L^2_1$-bound of the approximate solutions we construct, does not follow directly from Theorem 4.20 (Gauge fixing on the sphere).
Identify $\bar B$ with the upper hemisphere of the 4-sphere $S$, and extend the system $\pg$ to a system for the $Spin^{U(2)}(4)$-bundle $P^u_s$. The extended system will be denoted by the same symbol $\pg$, and we can assume that $\pg$ has the form $(\qg,K)$ with $\qg$ close to the system $\qg_s$ constructed in the proof of Theorem 4.7, so that Theorem 4.14 and Corollary 4.15 applies. We shall use these results in the particular case $g=g_c$; with respect to this metric connections with $L^2$-small curvature can be brought in the Coulomb gauge, by 4.20.\
Step 1. For a sufficiently small positive number $R<1$ we use Lemma 4.18 to get a trivialization of $E_0|_{{\cal N}'_R}$, such that the $L^4$-norm of the corresponding connection matrix is controlled by $\nr F_A\nr_{L^2({\cal N}_R)}$. By the same gluing procedure we get a bundle $E^R$ on the punctured sphere $S^{\bullet}$, trivialized on $S\setminus
\bar B(\frac{4R}{6})$. We cut off the pair $u$ this time towards the outer boundary of the ball, and we get a pair $u^R=(A^R,\Psi^R)$. It holds $$\lim\limits_{R\rightarrow 0} \nr sw_\pg(u^{R})\nr_{L^2}=
\lim\limits_{R\rightarrow 0}
\nr F_{A^{R}}\nr_{L^2}=\lim\limits_{R\rightarrow 0}
\nr \Psi^{R}\nr_{L^4}=0 \ , \eqno{(2)}$$ The first two relations follow as in the proof of Lemma 4.19, since both maps $sw_\pg(\cdot)$, $F_{\cdot}$ can be written as the sum of a first order operator and a quadratic map, hence the perturbations produced by of the two cut-off operations can be estimated in terms of the $L^2$-norm of the curvature restricted to the corresponding annuli.
To get the third formula, it is enough to notice that the pointwise norm of the spinor is invariant under bundle isomorphisms, and that the $L^4$-norm of $\Psi|_{B^\bullet(R)}$ can be estimated in terms of $\nr
F_{A|_{B^\bullet(R)}}\nr_{L^2}^\frac{1}{2}$.
Suppose now that $r<R<1$ and use the same procedure (to modify the bundle and cut off the solution), but this time in both directions.
We get $SU(2)$-bundles, $E_r^R$ on the sphere, which come with trivializations over $B(\frac{5r}{6})$, $S\setminus \bar B(\frac{4R}{6})$, and with an isomorphism $$E_r^R|_{B(\frac{5R}{6})\setminus \bar B(\frac{4r}{6})}
\textmap{\simeq\ \rho_{r,R}}
E_s|_{B(\frac{5R}{6})\setminus \bar B(\frac{4r}{6})}\ ,$$ as well as cut-off pairs $$u_r^R=(A_r^R,\Psi_r^R)\in {\cal A}(E_r^R)
\times A^0(S^+_s\otimes E_r^R)\ .$$ With this construction, it holds $$\lim\limits_{r\rightarrow 0} \nr sw_\pg(u_r^R)\nr_{L^2}=\nr
sw_\pg(u^R)\nr_{L^2}\ ,\ \ \lim\limits_{r\rightarrow 0}
\nr F_{A_r^R}\nr_{L^2}=\nr F_{A^R}\nr_{L^2}\ ,$$ $$\lim\limits_{r\rightarrow 0} \nr
\Psi_r^R\nr_{L^4}=\nr \Psi ^R\nr_{L^4} \ . \eqno{(3)}$$ Note that the double gluing-procedure we used could apriori give rise to a - $SU(2)$-bundle $E_{r,R}$ on the sphere. But since the curvature $F_{A^R_r}$ can be made as small as we please, it follows that all the bundles $E_{r,R}$ become trivial, if $R$ is small.
Step 2. Using (2), (3) and Theorem 4.20 it follows that, once $R$ is small, there exists an $SU(2)$-isomorphism $\theta_r^R:E_s\map E_r^R$ such that $B^R_r:=\theta_{r,R}^*(A^R_r)$ satisfies $$d_{g_c}^*(B^R_r)=0\ , \ \ \nr B^R_r\nr_{L^2_1}\leq M_c \nr
F_{A^R_r}\nr_{L^2} \eqno{(4)}$$ Put $\Phi^R_r:=(\theta^R_r)^*(\Psi_r^R)$, $v^R_r:=(B^R_r,\Phi^R_r)$.
Step 3. Using (2), (3), (4) and the boundedness of the embedding $L^2_1\subset L^4$, it follows that, if $R$ is small enough, the $L^4$-norm of the pair $v^R_r$ can be made smaller as the constant $\alpha$ in the Regularity Theorem 4.14, so that we get an estimate of the form $$\nr v^R_r\nr_{L^2_1}\leq c \nr sw_\pg( v^R_r) \nr_{L^2}=
\nr sw_\pg(u^R_r)\nr_{L^2}\
.\eqno{(5)}$$ The relations (2), (3) imply now that , choosing $R$ small, we can assure that $$\nr v^R_r\nr_{L^2_1}\leq \alpha_1\ , \eqno{(6)}$$ where $\alpha_1$ is the constant in Corollary 4.15. From this point the proof goes further like in the instanton case: We choose $R$ sufficiently small such that all the mentioned properties are fulfilled, and we let $r$ tend to 0. Using the $L^2_1$-boundedness obtained in (6) it follows that we can find a sequence $r_i\rightarrow 0$ such that $v_i=(B_i,\Phi_i):=v^R_{r_i}$ converges weakly in $L^2_1$ to an $L^2_1$-pair $v=(B,\Phi)$.
Step 4. We want to prove that $v$ is smooth. The weak limit $v$ must also satisfies $\nr
v
\nr_{L^2_1}\leq \alpha_1$ by the weak-semicontinuity of the norm in reflexive Banach spaces. Therefore, by Corollary 4.15, we only have to prove that the $L^2$-section $sw(v)$ is smooth.
But on any small ball $D$ , $\bar D\subset S^{\bullet}$, the pairs $v_i=(B_i,\Phi_i)$ remain in the same gauge equivalence class. Recall now from \[DK\] that the Sobolev norms of any connection $H$ in Coulomb gauge can be estimated in terms of the gauge invariant expressions $$\nr F_H\nr_{L^\infty}\ , \ \nr \nabla_H^{(i)} F_H\nr_{L^2} \ ,$$ as soon as its $L^4$-norm is sufficiently small. Using the estimate (4) and the scale invariance of the $L^4$-norm on 1-forms, this condition will be also fulfilled (for all small balls $D$), if $R$ is sufficiently small.
On the other hand one can easily bound the Sobolev norms of a spinor $\Xi$ in terms of the gauge invariant expressions $\nr \nabla^{(i)}_H\Xi\nr_{L^2}$ and the Sobolev norms of the connection $H$.
Therefore, taking a subsequence if necessary, we can assume that $v_i$ converges in the Fréchet ${\cal C}^{\infty}$-topology on $S^{\bullet}$, so that $sw(v)$ is smooth on the punctured sphere.
But, by Lemma 4.19, $\lim\limits_{i}
\nr sw_\pg(v_i|_{B(\frac{4R}{6})})\nr _{L^2}\rightarrow
0$, so $sw(v)$, which is the limit of $sw(v_i)$ in the distribution sense, vanishes in a neighbourhood of $0$.
On the other hand, for any ball $D$, $\bar D\subset B^\bullet(\frac{4R}{6})$, the isomorphism $\theta^R_{r_i}$ intertwines the connection matrices $A$, $B_i$, and $B_i$ converges in the ${\cal C}^{\infty}$ topology on such a ball. Therefore a subsequence $\theta^R_{r_{i_n}}$ converges in the ${\cal C}^{\infty}$ topology on $B^\bullet(\frac{5R}{6})$ to a smooth bundle isomorphism $\theta$, such that $$\theta^*(A|_{B^\bullet(\frac{5R}{6})})=B|_{B^\bullet(\frac{5R}{6})}\ .$$ Taking the limit of $[\theta^R_{r_{i_n}}]^*(\Psi|_{{B(\frac{5R}{6})\setminus \bar
B(\frac{4r_{i_n}}{6})})})=\Phi^R_{r_{i_n}}|_{{B(\frac{5R}{6})
\setminus \bar
B(\frac{4r_{i_n}}{6})})}$ for $n\rightarrow\infty$, we also get $$\theta^*(\Psi|_{B^\bullet(\frac{5R}{6})})=
\Phi|_{B^\bullet(\frac{5R}{6})})$$
Compactified moduli spaces
--------------------------
Let $X$ be a closed oriented 4-manifold. For a $Spin^{U(2)}(4)$-bundle $P^u$ with $P^u\times_\pi\R^4\simeq \Lambda^1$ and a system of data $\pg=(\gamma,C,a,\beta,K)$ for $P^u$ denote by ${\cal M}_\pg(P^u)$ the moduli space of pairs $(A,\Psi)\in{\cal A}(\bar\delta(P^u))\times A^0(\Sigma^+(P^G))$ solving the $PU(2)$-monopole equations associated with $\pg$.
By Proposition 2.1, the data of a $Spin^{U(2)}(4)$-bundle $P'^u$ on $X$ with $\det(P'^u)\simeq \det(P^u)$, $P'^u\times_\pi\R^4\simeq P^u\times_\pi\R^4$ is equivalent via the map $\bar\delta$ to the data of $PU(2)$-bundle $\bar P'$ whose Pontrjagin class satisfies $$p_1(\bar P')\equiv (w_2(X)+ \bar c_1(\det(P^u)))^2\ {\rm mod}\ 4 \ .$$ For every number $l\in \N$ we fix:\
1. A $Spin^{U(2)}(4)$-bundle $P_l^u$ with $$l=\frac{1}{4}\left(p_1(\bar\delta(P_l^u))-
(p_1(\bar\delta(P^u))\right)$$ 2. Identifications $$P_l^u\times_\pi\R^4\textmap{\simeq}P^u\times_\pi\R^4\ ,\ \
\det(P_l^u)\textmap{\simeq}\det(P^u). \eqno{(1)}$$ These bundle isomorphisms allow us to identify the spaces of perturbations-data associated with the bundles $P^u$, $P^u_l$.
An ideal $PU(2)$-monopole of type $(P^u,\pg)$ is a pair $([A',\Psi'],\{x_1,\dots,x_l\})$ consisting of an element $\{x_1,\dots,x_l\}$ in a symmetric power $S^l(X)$ of $X$ and a monopole $[A',\Psi']\in{\cal M}_\pg(P_l^u)$.
We denote by $I{\cal M}_\pg(P^u)$ the space of ideal monopoles of type $(P^u,\pg)$.\
Let $\delta_{x}$ be the Dirac measure associated with a point $x\in X$. If $\pg=(\gamma,A,a,\beta,K)$, we always use the metric $g_\gamma$ to compute the norms and to define (anti-)self-duality for 2-forms.
The map $F:I{\cal M}_\pg(P^u)\map [{\cal C}^0(X,\R)]^*$, defined by $$F([A',\Psi'],\{x_1,\dots,x_l\})=|F_{A'}|^2+8\pi^2
\sum\limits_{i=1}^l\delta_{x_i} \ ,$$ is bounded with respect to the strong topology in the dual space $ [{\cal C}^0(X,\R)]^*$.
Let $\varphi\in{\cal C}^0(X,\R)$ with $\sup\limits_X|\varphi|\leq 1$. Then $$\begin{array}{ll}|\langle F([A',\Psi'],\{x_1,\dots,x_l\}),
\varphi\rangle|\leq& \left[\nr
F_{A'}^-\nr_{L^2}^2-
\nr F_{A'}^+\nr_{L^2}^2\right] +2\nr F_{A'}^+\nr_{L^2}^2 +8\pi^2 l
\\ \\
&=-2\pi^2 p_1(\bar\delta(P^u))+2 C\nr \Psi'\nr_{L^4}^4\ ,
\end{array}$$ where $C$ is a universal positive constant. The assertion follows from the apriori ${\cal
C}^0$-boundedness of the spinor component of a solution (Proposition 4.11).
Let $\mg'=([A',\Psi'],s')$ be an ideal monopole of type $(P^u,\pg)$ with $s'\in S^{l'}(X)$ and $[A',\Psi']\in{\cal M}_\pg(P^u_{l'})$. For a positive number $\varepsilon$ we define $U(\mg',\varepsilon)$ to be the set of ideal monopoles $\mg''=([A'',\Psi''],s'')$ of type $(P^u,\pg)$ with $s''\subset s'$, and which have the following property:
There exists an isomorphism of $Spin^{U(2)}(4)$-bundles $$\varphi:P_{l''}^u|_{X\setminus s'}\map P_{l'}^u|_{X\setminus s'}$$ which is compatible with the identifications (1) such that $$d_1(\varphi^*(A',\Psi'),(A'',\Psi''))<\varepsilon\ ,$$ where $d_1$ is a metric defining the Fréchet ${\cal C}^{\infty}$-topology in the product $${\cal A}(\bar\delta(P_{l''}^u|_{X\setminus
s'}))\times A^0(\Sigma^+(P_{l''}^u|_{X\setminus s'})))\ .$$
Let $M>0$ be a bound for the map $F$ defined above. The weak topology in the ball of radius $M$ in $[{\cal C}^0(X,\R)]^*$ is metrisable (see \[La\], Theorem 9.4.2). Let $d_2$ be a metric defining this topology.
We endow $I{\cal M}_\pg(P^u)$ with a metric topology by taking as basis of open neighbourhoods for an ideal monopole$\mg'$ of type $(P^u,\pg)$ the sets of the form $U(\mg',\varepsilon)\cap F^{-1}(B_{d_2}(F(\mg'),\varepsilon))$, $\varepsilon>0$.
With respect to the metric topology defined above the moduli space ${\cal M}_\pg(P^u) \subset
I{\cal M}_\pg(P^u) $ is an open subspace with compact closure $\overline{{\cal M}_\pg(P^u)}$.
The first assertion is obvious. For the second, we use the same argument as in the instanton case, but we make use in an essential way of the ${\cal C}^0$-boundedness of the spinor:
Let $\mg_n$ a sequence of ideal monopoles. It is easy to see that we can reduce the general case to the case where $\mg_n=[A_n,\Psi_n]\in {\cal M}_\pg(P^u)$. By Lemma 4.23, the sequence of measures $\mu_n:=F(\mg_n)$ is bounded, so after replacing $\mg_n$ by a subsequence, if necessary, it converges weakly to a (positive) measure $\mu$ of total volume $\mu(1)\leq M$. The set $$S_\varepsilon:=\{x\in X|\exists n\in\N\ \forall m\geq n\
(\mu_m(D)\geq
\varepsilon^2\ {\rm for\ every}\ {\rm geodesic\ ball}\ D\ni x)\}$$ contains at most $\frac{M}{\varepsilon^2}$ points, so it is finite for every positive number $\varepsilon$. Choosing the constant $\varepsilon$ provided by the “Global compactness” theorem (Corollary 4.10), it follows by a standard diagonal procedure that there exists a subsequence $(\mg_{n_m})_m$ and gauge transformations $f_m$ on $X\setminus S_\varepsilon$, such that $f_m^*(\mg_{n_m})$ converges to a solution $(A_0,\Psi_0)$ of the monopole equations $SW_\pg$ restricted to $X\setminus
S_\varepsilon$. By the “Removable Singularities” theorem, we can extend this solution to a global solution $(\tilde A_0,\tilde \Psi_0)$ of the monopole equations associated with $\pg$ and a new $Spin^{U(2)}(4)$-bundle $P'^u$, which comes with identifications $$P'^u\times_\pi\R^4\textmap{\simeq}P^u\times_\pi\R^4\ ,\ \
\det(P'^u)\textmap{\simeq}\det(P^u).$$ We have $$|F_{\tilde A_0}|^2 =|F_{A_0}|^2=\mu-8\pi^2
\sum_{x\in S_\varepsilon}\lambda_x\delta_x$$ with positive numbers $\lambda_x$. It remains to prove that the $\lambda_x$ are integers. Since $F(\mg_{n_m})\rightarrow\mu$, we have for small enough $r>0$ $$\lambda_x=\lim\limits_{m\rightarrow\infty}\frac{1}{8\pi^2}
\int\limits_{B(x,r)}|F_{A_{n_m}}|^2-|F_{\tilde A_0}|^2=$$ $$\lim\limits_{m\rightarrow\infty}\frac{1}{8\pi^2}
\int\limits_{B(x,r)}-\tr (F_{A_{n_m}}^2)+\tr (F_{\tilde A_0})^2+
2 \left(| (\Psi_{n_m}\bar\Psi_{n_m})_0|^2-|
(\tilde \Psi_0\bar{\tilde \Psi_0})_0|^2\right) \ .$$
As in the instanton case we get $$\lim\limits_{m\rightarrow\infty}\frac{1}{8\pi^2}
\left[\int\limits_{B(x,r)}-\tr (F_{A_{n_m}}^2)+
\tr (F_{\tilde A_0})^2\right]_{{\rm mod}\
\Z}=\lim\limits_{m\rightarrow\infty}
(\tau_{S(x,r)}(A_{n_m})-\tau_{S(x,r)}({\tilde A_0}))$$ $$=0\ {\rm in}\
\qmod{\R}{\Z}$$ by the convergence $f_m^*(A_{n_m}|_{X\setminus S_\varepsilon})\rightarrow {\tilde
A_0}|_{X\setminus S_\varepsilon}$. Here $\tau_S(B)$ denotes the Chern-Simons invariant of the connection $B$ on a 3-manifold $S$ (\[DK\]).
On the other hand, by the apriori ${\cal C}^0$-bound of the spinor component on the space of monopoles, the term $\int\limits_{B(x,r)}2
\left(| (\Psi_{n_m}\bar \Psi_{n_m})_0|^2-
| (\tilde \Psi_0\bar{\tilde \Psi_0})_0|^2\right)$ can be made as small as we please by choosing $r$ sufficiently small. This shows that the $\lambda_x$ are integers, and that $$\sum_{x\in S_\varepsilon}\lambda_x=
\frac{1}{4}\left(p_1(\bar\delta(P'^u))-
(p_1(\bar\delta(P^u))\right)\ ,$$ which completes the proof.
0 cm
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Author’s address :\
Mathematisches Institut, Universität Zürich, Winterthurerstr. 190,\
CH-8057 Zürich, [ e-mail]{}: teleman@math.unizh.ch
[^1]: This gap as well as the difficulty of the problem was pointed out by the author during the Workshop “4-dimensional manifolds”, Oberwolfach, March 1996.
[^2]: In order to have [*well defined*]{} invariants, one needs a [*smooth*]{} parameterized moduli space ( \[DK\], p 143, 149). Moreover, the Kählerian parameters are [*all*]{} -generic in our sense; on the other hand, computations needed in order to get a proof of the Van de Ven conjecture using $Spin^c$-invariants, must be done in the Kähler case.
[^3]: We used here the following simple remark: The space of $L^2_{l-1}$-sections in the space of $L^2_{l-2}$ sections in a bundle has infinite codimension. Note that $L^2_{l-1}$ is nonetheless dense in $L^2_{l-2}$.
[^4]: Such a metric can be obtained as follows: consider a plane [*convex*]{} curve with symmetry axis $Oy$, which is horizontal in a neighbourhood of its upper intersection point with $Oy$. Then rotate this curve around the $Oy$-axis in the 5-dimensional space [R]{}$^4\times Oy$ . The hypersurface obtained in this way is also conformally flat, by a theorem of E. Cartan (see \[GHL\], p. 157, \[Ch\], Th. 4.2, p. 162)
[^5]: Note that in \[DK\] it is stated a slightly weaker form of this theorem (Proposition 2.3.13 p. 63): The hypothesis requires that $A$ can be joined to the flat connection by a path of connections with $L^2$-small curvature . However, the second proof of this result, which is given in section 2.3.10, [*does not*]{} use this additional assumption. I am grateful to Peter Kronheimer for pointing me out this important detail. On the other hand, note that this second proof works only for the standard constant curvature metric, and can be generalized to conformally flat metrics with non-negative sectional curvature. Since our regularity theorem works for solutions whose connection component is in Coulomb gauge with respect to [*any*]{} metric, not necessary close to the metric defined by the $Spin^{U(2)}$-structure, we don’t need this generalization
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abstract: 'Several dynamical scenarios have been proposed that can lead to prompt mass segregation on the crossing time scale of a young cluster. They generally rely on cool and/or clumpy initial conditions, and are most relevant to small systems. As a counterpoint, we present a novel dynamical mechanism that can operate in relatively large, homogeneous, cool or cold systems. This mechanism may be important in understanding the assembly of large mass-segregated clusters from smaller clumps.'
---
Early mass segregation may be critical to the long-term survival of a stellar system (Vesperini et al. 2009a). It also defines the early cluster environment within which stars move and interact. In recent years, several dynamical studies have explored routes to early mass segregation that do not simply require that a cluster formed in that state. [@McMillan_etal2007] found that mergers of mass-segregated “clumps” tend to preserve that segregation in the final merger product, so that, if the clumps are formed segregated, or have time to segregate before they merge, the result is a strongly mass-segregated cluster. [@Allison_etal2009] found similar behavior, starting from fractal clumpy initial conditions in small, cool model clusters.
Ultimately, these scenarios rely on normal relaxation processes in small stellar systems. However, as illustrated in Figure 1, rapid segregation is also possible in significantly larger systems. The initial conditions of the simulation shown here consist of a cold (virial ratio $q=0.001$), homogeneous sphere with a [@Kroupa2002] mass distribution. No segregation is seen before the “bounce” at $t\sim1.5$ initial dynamical times, while immediately afterward the highest mass groups are clearly ordered by radius. This behavior was also noted by [@Vesperini_etal2006] and [@Vesperini_etal2009b].
The phenomenon of rapid segregation cannot be due to enhanced relaxation around the high density bounce. This would only be possible if the system were still cold at that time, and our simulations clearly indicate that this is not the case. Instead, as shown in Figure 2, we find that the system fragments as it collapses, as discussed in detail by [@Aarseth_etal1988], and the fragments mass segregate quite early on during the collapse process. Significant segregation within the clumps is already established by $t\sim1$, well before the bounce, and is preserved when the clumps subsequently merge at $t=1.5$, essentially as described by [@McMillan_etal2007].
The phenomenon persists as we vary the initial system parameters, and is still measurable even for fairly “warm” initial conditions ($q\sim0.1$), with and without initial clumping (fractal dimension $d\sim2-3$), and for large systems, up to $N\sim10^5$. Thus it may provide the basis of a viable mechanism for extending earlier dynamical segregation scenarios to substantially larger systems.
This work was supported in part by NSF grants AST-0708299 and AST-0959884, and NASA grant NNX08AH15G.
Aarseth, S. J., Lin, D. N. C., & Papaloizou, J. C. B. 1988, ApJ, 324, 288
Allison, R. J., Goodwin, S. P., Parker, R. J., de Grijs, R., Portegies Zwart, S. F., & Kouwenhoven, M. B. N. 2009, ApJL, 700, L99
Kroupa, P. 2002, Science, 295, 82
McMillan, S. L. W., Vesperini, E., & Portegies Zwart, S. F. 2007, ApJL, 655, L45
Vesperini, E., McMillan, S. L. W., & Portegies Zwart, S. F. 2006, Joint Discussion 14, 26th General Assembly of the IAU
Vesperini, E., McMillan, S. L. W., & Portegies Zwart, S. F. 2009a, ApJ, 698, 615
Vesperini, E., McMillan, S. L. W., & Portegies Zwart, S. F. 2009b, in Globular Clusters – Guides to Galaxies, ESO Astrophysics Symposia, T. Richtler and S. Larsen eds. (Springer: Berlin), p. 429
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abstract: 'Recent studies have shown that the efficiency of deep neural networks in mobile applications can be significantly improved by distributing the computational workload between the mobile device and the cloud. This paradigm, termed *collaborative intelligence*, involves communicating feature data between the mobile and the cloud. The efficiency of such approach can be further improved by lossy compression of feature data, which has not been examined to date. In this work we focus on collaborative object detection and study the impact of both near-lossless and lossy compression of feature data on its accuracy. We also propose a strategy for improving the accuracy under lossy feature compression. Experiments indicate that using this strategy, the communication overhead can be reduced by up to 70% without sacrificing accuracy.'
address: 'School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada'
bibliography:
- 'ref.bib'
title: DEEP FEATURE COMPRESSION FOR COLLABORATIVE OBJECT DETECTION
---
Deep feature compression, collaborative intelligence, compression-augmentation, object detection
Introduction {#sec:intro}
============
Mobile and Internet-of-Things (IoT) [@xia2012internet] devices are increasingly relying on Artificial Intelligence (AI) engines to enable sophisticated applications such as personal digital assistants [@PDA_SPM2017], self-driving vehicles, autonomous drones, smart cities, and so on. The AI engines themselves are generally built on deep learning models. The most common way of deploying such models is to place them in the cloud and have the sensor data (images, speech, etc.) uploaded from the mobile to the cloud for processing. This is referred to as the *cloud-only* approach. More recently, with smaller graphical processing units (GPUs) making their way into mobile/IoT devices, some deep models might be able to run on the mobile device, an approach referred to as *mobile-only*.
A recent study [@kang2017neurosurgeon] has examined a spectrum of possibilities in between the cloud-only and mobile-only extremes. Specifically, they considered splitting a deep network into two parts: the front end (consisting of an input layer and a number of subsequent layers), which runs on the mobile, and the back end (consisting of the remaining layers), which runs on the cloud. In this approach, termed *collaborative intelligence*, the front end computes features up to some layer in the network, then these features are uploaded to the cloud for the remainder of the computation. The authors examined the energy consumption and latency associated with performing computation in this way, for various split points in typical deep models. Their findings indicate that significant savings can be achieved in both energy and latency if the network is split appropriately. They also proposed an algorithm called *Neurosurgeon* to find the optimal split point, depending on whether energy or latency is to be minimized.
The reason why collaborative intelligence can be more efficient than cloud-only and mobile-only approaches is that the feature data volume in deep convolutional neural networks (CNNs) typically decreases as we move from the input to the output. Executing initial layers on the mobile will cost some energy and time, but if the network is split appropriately, we will end up with far less data to be uploaded to the cloud, which will save both transmission latency on the uplink and the energy used for radio transmission. Hence, on the balance, there may be a net benefit in energy and/or latency. Based on [@kang2017neurosurgeon], depending on the resources available (GPU or CPU on the mobile, speed and energy for wireless transmission, etc.), optimal split points for CNNs tend to be deep in the network.
A recently released study [@jointdnn] has extended the approach of [@kang2017neurosurgeon] to include model training and additional network architectures. While the network is again split between the mobile and the cloud, in the framework proposed in [@jointdnn] the data can move both ways between the mobile and the cloud in order to optimize efficiency of both training and inference.
While [@kang2017neurosurgeon; @jointdnn] have established the potential benefits of collaborative intelligence, the issue of efficient transfer of feature data between the mobile and the cloud is largely unexplored. Specifically, [@kang2017neurosurgeon] does not consider feature compression at all, while [@jointdnn] uses 8-bit quantization of feature data followed by lossless compression, but does not examine the impact of such processing on the application. Feature compression can further improve the efficiency of collaborative intelligence by minimizing the latency and energy of feature data transfer. The impact of compressing the input has been studied in several CNN applications [@dodge2016understanding; @quality_model_for_od_using_compressed_video; @hevc_for_object_detection] and the effects vary from case to case. However, the impact of feature compression has not been studied yet, to our knowledge.
In this work, we focus on a deep model for object detection and study the impact of feature compression on its accuracy. Section \[sec:prev\_work\] presents preliminaries, while Section \[sec:proposed\] describes the proposed methods. Experimental results and conclusions are presented in Sections \[sec:experiments\] and \[sec:conclusion\], respectively.
Preliminaries {#sec:prev_work}
=============
Object detection has been transformed in recent years with the advent of deep models that are able to simultaneously detect, localize, and classify objects in an image. Examples of such detectors include R-CNN [@RCNN_obj], SSD [@SSD], and YOLO [@YOLO1]. This work focuses on YOLO. One of the major innovations of these detectors was that they were trained using a cost function composed of both bounding box error and object class error terms. The YOLO loss function is [@YOLO1]: $$\begin{gathered}
%\begin{split}
%\begin{gathered}
\lambda_{coord} \sum_{i=0}^{S^2} \sum_{j=0}^B \mathbbm{1}_{ij}^{obj} \left[(x_i - \hat{x_i})^2 + (y_i - \hat{y_i})^2 \right] \\
+ \lambda_{coord} \sum_{i=0}^{S^2} \sum_{j=0}^B \mathbbm{1}_{ij}^{obj} \left[(\sqrt{w_i} - \sqrt{\hat{w_i}})^2 + (\sqrt{h_i} - \sqrt{\hat{h_i}})^2\right] \\
+\sum_{i=0}^{S^2} \sum_{j=0}^B \mathbbm{1}_{ij}^{obj} (C_i - \hat{C_i})^2 \\
+\lambda_{noobj} \sum_{i=0}^{S^2} \sum_{j=0}^B \mathbbm{1}_{ij}^{noobj} (C_i - \hat{C_i})^2 \\
+\sum_{i=0}^{S^2} \mathbbm{1}_{i}^{obj} \sum_{c \in classes} ({p_i}(c) - \hat{p_i}(c))^2
%\end{gathered}
%\end{split}
\label{eq:loss}\end{gathered}$$ where $(x_i,y_i)$ is the center of the ground truth bounding box, $w_i$ and $h_i$ are its width and height, $(\hat{x_i},\hat{y_i})$ is the center of the predicted bounding box whose width and height are $\hat{w_i}$ and $\hat{h_i}$, respectively. $C_i$ and $\hat{C_i}$ are the ground truth and predicted confidence scores corresponding to cell $i$, ${p_i}(c)$ and $\hat{p_i}(c)$ are the ground truth and predicted conditional probabilities for the object class $c$ in cell $i$, $\mathbbm{1}_{ij}^{obj}$ is equal to $1$ if the $j$-th bounding box in cell $i$ is responsible for prediction (i,e. box $j$ has the largest Intersection-over-Union among all boxes in cell $i$), and $\mathbbm{1}_{ij}^{noobj}=1-\mathbbm{1}_{ij}^{obj}$. The scaling factors used are $\lambda_{coord}=5$ and $\lambda_{noobj}=0.5$.
Our experiments in this work are based on the recent version of YOLO called YOLO9000 [@YOLO2]. Fig. \[fig:complexity\_and\_volume\] shows the feature data volume (number of feature samples) at the output of each layer of this model, as well as the cumulative computational cost (normalized execution time) as we move from the input layer towards the output. Computational cost was measured on a desktop machine with a Titan X GPU and Intel i7-6800K CPU over the images from a dataset described in Section \[sec:experiments\]. As seen in the figure, the feature data volume is fairly small starting with max-pooling layer max\_7. Hence, this layer, or other downstream layers seem to be good points to split the network. Note that max-pooling (and other pooling) layers reduce the data volume, so from the point of view of data size, it is always advantageous to split the network at the output of the max-pooling layer rather than at its input.
![Cumulative computation complexity and layer-wise output data volume[]{data-label="fig:complexity_and_volume"}](complexity_and_volume.png){width="\textwidth"}
![Three ways for mobile-cloud collaborative intelligence: (a) Lossless transfer, (b) quantization followed by lossless compression, and (c) lossy compression.[]{data-label="fig:proposed_network"}](neuro_surgeon.jpg){width="\textwidth"}
(a)
![Three ways for mobile-cloud collaborative intelligence: (a) Lossless transfer, (b) quantization followed by lossless compression, and (c) lossy compression.[]{data-label="fig:proposed_network"}](paritioned_network_with_lossless.jpg){width="\textwidth"}
(b)
![Three ways for mobile-cloud collaborative intelligence: (a) Lossless transfer, (b) quantization followed by lossless compression, and (c) lossy compression.[]{data-label="fig:proposed_network"}](paritioned_network_with_lossy.jpg){width="\textwidth"}
(c)
If we were to split the network at the output of some layer and transfer its feature data losslessly (as 32-bit floating point numbers) to the next layer (in the cloud), the accuracy would clearly stay the same as without the split[^1]. This is the approach taken in [@kang2017neurosurgeon], and is illustrated in Fig. \[fig:proposed\_network\](a). But this is inefficient because the data likely contains some redundancy.
A more efficient approach would be to compress the data prior to upload to the cloud. To achieve this, we could quantize the data, say to 8 bits per sample, then encode the quantized data losslessly. This is the approach taken in [@jointdnn] with a lossless PNG encoder. It is illustrated in Fig. \[fig:proposed\_network\](b), where the quantization layer is called the Q-layer. This approach is *near-lossless* because there is some quantization involved, and due to this quantization the accuracy of inference may be affected. An even more efficient approach to data transfer is to employ lossy compression after the Q-layer (Fig. \[fig:proposed\_network\](c)), but this will have an even greater impact on the accuracy. These issues are examined in Section \[sec:experiments\].
Proposed methods {#sec:proposed}
================
Quantization {#ssec:qlayer}
------------
In order to leverage existing codecs, the feature data is first quantized to 8-, 10-, or 12-bit precision in a Q-layer, which is inserted at the split point. Let $\mathbf{V} \in \mathbb{R}^{N \times M \times C}$ be the tensor containing the feature data at the point of split, with $N$ rows, $M$ columns, and $C$ channels. Let $\textup{min}(\mathbf{V})$ and $\textup{max}(\mathbf{V})$ be the minimum and maximum value in $\mathbf{V}$, respectively. Quantization with $n_{bit}$-precision and the corresponding inverse quantization in the inverse Q-layer are performed as $$\widetilde{\mathbf{V}} = \textup{round} \left (\frac{\mathbf{V}-\textup{min}(\mathbf{V})}{\textup{max}(\mathbf{V})-\textup{min}(\mathbf{V})}\cdot (2^{n_{bit}}-1)\right )
\label{eq:quantized_vector}$$
$$\widehat{\mathbf{V}} = \frac{\widetilde{\mathbf{V}}\cdot (\textup{max}(\mathbf{V})-\textup{min}(\mathbf{V}))}{2^{n_{bit}}-1}+\textup{min}(\mathbf{V})
\label{eq:dequantized_vector}$$
where $\widetilde{\mathbf{V}}$ is the quantized feature tensor, $\widehat{\mathbf{V}}$ is the de-quantized feature tensor, and $\textup{round}(\cdot)$ represents rounding to the nearest integer. $\textup{min}(\mathbf{V})$ and $\textup{max}(\mathbf{V})$ need to be stored as 32-bit floats (8 Bytes total) and transferred to the cloud for de-quantization. This is taken into account when computing total bits in the experiments. Note that in some cases, such as when the previous activation layer is sigmoid or ReLU (assumed in [@jointdnn]), we can consider $\textup{min}(\mathbf{V})=0$ and avoid transmitting it, but for more general activation layers such as Leaky ReLU (which is used in YOLO9000) this parameter is required.
Compression {#ssec:tiling}
-----------
![Combining feature channels into an image by (a) tiling and (b) quilting.[]{data-label="fig:tiled_map"}](tiling_example.jpg){width="\textwidth"}
(a)
\[fig:tiling\]
![Combining feature channels into an image by (a) tiling and (b) quilting.[]{data-label="fig:tiled_map"}](quilting_example.jpg){width="\textwidth"}
(b)
\[fig:quilting\]
Quantized feature tensor $\widetilde{\mathbf{V}}$ can be encoded by a number of existing codecs. If we interpret the $N \times M \times C$ tensor as $C$ frames of size $N\times M$, we could employ a video codec to compress it. We could combine groups of feature channels into larger frames, to end up with less than $C$ frames with larger resolution. Finally, all channels could be combined into a single image. Even in this case there are a number of possibilities, such as tiling (Fig. \[fig:tiled\_map\](a)), where the entire channel is placed in the image as a tile, followed by another tile, and so on, and quilting (Fig. \[fig:tiled\_map\](b)), where neighboring samples come from different channels. We tested a number of such methods and found that simple tiling by channel index provided the best results, so we use that method from here on. Hence, tiled feature channels are compressed as a still image.
For compression, we employ high efficiency video coding (HEVC) [@hevc] standard, specifically HEVC Range extension (RExt) [@flynn2016overview] which supports 4:0:0 sample format with various bit-depths. HM16.12 [@HM16.12] in the experiments and all coding tools and configuration follow common test condition [@hevc_ctc]. RDOQ tool is turned off and the coding tree unit (CTU) size is set to 16$\times$16, because the feature channel resolution is relatively small deep in the network.
Compression-augmented training {#ssec:training}
------------------------------
As will be seen in Section \[sec:experiments\], Q-layer quantization followed by lossless compression has little effect on the accuracy. However, lossy compression may affect the accuracy, especially when the quantization parameter (QP) is high. This loss in accuracy can be somewhat compensated by compression-augmented training. Instead of using the network parameters (weights) supplied with the model, we re-train the model by considering lossy compression at the point of split. During training, at each forward pass through the network, the feature data at the split point is tiled and compressed using a randomly chosen QP value. In our experiments we used QP in the range \[Lossless, 22, 27, 32, 37\]. After compression, the decompressed data is passed further down the network.
This kind of compression augmentation can be interpreted as a form of regularization, where quantization noise is inserted into an intermediate layer deep in the network. It encourages the network to learn the downstream weights (from the split point) that provide good accuracy when processing decompressed features, and also to learn upstream weights that generate features that are robust to compression.
Experiments {#sec:experiments}
===========
Following [@YOLO2], a total of 16,551 images from VOC2007 and VOC2012 datasets [@pascal-voc-2007; @pascal-voc-2012] are used for training and another 4,952 images from VOC2007 for testing. Twenty different object classes are represented in the dataset.
![mAP vs. KBPI for lossless deep feature compression[]{data-label="fig:lossless_comp"}](lossless_comparison_rev1.png){width="\textwidth"}
We first test the impact of lossless compression (after the Q-layer) on accuracy. As is common with multi-class object detectors [@VOC], we use mean Average Precision (mAP) as a measure of accuracy, and look at its variation with 8-bit, 10-bit and 12-bit quantization in the Q-layer. The compression of feature data is quantified using average Kbits per image (KBPI). Fig. \[fig:lossless\_comp\] presents mAP versus KBPI for various split points in the network. Vertical bars show the standard deviation of mAP at a given average KBPI, while horizontal bars show the standard deviation of KBPI for the corresponding average mAP. The red square indicates the operating point achieved by the cloud-only approach, without network splitting and uploading the input JPEG images to the cloud.
As seen in the figure, when the split point is close to the input (e.g. max\_3, conv\_6 or conv\_10 layers), the data volume is too large, and even with lossless compression of feature data, it is more efficient to simply upload input images to the cloud. But as we move down the network, it becomes more advantageous to upload feature data. Meanwhile, the mAP does not change much - scores around 0.7465-0.7475 are achieved for all the cases. Hence, lossless compression of deep features (following 8-, 10-, or 12-bit quantization) has only a minor influence on accuracy, but also provides limited (if any) bit savings for data transfer to the cloud.
![mAP vs. KBPI for lossy deep feature compression[]{data-label="fig:rate_map_curve"}](rate_mAP_curve_rev1.png){width="\textwidth"}
Lossy compression offers significant bit savings, but care must be taken to minimize the loss of accuracy. In order to evaluate the impact of lossy compression, we show mAP vs. KBPI curves in Fig. \[fig:rate\_map\_curve\]. The green curve corresponds to compressing the input image, as the default cloud-only approach. The blue curves correspond to splitting the network at the output of max\_11 layer, and red curves correspond to the split after the max\_17 layer. In each case, the solid line corresponds to using default YOLO9000 weights while the dashed line corresponds to using the weights obtained by compression-augmented training, starting from the pre-trained weight, “Darknet19 448x448”, for ImageNet classification [@ILSVRC15] and following the training procedure in [@darknet]. As seen in the figure, lossy compression can provide significant bit savings over the cloud-only approach, while compression-augmented training further extends the range of useful compression levels for a given mAP.
To quantify the differences between various cases, we adopt a Bjontegaard Delta (BD) approach [@Bjontegaard]. Specifically, we use the BD calculation to compute BD-KBPI-mAP, which indicates the average difference in KBPI at the same mAP. The results are shown in Table \[tbl:performance\], where the default case against which the comparison is made is the cloud-only approach. As shown in the table, compressing features at the output of max\_11 (max\_17) while using default weights would give an average saving of 6% (60%) at the same mAP compared to cloud-only approach. Meanwhile, the weights obtained through compression-augmented training would provide an additional bit saving of 39% (10%), for the total of up to 45% (70%) bit savings.
Conclusions {#sec:conclusion}
===========
We studied deep feature compression for collaborative object detection between the mobile and the cloud. We examined the impact of compression on detection accuracy and showed that lossless compression of 8-bit (or higher) quantized data does not have much impact on the accuracy. Lossy compression provides higher bit savings, but also affects the accuracy. To compensate for this, we proposed compression-augmented training, which is able to extend the range of useful compression levels for a desired accuracy.
[^1]: If the data is not corrupted during transmission, as assumed in [@kang2017neurosurgeon; @jointdnn].
|
---
abstract: 'With the use of a background Milky-Way-like potential model, we performed stellar orbital and magnetohydrodynamic (MHD) simulations. As a first experiment, we studied the gaseous response to a bisymmetric spiral arm potential: the widely employed cosine potential model and a self-gravitating tridimensional density distribution based model called PERLAS. Important differences are noticeable in these simulations, while the simplified cosine potential produces two spiral arms for all cases, the more realistic density based model produces a response of four spiral arms on the gaseous disk, except for weak arms -i.e. close to the linear regime- where a two-armed structure is formed. In order to compare the stellar and gas response to the spiral arms, we have also included a detailed periodic orbit study and explored different structural parameters within observational uncertainties. The four armed response has been explained as the result of ultra harmonic resonances, or as shocks with the massive bisymmetric spiral structure, among other. From the results of this work, and comparing the stellar and gaseous responses, we tracked down an alternative explanation to the formation of branches, based only on the orbital response to a self-gravitating spiral arms model. The presence of features such as branches, might be an indication of transiency of the arms.'
author:
- |
Angeles Pérez-Villegas$^{1,2}$ [^1], Gilberto C. Gómez$^{1}$ [^2] and Bárbara Pichardo$^3$ [^3]\
$^{1}$Centro de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Apdo. postal 3-72, Morelia Mich. 58089, México\
$^{2}$ Max-Planck-Institüt für Extraterrestrische Physik, Gießenbachstraße, D-85741 Garching, Germany\
$^{3}$Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo. postal 70-264 Ciudad Universitaria, D.F. 04510, México
date: 'Accepted . Received ; in original form '
title: The Galactic Branches as a Possible Evidence for Transient Spiral Arms
---
\[firstpage\]
MHD – Galaxy: disc – Galaxy: strcuture – Galaxy: kinematics and dynamics – galaxies: spiral – galaxies: structure
Introduction {#sec:Intro}
============
Spiral arms are one of the the most striking, beautiful and scientifically challenging structures of disc galaxies. They have fascinated and intrigued astronomers for centuries. As a rough initial approximation to treat this complex problem, the spiral arms had to be assumed as almost massless and/or with extremely modest pitch angles, in order to obtain a solution as a linear perturbation to the axisymmetric background potential. The proposed solution to the permanence of spiral arms in galaxies was based on the spiral density wave linear theory (@LS64, inspired in the 60’s work of B. Lindblad and P.O. Lindblad). The analytical solution at first order of the theory, known as the tight-winding approximation (TWA), that represents a weak potential model, extremely idealized as a smooth, even negligible perturbation to the background potential, was modeled as a simplified cosine function to represent the spiral arms gravitational potential. Probably followed by this initial attempt of solving the nature of spiral arms, a lot of work has tackled repeatedly the problem of modeling them, as a simplified periodic function that disregards the importance of their dynamical effects. However, spiral arms have proven to be an influential feature on galactic modern models (such as PERLAS and N-body models, based on three dimensional density structures), far beyond of a simple smooth perturbation.
If we compare the spiral arms with the galactic bar, the latter has a mass between 10 and 20% of the disc mass [@Matsumoto_et82; @Dwek95; @Zhao96; @WeSell99], unlike the spiral arms that have a mass smaller in general than 5% of the disc mass [@PMME03]. Consequently, the largest influence on the disc of strongly barred galaxies is, of course, due to the central bar. Therefore one would be tempted to oversimplify the scenario, by assuming that spiral arms are not influential at all on the dynamics of the disc. However, there are determinations that suggest that the majority of spiral galaxies, or at least those with clearly delineated spiral arms, are rather far from being linear [@Antoja11; @Antoja10; @Antoja09; @VSK06; @Roca_et13; @Roca_et14; @PMME03; @Kawata_et14; @SellCal14; @Sell11].
Based on all those rigorous studies, spiral arms seem to merit an extra effort to model them beyond a simple perturbing term. This fact is even more significant in the case of the interstellar gas, which is considerably more sensitive to the details of the potential, and in general, responds strongly even if the mass of the spiral arms is only a small fraction of the axisymmetric background [@KKK14; @Gomez_et13; @ShuMR73].
**For almost a century, formal studies of spiral arms have been carried on. Yet, the spiral arms morphology, origin and nature are still poorly understood. With respect to their morphology, there is plentiful theoretical and observational literature on structures related to spiral arms, such as spurs, branches, feathers and beads, and their plausible explanations [@W70; @L70; @ShuMR73; @S76; @ED80; @SP01; @LeeShu12; @Kim_Ost06; @Chakrabarti_03; @Kim_Ost02; @Dobbs_et11; @SO06; @CS08; @B88]. Regarding spurs and feathering, interesting scenarios to explain these features include plain hydrodynamic simulations, where the Kelvin-Helmholtz instability creates spurs produced by shocks with the spiral arms (@WK04; an extension to self-gravitating three dimensional models of this work is presented in @Kim_Ost06). Other scenarios include the use of global MHD simulations, adding gas self-gravity and magnetic fields that produce differential compression of gas flowing through the arms resulting in the formation of sheared structures in the interarm regions that resemble spurs and feathers [@SO06; @Kim_Ost02], or through gravitational or magneto-Jeans instabilities [@B88].**
On the other hand, features such as branches are significantly longer than spurs that emerge almost parallel to the main spiral arms but with different pitch angles (usually smaller), and are generally associated with resonances. It is worth to note that, in the literature, the names spurs, feathers and branches are assigned to slightly different structures. In this work we will adopt the definition given by @Feitz_Sch82 and @Chakrabarti_03, where the “spurs” and “feathers” are small structures that arise almost perpendicularly from the main spiral arm, and “branches” are narrow structures that arise almost parallel to the main spiral arm, with smaller pitch angles and extension longer than spurs (examples of branches can be seen in galaxies like NGC 309, NGC 1637, NGC 2997, NGC 6946, among others). Regarding such structures, theory and gas simulations have already shown a bifurcation of spiral arms produced by ultraharmonic resonances induced by the main spiral arm on the background gas flow [@ShuMR73; @PH94; @PH97; @Chakrabarti_03; @AL92]. In this case, the bifurcation of the arms is expected due to the topology of the stable periodic orbits at the 4/1 resonance. Other interpretations also include the possibility that branches are the response of the gas produced by strong shocks against the main massive spiral arms, i.e. the ones made mainly of older and smaller stars [ **[@Fujimoto68; @Roberts69; @ShuMR73; @BW67; @RHV79; @Martos_et04; @KimKim14; @Chakrabarti_03; @Yanez_et_2008].**]{}
Specifically, regarding the Milky Way Galaxy, its general structure has been extensively studied and the latest determinations seem to agree with two grand design symmetric spiral arms, seen on IR and optical, and some other weaker arms, detectable mainly in optical observations [@Drimmel_Spergel01; @Benjamin_etal05; @Drimmel00; @Drimmel_Spergel01; @Vallee13; @Vallee02]. The weaker arms ([**branches**]{}) have been explained through hydrodynamic simulations as the response to the two-armed stellar pattern. For example, @EngGer99 and @Fux99 found that the gas response to a barred potential can produce a four-armed spiral structure. Likewise, from hydrodynamic and magnetohydrodynamic (MHD) simulations, the gas responds to an imposed two-armed spiral potential with four spiral arms [@Gomez_et13; @Martos_et04; @ShuMR73]. In this context, the gas component shocks at the position of the spiral arms, producing a density enhancement and star formation [@Fujimoto68; @Roberts69; @Moore_et12; @Seo_Kim14] that, depending on the relative velocity between the gas and the spiral arm, dwell upstream or downstream or in the spiral arm. In particular, @Roberts69 showed that the nonlinear response of the gas to a stationary stellar spiral arm potential may produce two shocks, which then would be associated to a doubling of the spiral arm number as seen in the gas component. @ShuMR73 demonstrated that shocks in galaxies arise necessarily if the spiral arm strength exceeds a certain critical value. Additionally, they found a range of values for the wave frequency that generates an ultraharmonic resonance which can provoke a secondary compression of the interstellar gas. This effect has been related to the origin of the Carina arm in the Milky Way [@ShuMR73], for example. However, since branches have been traditionally explained as shocks induced by the spiral arms, it is important to mention other observational work that shows there is little difference (or non at all) in the star formation efficiency (i.e. no shocks) between the spiral arms regions and the rest of the disk [@FRWL10; @Eden_et13; @Foyle_et11; @Dobbs_et11].
Finally, regarding their nature, a problem of great interest is whether spiral arms are a transient or a long-lived feature. On this matter, recent numerical simulations show that spiral arms are transient and recurrent structures [@Dobbs_Bonnell06; @Wada11; @Roskar12; @Donghia13; @Perez-Villegas_et13; @Perez-Villegas_et12; @Sell11; @SellCal14; @Kawata_et14; @Foyle_et11]. However, in a study by @Scarano_Lepine13 on a sample of 27 galaxies, the authors concluded that the break found in the radial metallicity distribution near to corotation resonance (CR), implies that the spiral structure is a rather long lived feature. Other authors, employing different techniques, seem to find observational proofs of long-lasting spiral arms [@Martinez-Garcia13; @Donn_Thom94; @Zhang98].
In this work we explore the gas response to a bisymmetric spiral arm potential. For this purpose we employed two different models, a cosine potential (generally used in literature) and the three-dimensional, density distribution based potential PERLAS, applied to the particular case of a Milky Way-like galaxy. With this study we find an alternative explanation to the formation of branches in disk galaxies, and their relation to transient or long-lived spiral arms.
This paper is organized as follows. In Section \[sec:method\], the galactic potential used to compute the stellar orbits and the MHD initial simulation set-up are described. In Section \[results\], we present first a comparison between the models of the spiral arms: PERLAS and the cosine potential; second, we present the gas response to spiral arms PERLAS model changing the structural parameters such as the pitch angle and the mass of the spiral arms, and their connection with the presence of galactic branches as a signature of transient spiral arms. Finally, we present a discussion and our conclusions in Section \[sec:conclusions\].
Methodology and Numerical Implementation {#sec:method}
========================================
Motivated by the fact that the gas in galaxies is dynamically colder than the stellar disk, in addition to being collisional, we can expect it to be much more sensitive to details of the given potential than stars. Therefore, any differences between the cosine and PERLAS potentials might be magnified in gas with respect to the stellar response. [**And so,**]{} for purposes of comparison, we produce the whole study employing both potentials. In this section we introduce briefly the cosine and PERLAS potential models and the MHD setup applied to a Milky Way like Galaxy.
In all cases, the spiral arm models are superimposed to the axisymmetric background potential of @AS91, which includes a Miyamoto-Nagai bulge and disk, and a supermassive spherical halo. Table \[tab:param\] presents the basic parameters of the axisymmetric background potential.
Parameter Value Reference
------------------------------ ---------------------------------- --------------------
[*Axisymmetric Components*]{}
$R_0$ 8.5 kpc 1
$\Theta_0$ 220 km s$^{-1}$ 1
Bulge mass $1.41\times10^{10}$ M$_\odot$ 1
Disk mass $8.56\times10^{10}$ M$_\odot$ 1
Halo mass $8.002\times10^{11}$ M$_\odot$ 1
[*Spiral Arms*]{}
Locus Logarithmic 2, 3
Arms number 2 4
Pitch angle$(i)$ $15.5\pm3.5\deg$ 4
$M_{arms}/M_{disk}$ $0.03\pm0.02$ 3
Mass $2.7-5.4 \times10^{9}$ M$_\odot$ 3
Inner limit $3.3\kpc$ ILR position based
Outer limit 12 CR position based
Scale-length $2.5\kpc$ Disk based
Pattern speed ($\Omega_{P}$) $-20 \kmskpc$ 5
\[tab:param\]\
References —(1) @AS91; (2) @GP98; (3) @PMME03; (4)@Drimmel00; (5) @Martos_et04.
Spiral arm models {#sec:sp_models}
-----------------
### Cosine Potential {#cosine}
As mentioned before, a large majority of the investigations on the gas associated to spiral arms model them as a linear perturbation of the axisymmetric background, represented by
$$\label{eq:cosine}
\Phi_{sp}(R,\phi)=f(R)\,\cos\left[2\phi+g(R)\right],$$
where $R,\phi$ are cylindrical coordinates, $f(R)$ is the amplitude function of the perturbation, given by @CG86 as $f(R)=-ARe^{\epsilon_s R}$, where $A$ is the amplitude and $\epsilon_s$ is the inverse of the scale-length. Finally, $g(R)$ describes the geometry of the spiral pattern ([*locus*]{}), given by @RHV79, as
$$\label{eq:locus}
g(R)=-\frac{2}{N\tan i_p}\ln[1+(R/R_s)^N],$$
where $i_p$ is the pitch angle, $R_s$ is the start position for the spiral arms, and $N$ is a constant that shapes the starting point of the spiral arms, [**so that**]{} $N\rightarrow 0$ represents a $180\deg$ tip transition from the bar to the spiral arms, while $N\rightarrow\infty$ means a $90\deg$ tip transition from the bar to the spiral arms. In the present work, we set it equal to 100 (for further details, see @PMME03).
### PERLAS model {#perlas}
In contrast, the PERLAS spiral arms model [@PMME03] is a bisymmetric, three-dimensional, self-gravitating stationary model based on an adjustable mass distribution, rather than a local arm approximation, as the cosine potential. Several studies have shown that there are differences in the stellar orbital dynamics when the spiral arms are modeled with the cosine potential or with the PERLAS model [@PMME03; @Antoja09; @Antoja11]. Furthermore, chaotic orbital studies have demonstrated that a more detailed spiral arm potential (the PERLAS model) induces an important fraction of chaos, enough to destroy the spiral structure [@Perez-Villegas_et13; @Perez-Villegas_et12], when all chaotic behavior was originally attributed to effects produced by the bar, such as overlapping of resonances [@Contopoulos_H12; @Contopoulos67].
PERLAS is constructed by a superposition of individual oblate inhomogeneous spheroids along a given locus (the same than in the cosine potential, equation \[eq:locus\]) and superimposed to the axisymmetric background. In this model, the spiral arms have a well defined mass, unlike the cosine potential, where the spiral arms are treated as a periodic function of the potential not straightforward translatable to mass. The mass assigned to build the spiral arms in the PERLAS model is subtracted from the disk mass. Thus, the inclusion of PERLAS to the Galactic model does not modify the total mass of the original axisymmetric background. In Table \[tab:param\] we present the parameters of spiral arms that we used in our simulations. For further details about the PERLAS model, we refer the reader to @PMME03.
Force fitting {#sec:fit}
-------------
In order to compare PERLAS with the cosine potential, we need to fit the spiral arm strength to make them as similar as possible. We achieve this by adjusting the amplitude of the cosine potential (the factor $A$ in eq. \[eq:cosine\]) so that the resulting radial and azimuthal forces approximate the ones obtained using the PERLAS model.
Figures \[fig:frad\] and \[fig:fazi\] show the resulting fit for the case of a Milky Way-like galaxy, where the pitch angle is $15.5\deg$ and spiral arm mass (in PERLAS) is $3\%$ of the disk mass. In this case, the cosine amplitude $A$ is $650\km^2\sec^{-2}\kpc^{-1}$. In Figure \[fig:frad\], each panel represents different radial lines, starting in $0\deg$ up to $75\deg$. In the Figure \[fig:fazi\], each panel shows the computed azimuthal force at different radii, from $4$ to $10\kpc$. Continuous lines represent PERLAS, dotted lines represent our fit to the cosine potential.
{width="\textwidth"}
{width="\textwidth"}
Spiral arm strength {#sec:strength}
-------------------
The spiral strength is related to the pitch angle and the mass of the arm. In the PERLAS model, the spiral arm mass is a small fraction of the disk mass. To quantify the strength of the spiral arms, we calculated the $Q_T(R)$ parameter [@SaTu80; @ComSan81], frequently used to quantify the strength of bars and spiral arms [@BuBl01; @LS02]. The value of $Q_T(R)$ is given by
$$Q_T(R)=\frac{F_T^{\rm max}(R)}
{|\langle F_R(R) \rangle|},
\label{eq:Q_T}$$
where $F^{\rm max}_T
=|\left(\partial\Phi(R,\theta)/\partial\theta\right)/R|_{\rm max}$ represents the maximum amplitude of the tangential force at radius $R$, and $\langle F_R(R)\rangle$, is the average axisymmetric radial force. Figure \[fig:QT\] shows $Q_T(R)$ for the PERLAS (solid line) and cosine (dotted line) potential models, for the case where the spiral arms mass of the PERLAS model has a 3% of the disk mass.
![$Q_T(R)$ parameter for the spiral arms of a Milky Way-like galaxy for the PERLAS ([*solid line*]{}) and cosine ([*dotted line*]{}) potential models.[]{data-label="fig:QT"}](Q_T_3.eps){width="50.00000%"}
As opposed to Figures \[fig:frad\] and \[fig:fazi\], where the force amplitude is almost the same, Figure \[fig:QT\] shows that there is a difference between both potentials. While the maximum value of $Q_T$ for PERLAS is $\sim 0.096$, for cosine is only $\sim
0.035$. Therefore, if the arm strength is measured using equation (\[eq:Q\_T\]) instead of the arm force, it is necessary to increase the cosine model amplitude. Figure \[fig:QT\_am20\] shows the $Q_T$ parameter corresponding to [**an**]{} increased cosine amplitude, in this case, $A=2000\km^2\sec^{-2}\kpc^{-1}$. In this case, since $Q_T$ for the cosine model is larger than PERLAS along all the radial range, this case should be considered as an example to test the gaseous disk response to an extreme cosine model, in order to verify if at this forcing the cosine can reproduce what PERLAS does. In this way we are bracketing the cosine potential within the values of the force for the PERLAS model.
![Similar to fig. \[fig:QT\], with the amplitude of the cosine model increased. []{data-label="fig:QT_am20"}](Q_T_m3_amp20.eps){width="50.00000%"}
The MHD set-up
--------------
The initial set-up of the simulations consisted on a gaseous disk, with density profile given by $n(r) = n_0 \rm{exp}
[-(r - r_0)/r_d]$, where $n_0 = 1.1\pcc$, $r_0 = 8\kpc$ and $r_d = 15\kpc$. The gas follows an isothermal equation of state with temperature $T = 8000\degK$. Additionally, the gas is permeated by a magnetic field, initially in the azimuthal direction, with an intensity given by $B(r) = B_0 \exp [-(r - r_0)/r_B]$, where $B_0 = 5 \muG$ and $r_B = 25\kpc$. The disk is set up in rotational equilibrium between the centrifugal force, the thermal and magnetic pressures, magnetic tension and the background axisymmetric potential [@AS91]. This equilibrium is perturbed by the spiral arm potentials under study, both rotating with a pattern speed $\Omega_p= 20\kmskpc$.
We employed the [zeus]{} code [@SN92a; @SN92b] to solve the MHD equations, which is a finite difference, time explicit, operator split, Eulerian code for ideal MHD. We used a 2D grid in cylindrical geometry, with $R \in [1.5, 22]\kpc$ and a full circle in the azimuthal coordinate, $\phi$, using $750 \times 1500$ grid points. Both boundary conditions in the radial direction were outflowing. All calculations are performed in the reference frame of the spiral arms. No self-gravity of the gas was considered.
Results
=======
We present in this section a comparison between the gas response to the simplified cosine potential for the spiral arms, and the density distribution based model PERLAS. In the limit for weak and/or small pitch angles (approximately linear regime), both models behave very similar as expected, however, for stronger arms (more massive or larger pitch angles), from this comparison we find severe differences in the gas behavior and formation of spiral arms and branch-like structures. The deviation of the response between both models is induced by the basic differences of these potentials. We then present an interpretation on the presence of galactic branches prior to the ultraharmonic 4:1 resonance and connect them to a possible signature of transient spiral arms.
Gas Response Comparison: PERLAS [*vs.*]{} cosine Potential Models {#sec:comparison}
-----------------------------------------------------------------
In order to compare both potentials, we fit the cosine potential with PERLAS to make it as similar as possible in the force amplitude (§\[sec:fit\]). In all cases, the axisymmetric gaseous disk, initially in rotational equilibrium, was perturbed by the spiral potential either PERLAS or the cosine.
As mentioned already, the cosine potential represents a simple solution from the density wave linear theory, self-consistent for tightly wound spiral arms (TWA, i.e. where the perturbation is very small, which means small pitch angles or with very reduced masses). The Milky Way Galaxy and the most of spiral galaxies are actually far from this regime. Thus, it should not be surprising that the gaseous disk [@Gomez_et13] and the stellar orbits [@Perez-Villegas_et12] show a different structure when subjected to a self-gravitating, more realistic model instead of a local approximation. With this in mind, the gaseous disk response to both models should be similar if we focus on a region of the parameter space where both potentials are valid, i.e., if we set PERLAS and the cosine potential so that it is approximately in the linear regime, with a very small pitch angle and spiral arm mass. In Figure \[fig:gas\_cos\_perlas\] we show the linear regime for both potentials. The density distribution with the cosine potential is presented in the left panel and PERLAS model in the right panel, the pitch angle is $6\deg$, the spiral arms mass is 1% of the disk mass, $A=100$ km$^2$ s$^{-2}$ kpc$^{-1}$. Indeed, the gas response to the potentials is similar as expected, forming two spiral arms in both cases.
{width="40.00000%"} {width="40.00000%"}
For the specific case of a Milky Way-like galaxy, we constructed a model that reproduces some of the observational parameters for the background and spiral arms potential to compare them with the cosine potential for the spiral arms (force fitted with PERLAS). The spiral arms pattern angular speed is $20 \kmskpc$, on a logarithmic locus with a pitch angle of $15.5 \deg$. The mass of spiral arm for PERLAS model is 3% of the disk mass, which is equivalent to a cosine amplitude of $A=650 \km^2 \sec^{-2} \kpc^{-1}$, when the non-axisymmetric force is employed for the fitting (see §\[sec:fit\]), and $A=2000\km^2\sec^{-2}\kpc^{-1}$, when the arm strength (i.e. the $Q_T$ parameter) is considered for the fitting instead (see §\[sec:strength\]). We [**follow the evolution of the system**]{} for $5\Gyr$.
Figure \[fig:gas\_coseno\] shows the resulting density distribution of the gaseous disk when is perturbed by the cosine potential with a small amplitude ($A=650 \km^2 \sec^{-2} \kpc^{-1}$). After the simulation starts, the gas very rapidly settles into a spiral pattern, forming two spiral arms at $30 \Myr$ (top left panel). $300 \Myr$ into the simulation (top right panel), the two spiral arms are better defined, and the gas is forming a secondary structure. As simulation progresses ($1.5$ and $3 \Gyr$, bottom panels), the gas continues responding to the spiral arms and even more substructure forms, but the large scale density response consists of two arms only. The pitch angle of the formed arms is $\sim 15.5\deg$, equal to the imposed potential at the beginning of the simulation. In the last panel, a gas instability at the corotation radius is seen, as reported previously [@Gomez_et13; @Martos_et04].
{width="40.00000%"} {width="40.00000%"}\
{width="40.00000%"} {width="40.00000%"}
{width="40.00000%"} {width="40.00000%"}\
{width="40.00000%"} {width="40.00000%"}
In contrast, Figure \[fig:gas\_perlas\] shows the density distribution of the gaseous disk when is perturbed by the PERLAS spiral arm potential. The gas very rapidly settles into a spiral pattern, as in the cosine potential case, forming two spiral arms at 30 Myr. At $300 \Myr$ and later, the gas forms four spiral arms instead of the two arms in the cosine potential simulation. Even though the simulation develops four spiral arms, these are associated in two [**pairs, each composed of**]{} a strong arm and a weak arm. The strong arms have a pitch angle of $\sim 15\deg$, and the weak arms have a pitch angle of $\sim 7\deg$. This doubling of the spiral arms in the gas response has been seen in other [**MHD**]{} simulations using the PERLAS model (@Gomez_et13 [@Martos_et04]; see also §\[sec:response\]).
{width="40.00000%"} {width="40.00000%"}\
{width="40.00000%"} {width="40.00000%"}
In §\[sec:strength\], we noted that the spiral arm strength, as measured by eq. (\[eq:Q\_T\]), was very different for both potentials, even if the force amplitudes were almost the same. Therefore, we decided to perform a second experiment with a larger amplitude $A=2000 \km^2 \sec^{-2} \kpc^{-1}$ for the cosine potential, that would be equivalent to a factor of $\sim 3$ times the force of PERLAS. This much larger amplitude for the cosine potential is stronger than PERLAS at any radii, as shown in Figure \[fig:QT\_am20\]. The purpose of this overestimated experiment was to test if the four spiral arms formed with PERLAS were due to the strength of spiral arms only. Figure \[fig:gas\_cos\_aum\] shows the gas density distribution resulting from perturbing the disk with the larger force amplitude of the cosine potential. It is readily seen that the effect of the spiral potential on the gas disk is larger and generates more substructure, but the gas still responds forming two spiral arms, unlike the density based potential PERLAS that forms four arms for larger pitch angles, i.e. for stronger arms. Consequently, this difference does not come from the force amplitude, but it seems rather originated from the self-gravitating nature and specific details of the potential that a local approximation for the forces given by the cosine potential is unable to reproduce. We further explain this in the next section.
Branch Formation and their Relation with the Transient Nature of Spiral Arms {#sec:response}
----------------------------------------------------------------------------
In §\[sec:comparison\] we showed that the intrinsic differences in the nature of the cosine and PERLAS potentials induce a distinct gaseous arm structure as compared to a disk perturbed by a cosine arm potential. While employing the spiral arm cosine potential, a bisymmetric gaseous structure seems to be an invariable outcome, for the PERLAS potential a two or four armed structure could be obtained. This result points to the necessity of understanding how the gas responds to the PERLAS model when we vary the structural parameters of the spiral arms, such as its mass and pitch angle, considering the uncertainties in the determination of these parameters for the Milky Way’s spiral arms. Therefore, we developed a set of simulations varying spiral arm pitch angles ($i$) and spiral arms masses. We present here some of them ranging from $12\deg$ to $19\deg$, and the mass of the spiral arms ($M_{sp}$), from 1.75% to 5% of the total disk mass ($M_D$).
Figure \[fig:periodic\_gas\] shows a mosaic of simulations. The panels in the left column are the stable periodic stellar orbits and the panels in the right column are the gas density distribution after $1.2\Gyr$. [**The maximum density response (where the orbits crowd producing a density enhancement) quantifies the orbital support to a given spiral perturbation through periodic orbits. We computed the stellar periodic orbits in order to explore the orbital support to the imposed PERLAS spiral arm potential (open squares in the left column-panels of Figure \[fig:periodic\_gas\]). To estimate the density response, we employ the method of @CG86. This method assumes that stars in circular orbits in an axisymmetric potential, with the same sense of rotation of the spiral arms, are trapped around the corresponding periodic orbit in the presence of the spiral arms. For this purpose, we calculated a set of central periodic orbits (between 50 and 60) and found the density response along them using the conservation of the mass flux between any two successive periodic orbits. With this information we seek the position of the maximum density response along each periodic orbit (filled squares in the left column-panels of fig. \[fig:periodic\_gas\]). These positions are compared with the imposed spiral arms (PERLAS model). The method implicitly considers a small dispersion (with respect to the central periodic orbit) since it studies a region where the flux is conserved. On the other hand, this dispersion is based on parameters for the galaxies where dynamics is quite ordered, orbits follow their periodic orbit closely, in such a way that we consider this study a good approximation. This method to estimate the density response has been widely used in literature [@ConG88; @AL97; @Yano03; @PMME03; @VSK06; @TEV08; @Perez-Villegas_et12; @Perez-Villegas_et13; @JunLep13]. We refer the reader to the work of @CG86 for more details.**]{} [**With this in mind, a model in Figure \[fig:periodic\_gas\]**]{} where the open and filled squares coincide would represent a stable, approximately orbitally self-consistent system, while a lack of coincidence would mean that the spiral is unlikely [**to be**]{} long-lasting [@Perez-Villegas_et12; @Perez-Villegas_et13].
For $M_{sp}=0.0175M_D$ and $i=12\deg$ (upper left panel of Figure \[fig:periodic\_gas\]), the stellar density response follows approximately the imposed spiral potential prior to the 4/1 resonance. After that, the stellar response forms a slightly smaller pitch angle than the imposed spiral. In the MHD simulation, the gaseous disk responds to the two-arm potential with the now familiar four spiral armed structure, where the pitch angle of the stronger pair of arms corresponds to that of the imposed pitch angle potential, while the other pair of gaseous arms has a systematically smaller pitch angle, corresponding closely to the regions of periodic orbits crowding. Now, with the same spiral arm mass but a pitch angle of $i=19\deg$ (second row of fig. \[fig:periodic\_gas\]), the stellar density response (i.e. periodic orbits) forms again a smaller pitch angle compared with the imposed, while in the gas, the four spiral arms seem stronger and the difference in the pitch angle between the gaseous and imposed arms is larger than the previous case. For $M_{sp}=0.05M_D$ and $i=12\deg$ (third row), the stellar density response closely follows the imposed spiral arm potential prior to the 4/1 resonance. After that, the stellar response forms a slightly smaller pitch angle than the imposed spiral, while the gas responds with four spiral arms, but the second pair is very weak but with a significantly smaller pitch angle than the imposed spiral arm potential. Finally, with the same spiral arm mass (5% of the disk mass) but a pitch angle of $i=19\deg$ (fourth row), we found no periodic orbits beyond 4/1 resonance and the stellar density response forms a pitch angle smaller than $19\deg$. The gaseous disk responds with well defined four spiral arms that extend up to the corotation radius. Notice that in this last simulation, there is not much that can be said about the (stellar or gaseous) orbital support since periodic orbits tend to disappear due to the strong forcing of the imposed spiral arms, meaning that spiral arms would rather be transient by construction in this case and the MHD gaseous disk behavior is difficult to predict from periodic orbit computations. However, such as the stellar arms constructed in this case, the gaseous spiral would be transient in likely even shorter timescales than in the case where periodic orbits exist but settle down systematically in smaller pitch angles than the original imposed spiral arms in the region where periodic orbits do exist. Notice that, in general, the arms that should eventually disappear in this scenario, are the stronger stellar imposed spiral arms (see second row of fig. \[fig:periodic\_gas\]; @Perez-Villegas_et13).
Therefore, the stellar response density maxima represent the regions of the arms where stars would crowd for long timescales, this is, where the existence of stable, long-lasting spiral arms would be more likely. On the other hand, if the stellar density response forms a spiral arm with a different pitch than the imposed angle, then the imposed spiral arms triggered on the disk (by a bar, an interaction, etc.), would rather be structures of transient nature since those are not orbitally supported [@Perez-Villegas_et13]. Likewise, for the case of the spiral arms in the Milky Way, the values frequently seen in literature for the pitch angle, range from $\sim
11\deg$ to $19\deg$. These values and the knowledge of the galactic type could provide some information about their nature, i.e. whether they are long-lasting or transient structures. [**Following the pitch angle restrictions found by @Perez-Villegas_et13, the larger pitch angle values reported in literature for the Milky Way galaxy would imply that its spiral arms are a transient feature.**]{} The formation of four spiral arms in the gas response is, in this scenario, another piece of evidence of a transient nature of the spiral arms in the Milky Way galaxy, as we claim it is the secondary pair of arms in Figure \[fig:periodic\_gas\], that coincides [**more**]{} closely with the stellar density response [**from**]{} periodic orbits, as expected from @Gomez_et13. Summarizing, the first (imposed) pair of massive spiral arms formed in the disk, with a larger pitch angle, triggers a second pair of arms (traced approximately by the periodic orbits), with smaller pitch angles [@Perez-Villegas_et13]. In this outline, the gas responds forming a second pair of arms aligned with the locus of the orbital crowding. This lighter structures would likely be preferentially formed by young stars and gas than by evolved stars because of their transient nature, i.e. similar to what we call “branches”. Finally, in this framework, the presence of clear and strong branches in spiral galaxies, with smaller pitch angles than the corresponding couple of massive spiral arms on a galaxy, would be a signature of the transient nature of the spiral arms in a given galaxy. On the other hand, spiral galaxies without evidence of branches could indicate the presence of a longer lasting spiral arm structure.
{width="27.00000%"} {width="31.00000%"}\
{width="27.00000%"} {width="31.00000%"}\
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{width="27.00000%"} {width="31.00000%"}
Discussion and Conclusions {#sec:conclusions}
==========================
With the use of a detailed three dimensional, density-distribution based potential for the spiral arms, combined with MHD simulations on a Milky Way-like galactic disk, we have studied the stellar orbital and gaseous response to the galactic potential. As a first experiment, we constructed a simple cosine potential (as the ones commonly employed in literature) that reproduced approximately what the density based potential PERLAS exerts on the stellar and gaseous dynamics.
The first set of simulations compare the gas response when the disk is perturbed by both spiral arm potential models. We also varied the structural parameters of the spiral arms in the PERLAS model, such as the pitch angle ($12\deg$ to $19\deg$) and the mass of spiral arms (1.75% to 5% of the stellar disk mass) in order to understand how these parameters affect the gaseous disk dynamics. Additionally, we constructed stellar periodic orbits and calculate the stellar response density maxima.
With these exercises we found that only in the case of the PERLAS model the gas and stellar density response (based on the existence of periodic orbits) is a consistent four-armed spiral structure: a couple of strong gaseous arms located at the position of the imposed stellar arms, and a second pair of weaker gaseous arms located at the position of the stellar orbit crowding. [**Since the potential and the stellar density response do not coincide, the spiral arms are**]{} prone to destruction. [**So, the presence of the second (gaseous) pair is interpreted as a sign of this transiency.**]{}
Our main conclusions can be summarized as follows:
- [**We performed a**]{} study of the gas response to [**two**]{} galactic potentials: the density distribution based model PERLAS and the widely employed in literature cosine potential. We verified that the gas response (a two armed structure) is the same for both models close to the linear regime only, i.e. [**for low-mass spiral arms and pitch angles smaller than $\sim 10\deg$.**]{} In the general case however (i.e. large pitch angle or arm mass), even when the cosine potential has been fitted as close as possible to the PERLAS model, they produce quite distinct outcomes on the gas response. In the case of the cosine potential, the gas responds invariably forming two spiral arms while, with the PERLAS model, the gas responds with four gaseous spiral arms (in the general strong arm case).
- We increased the strength of the spiral arms represented by the cosine potential up to a point the arms were equivalent and even beyond the mass of a strong bar as an experiment to try to reproduce the gas response provided by the PERLAS model. However, the answer was always a bisymmetric structure. We conclude that the spiral arm strength is not responsible for the four-arm gas response, but rather it is the product of the forcing generated by the whole density distribution better represented by the PERLAS model that, in turn, forces the periodic orbit response to shift its crowding regions inside the imposed locus of the massive spiral arms.
- Using the PERLAS model, we changed the structural parameters of the spiral arms according to the observational and theoretical uncertainties in the determination of the Milky Way’s spiral arms in literature. Within these values, for the general case, the stellar response density maxima systematically forms spiral arms with a smaller pitch angle than the imposed spiral, meaning that the spiral arms might be a transient feature in the Milky Way Galaxy [@Perez-Villegas_et13]. The presence of a second pair of lighter spiral arms (“branches”), with smaller pitch angles induced by the first more massive stellar spiral arms, might be evidence of the lack of support to the stronger arms on a galactic disk, and therefore evidence of the transient nature of spiral arms in a given galaxy. Applying this scenario to the Milky Way, for the stronger spiral arm values reported in literature (i.e. pitch angles larger than $\sim 10\deg$, and masses larger than $\sim 2\%$ of the disk), the spiral arms in the Milky Way would be of transient nature.
- Although in this work we applied the models to Milky Way-like galactic discs, it is worth noticing that the results are general. This means that the presence of branches with smaller pitch angles than the main arms might be the “smoking gun” that proves transiency of spiral arms in any given galaxy.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Edmundo Moreno for enlightening discussions that helped to improve this work. APV acknowledges the support of the postdoctoral Fellowship of DGAPA-UNAM, México. This work has received financial support from DGAPA PAPIIT grants IN111313 and IN114114.
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\[lastpage\]
[^1]: E-mail: a.perez@crya.unam.mx
[^2]: E-mail: g.gomez@crya.unam.mx
[^3]: E-mail: barbara@astro.unam.mx
|
---
abstract: 'We describe the topology of the space of all geometric limits of closed abelian subgroups of $\operatorname{PSL_2({\mathbb{C}})}$. Main tools and ideas will come from the previous paper [@BC1].'
author:
- |
Hyungryul Baik & Lucien Clavier [^1]\
Department of Mathematics\
310 Malott Hall, Cornell University\
Ithaca, New York 14853-4201 USA
bibliography:
- 'biblio.bib'
title: 'The Space of Geometric Limits of Abelian Subgroups of $\operatorname{PSL_2({\mathbb{C}})}$'
---
Introduction {#intro}
============
Matrix representations
======================
$\operatorname{\mathbb{C}\mathbb{P}^1}$ as a quotient {#cp1}
-----------------------------------------------------
Matrix representations of elliptic and hyperbolic isometries {#matrixEH}
------------------------------------------------------------
Matrix representations of parabolic isometries {#parabolics}
----------------------------------------------
Reduction lemma {#sec:redlem}
===============
Case $R_\infty<\infty$: the Chabauty space of ${\mathbb{C}}^\ast$ {#sec:cstar}
=================================================================
Case $R_\infty<\infty$: how the non-parabolic subgroups accumulate onto $\operatorname{\bf{P}}_2$ {#sec:bcstar}
=================================================================================================
Local models for $\operatorname{\bf{C}}_2$ {#sec:pieces}
==========================================
Summary statement {#sec:future}
=================
[^1]: We really appreciate that John H. Hubbard let us know about this problem and explained how we could approach at the beginning. He also has provided us a lot of advices through enlightening discussions. We also thank to Bill Thurston for the helpful discusstions.
|
---
abstract: 'The Doppler-free spectroscopy of atomic thallium ($\rm{^{203}Tl}$ and $\rm{^{205}Tl}$) $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ transitions have been observed using two counter-propagating laser beams perpendicular to the atomic beam, and the absolute frequencies of these transitions have been measured to an accuracy of $\rm<$1 MHz (1 ppb) using optical frequency comb. Our results improved the hyperfine splittings by a factor of 7$\rm{\sim}$8. The isotope shifts are in agreement with the previous experimental results.'
author:
- 'Yi-Wei Liu'
- 'Yu-Hung Lien'
- 'Wei-Ling Cheng'
- 'Chia-Hui Ho'
- 'Jow-Tsong Shy'
- 'Hsiang-Chen Chui'
bibliography:
- 'tl.bib'
title: 'Absolute frequency measurement of atomic thallium $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ transitions using optical frequency comb'
---
Introduction
============
High precision measurement in atomic system shows very promising in testing new physics beyond the standard model. The linear Stark effect measurement in atomic thallium sets the current upper limit of electron’s electric dipole moment (EDM)[@2002:Commins], whose existence implies the CP violation. Meanwhile, atomic thallium also plays an important role in parity non-conservation (PNC) experiment. The PNC effect has been observed in atomic thallium system using $\rm{6P_{1/2}\rightarrow6P_{3/2}}$ transition in 1995 [@1995:Vetter; @1995:Edwards]. The optical rotation measurement of thallium reached 1% of experimental uncertainty. Combining the theoretical calculation, it leads to the weak charge of thallium nucleus, which can be compared with the predication of the standard model. However, the dominating uncertainty is the theoretical calculation, which is as large as 3%. Such a large uncertainty is due to its complicate atomic structure. In comparison with the most recent PNC experiment with cesium, whose atomic structure is better understood, an experimental accuracy of 0.5% has been achieved, and the accuracy of theoretical calculation is only 1% [@1997:Wood; @2001:KozlovPRL; @2001:Dzuba]. This result leads to the most accurate measurement of the weak charge of the cesium nucleus.
In atomic thallium, strong correlation between three valance-electrons can not be treated accurately using many-body-perturbation theory (MBPT), which is the only calculation needed to be included in the case of cesium, as pointed out by Kozlov [@2001:Kozlov] and Dzuba [@1996:Dzuba]. The calculation combining MBPT and the configuration interaction (CI) [@1996:Dzuba] was developed to solve the correlation problem. To test the accuracy of this theoretical approach, various observables, such as transition energies, hyperfine splittings, transition amplitudes, and polarizabilites, should be calculated and compared with experimental results. Therefore, experimental measurements with high precision would be important for the improvement of theoretical calculation.
In this paper, we report precision our measurements of atomic thallium $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ transition frequencies and isotope shifts using an optical frequency comb (OFC), which is a versatile optical frequency measuring tool [@2000:Hall]. The absolute frequencies in the region between 500-1000 nm can be measured to high accuracies with OFC. A self-referenced Ti:Sapphire frequency comb linked to a master atomic clock can reach to an accuracy of kHz, or better. Such a high precision experiment was possible only using a complicate frequency chain before. Now, in most of cases with a compact OFC system, the accuracy of optical transition frequency measurement is only limited by the linewidth of the observed atomic or molecular transitions and signal-to-noise ratio of the spectrum, rather than the frequency measuring tool itself.
In atomic thallium, the ground state 377 nm resonance transition has been studied in details using both gas cell [@2000:Majumder] [@2002:Liu] and atomic beam [@1993:Hermann]. However, the large linewidthes, due to laser or the Doppler broadened features, limited the accuracy of measurements. The Doppler-broadened spectrum of this transition has been performed in both of direct absorption [@2000:Majumder] and fluorescence [@2002:Liu] experiments with a gas cell. However, to our knowledge, no Doppler-free saturation spectroscopy has been observed yet. Hence, the large Doppler linewidth (600 MHz) limits the accuracy of isotope shift and hyperfine splitting measurement in $\rm{7S_{1/2}}$.
The major difficulty of the cell experiment to acquire Doppler-free spectrum is due to the large branch ratio of $\rm{7S_{1/2}\rightarrow6P_{3/2}}$ decay route (in comparison with $\rm{7S_{1/2}\rightarrow6P_{1/2}}$, see Fig. \[fig:level\]), and the long lifetime of $\rm{6P_{3/2}}$ metastable state. The exciting 377 nm laser causes an optical pumping effect to trap population in the $\rm{6P_{3/2}}$ metastable state, then a long recovering time of the $\rm{6P_{1/2}}$ ground state population. This implies a low effective saturation intensity of $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ transition, therefore a strong power broadening effect. To perform the Doppler-free saturation spectroscopy with gas cell, even the minimum required power for reasonable signal strength could induce a large power-broadened linewidth close to Doppler width. Lowering laser intensity will not reduce power broadening only, but also the signal strength. A detailed study of such optical pumping effect in atomic thallium can be found in [@2002:Liu].
In the atomic beam experiment, the finite laser-atom interaction time (transit time) becomes an effective shorter recover time of ground state population [@2003:Demtroder], and then reduces power broadening. Although, there is a trade-off of transit time broadening. In the interaction region of 1 mm, the transit time broadening of atom with a velocity of 300 m/s, is $\rm{<}$ 1 MHz. Using atomic beam spectroscopy with an unidirectional laser beam, the Doppler shift becomes a serious systematic error, because of the imperfect perpendicular between laser and atomic beam. In our experiment, we employed two counter propagating laser beams and observed the Lamb dip to eliminate such a problem. The observed linewdith of Doppler-free spectrum is only $\rm{\sim20~MHz}$ without the first order Doppler shift .
Experiment
==========
![\[fig:level\]The partial level diagram of atomic thallium (not scaled).](level){width="\linewidth"}
The experimental setup is illustrated as Fig. \[fig:setup\]. A collimated atomic beam was irradiated perpendicularly by the frequency-doubled 755 nm Ti:Sapphire laser. The atomic beam source was an effusion oven containing thallium. The transition was detected by measuring 535 nm fluorescence. A small portion of 755 nm laser beam was picked up using a blank glass plate, and sent to OFC through a single mode optical fiber. A reference cavity with a FSR= 300 MHz was used to monitor the laser mode while laser was scanning. The absolute frequency of laser was also measured using a home-made wavemeter with GHz accuracy.
![\[fig:setup\]The experimental setup of absolute frequency measurement of thallium $6P_{1/2}\rightarrow7S_{1/2}$ transition](setup){width="\linewidth"}
377 nm Laser System
-------------------
The 377 nm UV light was generated using a LBO crystal (Breswter’s angle cut, $\rm{3\times3\times10~mm^3}$) in an enhancement cavity pumped by a Ti:Sapphire laser (Coherent MBR-110) of 755 nm. The enhancement cavity was a ring configuration with astigmatism compensation. Cavity resonance was locked on the fundamental laser frequency using the technique of polarization rotation [@1980:Couillaud]. The typical conversion efficiency was 15%/W with an input coupler of T= 3%. 500 $\rm{\mu W}$ of the 377 nm laser power was used in our experiment. The intensity noise was $\rm{<}$ 5% due to the acoustic vibration in the frequency of $\rm\sim$ kHz. It was collimated to a beam size of 3 mm${\rm \times}$ 10 mm using an AR coated lens. The major axis of the elliptical beam shape was parallel to the atomic beam.
Atomic Beam and Signal Detection
--------------------------------
The collimated atomic beam with reduced transverse velocity component was used to reduce the Doppler broadening effect and to increase the spectral resolution. The thermal atomic beam, as Fig. \[fig:oven\], was generated from thallium bulk, which was heated to $\rm{450^{\circ}C}$. The temperature near the front exit hole was kept $\rm{50^{\circ}C}$ higher than the rest part of oven to prevent condensation using another heater. The most probable velocity of thallium atom was $\sim 300$ m/s. The atomic thallium beam was collimated using two 2 mm apertures separated by 20 mm. The residual Doppler width, due to the beam divergence, was estimated to be 30 MHz in our system. These apertures, between oven and experimental chambers, were also for differential pumping. The experimental chamber was pumped using a small turbo pump (50 l/s) to maintain a pressure ${\rm<10^{-5}}$ torr. The oven chamber was pumped using a larger turbo pump (170 l/s) to reach a pressure ${\rm\sim10^{-5}}$ torr. The higher chamber pressure was caused by the high temperature of oven. The flange joining these two chambers was water-cooled to prevent heat-transferring from oven to interaction chamber.
Two counter-propagating laser beams were used to eliminate the linear Doppler shift and the Lamb dip can be observed. Two view ports for laser injection are AR coated UV fused silica. Laser induced fluorescence (LIF) was collected using a 1 inch f= 30 mm lens, passed through a dielectric 535 nm filter, and detected using a photomultiplier. The laser intensity was modulated using an optical chopper with a frequency of 2 kHz to reduce noise.
![\[fig:oven\]The atomic beam apparatus.](oven){width="\linewidth"}
Optical Frequency Comb
----------------------
The optical frequency comb system was based on a mode-locked Ti:Sapphire laser (Giga Jet 20) with 1 GHz repetition rate, and pumped by a 5W 532 nm DPSS Laser (Millennia V). It was a self-referenced configuration linked to a rubidium clock (SRS PRS10) using the technique of optical frequency synthesizer [@2000:Hall], as shown in Fig. \[fig:comb\]. All of the electronic universal counters and the frequency synthesizers were externally referenced to the same rubidium frequency standard. A 760 nm bandpass filter was used to filter out the unnecessary frequency comb lines and to avoid optical damage of the photodiode (Fig. \[fig:setup\]). The frequency fluctuation of the stabilized repetition frequency and offset frequency were 4 mHz, and 11 mHz, respectively. The total frequency fluctuation is few kHz in optical frequency, and the accuracy , which was limited by the rubidium master clock and phase noise from frequency synthesizer, was $10^{-11}$ [@comb2]. The Allen deviation of stabilized repetition rate is shown in Fig. \[fig:comb-stability\]. It could be improved by linking the master clock to 1pps signal from GPS receiver. However, This is not necessary in our experiment, since the required accuracy is few tens kHz.
We calibrated this frequency comb system by measuring the well-known $\rm{a_{10}}$ hyperfine component of molecular iodine R(56)32-0 transition at 532 nm. The result is in good agreement with the established international standard.
![\[fig:comb\]The self-reference optical frequency comb with a mode-locked Ti:Sapphire laser and a 100 cm photonic crystal fiber.](comb){width="\linewidth"}
![\[fig:comb-stability\]The Allen deviation of repetition rate (circle) and 1 GHz frequency from HP8643A frequency synthesizer (triangle).](comb-stability){width="\linewidth"}
Frequency measurement
---------------------
The 755 nm laser and frequency comb laser were combined using a polarizing beam splitter (PBS), and projected to $\rm{45^\circ}$ polarization using a polarizer. The beat frequency, which was detected using an avalanche photodiode (APD), and the fluorescence signal were simultaneously recorded by a computer. We performed a very slow scan, which is 100 MHz/min, to minimize the lag between the frequency reading and fluorescence signal output. Although such a lag effect was cancelled out by backward scanning. A single scan consists of 1000–2000 data points within the frequency range of 100–200 MHz in terms of laser frequency.
The laser frequency can be calculated with a simple equation: $$f_{unknown}=N\times f_{r}\pm f_{o}\pm f_{b}\ ,$$ where the repetition rate of comb laser $f_{r}$ is $\sim$1 GHz. $N$ was a large integer, $\sim 4\times10^{5}$. The offset frequency of comb laser $f_{o}$ and the measured beat frequency $f_{b}$ were typically several hundreds MHz. Firstly, $N$ was roughly determined by the home-made wavemeter with an accuracy of GHz. Then, comparing several scans of the same transition with various $f_{o}$, a series of corresponding measurements of $f_{b}$ was obtained. Since each set of $f_{o}$, $f_{b}$, and $N$ should result in the same frequency, the signs of $f_{o}$ and $f_{b}$, and $N$ can be accurately determined. To improve the statistic accuracy, the final spectrum was a histogram combining several different scans. The errorbar of the histogram was given by the standard deviation of signal strength of the same laser frequency bin. Figure \[fig:histogram\] shows the histogram of all (20) measurements of $\rm{^{203}Tl}$: $\rm{6P_{1/2}(F=0)\rightarrow 7S_{1/2}(F=1)}$ transition. The binning of frequency is 0.5 MHz. Using 0.1 MHz binning, there was no improvement or degradation of the resulted transition frequency.
The spectrum was fitted to a combination of two Viogt and a Lorentzian functions: $$\label{eq:shape}
\begin{split}
S&=A_1\times\rm{Voigt}(\omega_0-\omega_{shift},w_{L},w_{G1})\\
&+A_2\times\rm{Voigt}(\omega_0+\omega_{shift},w_{L},w_{G2})\\
&-A_3\times\rm{Lorentzian}(\omega_0,w_{L})\\
&+C
\end{split}$$ The negative sign of $A_3$ represents the central dip of the spectrum. ${\rm w_{G1}}$ and ${\rm w_{G2}}$ are the Gaussian widths of two Voigt functions. ${\rm w_{L}}$ is the homogenous broadening, including laser linewidth, transit time broadening, power broadening, natural linewidth, and so forth. Two Viogt functions refer to two fluorescence signals excited from two counter-propagating light. The splitting of these two profiles ($\rm{\omega_{shift}}$) is due to that the laser beams are not exactly perpendicular to the atomic beam with a offset angle $\rm{\Delta\phi}$. We experimentally scanned $\rm{ \Delta\phi}$ from +40 mrd to -40 mrd to confirm that the central dip was a Doppler-free saturation dip, rather than just a signal decreasing in a profile combining two separated Doppler broadened spectrum due to forward and backward laser beams. The center of the Doppler-free dip (Lorentzian, from atoms with $v=0$), $\omega_0$, was independent on $\rm{\omega_{shift}}$, and a sensitive marker for the exact center of the atomic transition. The fitting program is written on ROOT platform (CERN) using the built-in Voigt function.
The frequency uncertainty is estimated by: $$(\Delta f_{unknown})^{2}=(N\times\Delta f_{r})^{2}+(\Delta f_{o})^{2}+(\Delta f_{b})^{2}\ .$$ The frequency fluctuation of the stabilized repetition frequency was 4 mHz, so the uncertainty of the absolute frequency measurement contributed by the first term ($N\times\Delta f_{r}$) is 2 kHz. The stabilized offset frequency fluctuation ${\Delta f_{o}}$ is only 11 mHz. Therefore, the predominant uncertainty of the absolute frequency measurement in this work is due to the uncertainty of beat frequency, $\Delta f_{b}$, which is the statistic uncertainty from the fitting to histogram data.
![\[fig:eline\]The histogram of $\rm{6P_{1/2}(F=0) \rightarrow 7S_{1/2}( F=1)}$ ($\rm{^{203}Tl}$) transition. The residual fitting error is $\rm{\times 10}$, and shows that the theoretical model (Eq. \[eq:shape\]) is in agreement with the experimental result. The frequency axis shown in this figure is the frequency of the fundamental laser, rather than the doubled UV light.[]{data-label="fig:histogram"}](eline){width="\linewidth"}
Results
=======
The absolute transition frequency
---------------------------------
Figure \[fig:eline\] shows one of transitions with fitting curve, the laser frequency we measured is infrared, rather than atomic transition frequency—UV light. This histogram with 500 kHz binning combined 5 forward-backward scans at different time. The accuracy of measured absolute frequency is sub-MHz, as the signal-to-noise ratio (S/N) is $\sim50$, and the FHWM of Lamb dip ($\rm{w_L}$) is $\rm<$30 MHz. We found that $\rm{\omega_{shift}\sim 15~MHz}$ and $\rm{ \Delta\phi\sim38~mrd}$, which is consistent with our experimental setup. The final absolute frequency measurements of all six components of thallium $6P_{1/2}-7S_{1/2}$ transitions, including three hyperfine transitions and two isotopes, are listed in Table. \[table:freq\]. They are labeled as $\rm{a-f}$ for identification, and in comparison with the previous experimental measurement and the theoretical calculation.
Line Transition Laser frequency (MHz)
---------------------- ---------------------------------------- -----------------------
a $\rm{^{203}Tl}$ F=1$\rm{\rightarrow}$0 793 761 855.9 (2)
b $\rm{^{205}Tl}$ F=1$\rm{\rightarrow}$0 793 763 376.9 (3)
c $\rm{^{203}Tl}$ F=1$\rm{\rightarrow}$1 793 774 036.3 (3)
d $\rm{^{205}Tl}$ F=1$\rm{\rightarrow}$1 793 775 672.9 (2)
e $\rm{^{203}Tl}$ F=0$\rm{\rightarrow}$1 793 795 141.6 (3)
f $\rm{^{205}Tl}$ F=0$\rm{\rightarrow}$1 793 796 983.4 (5)
Experiment [@Kurucz] $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ 793 775.5 (GHz)
Theory[@2001:Kozlov] $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ 793 100(10) (GHz)
: \[table:freq\]The transition frequencies of $\rm{6P_{1/2}\rightarrow7S_{1/2}}$.
$\rm{^{205}Tl}$ $\rm{6P_{1/2}}$ $\rm{^{203}Tl}$ $\rm{6P_{1/2}}$ $\rm{^{205}Tl}$ $\rm{7S_{1/2}}$ $\rm{^{203}Tl}$ $\rm{7S_{1/2}}$ IS of $\rm{6P_{1/2}\rightarrow7S_{1/2}}$
---------------------------------- --------------------------------- --------------------------------- --------------------------------- --------------------------------- ------------------------------------------
This work 21310.5(3) 21105.3(3) 12296.0(2) 12180.3(2) 1659.0(6)
Ref.[@2000:Majumder] 12294.5(15) 12180.5(18) 1659.0(6)
Ref.[@1993:Hermann] 12297.2(16) 12181.6(22)
Ref.[@1985:Neugart] 12284.0(60) 12172.0(60)
Ref.[@1962:Shuler] 12318(36) 12225(42)
Ref.[@1956:Lurio] 21310.835(5) 21105.447(5)
Theory[@2001:Kozlov] 21663 12666
Theory[@1995:Mrtensson-Pendrill] 21300 12760
Theory[@1998:Dzuba] 21623 12307
HFS, IS and the Mean Square Isotopic Change $\lambda_{c,m}$
-----------------------------------------------------------
Our hyperfine splitting and isotope shift measurements are in good agreement with the previous experimental results as shown in Table \[table:HFS\]. Our measurement of $\rm{6P_{1/2}}$ hyperfine splitting, using purely optical method, is consistent with the magnetic resonance experiment in 1956 [@1956:Lurio], however our result is less accurate. For the $\rm{7S_{1/2}}$ hyperfine splitting, our measurement improved the accuracy by a factor of 7$\rm\sim$8, in comparison with the most recent measurements using gas cell [@2000:Majumder] and the Doppler-broadened spectroscopy with atomic beam [@1993:Hermann]. The $6P_{1/2}\rightarrow7S_{1/2}$ transition isotope shift between $\rm{^{205}Tl}$ and $\rm{^{203}Tl}$ ($\delta\nu_{205-203}=1659.0(6)$ MHz) was also measured. This result is in very agreement with [@2000:Majumder].
The hyperfine anomaly, $\Delta=[(A_{205}/A_{203})(g_{203}/g_{205})-1]$, can be deduced from our measured hyperfine constants $A_{205}$ and $A_{203}$. Using the most precise magnetic moments $g_{203}=1.62225787$ and $g_{205}=1.63831461$ measured in [@1989:Raghavan], $\Delta=-3.34(23)\times10^{-4}$ was deduced. The hyperfine anomaly can be used to infer a parameter for the mean square isotopic change $\lambda_{c,m}$, which is related to the magnetic moments and charge distributions in the isotopes. For atomic thallium, the ratio of $\rm{\Delta/\lambda_{c,m}}$ has been theoretically calculated by [@1995:Mrtensson-Pendrill]. They gave $\rm{\Delta/\lambda_{c,m}=-7.62\times10^{-4}~fm^2}$ for $\rm{7S_{1/2}}$. Combining our hyperfine anomaly measurement of $\rm{7S_{1/2}}$, we infer $$\lambda_{c,m}=0.44(3)\ \rm{fm^{2}}$$ using our experimental results. This parameter inferred from the results of the $\rm{7S_{1/2}}$ state HFS reported in [@2000:Majumder] is given as $\lambda_{c,m}=0.61(20)\ \rm{fm^{2}}$ and $\lambda_{c,m}=0.45(24)\ \rm{fm^{2}}$ in [@1993:Hermann]. $\lambda_{c,m}=0.42\ \rm{fm^{2}}$ can also be extracted from the much more precise ground state $\rm{6P_{1/2}}$ HFS measured by [@1956:Lurio] and [@1995:Mrtensson-Pendrill]. All of theses measurements are summarized in Fig. \[fig:fm\]. Our result is the most precise value based on $\rm{7S_{1/2}}$ HFS.
![Comparison of the mean square isotopic change based on various experimental results :\
A. 1956-HFS of $\rm{6P_{1/2}}$ [@1956:Lurio] B. 1990-HFS of $\rm{7S_{1/2}}$ [@1993:Hermann]\
C. 1995-HFS of $\rm{6P_{1/2}}$ [@1995:Mrtensson-Pendrill] D. 2000-HFS of $\rm{7S_{1/2}}$ [@2000:Majumder]\
E. HFS of $\rm{7S_{1/2}}$ (this work)[]{data-label="fig:fm"}](fm){width="\linewidth"}
Conclusion
==========
The Doppler-free spectroscopy of atomic thallium $\rm{6P_{1/2}\rightarrow7S_{1/2}}$ was observed using atomic beam. For the first time, the absolute transition frequencies of six components of $\rm{^{203}Tl}$ and $\rm{^{205}Tl}$ were measured to sub-MHz accuracy. These results improve the measurement of hyperfine splittings of $\rm{7S_{1/2}}$ state. The mean square isotopic change $\lambda_{c,m}$ deduced from the hyperfine anomaly of $\rm{7S_{1/2}}$ state has also been improved. Our result is in very good agreement with the most precise value from the hyperfine anomaly of $\rm{6P_{1/2}}$.
The optical pumping effect trapping the population on $\rm{6P_{3/2}}$ state can be used for the future laser cooling of atomic thallium through $\rm{6P_{3/2}\rightarrow5D_{5/2}}$ transition using a 351 nm laser. Similar cooling schemes have been realized in other atomic species of AIII group [@1995:SALee][@2000:SALee]. Laser cooled atomic thallium beam could further improve various experiments on testing fundamental atomic physics using atomic thallium.
This work was supported by the National Science Council of Taiwan under Grant No.95-2112-M-007-005- and No. 94-2112-M-007-011-.
|
---
abstract: |
The energy spectrum of magnetohydrodynamic turbulence attracts interest due to its fundamental importance and its relevance for interpreting astrophysical data. Here we present measurements of the energy spectra from a series of high-resolution direct numerical simulations of MHD turbulence with a strong guide field and for increasing Reynolds number. The presented simulations, with numerical resolutions up to $2048^3$ mesh points and statistics accumulated over 30 to 150 eddy turnover times, constitute, to the best of our knowledge, the largest statistical sample of steady state MHD turbulence to date. We study both the balanced case, where the energies associated with Alfvén modes propagating in opposite directions along the guide field, $E^+(k_\perp)$ and $E^-(k_\perp)$, are equal, and the imbalanced case where the energies are different. In the balanced case, we find that the energy spectrum converges to a power law with exponent $-3/2$ as the Reynolds number is increased, consistent with phenomenological models that include scale-dependent dynamic alignment. For the imbalanced case, with $E^+>E^-$, the simulations show that $E^-\propto k_{\perp}^{-3/2}$ for all Reynolds numbers considered, while $E^+$ has a slightly steeper spectrum at small $Re$. As the Reynolds number increases, $E^+$ flattens. Since $E^\pm$ are pinned at the dissipation scale and anchored at the driving scales, we postulate that at sufficiently high $Re$ the spectra will become parallel in the inertial range and scale as $E^+ \propto
E^-\propto k_{\perp}^{-3/2}$. Questions regarding the universality of the spectrum and the value of the “Kolmogorov constant” are discussed.
author:
- 'Jean Carlos Perez$^{1,2}$, Joanne Mason$^3$, Stanislav Boldyrev$^2$, Fausto Cattaneo$^3$'
title: On the energy spectrum of strong magnetohydrodynamic turbulence
---
Introduction
============
Astrophysical plasmas are typically magnetized and turbulent, with turbulent fluctuations spanning a tremendous range of scales in which the energy spectrum follows a power law scaling [e.g., @armstrong_rs95; @Goldstein_rm95]. Incompressible magnetohydrodynamics (MHD) provides the simplest theoretical framework for studying magnetized plasma turbulence. The precise form of the MHD turbulence spectrum is crucial for a variety of processes in astrophysical systems with extended inertial intervals, such as plasma heating and wave-particle interactions, which are sensitive to small variations in the spatial scaling of the fluctuations [e.g., @chandran_etal09; @chandran_etal10; @ng_b10]. The incompressible MHD equations take the form $$\begin{aligned}
\label{mhd-elsasser}
\left( \frac{\partial}{\partial t}\mp{\mathbf{V}}_A\cdot\nabla \right)
{\mathbf{z}}^\pm+\left({\mathbf{z}}^\mp\cdot\nabla\right){\mathbf{z}}^\pm &=& -\nabla
P +\nu\nabla^2 {\mathbf{z}}^{\pm}+{\mathbf{f}}^\pm, \nonumber \\ \nabla \cdot
{{\mathbf{z}}}^{\pm}&=&0
\end{aligned}$$ where ${\mathbf{z}}^\pm={\mathbf{v}}\pm{\mathbf{b}}$ are the Elsässer variables, ${\mathbf{v}}$ is the fluctuating plasma velocity, ${\mathbf{b}}$ is the fluctuating magnetic field (in units of the Alfvén velocity), ${\mathbf{V_A}}={\bf
B}_0/\sqrt{4\pi\rho_0}$ is the Alfvén velocity based upon the uniform background magnetic field ${\mathbf{B_0}}$, $P=(p/\rho_0+b^2/2)$, $p$ is the plasma pressure, $\rho_0$ is the background plasma density, $\nu$ is the fluid viscosity (which, for simplicity, we have taken to be equal to the magnetic diffusivity) and ${\mathbf{f}}^\pm$ represent forces that drive the turbulence at large scales. It can be shown that in the limit of small amplitude fluctuations, and in the absence of forcing and dissipation, the system describes non-interacting linear Alfvén waves with dispersion relation $\omega^\pm({\mathbf{k}})=\pm k_\|
V_A$. The incompressibility condition requires that these waves be transverse. Typically they are decomposed into shear Alfvén waves (with polarizations perpendicular to both ${\mathbf{B_0}}$ and to the wave-vector ${\mathbf{k}}$) and pseudo-Alfvén waves (with polarizations in the plane of ${\mathbf{B_0}}$ and ${\mathbf{k}}$ and perpendicular to ${\mathbf{k}}$).
Nonlinear interactions (or collisions) between counter-propagating Alfvén wave packets distort the packets, splitting them into smaller ones until a scale is reached when their energy is converted into heat by dissipation [@kraichnan_65]. The efficiency of the nonlinear interaction is controlled by the relative size of the linear and nonlinear terms in equation (\[mhd-elsasser\]): The regime in which the linear terms dominate over the nonlinear terms is known as [*weak*]{} MHD turbulence, otherwise the turbulence is called [ *strong*]{}. The Fourier energy spectrum of MHD turbulence can be derived analytically only in the limit of weak turbulence [e.g., @ng97; @newell_nb01; @kuznetsov2001; @galtier_nnp02; @lithwick_g03; @galtier_c06; @chandran08a; @boldyrev_p09]. However, it has been demonstrated both analytically and numerically that the energy cascade occurs predominantly in the plane perpendicular to the guiding magnetic field [@newell_nb01; @galtier_nnp02; @shebalin83; @maron_g01], which ensures that even if the turbulence is weak at large scales it encounters the strong regime as the cascade proceeds to smaller scales. Although weak turbulence may exist in some astrophysical systems [e.g., @bhattacharjee_n01; @saur_etal02; @melrose06; @chandran08a], magnetic turbulence in nature is typically strong, for which an exact analytic treatment is not available. In this case, high-resolution, well-optimized numerical simulations play a significant role in guiding our understanding of the turbulent dynamics. This provides the motivation for the present work.
The ideal MHD system conserves the Elsässer energies $E^+=\frac{1}{4}\int (z^+)^2 d^3 x$ and $E^-=\frac{1}{4}\int (z^-)^2
d^3 x$ (equivalently, the total energy $E=E^v+E^b=\frac{1}{2} \int
(v^2 + b^2)d^3 x=E^+ + E^-$ and the cross-helicity $H^C=\int ({\bf
v}\cdot {\bf b}) d^3 x= E^+ - E-$ are conserved). The energies $E^+$ and $E^-$ cascade in a turbulent state toward small scales due to the nonlinear interactions of oppositely moving ${\mathbf{z^+}}$ and ${\mathbf{z^-}}$ Alfvén packets. MHD turbulence is called [*balanced*]{} when the energies carried by oppositely moving fluctuations $E^\pm$ are equal, and it is called [*imbalanced*]{} when they are not the same. MHD turbulence in nature and in the laboratory is typically imbalanced. For instance, this is the case when the turbulence is generated by spatially localized sources, as is the case in the solar wind where more Alfvén waves propagate away from the Sun than towards it. The independent conservation of the two Elsässer energies (compared to only one conserved energy in hydrodynamics) has a profound consequence for the MHD dynamics [e.g., @dobrowolny80; @grappin_pl83; @meneguzzi96; @boldyrev_05; @boldyrev_06; @mason_etal06; @zhou2004; @matthaeus_etal08; @perez_b09; @boldyrev_etal09].
In this work we present the results of a series of direct numerical simulations of MHD and Reduced MHD for balanced and moderately imbalanced turbulence and investigate how the scalings of the Elsässer spectra behave as the Reynolds number is increased. We also present the first high-resolution direct comparison of simulations of MHD vs RMHD turbulence, demonstrating that the latter model completely captures the turbulence dynamics of strong MHD turbulence at roughly half the computational cost of a full MHD simulation.
This paper is organized as follows. In section \[sec:pheno\] we briefly describe the most recent phenomenological efforts to understand scaling laws in MHD turbulence, particularly in the imbalanced case. In section \[sec:numerics\] we describe the numerical set up and the parameter regime for our simulations. In section \[sec:spectrum\] we show measurements of the energy spectrum from a series of numerical simulations with varying Reynolds numbers. In section \[sec:alignment\] we show measurements of scale-dependent dynamic alignment and establish its relation with the $-3/2$ scaling of the energy spectrum. In section \[sec:kolmogorov\] we discuss the approach to the universal regime and the universality of Kolmogorov’s constant in MHD. We show that dynamic alignment introduces a new robust scale-dependent quantity that enters the definition of the energy spectrum and uniquely sets the Kolmogorov constant. We propose that this new quantity is a consequence of cross-helicity conservation. Finally, in section \[sec:discussion\] we discuss our results.
MHD turbulence phenomenology {#sec:pheno}
============================
For strong MHD turbulence, Goldreich & Sridhar [@goldreich_s95] argued that the pseudo-Alfvén modes are dynamically irrelevant for the turbulent cascade (since strong MHD turbulence is dominated by fluctuations with $k_\perp \gg k_\|$, the polarization of the pseudo-Alfvén fluctuations is almost parallel to the guide field and they are therefore coupled only to field-parallel gradients, which are small since $k_\| \ll k_{\perp}$). If one filters out the pseudo-Alfvén modes by setting ${\mathbf{z}}^\pm_\|=0$, it can be shown that the resulting system is equivalent to the Reduced MHD model: $$\begin{aligned}
{\left(}\frac{\partial}{\partial t}\mp{\mathbf{V}}_A\cdot\nabla_\|{\right)}{\mathbf{z}}^\pm+\left({\mathbf{z}}^\mp\cdot\nabla_\perp\right){\mathbf{z}}^\pm =
-\nabla_\perp P\nonumber\\ +\nu\nabla^2{\mathbf{z}}^\pm +{\mathbf{f}}_\perp^\pm,
\label{rmhd-elsasser}\end{aligned}$$ We note that in RMHD the fluctuating fields have only two vector components, but that each depends on all three spatial coordinates. Moreover, because the ${\mathbf{z}}^\pm$ are assumed incompressible ($\nabla\cdot{\mathbf{z}}^\pm=0$), each field has only one degree of freedom which is more commonly expressed in terms of stream functions in the more standard form of the RMHD equations [@kadomtsev_p74; @strauss_76].
Conservation of both the Elsässer energies means that once an imbalance has been created it cannot be destroyed by the MHD dynamics. It is also well known that decaying MHD turbulence, affected only by the dissipation, becomes increasingly more imbalanced with time [e.g., @dobrowolny80; @zhou2004; @matthaeus_etal08]. Several analytic and numerical studies have shown that imbalance is also an inherent property of *driven* MHD turbulence even if the turbulence is forced without introducing a net imbalance at the largest scales – the turbulent domain spontaneously fragments into local imbalanced domains where the cross helicity is either positive or negative [@grappin_pl83; @meneguzzi96; @zhou2004; @boldyrev_05; @boldyrev_06; @mason_etal06; @perez_b09; @boldyrev_etal09].
In imbalanced domains, the directions of the magnetic and velocity fluctuations are not independent, rather, they are either aligned or counter-aligned to a certain degree [^1]. The organization of such a domain is the following: the [*directions*]{} of both the magnetic and velocity fluctuations vary within a small angle (comparable to the alignment angle) throughout the domain, while their amplitudes change predominantly in the direction normal to their polarizations. Such positively and negatively aligned domains appear to be the building blocks of MHD turbulence, whether it is balanced overall or not.
The origin of such domains can be qualitatively understood from the conservation of energy and cross-helicity in an ideal MHD system. When small dissipation is present and the system is unforced, it can be argued that energy decays faster than cross-helicity. This selective decay would eventually lead to Alfvénization of the flow, that is, to progressively stronger alignment (or counter-alignment, depending on the initial state) between the directions of the magnetic and velocity fluctuations, e.g., [@dobrowolny80; @matthaeus80; @grappin_pl83; @biskamp_03]. In a perfectly aligned (counter-aligned) state either ${\bf z}^+$ or ${\bf z}^-$ is identically zero, and the nonlinear interaction vanishes. In a driven state, characterized by strong nonlinear interaction and a constant energy flux over scales, the alignment cannot be perfect. Rather, it turns out that alignment depends on the scale, the smaller the scales the better the alignment. Below we will demonstrate this phenomenon in numerical simulations. From a more qualitative point of view, one can argue that whenever a partly aligned domain appears, nonlinear interaction inside such a domain gets reduced, and its evolution time increases compared to non-aligned domains. Therefore aligned domains persist longer, which explains the tendency of a turbulent flow to exhibit such self-organization. These aligned domains are the domains where the essential energy of the turbulence is contained, and they are typically well seen in numerical simulations. Solar wind observations also show that globally balanced turbulence is made up of locally imbalanced patches at all scales [@podesta09; @podesta_b10a; @chen12].
In the aligned or imbalanced domains, the Elsässer energies are unequal, and one can ask whether their spectra have to be the same. This raises questions of whether MHD turbulence is universal and scale-invariant. Indeed, if imbalanced domains have different spectra that depend on the degree of imbalance, their superposition may not have a universal scaling.
Phenomenological treatment of strong imbalanced MHD turbulence is complicated by the fact that one can formally construct two time scales for the nonlinear energy transfer: The times of nonlinear deformation of the $z^\pm$ packets at some spatial scale $\lambda$ are $\tau^\pm\sim \lambda/z_\lambda^\mp$, which can be significantly different in the case of strong imbalance, [e.g., @dobrowolny80; @matthaeus_etal08]. In recent years, several phenomenological models attempting to accommodate this difference have been proposed. However, the theories have generated conflicting predictions because they use different assumptions regarding the physics of the nonlinear energy cascade. For example, the theory by Lithwick [*et al.*]{} [@lithwick_gs07] concludes that in the imbalanced regions the Elsässer spectra have the scalings $E^+(k_{\perp})\propto E^-(k_{\perp})\propto
k_{\perp}^{-5/3}$; the same spectra were also suggested by Beresnyak and Lazarian [@beresnyak_l08]. The theory by Chandran [@chandran08] proposes that the spectra of $E^+(k_{\perp})$ and $E^-(k_{\perp})$ are different depending of the degree of imbalance, while the theories by Perez and Boldyrev [@perez_b09] and Podesta and Bhattacharjee [@podesta_b10b] find that the spectra of $E^+(k_{\perp})$ and $E^-(k_{\perp})$ have different amplitudes but the same scalings $E^+(k_{\perp})\propto E^-(k_{\perp})\propto
k_{\perp}^{-3/2}$.
One would expect that numerical simulations could clarify the picture. However, the first numerical simulations of strongly imbalanced MHD turbulence [e.g., @rappazzo_etal07; @beresnyak_l08; @perez_b09] also produced conflicting results regarding which power law $E^\pm$ should follow. The conflicting numerical findings apparently reflect the fact that imbalanced MHD simulations require significantly more computational effort compared to the balanced cases [@perez_b10_2]. This happens since in the imbalanced domains the nonlinear interaction is depleted and the Reynolds and magnetic Reynolds numbers are reduced. This can be formally seen from the fact that, in a strongly imbalanced domain with $z^+ \gg z^-$ , the $z^+$ field is advected by a low-amplitude $z^-$ field, and therefore $z^+$ becomes directly affected by the dissipation at smaller wave-vectors (compared with the balanced case), which reduces its inertial interval. Now, $z^-$ is advected by a strong $z^+$, but $z^+$ is significantly affected by the dissipation, so the inertial interval of $z^-$ becomes spoiled as well.
In order to produce large inertial intervals simultaneously for both Elsässer fields when strongly imbalanced domains are present in the flow, one therefore needs to have a significantly higher Reynolds number as compared to the balanced case. However, as one increases the Reynolds number, one needs to increase the numerical resolution in order to appropriately resolve the small scales and to make sure the numerical run is stable. Therefore, the larger the imbalance, the larger the numerical resolution required to describe correctly the Elsässer spectra. Fortunately, it has been argued that Reduced MHD can be used to investigate the universal properties of MHD turbulence, which offers the advantage that an RMHD simulation can be achieved at half the cost of an MHD simulation.
Numerical Setup {#sec:numerics}
===============
We solve the MHD equations (\[mhd-elsasser\]) and their RMHD counterpart (\[rmhd-elsasser\]) in a periodic, rectangular domain with aspect ratio $L_{\perp}^2 \times L_\|$, where the subscripts denote the directions perpendicular and parallel to ${\mathbf{B_0}}$, respectively. We set $L_{\perp}=2\pi$, $L_\|/L_\perp=6$ or $10$ and ${\mathbf{B_0}}=5{\mathbf{e_z}}$. A fully dealiased 3D pseudo-spectral algorithm is used to perform the spatial discretization on a grid with a resolution of $N_{\perp}^2\times N_\|$ mesh points. We note that the domain is elongated in the direction of the guide field in order to accommodate the elongated wave-packets and to enable us to drive the turbulence in the strong regime while maintaining an inertial range that is as extended as possible (see [@perez_b10]). This is a physical requirement that should be satisfied no matter what model system, full MHD or reduced MHD, is used for simulations.
In the case of reduced MHD though, when the $z^{\pm}_\|$ components are explicitly removed, the resulting system (\[rmhd-elsasser\]) is invariant with respect to simultaneous rescaling of the background field $B_0$ and the field-parallel spatial dimension of the system, if one neglects the dissipation terms. Therefore, for any strength of the background field $B_0\gg 1$, one can rescale the field to $B_0=1$ and the field-parallel box size to $L_\|=L_\perp$, that is, conduct the simulations in a cubic box. We should note however that the dissipation terms in (\[rmhd-elsasser\]) are not invariant and they should be changed accordingly under such rescaling.
To save on computational cost we have reduced the field-parallel numerical resolution for some simulations, i.e., the numerical grid is anisotropic with $L_{\|}/N_{\|}> L_{\perp}/N_{\perp}$. This is appropriate since the energy cascade proceeds much faster in the field-perpendicular direction, and the energy spectra decline relatively slowly in the field-perpendicular direction and relatively fast in the field-parallel direction. Energies at large $k_\|$ are therefore reduced and a lower field-parallel resolution is not expected to alter the behavior of the spectra in the inertial interval. An isotropic resolution with the value imposed by the field-perpendicular dynamics would therefore be wasteful.
We should however caution that a reduced resolution (or, equivalently, unreasonably high Reynolds number for a given resolution) may contaminate the [*dissipative*]{} physics, even if the inertial interval is unaffected. For example, if the precise scaling behavior in the dissipation interval is of interest, as is the case for extended scaling laws such as the dynamic alignment angle, somewhat smaller Reynolds numbers may need to be chosen. As a general rule, whether the numerical simulations are conducted to investigate the inertial or the dissipation interval, a resolution study must be performed in order to establish the optimal Reynolds number for a given task. In particular, it has to be verified that increasing the numerical resolution while keeping the physical parameters such as Reynolds number, forcing mechanism, etc. unchanged does not affect the studied spectra, e.g., [@mason_etal11]. This point will be illustrated below in the balanced case.
The turbulence is driven at the largest scales by colliding Alfvén modes [^2]. We drive both Elsässer populations by applying statistically independent random forces ${\mathbf{f^+}}$ and ${\mathbf{f^-}}$ in Fourier space at wave-numbers $2\pi/L_{\perp} \leq k_{\perp} \leq 2 (2\pi/L_{\perp})$, $k_\| =
2\pi/L_\|$. The forces have no component along $z$ and are solenoidal in the $xy$-plane. All of the Fourier coefficients outside the above range of wave-numbers are zero and inside that range are Gaussian random numbers with amplitudes chosen so that $v_{rms}\sim 1$. The individual random values are refreshed independently on average approximately $10$ times per turnover of the large-scale eddies. The variances $\sigma_{\pm}^2=\langle |{\mathbf{f}}^{\pm} |^2\rangle$ control the average rates of energy injection into the $z^+$ and $z^-$ fields. We take $\sigma^+>\sigma^-$ and in the statistically steady state we measure the degree of imbalance through the parameter $h=(E^+-E^-)/(E^++E^-)=H^C/E$. Thus $h=0$ corresponds to balanced turbulence and $h=1$ defines maximally imbalanced turbulence. Time is normalized to the large scale eddy turnover time $\tau_0=L_\perp/(2\pi
v_{rms})$. The field-perpendicular Reynolds number is defined as $Re_{\perp}=v_{rms}(L_\perp/2\pi)/\nu \approx 1/{\nu}$. In order to accommodate the reduced field-parallel resolution we have also modified the diffusion operator in equations (\[mhd-elsasser\]) and (\[rmhd-elsasser\]), i.e., we have replaced $\nu \nabla ^2$ with $\nu (\partial_{xx}+\partial_{yy})+ \nu_\|\partial_{zz}$.
The system is evolved until a stationary state is reached, which is confirmed by observing the time evolution of the total energy of the fluctuations. The data are then sampled in intervals of the order of the eddy turnover time. All results presented correspond to averages over 30-150 samples for each run. As shown in Table \[tab:simlist\], we conduct a number of MHD and RMHD simulations in the balanced and imbalanced regime in order to investigate the scaling of the energy spectra as the field-perpendicular Reynolds number increases.
Case Regime $N_{\perp}$ $N_{\|}$ $h$ $L_{\|}/L_{\perp} $ $Re_{\perp}$ $\nu_{\|}$
----------------- -------- ------------- ---------- ------ --------------------- -------------- ------------ --
RB1a RMHD 512 256 0 6 2400 $\nu$
RB1b RMHD 512 512 0 6 2400 $\nu$
RB1c RMHD 512 512 0 6 1800 $\nu$
RB2a RMHD 1024 256 0 6 6000 $2.5\nu$
RB2b RMHD 1024 1024 0 6 6000 $\nu$
RB2c RMHD 1024 1024 0 6 3200 $\nu$
RB2d RMHD 1024 1024 0 6 1800 $\nu$
RB3a RMHD 2048 512 0 6 15000 $2.5\nu$
RB3b RMHD 2048 2048 0 6 15000 $\nu$
RB3c RMHD 2048 2048 0 6 9000 $\nu$
RB3d RMHD 2048 2048 0 6 5700 $\nu$
RI1 RMHD 512 256 0.45 10 2200 $\nu$
RI2 RMHD 1024 256 0.5 10 5600 $2.5\nu$
RI3 RMHD 2048 512 0.5 10 14000 $2.5\nu$
MB1 MHD 512 256 0 10 2200 $\nu$
MB2 MHD 1024 256 0 10 5600 $2.5\nu$
MI1 MHD 512 256 0.5 10 2200 $\nu$
MI2 MHD 1024 256 0.5 10 5600 $2.5\nu$
MI3 MHD 2048 512 0.5 10 14000 $2.5\nu$
\[tab:simlist\]
: Summary of the numerical runs with different numerical resolutions and different Reynolds numbers. There is no particular scheme used to choose the Reynolds number for a given resolution other than to ensure that the studied scaling properties are well demonstrated and the numerical runs are stable.
![Total field-perpendicular energy spectrum in balanced RMHD as the Reynolds number increases (Cases RB1a, RB2a, RB3a). []{data-label="fig:balanced_rmhd"}](fig1.eps){width="\columnwidth"}
Measurements of the Energy Spectrum {#sec:spectrum}
===================================
The field-perpendicular energy spectrum is obtained by averaging the angle-integrated Fourier spectrum, $$E(k_\perp)=0.5\langle|{\bf v}({\bf k_\perp})|^2\rangle
k_\perp+0.5\langle|{\bf b}({\bf k_\perp})|^2\rangle k_\perp,$$ over field-perpendicular planes in all samples. Identifying the inertial range in numerical simulations with limited resolution is generally difficult, due to the relatively modest separation between the forcing and dissipation scales that current super-computers can afford. For instance, a measurement of the turbulence spectrum for a single Reynolds number is not enough to ensure that the simulated turbulence has converged to the asymptotic universal scaling. Instead, one carries out a set of numerical simulations with increasing resolution and Reynolds number. The spectra are then compensated by the different phenomenological predictions and the preferred model is distinguished by the best fit. In Figures \[fig:balanced\_rmhd\] to \[fig:imbalanced\_mhd\] the inertial range is identified by the flat regions of the spectra compensated by $k^{3/2}$, which extend further to the right with increasing Reynolds number (and resolution).
Figures \[fig:balanced\_rmhd\] and \[fig:balanced\_mhd\] show the total field-perpendicular energy spectrum $E(k_{\perp})$ in the balanced regime for the RMHD and MHD cases, respectively. The RMHD and MHD spectra are remarkably similar, confirming that the pseudo-Alfvén modes are dynamically insignificant and that the RMHD approximation is valid. In both cases the total energy spectrum remains of the form $E (k_{\perp} )\sim k_{\perp}^{-3/2}$ as the Reynolds number increases, with the inertial range starting at $k
\approx 4$ and extending up to $k \gtrsim 30$ in the highest Reynolds number case. In neither RMHD or MHD is there any evidence of a build up of energy close to the dissipative wave-numbers–often referred to as a bottleneck effect– with both spectra falling off smoothly in the dissipative range.
Figures \[fig:imbalanced\_rmhd\] and \[fig:imbalanced\_mhd\] show the field-perpendicular Elsässer spectra in the imbalanced regime for the RMHD and MHD cases, respectively. Again the behavior of both spectra in the RMHD and MHD regimes are very similar. In both cases, it is seen that while $E^-$ keeps the scaling $E^-(k_{\perp}) \sim
k_{\perp}^{-3/2}$ as the Reynolds number increases, the scaling of $E^+(k_{\perp})$ is more difficult to pin down. Indeed, both the RMHD and MHD results for $Re=2200$ yield a steeper spectrum for $E^+(k_{\perp})$, with an exponent possibly nearer to $-5/3$ than $-3/2$. However, we believe that there is no real significance to the value of $-5/3$ here, the exponent is simply steeper than $-3/2$. Indeed, in both cases, as the Reynolds number is increased $E^+(k_{\perp})$ appears to flatten, which means that $E^+(k_\perp)$ has not fully established the universal scaling behavior yet. Since $E^+(k_{\perp})$ and $E^-(k_{\perp})$ are pinned (i.e., converge to each other) at the dissipation scales and are anchored (i.e., independent of the Reynolds number) at the driving scales, we postulate that at sufficiently high $Re$ (where the inertial range is extensive) the spectra will become parallel in the inertial range and attain the scaling $E^\pm(k_{\perp}) \sim k_{\perp}^{-3/2}$. Numerical tests of this prediction must await a significant increase in computational power.
Measurements of dynamic alignment {#sec:alignment}
=================================
An important test that can be performed in the presented simulations concerns the so-called dynamic alignment angle. This angle is [ *defined*]{} by the following ratio of the two specially constructed structure functions [@mason_etal06]: $$\theta(l) = \frac{\left\langle|\delta {\mathbf{v}}_\perp({\mathbf{l}})\times\delta {\mathbf{b}}_\perp({\mathbf{l}})|\right\rangle}{\left\langle|\delta {\mathbf{v}}_\perp({\mathbf{l}})||\delta
{\mathbf{b}}_\perp({\mathbf{l}})|\right\rangle},
\label{daeq}$$ where $\delta {\mathbf{v}}_\perp({\mathbf{l}})$ and $\delta {\mathbf{b}}_\perp({\mathbf{l}})$ are the field-perpendicular velocity and magnetic field increments, respectively, corresponding to the field-perpendicular scale separation ${\bf l}$. (We note that in definition (\[daeq\]) we have assumed that the angle is small, and hence no distinction between $\theta(l)$ and $\sin \theta(l)$ is made. Hereafter, by $\theta(l)$ we will always understand the quantity (\[daeq\])).
As proposed in [@boldyrev_05; @boldyrev_06] the alignment angle $\theta(l)$ has a nontrivial scaling with $l$, which may explain the observed $-3/2$ scaling exponent of the energy spectrum. As discovered in [@mason_etal11], the scale dependent dynamic alignment exists not only in the inertial interval, but it also extends into the dissipation range and is limited only by the grid size of the numerical scheme. We will demonstrate that the alignment angle scaling provides a sensitive test probing the turbulent cascade deep in the dissipation interval. In particular we will see that if the simulated dissipation range is under-resolved (e.g., as a result of the use of too large a Reynolds number or strongly anisotropic resolution), the dynamic alignment can be easily spoiled at the dissipation scales even if it is present in the inertial interval.
The measurements of the alignment angle are presented in Figure \[fig:alignment\_balanced\_rmhd\]. The first panel shows three simulations (RB2d,c,b in Table \[tab:simlist\]) performed at the same numerical resolution of $1024^3$ but with different Reynolds numbers $Re=1800,\, 3200, \, 6000$. Plots for $Re=1800, \, 3200$ show a remarkable property of the alignment scaling: It extends deep down into the dissipation region, practically up to the scale of the numerical discretization, [*independently*]{} of the Reynolds number (see also [@mason_etal11]). However, this behavior is spoiled if the Reynolds number is pushed to very high values, at which the dissipation interval becomes under-resolved. In this case, the scaling starts to degrade at large wave-numbers, as is seen in the case $Re=6000$.
The alignment scaling is however restored back to its original value if the numerical resolution is increased to $2048^3$, so that the dissipation scales become well resolved again. This is seen from comparison of the plot for RB2b ($1024^3$, $Re=6000$) in the first panel of Figure \[fig:alignment\_balanced\_rmhd\] with the plot for RB3d ($2048^3$, $Re=5700$) in the second panel. Further increase of the Reynolds number in the second panel of this figure demonstrates that the alignment scaling is stable up to $Re=9000$ (RB3c, $2048^3$), however, it starts to degrade at large wave-numbers for higher Reynolds numbers $Re=15000$ (RB3b, $2048^3$), in complete analogy with the behavior depicted in the upper panel of Fig. \[fig:alignment\_balanced\_rmhd\] at smaller resolution. The alignment angle is spoiled even more in the run RB3a ($2048^2\times
512$, $Re=15000$) in the same figure where we simultaneously decrease the field-parallel numerical resolution, making the dissipation interval even more under-resolved. Note however, that in both the first and the second panels of Fig. \[fig:alignment\_balanced\_rmhd\], the heaviest distortion of the alignment behavior occurs in the dissipation region, while the inertial interval (approximately contained between the two vertical lines) is relatively unaffected. This may explain why an under-resolved dissipation interval is not manifest in the scaling of energy spectra, as seen in Figure \[fig:2048\_spectra\_comparison\].
Fig. \[fig:alignment\_balanced\_rmhd\_resolved\] shows three well resolved simulations with numerical resolutions increasing from $512^3$ to $1024^3$ to $2048^3$. We observe that the scaling interval of the alignment angle becomes progressively longer and its scaling index stays close to the predicted value $1/4$ [@boldyrev_05] with little or practically no dependence on the Reynolds number [^3]. This means that we observe a truly universal scaling behavior of the dynamic alignment. The lower panel of Fig. \[fig:alignment\_balanced\_rmhd\_resolved\] shows the same curves where the spatial scale is normalized by the dissipation length. We observe that the flattened parts of the curves at small scales do not overlap under such rescaling, which supports our observation mentioned above that the extent of the scaling interval is not defined solely by the dissipation scale, but rather depends on the numerical discretization step.
Energy spectrum: Kolmogorov constant and dissipation scale {#sec:kolmogorov}
==========================================================
\[energy\_scaling\] For a more complete study of the energy spectrum, one can also evaluate the amplitude of the spectrum and the dissipation scale for each simulation and verify that they agree with a given phenomenology. Since our spectral scaling conforms to the phenomenology of Boldyrev [@boldyrev_05; @boldyrev_06], we now study in more detail the scaling associated with this model. First, we need to derive the expression for the energy spectrum, which is done in the following way [@boldyrev_05]. The time of nonlinear interaction at field-perpendicular scale $\lambda$ in this model is $\tau\sim \lambda/(v_\lambda \theta_\lambda)$, where $v_\lambda$ denotes the typical (rms) velocity fluctuations, $\theta_\lambda =\theta_0(\lambda/L_\perp)^{1/4}$ is the scale-dependent alignment angle between magnetic and velocity fluctuations, which was studied in the previous section, and $\theta_0$ is the typical alignment angle at the outer scale (forcing scale) $L_\perp$. The rate of energy cascade is then evaluated as $\epsilon=v_\lambda^3\theta_\lambda/\lambda$, from which it follows that $E(k_\perp)\sim \epsilon^{2/3}(\theta_0/L_\perp^{1/4})^{-2/3}k_\perp^{-3/2}.$ One however notices that the amplitude of the energy spectrum is not uniquely defined in this equation, since the outer-scale quantities $\theta_0$ and $L_\perp$ essentially depend on the forcing routine. This is understood from the following example. Assume that the large-scale force drives only unidirectional Alfvén waves $z^+$, for which ${\bf v}$ is perfectly aligned with ${\bf b}$ and $\theta_0=0$. Then the wave energy will grow without bound, since the nonlinear interaction leading to the energy cascade and eventual dissipation at small scales is absent.
Even when a particular forcing routine is specified, the definitions of the values of $\theta_0$ and $L_\perp$ are still subjective since they essentially rely on the outer-scale properties of turbulence rather than on the measurements of the inertial interval. We now propose that this problem can be remedied in an efficient way. For that we notice that there exists a well-defined quantity that is remarkably stable (scale-independent) in the inertial interval: $$\begin{aligned}
\Lambda^{-1/4}=\theta(l)/l^{1/4},
\label{eq:Lambda}\end{aligned}$$ where $\theta(l)$ is defined in (\[daeq\]), see the discussion in the preceding section. In this definition one can use [*any*]{} scale $l$ from the inertial interval or dissipation interval if the numerical simulations are well resolved. A somewhat simpler rule can be used in numerical (or observational) studies, where one does not have to know a priori what scales correspond to the inertial interval and does not have the luxury of having the plot in Fig. \[fig:alignment\_balanced\_rmhd\_resolved\] available. In this case $l$ in formula (\[eq:Lambda\]) can be chosen to be the Taylor micro-scale based on either the magnetic or the velocity fluctuations, $l=v_{rms}/|\nabla \times {\bf v}|_{rms}$ or $l=b_{rms}/|\nabla \times
{\bf b}|_{rms}$, assuming the magnetic Prandtl number is of order one. We therefore propose the following normalization of the energy spectrum: $$\begin{aligned}
E(k_\perp)=C_k \epsilon^{2/3}\Lambda^{1/6}k_\perp^{-3/2},
\label{eq:en_spectrum}\end{aligned}$$ where $\Lambda$ is [*defined*]{} by (\[eq:Lambda\]). The scale $\Lambda$ that is defined solely through the inertial-interval quantities, incorporates the essential information about the cross-helical structure of MHD turbulence. It is not uniquely defined by the outher scale of the turbulence, rather it also depends on the large-scale driving mechanism. Therefore, the inertial-interval energy spectrum is defined by the [*two*]{} quantities $\epsilon$ and $\Lambda$, characterizing the energy cascade rate and the level of cross-helical organization of the flow. The presence of the two quantities characterizing the spectrum of MHD turbulence (as oppose to only one quantity in hydrodynamic turbulence) is the manifestation of the two conserved quantities cascading toward small scales in MHD turbulence: energy and cross-helicity.
We expect that the constant $C_k$ in (\[eq:en\_spectrum\]) may be “universal," that is, largely independent of the character of the driving, analogous to the Kolmogorov constant in hydrodynamical turbulence. This constant can be measured in our simulations in the following way. First, we specify $l$ that we use to measure the alignment scale $\Lambda$ in (\[eq:Lambda\]). According to our plots in Figs. \[fig:alignment\_balanced\_rmhd\] and \[fig:alignment\_balanced\_rmhd\_resolved\], we may choose $l=0.07L_\perp$, say, as a scale belonging to the inertial interval and not yet affected by the numerical resolution effects. Then, for simulations RB1a, RB2a, RB3a we find $\Lambda=1.34L_\perp,1.41L_\perp,1.48L_\perp$, respectively.
The dissipation rate can be evaluated based on the energy spectrum (\[eq:en\_spectrum\]) as follows: $$\begin{aligned}
\epsilon = \int E(k_\|, k_\perp)(\nu k_\perp^2 + \nu_\| k_\|^2)
dk_\|dk_\perp .
\label{epsilon}\end{aligned}$$ Our numerical results confirm that the integral of $\nu_\| k_\|^2$ leads to a negligible correction to the dissipation rate, and therefore it can be omitted, and we can use the field-perpendicular spectrum $E(k_\perp)=\int E(k_\|, k_\perp) dk_\|$. Then, for simulations RB1a, RB2a, RB3a we find: $\epsilon=0.15,0.15,0.16$.
The dissipation scale can be found (or defined) based on the energy spectrum. Omitting the dimensionless constants, we then accept, by definition, $$\begin{aligned}
\eta = \epsilon^{-2/9} \Lambda^{1/9} \nu^{2/3}.
\label{eta}\end{aligned}$$ We can demonstrate that our simulations agree with this scaling by plotting the energy spectra in the balanced case (RB1a,2a,3a) versus the wave-vector normalized with the dissipation scale , where we measure the dissipation rate directly from the simulations via (\[epsilon\]), and the alignment scale from (\[eq:Lambda\]). The top frame in figure \[fig:balanced\_rmhd\_convergence\] shows that in this case the dissipative region starts around $k\eta\approx
0.1$, independent of the Reynolds number. The extent of the inertial range, defined as the ratio between the scale $l_0$ at the beginning of the inertial range (from figure \[fig:balanced\_rmhd\], $k = 4$ and hence $l_0 \approx
L_\perp/8$) and the dissipation scale $l_d\approx L_\perp/(2k_d)=5\eta L_\perp$, where $k_d=0.1/\eta$ from figure \[fig:balanced\_rmhd\_convergence\]), increases up to one decade in the RB3a case [^4]. Note that with the wave vector normalized with the single parameter $\eta$, the [*whole spectra*]{} collapse onto each other, thus providing additional evidence that the universal functional behavior of the spectrum is obtained in our simulations. The lower plot in figure \[fig:balanced\_rmhd\_convergence\] shows that the length of the inertial range increases as $l_0/l_d\sim Re^{2/3}$, also in good agreement with the estimate for the dissipation scale (\[eta\]). The “Kolmogorov constant” $C_k$ can be evaluated from the upper plot as $C_k\approx 2$.
Discussion {#sec:discussion}
==========
We have presented results from state-of-the-art direct numerical simulations of balanced and imbalanced driven MHD turbulence. The simulations are achieved at the extremely large numerical resolution of $2048^3$ and the longest running time, with many runs spanning more than a hundred eddy turnover times in the steady state. The simulations were performed using two pseudo-spectral codes, one solving the MHD equations and the other solving the RMHD equations. In theories and simulations of MHD turbulence, it has long been argued that RMHD provides a correct and accurate framework for investigating the universal properties of MHD turbulence both in the weak and strong turbulence regimes. We have presented a direct comparison of high-resolution numerical simulations of MHD vs RMHD turbulence using two independently developed pseudo-spectral codes with identical parameters. It is shown that in the strong turbulence regime, in both the balanced and imbalanced state, the energy spectrum of the Elsässer variables in MHD and RMHD are in remarkable agreement (for details of a lower resolution comparison, including the individual velocity and magnetic spectra and the alignment angle, see [@mason_etal12]). These results are of essential value for MHD turbulence research, as simulating MHD turbulence can be accomplished using RMHD codes that generally incur a smaller computational cost.
In the balanced case, the simulated energy spectra of $E^+$ and $E^-$ show a clearly identifiable inertial range, consistent with a slope of $k_\perp^{-3/2}$ for both $E^+$ and $E^-$. It is observed from Figures \[fig:balanced\_rmhd\] and \[fig:balanced\_mhd\] that the compensated energy spectra show a flat region that extends as the Reynolds number is increased. This is consistent with previous, lower resolution simulations of strongly magnetized MHD turbulence, e.g., [@maron_g01; @haugen_04; @muller_g05; @mininni_p07; @mason_cb08; @chen_11; @tobias_etal2011]. In the imbalanced case, the interpretation of the numerical results is not as straightforward. Figures \[fig:imbalanced\_rmhd\] and \[fig:imbalanced\_mhd\] show that the energy spectrum of $E^-$ remains reasonably close to $k_\perp^{-3/2}$, only slightly changing its overall amplitude for small Reynolds numbers. As for the $E^+$ spectrum, the compensated spectrum shows a slope slightly steeper than $-3/2$ which however flattens as the Reynolds number increases. Another observation from the large Reynolds number imbalanced numerical simulations is that the spectra of $E^+$ and $E^-$ are “anchored” at large scales and “pinned” at the dissipation scale. From these results we propose that the energy spectra of $E^+$ becomes asymptotically closer to $k_\perp^{-3/2}$ as the Reynolds number is increased. Much higher resolutions, exceeding the capabilities of today’s supercomputers, are required to conclusively demonstrate this conjecture.
Finally, during the refereeing process, our attention was drawn to recent publications by the group of Beresnyak & Lazarian [@beresnyak_l2010; @beresnyak_11; @beresnyak_12], in which the authors address issues similar to the ones contained in this paper. Most of the [conclusions]{} of those papers appear to be at odds with ours (and with similar results or other groups, e.g., [@maron_g01; @haugen_04; @muller_g05; @mininni_p07; @mason_cb08; @chen_11]). We however note that the actual [numerical results]{} presented in [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] agree with ours in the range of scales that we study, while they differ from ours at very large wavenumbers, e.g., $k\gtrsim 50$ in the runs with highest resolution. Beresnyak & Lazarian suggest that the true inertial interval exists only at these large wavenumbers where they perform their measurements of the scaling relations. The formal cause of the disagreement of our conclusions with those by Beresnyak & Lazarian is thus the numerical measurements being performed in essentially different regions of the phase space.
The question however remains as to what causes the results of the numerical simulations by Beresnyak & Lazarian to disagree with ours at small, sub-inertial scales. According to our analysis, the answer is the following: the $k$-space intervals on which references [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] base their conclusions are significantly affected by numerical effects due to the numerical setup they use. It is not appropriate in their simulations to use those intervals for addressing either the inertial or the dissipation regimes. We however note that the dissipation-range dynamics and the behavior of the numerical solution of the MHD equations close to the numerical cutoff is an interesting and not well studied question. It is therefore worth addressing the differences between our simulations and those by [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] in more detail. Since such analysis is not the main objective of the present work, we have presented the corresponding discussion in the Appendix.
This work was supported by the NSF/DoE partnership grant NSF-ATM-1003451 at the University of New Hampshire, the NSF sponsored Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas at the University of Chicago and the University of Wisconsin - Madison, the US DoE awards DE-FG02-07ER54932, DE-SC0003888, DE-SC0001794, the NSF grants PHY- 0903872 and AGS-1003451, and the DoE INCITE 2010 Award. This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357. The studies were also supported by allocations of advanced computing resources provided by the NSF TeraGrid allocation TG-PHY110016 at the National Institute for Computational Sciences and the PADS resource (NSF grant OCI-0821678) at the Computation Institute, a joint institute of Argonne National Laboratory and the University of Chicago.
APPENDIX: Numerical study of MHD tubrulence at subrange scales {#sec:appendix .unnumbered}
==============================================================
In this Appendix we comment on the numerical reconstruction of solution of the MHD equations at small scales, that is, scales within the dissipation range and close to the numerical cutoff in $k$-space (the dealiasing cutoff in a pseudo-spectral code). Recent publications by Beresnyak & Lazarian [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] found that the energy spectrum in this region (roughly corresponding to $k\gtrsim 50$ in their highest-resolution runs) can have a peculiar structure that is inconsistent with the structure and the scaling found in our numerical simulations. Much confusion was created by the suggestion by [@beresnyak_11; @beresnyak_12] that this high-$k$ region is, in fact, the true inertial interval of MHD turbulence, while the region that is studied in our works (corresponding to $4 \lesssim k \lesssim 30$ in highest-resolution runs) is a “non-converged" forcing-dominated region.
It is therefore useful to address the small-scale numerical solution of the MHD equations in more detail and to relate our findings and conclusions to those presented in the recent works by Beresnyak and Lazarian [@beresnyak_l2010; @beresnyak_11; @beresnyak_12]. These references claim that the energy spectral index of MHD turbulence is $-5/3$ and that there is no conclusive evidence for dynamic alignment in the numerical results. In discussing what could lead to such (in our opinion, erroneous) conclusions it is useful to distinguish two factors. One is related to differences that arise because the simulations by Beresnyak & Lazarian that allegedly are identical to ours, in fact are not identical at all because of differences in the details of the numerical setup. The other is related to the methods that are used to analyze the results and, ultimately, support one claim or another. Both play a role in the origin of the disagreement.
First, we concentrate on issues that result from the different setup of the numerical simulations. In our previous publications (e.g., [@perez_b10_2; @mason_etal12]) we have discussed at length those aspects of the simulation design that are essential for accurately capturing the physics of the strong turbulent cascade. It is not necessary to repeat those discussions here, however, it is important to point out that many of the simulations of Beresnyak and Lazarian [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] differ from ours through their choice of numerical hyper-dissipation, significantly smaller viscosities for a given numerical resolution, and a considerably smaller statistical ensemble from which averages are computed. Each of these factors is potentially detrimental for the observation of the correct scaling behavior. For example, the measurements of the alignment angle that are shown in Figure 3 of reference [@beresnyak_11] and Figure 2 of reference [@beresnyak_12] lead Beresnyak to conclude that dynamic alignment is not present in MHD turbulence as the alignment angle saturates, i.e. flattens as a function of $l$ at small $l$, when the Reynolds number increases. However, those plots exhibit a behavior that is qualitatively similar to that displayed in our Figure \[fig:alignment\_balanced\_rmhd\], where insufficient numerical resolution is demonstrated to affect the alignment angle at small scales. It therefore reasonable to conclude that the observed flattening of the alignment angle in the simulations of references [@beresnyak_11; @beresnyak_12] is an artifact of unresolved dissipation scales and, possibly, part of the inertial-range scales, rather than a physical effect.
The influence of hyper-dissipation may be similarly assessed from comparing the energy spectra obtained in our work with the energy spectra obtained in, say, reference [@beresnyak_11]. Our spectra in Figures \[fig:balanced\_rmhd\], \[fig:balanced\_mhd\], \[fig:2048\_spectra\_comparison\] & \[fig:balanced\_rmhd\_convergence\] exhibit an extended interval with the scaling $k^{-3/2}$, identified as the inertial interval, followed by a steep decline, identified as the dissipation range. The spectra in Figure 2 of reference [@beresnyak_11] also show an extended interval with the scaling $k^{-3/2}$ (interpreted in reference [@beresnyak_11] as a “non-converged” range) followed at large wavenumbers by a very short steepening (interpreted in [@beresnyak_11] as the “inertial interval”) and then flattening and ultimate cut-off. In our view, such an interpretation is incorrect; the spectral behavior observed in reference [@beresnyak_11] close to the dissipation region is not a property of the inertial interval, but rather is evidence of the so-called bottleneck effect that is expected when numerical hyper-dissipation is present. Indeed, as discussed in references [@frisch_etal2008; @cichowlas_2005], an energy spectrum abruptly terminated in $k$-space by hyper-dissipation or by other Galerkin-type truncation mechanisms, exhibits an inertial interval followed by a pseudo-dissipation region (steepening of the spectrum), then by a partly thermalized region (a rise in the spectrum), and then by a far dissipation range (ultimate cut-off). The measurements presented in References [@beresnyak_11; @beresnyak_12] are consistent with such spectral behavior, which motivates a natural explanation of their results as an inertial interval with the $-3/2$ scaling, modified by a substantial bottleneck effect close to the dissipation scales. Moreover, a thermalization brought about by sharp termination of the spectrum in the $k$-space tends to decorrelate small-scale fluctuations, which otherwise would remain strongly aligned throughout the dissipation interval, cf. our Figure \[fig:alignment\_balanced\_rmhd\_resolved\]. This is also consistent with the significant loss of dynamic alignment at small scales that is observed in references [@beresnyak_11; @beresnyak_12].
A more detailed comparison of our results can be made with those MHD simulations by Beresnyak [@beresnyak_12] that employ a physical Laplacian dissipation (simulations R8 & R9 in [@beresnyak_12]). By evaluating the Reynolds numbers for those calculations in the same way that it is done in our work, $Re=v_{rms}L/(2\pi \nu)$ with $v_{rms}\approx 1$, we find that simulation R8, with a resolution $768^3$ mesh points, is performed at the Reynolds number $Re\approx 8000$, while calculation R9 (resolution $1536^3$) is performed at $Re\approx 20000$. According to our results in Figure \[fig:alignment\_balanced\_rmhd\], in the simulations with a resolution of $1024^3$ mesh points the dissipation interval is under-resolved already at $Re\approx 6000$, while in the $2048^3$ simulations the dissipation interval is under-resolved at $Re\approx 15000$. Thus, the runs R8 & R9 of reference [@beresnyak_12] that are most similar to ours have [*lower*]{} numerical resolutions while [*higher*]{} Reynolds numbers. Therefore, they have essentially unresolved dissipation intervals and, possibly, parts of the inertial intervals.
The lack of resolution at the bottom of the inertial intervals in the simulations R8 & R9 can also be seen from the alignment-angle curves shown in Figure 2 of [@beresnyak_12]. Under the rescaling applied in that figure, the curves should approach each other in the inertial interval, as they do in our Fig. \[fig:alignment\_balanced\_rmhd\_resolved\], lower panel. In contrast, one can see only a short region of in Figure 2 of [@beresnyak_12] (runs R8 & R9) where the curves approach each other, approximately within the range $20 \lesssim l/\eta \lesssim 40$. Apparently, this is the only piece of the inertial interval that is resolved, and in this interval the scaling exponent of the angle indeed approaches $1/4$, see Figure 3 in reference [@beresnyak_12], as expected according to our results.
We now turn to the second factor that contributes to the differing conclusions drawn by the Beresnyak & Lazarian group, namely the method of analysis. We recall that the objective is to determine the scaling behaviour within the inertial range. Concerning the energy spectrum, we assess whether the numerical data preferentially supports $E(k_\perp)\propto k_\perp^{-3/2}$ or $E(k_\perp)\propto k_\perp^{-5/3}$ directly by compensating the numerical data by $k^{3/2}$ and by $k^{5/3}$ in turn. For the correct model, the inertial range then corresponds to the range of scales over which the compensated spectrum is flat. We always find that $k_\perp^{-3/2}$ provides the better fit, with the inertial range starting at $k_\perp\approx 4$ and extending up to $k_\perp\gtrsim 30$ at highest resolution. As the Reynolds number increases, numerical convergence is demonstrated by the fact that this region maintains its amplitude and scaling and increases in extent to larger wavenumbers, see, e.g. our Figure \[fig:balanced\_rmhd\].
In contrast, Beresnyak [@beresnyak_12] uses an [*indirect*]{} method to select the preferred spectral exponent. He uses the two phenomenological models that describe the inertial range characteristics to predict the dissipation scales ($\eta$), plots the compensated spectrum as a function of the dimensionless wavenumber $k\eta$, and identifies the preferred model as that which displays the better convergence properties at [*large*]{} wavenumbers $k\eta$ as the Reynolds number increases. Figure 1 of Beresnyak [@beresnyak_12] lead him to conclude that it is the $-5/3$ model that displays the better convergence properties at large $k$.
It can be shown, however, that the convergence at small scales observed in [@beresnyak_11; @beresnyak_12] is a simple artifact of the numerical setup adopted in [@beresnyak_11; @beresnyak_12], rather than a physical effect. To explain this, we note that any discrete numerical scheme solves only the corresponding discrete algebraic equations. If the numerical setup is done correctly, the numerical solution approximates the physical one independently of the discretization step. If, however, a special numerical setup is adopted where $\eta$ is rigidly tied to the grid size such that $\eta N$ is kept the same in all runs (as is done in [@beresnyak_11; @beresnyak_12]), then the numerical solution plotted as a function of $k\eta$ is always affected by the discretization in the same way, thus consistently reproducing the same small-scale numerical effects that are present in the setup. The convergence at large $k\eta$ is then the convergence among solutions of the given numerical scheme, which should not be confused with the convergence to the physical solution.
To illustrate this effect in our simulations we replot the spectra presented in Fig. \[fig:balanced\_rmhd\_convergence\] choosing the Kolmogorov normalization scale $\eta_{K41}= \nu^{3/4}\epsilon^{-1/4}$. Due to a particular choice of viscosities in our runs depicted in Fig. \[fig:balanced\_rmhd\_convergence\], in this case $\eta_{K41}$ happens to double every time the resolution decreases by a factor of $2$, thus ensuring that $\eta_{K41} N_\perp = {\rm const}$, see Fig. \[fig:balanced\_rmhd\_convergence\_k41\]. It is therefore not surprising that all the curves converge in the vicinity of the numerical dealiasing cutoff corresponding to $k\eta_{K41}\approx 0.8$, while they do not converge in the inertial interval and in the most of the dissipation interval. A similar, by design, convergence is present in Fig. 2 of [@beresnyak_11] and Fig. 1 of [@beresnyak_12]. Such convergence at very small scales is a spurious numerical effect, which does not reflect the convergence of the physical solutions, and which cannot give preference to any phenomenological model. When the viscosities in different runs do [*not*]{} conform to the special condition $\eta N_\perp={\rm const}$, the spurious convergence disappears, and the $-5/3$ model does not fit the data, while the $-3/2$ model still provides a good fit in the inertial and dissipation intervals, see Fig. \[fig:balanced\_rmhd\_convergence\_k41\_vs\_DA\].
We therefore conclude that the numerical simulations by Beresnyak & Lazarian group [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] are likely significantly affected by numerical effects at small scales where their measurements are performed. This is notwithstanding the statements made in [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] that the simulations are resolved in those works. These statements, in our opinion, are not supported by the factual numerical data presented in these papers. Until the effects of hyper-dissipation are better understood and numerical convergence is demonstrated in the settings of [@beresnyak_l2010; @beresnyak_11; @beresnyak_12], it is hard to assess fully the degree to which the numerical findings of [@beresnyak_l2010; @beresnyak_11; @beresnyak_12] can be compared with our results, or with similar results of other groups [e.g., @maron_g01; @haugen_04; @muller_g05; @mininni_p07; @chen_11].
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A. [Beresnyak]{}. *[Basic properties of magnetohydrodynamic turbulence in the inertial range]{}*. MNRAS **422**, 3495 (2012).
U. [Frisch]{}, S. [Kurien]{}, R. [Pandit]{}, W. [Pauls]{}, S. S. [Ray]{}, A. [Wirth]{}, and J.-Z. [Zhu]{}. *[Hyperviscosity, Galerkin Truncation, and Bottlenecks in Turbulence]{}*. Physical Review Letters **101**, 144501 (2008).
C. [Cichowlas]{}, P. [Bona[ï]{}ti]{}, F. [Debbasch]{}, and M. [Brachet]{}. *[Effective Dissipation and Turbulence in Spectrally Truncated Euler Flows]{}*. Physical Review Letters **95**, 264502 (2005).
[^1]: A simple geometrical consideration shows that a given degree of imbalance sets an upper boundary on the allowed alignment angle between the magnetic and velocity vectors. The larger the imbalance $z^+/z^-$, the smaller the allowed angle between magnetic and velocity fluctuations. The opposite is, however, not necessarily true.
[^2]: Turbulence may also be driven by driving ${\bf v}$ or ${\bf b}$ fluctuations at large scales; this does not affect the inertial interval, see [@mason_cb08].
[^3]: The actual values of the found numerical slopes are, in fact, slightly smaller than $1/4$ and, possibly, closer to $0.22$. According to our phenomenological picture this would correspond to the energy spectrum $-1.52$, which is indistinguishable, on a phenomenological level, from the predicted $-1.5$. Possible origins of the discrepancy may include not large enough Reynolds numbers or/and small intermittency effects.
[^4]: The correspondence $kl_k=\pi$ between eddy scale $l_k$ and wave-vector $k$ is assumed.
|
---
abstract: 'This gives an alternate proof of the result of [@chern3 Theorem 2.1]: The first Hilbert coefficient of parameter ideals in an unmixed Noetherian local ring is always negative unless the ring is Cohen–Macaulay.'
address:
- 'Department of Mathematics, New York City College of Technology-Cuny, 300 Jay Street, Brooklyn, NY 11201, U. S. A.'
- 'Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan'
- 'Department of Mathematics, Southern Connecticut State University, 501 Crescent Street, New Haven, CT 06515-1533, U. S. A.'
- 'Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan'
- 'Department of Information Technology and Applied Mathematics, Ton Duc Thang University, 98 Ngo Tat To Street, Ward 19, Binh Thanh District, Ho Chi Minh City, Vietnam'
- 'Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019, U. S. A.'
author:
- 'L. Ghezzi'
- 'S. Goto'
- 'J. Hong'
- 'K. Ozeki'
- 'T.T. Phuong'
- 'W. V. Vasconcelos'
title: Negativity Conjecture for the First Hilbert Coefficient
---
[^1]
Introduction
============
Let $R$ be a Noetherian local ring with the maximal ideal $\m$ of dimension $d >0$. Let $I$ be an $\m$–primary $R$–ideal. For sufficiently large $n$, the length $\l(R/I^{n+1})$ is of polynomial type : $$P_{I}(n)=\sum_{i=0}^{d} (-1)^i e_i(I) {{n+d-i}\choose{d-i}}.$$ The integers $e_i(I)$’s are called the [*Hilbert coefficients*]{} of $I$. The first Hilbert coefficient $e_1(Q)$ of a parameter ideal $Q$ codes structural information about the ring $R$ itself. In response to a question in [@chern1], the following was settled in [@chern3].
\[e1neg\][([@chern3 Theorem 2.1])]{} An unmixed Noetherian local ring $R$ is not Cohen-Macaulay if and only if $e_1(Q)<0$ for a parameter ideal $Q$.
Meanwhile, for any parameter ideal $Q$ of $R$, it was proved that $e_1(Q)\leq 0$ ([@chern3 Corollary 2.5], [@MV]). Hence the above theorem can be rephrased as follows :
\[e1zero\] An unmixed Noetherian local ring $R$ is Cohen–Macaulay if and only if $e_1(Q)=0$ for some parameter ideal $Q$ of $R$.
In the following section, we give an alternate proof.
The Proof
=========
[**Proof of Theorem \[e1neg\]**]{} We use a setup developed in [@chern2]. It is enough to show that if $R$ is not Cohen–Macaulay, then $e_1(Q) <0$. We may assume that the residue field is infinite.
By passing to $\m$–adic completion $\widehat{R}$, we may also assume that $R$ is complete. Then there exists a Gorenstein local ring $(S, \n)$ of dimension $d=\dim(R)$ such that $R$ is a homomorphic image of $S$. This means that there exists a canonical module $\omega_R = \Hom_S(R, S)$. Consider the natural homomorphism $$\varphi : R \lar \Hom_S(\omega_R, \; \omega_R) \simeq \Hom_S(\omega_R,\; S).$$ Because $R$ is unmixed, this map $\varphi$ is injective ([@Aoy 1.11.1]). Moreover $H^1_{\n}(R)$ has finite length. Indeed, let $A=\Hom_S(\omega_R, \; \omega_R)$. Then by applying local cohomology to $0 \rar R \rar A \rar D \rar 0$, we obtain $H^{1}_{\n}(R) \simeq H^{0}_{\n}(D)$ since $\depth(A) \geq 2$ ([@GNa], [@HU]).
By dualizing $S^n \rar \omega_R \rar 0$ into $S$, we obtain another injective map $$0 \lar \Hom_S(\omega_R,\; S) \lar S^n.$$ Composing these two maps, we obtain an embedding ${\ds R \hookrightarrow S^n}$.
Let $Q$ be a parameter ideal of $R$. Then there exists a parameter ideal $\q$ of $S$ such that $\q R =Q$ ([@chern2 Lemma 3.1]). Therefore the associated graded ring of $Q$ is isomorphic to the associated graded module of $\q$ with respect to the $S$–module $R$ : $$\gr_Q(R) \simeq \gr_{\q}(R),$$ which implies that $$e_1(Q) = e_1(\q, R),$$ where $e_1(\q, R)$ denotes the first Hilbert coefficient of $\q$ with respect to $S$–module $R$.
Consider the exact sequence of $S$–modules: $$0 \lar R \lar S^n \lar C \lar 0.$$ Let $y$ be a superficial element for $\q$ with respect to $R$ such that $y$ is a part of minimal generating set of $\q$ and that $y \not\in \Ass_S(C) \setminus \{ \n \}$. By tensoring the exact sequence of $S$–modules with $S/(y)$, we get $$0 \lar T =\Tor_1^S(S/yS, C) \lar R/yR \stackrel{\zeta}{\lar} S^n/y S^n \lar C/yC \lar 0.$$ Let $R\,'= R/yR$ and $S\,'=\im(\zeta)$ and consider the short exact sequence : $$0 \lar T \lar R\,' \lar S\,' \lar 0.$$ Then either $T=0$ or $T$ has finite length $\l(T) < \infty$.
Now we use induction on $d=\dim(R)$ to show that if $R$ is not Cohen–Macaulay, then $e_1(\q, S) <0$.
Let $d=2$ and $\q=(y, z)$. Then $T \neq 0$ so that $\l(T) < \infty$. Applying the Snake Lemma to $$\begin{CD}
0 @>>> T \cap z^n R\,' @>>> z^n R\,' @>>> z^n S\,' @>>> 0 \\
& & @VVV @VVV @VVV \\
0 @>>> T @>>> R\,' @>>> S\,' @>>> 0
\end{CD}$$ we get, for sufficiently large $n$, $$\l(R\,'/z^n R\,') = \l(T) + \l(S\,'/z^n S\,').$$ Computing the Hilbert polynomials, we have $$e_1(\q/y, R/yR) = -\l(T) <0$$ so that $$e_1(\q, R) = e_1(\q/y, R/yR) - \l(0:_R y) = -\l(T) - \l(0:_R y) < 0.$$
Now suppose that $d \geq 3$. From the exact sequence $$0 \lar T \lar R\,'=R/yR \lar S\,' \lar 0,$$ we have $$e_1(\q, R) = e_1(\q/(y) , R/yR ) = e_1(\q/(y) , S\,').$$ By an induction argument, it is enough to show that $S\,'$ is not Cohen–Macaulay since $\dim(S/yS)=d-1$.
Suppose that $S\,'$ is Cohen–Macaulay. Let $\n$ be the maximal ideal of $S/yS$. From the exact sequence $$0 \lar T \lar R\,'=R/yR \lar S\,' \lar 0,$$ we obtain the long exact sequence: $$0 \rar H_{\n}^0(T) \rar H_{\n}^0(R\,') \rar H_{\n}^0(S\,') \rar H_{\n}^1(T) \rar H_{\n}^1(R\,') \rar H_{\n}^1(S\,').$$By the assumption that $S\,'$ is Cohen–Macaulay of dimension $d-1 \geq 2$ and the fact that $T$ is a torsion module, we get
$$0 \rar T \stackrel{\simeq}{\lar} H_{\n}^0(R\,') \rar 0 \rar 0 \rar H_{\n}^1(R\,') \rar 0.$$
We may assume that $y$ is a nonzerodivisor on $R$. From the exact sequence $$0 \lar R \stackrel{\cdot y}{\lar} R \lar R/yR \lar 0,$$ we obtain the following exact sequence: $$0 \lar {T \simeq H_{\n}^0(R\,')} \lar H_{\n}^1(R) \stackrel{\cdot y}{\lar} H_{\n}^1(R) \lar {H_{\n}^1(R\,')=0}.$$ Since $H_{\n}^1(R) $ is finitely generated and ${\ds H_{\n}^1(R)=yH_{\n}^1(R)} $, we have ${\ds H_{\n}^1(R)=0}$. This means that $T=0$. Therefore $${\ds 0 \rar T=0 \rar R/yR \stackrel{\simeq}{\lar} S\,' \rar 0.}$$ Since $S\,'$ is Cohen–Macaulay, $R\,'=R/yR$ is Cohen–Macaulay. Since $y$ is regular on $R$, $R$ is Cohen–Macaulay, which is a contradiction.
[99]{}
[^1]: [AMS 2000 [*Mathematics Subject Classification:*]{} 13H10, 13H15, 13A30.]{}\
The first author is partially supported by a grant from the City University of New York PSC-CUNY Research Award Program-40. The second author is partially supported by Grant-in-Aid for Scientific Researches (C) in Japan (19540054). The fourth author is supported by a grant from MIMS (Meiji Institute for Advanced Study of Mathematical Sciences). The fifth author is supported by JSPS Ronpaku (Dissertation of PhD) Program. The last author is partially supported by the NSF
|
---
abstract: |
Deep neural networks have been shown to suffer from a surprising weakness: their classification outputs can be changed by small, non-random perturbations of their inputs. This *adversarial example phenomenon* has been explained as originating from deep networks being “too linear” [@goodfellow2014explaining]. We show here that the linear explanation of adversarial examples presents a number of limitations: the formal argument is not convincing; linear classifiers do not always suffer from the phenomenon, and when they do their adversarial examples are different from the ones affecting deep networks.
We propose a new perspective on the phenomenon. We argue that adversarial examples exist when the classification boundary lies close to the submanifold of sampled data, and present a mathematical analysis of this new perspective in the linear case. We define the notion of *adversarial strength* and show that it can be reduced to the *deviation angle* between the classifier considered and the nearest centroid classifier. Then, we show that the adversarial strength can be made arbitrarily high independently of the classification performance due to a mechanism that we call *boundary tilting*. This result leads us to defining a new taxonomy of adversarial examples. Finally, we show that the adversarial strength observed in practice is directly dependent on the level of regularisation used and the strongest adversarial examples, symptomatic of overfitting, can be avoided by using a proper level of regularisation.
author:
- |
Thomas Tanay, *Computer Science, UCL*\
[*thomas.tanay.13@ucl.ac.uk*]{}\
Lewis Griffin, *Computer Science, UCL*
bibliography:
- 'biblio.bib'
title: |
**A Boundary Tilting Perspective\
on the Phenomenon of Adversarial Examples**
---
Introduction
============
Tremendous progress has been made in the field of Deep Learning in recent years. Convolutional Neural Networks in particular, started to deliver promising results in 2012 on the ImageNet Large Scale Visual Recognition Challenge [@krizhevsky2012imagenet]. Since then, improvements have come at a very high pace: the range of applications has widened [@xu2015show; @mnih2015human], network architectures have become deeper and more complex [@szegedy2015going; @simonyan2014very], training methods have improved [@he2015deep], and other important tricks have helped increase classification performance and reduce training time [@srivastava2014dropout; @ioffe2015batch]. As a consequence, deep networks that are able to outperform humans are now being produced: for instance on the challenging imageNet dataset [@he2015delving], or on face recognition [@schroff2015facenet]. Yet the same networks present a surprising weakness: their classifications are extremely sensitive to some small, non-random perturbations [@szegedy2013intriguing]. As a result, any correctly classified image possesses *adversarial examples*: perturbed images that appear identical (or nearly identical) to the original image according to human observers — and hence that should belong to the same class — that are classified differently by the networks (see figure \[exampleAE\]). There seems to be a fundamental contradiction in the existence of adversarial examples in state-of-the-art neural networks. On the one hand, these classifiers learn powerful representations on their inputs, resulting in high performance classification. On the other hand, every image of each class is only a small perturbation away from an image of a different class. Stated differently, the classes defined in image space seem to be both well-separated and intersecting everywhere. In the following, we refer to this apparent contradiction as the *adversarial examples paradox*.
[0.45]{} ![Adversarial examples for two different models (from [@goodfellow2014explaining]).[]{data-label="exampleAE"}](panda.png "fig:"){width="\textwidth"}
[0.45]{} ![Adversarial examples for two different models (from [@goodfellow2014explaining]).[]{data-label="exampleAE"}](mnist.png "fig:"){width="\textwidth"}
In section \[sec: Previous Explanations\], we present two existing answers to this paradox including the currently accepted linear explanation of [@goodfellow2014explaining]. In section \[sec: Limitations with the Linear Explanation\], we argue that the linear explanation presents a number of limitations: the formal argument is unconvincing; we can define classes of images on which linear models do not suffer from the phenomenon; and the adversarial examples affecting logistic regression on the 3s vs 7s MNIST problem appear qualitatively very different from the ones affecting GoogLeNet on ImageNet. In section \[sec: The Boundary Tilting Perspective\], we introduce the boundary tilting perspective. We start by presenting a new pictorial solution to the adversarial examples paradox: a submanifold of sampled data, intersected by a class boundary that lies close to it, suffers from adversarial examples. Then we develop a mathematical analysis of the new perspective in the linear case. We define a strict condition for the non-existence of adversarial examples, from which we deduce a measure of *strength* for the adversarial examples affecting a class of images. Then we show that the adversarial strength can be reduced to a simple parameter: the *deviation angle* between the weight vector of the classifier considered and the weight vector of the nearest centroid classifier. We also show that the adversarial strength can become arbitrarily high without affecting performance when the classification boundary tilts along a component of low variance in the data. This result leads us to defining a new taxonomy of adversarial examples. Finally, we show experimentally using SVM that the adversarial strength observed in practice is controlled by the level of regularisation used. With very high regularisation, the phenomenon of adversarial examples is minimised and the classifier defined converges towards the nearest centroid classifier. With very low regularisation however, the training data is overfitted by boundary tilting, leading to the existence of strong adversarial examples.
Previous Explanations {#sec: Previous Explanations}
=====================
Low-probability “pockets” in the manifold
-----------------------------------------
In [@szegedy2013intriguing], the existence of adversarial examples was regarded as an intriguing phenomenon. No detailed explanation was proposed, and only a simple analogy was introduced:
> *“Possible explanation is that the set of adversarial negatives is of extremely low probability, and thus is never (or rarely) observed in the test set, *yet it is dense (much like the rational numbers)*, and so it is found virtually near every test case”* \[emphasis added\]
Using the mathematical concept of density, and the example of the rational numbers in particular, we can indeed define a classifier that suffers from the phenomenon of adversarial examples. Consider the classifier $\mathcal{C}$ operating on the real numbers with the following decision rule for a test number $x$:
- $x$ belongs to if it is positive irrational or negative rational.
- $x$ belongs to if it is negative irrational or positive rational.
On a test set selected at random among real numbers, $\mathcal{C}$ discriminates perfectly between positive and negative numbers: real numbers contain infinitely more irrational numbers than rational numbers and for whatever test number $x$ we choose at random among real numbers, $x$ is infinitely likely to be irrational, and thus correctly classified. Yet $\mathcal{C}$ suffers from the phenomenon of adversarial examples: since the set of rational numbers is dense in the set of real numbers, $x$ is infinitely close to rational numbers that constitute adversarial examples.
####
The rational numbers analogy is interesting, but it leaves one important question open: why would deep networks define decision rules that are in any way as strange as the one defined by our example classifier $\mathcal{C}$? By what mechanism should the low-probability “pockets” be created? Without attempting to provide a detailed answer, [@szegedy2013intriguing] suggested that it was made possible by the high non-linearity of deep networks.
Linear explanation
------------------
[@goodfellow2014explaining] subsequently provided a more detailed analysis of the phenomenon, and introduced the linear explanation — currently generally accepted. Their explanation relies on a new analogy:
> *“We can think of this as a sort of *‘accidental steganography’*, where a linear model is forced to attend exclusively to the signal that aligns most closely with its weights, even if multiple signals are present and other signals have much greater amplitude.”* \[emphasis added\]
Given an input $x$ and an adversarial example $\boldsymbol{\tilde{x}} = \boldsymbol{x} + \boldsymbol{\eta}$ where $\boldsymbol{\eta}$ is subject to the constraint $\|\boldsymbol{\eta}\|_\infty < \epsilon$, the argument is the following:
> *“Consider the dot product between a weight vector $\boldsymbol{w}$ and an adversarial example $\boldsymbol{\tilde{x}}$: $${\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{\tilde{x}} = {\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{x} + {\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{\eta}$$ The adversarial perturbation causes the activation to grow by ${\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{\eta}$. We can maximise this increase subject to the max norm constraint on $\boldsymbol{\eta}$ by assigning $\boldsymbol{\eta} = \epsilon\,\boldsymbol{sign(w)}$. If $\boldsymbol{w}$ has $n$ dimensions and the average magnitude of an element of the weight vector is $m$, then the activation will grow by $\epsilon\,m\,n$. Since $\|\boldsymbol{\eta}\|_\infty$ does not grow with the dimensionality of the problem but the change in activation caused by the perturbation by $\boldsymbol{\eta}$ can grow linearly with $n$, then for high dimensional problems, we can make many infinitesimal changes to the input that add up to one large change to the output.”*
The authors concluded that “a simple linear model can have adversarial examples if its input has sufficient dimensionality”. This argument was followed with the observation that small linear movements in the direction of the sign of the gradient (with respect to the input image) can cause deep networks to change their predictions, and hence that “linear behaviour in high-dimensional spaces is sufficient to cause adversarial examples”.
**Technical remarks:**
1. What norm should be used to evaluate the magnitude of a small perturbation? The image perturbations used to generate adversarial examples are typically measured with a norm that does not necessarily match perceptual magnitude. For instance, [@goodfellow2014explaining] use the infinity norm, based on the idea that digital measuring devices are insensitive to small perturbations whose infinity norm is below a certain threshold (because of digital quantization). This is a reasonable but arbitrary choice. We might consider other norms more adapted (such as 1- or 2-norm) — because for human observers, the magnitude of a perturbation does not only depend on the maximum change along individual pixels but also on the number of changing pixels. This is a fairly technical point of little importance in practice, except for determining the specific direction in which to move when looking for adversarial examples. We use the 2-norm, so that the direction we move in is simply the direction of the gradient. In other words, we create adversarial examples by adding the quantity $\epsilon\,\boldsymbol{w}/\|\boldsymbol{w}\|_2$ to the input image, instead of adding the quantity $\epsilon\,\boldsymbol{sign(w)}$, as one does for the infinity norm.
2. In previous works, the phenomenon of adversarial examples in linear classification was investigated using logistic regression [@szegedy2013intriguing; @goodfellow2014explaining]. In the present study, we use another standard linear classifier: support vector machine (SVM) with linear kernel. The two methods are largely equivalent but we prefer SVM for its geometrical interpretation, more adapted to the boundary tilting perspective we introduce in the following.
Limitations with the Linear Explanation {#sec: Limitations with the Linear Explanation}
=======================================
An unconvincing argument {#sec: An unconvincing argument}
------------------------
The idea of accidental steganography is a seducing intuition that seems to illustrate well the phenomenon of adversarial examples. Yet the argument is unconvincing: small perturbations do not provoke changes in activation that grow linearly with the dimensionality of the problem, *when they are considered relatively to the activations themselves*. Consider the dot product between a weight vector $\boldsymbol{w}$ and an adversarial example $\boldsymbol{\tilde{x}}$ again: ${\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{\tilde{x}} = {\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{x} + {\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{\eta}$. As we have seen before, the change in activation ${\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{\eta}$ grows linearly with the problem; but so does the activation ${\ensuremath{\boldsymbol{w}^{\scriptscriptstyle \top}}} \cdot \boldsymbol{x}$ (provided that the weight and pixel distributions in $\boldsymbol{w}$ and $\boldsymbol{x}$ stay unchanged), and the ratio between the two quantities stays constant.
####
We illustrate this by performing linear classification on a modified version of the 3s vs 7s MNIST problem where the image size has been increased to $200 \times 200$. We generated the new dimensions by linear interpolation and increased variability by adding some noise to the original and the modified datasets (random perturbations between $[-0.05,0.05]$ on every pixel). The results for the two image sizes look strikingly similar (see figure \[MNISTdim\]). Importantly, increasing the image resolution has no influence on the perceptual magnitude of the adversarial perturbations, even if the dimension of the problem has been multiplied by more than 50.
[0.49]{} ![Increasing the dimensionality of the problem does not make the phenomenon of adversarial examples worse. Whether the image size is $28 \times 28$ or $200 \times 200$, the weight vector found by linear SVM looks very similar to the one found by logistic regression in [@goodfellow2014explaining]. The two SVM models have an error rate of $2.7\%$. The magnitude $\epsilon$ of the perturbations has been chosen in both cases such that $99\%$ of the digits in the test set are misclassified ($\epsilon_{28} = 4.6, \epsilon_{200} = 31. \approx \epsilon_{28} \times 200/28$)[]{data-label="MNISTdim"}](AEs28.pdf "fig:"){width="\textwidth"}
[0.49]{} ![Increasing the dimensionality of the problem does not make the phenomenon of adversarial examples worse. Whether the image size is $28 \times 28$ or $200 \times 200$, the weight vector found by linear SVM looks very similar to the one found by logistic regression in [@goodfellow2014explaining]. The two SVM models have an error rate of $2.7\%$. The magnitude $\epsilon$ of the perturbations has been chosen in both cases such that $99\%$ of the digits in the test set are misclassified ($\epsilon_{28} = 4.6, \epsilon_{200} = 31. \approx \epsilon_{28} \times 200/28$)[]{data-label="MNISTdim"}](AEs200.pdf "fig:"){width="\textwidth"}
In sum, the dimensionality argument does not hold: high dimensional problems are not necessarily more prone to the phenomenon of adversarial examples. Without this central result however, can we still maintain that linear behaviour is sufficient to cause adversarial examples?
Linear behaviour is not sufficient to cause adversarial examples {#sec: Linear behaviour is not sufficient to cause adversarial examples}
----------------------------------------------------------------
According to the linear explanation of [@goodfellow2014explaining], linear behaviour itself is responsible for the existence of adversarial examples. If we take this explanation literally, then we expect all linear classification problems to suffer from the phenomenon. Yet we can find classes of images for which adversarial examples do not exist at all. Consider the following toy problem (figure \[toyExample\]).
Let $I$ and $J$ be two classes of images of size $100\times100$ defined as follow:
Class $I$.
: Left half-image noisy (random pixel values in $[0, 1]$) and right half-image black (pixel value: 0).
Class $J$.
: Left half-image noisy (random pixel values in $[0, 1]$) and right half-image white (pixel value: 1).
If we train a linear SVM on $5000$ images of each class, we achieve perfect separation of the training data with full generalisation to novel test data. When we look at the weight vector $\boldsymbol{w}$ defined by SVM, we notice that it correctly represents the feature separating the two classes: it ignores the left half-image (all weights near zero) and takes into consideration the entire right half-image (all weights near 1). As a result, adversarial examples do not exist. Indeed, if we take an image in one of the two classes and move in the gradient direction until we reach the class boundary, then we get an image that is also perceived as being between the two classes according to human observers (grey right half-image); and if we continue to move in the gradient direction until we reach a confidence level that the new image belongs to the new class equal to the confidence level that the original image belonged to the original class, then we get an image that is also perceived as belonging to the new class according to human observers.
![Toy problem of two classes $I$ and $J$ that do not suffer from the phenomenon of adversarial examples. When we follow the procedure that normally leads to the creation of adversarial examples, we get instead real instances of images that belong to the other class. We call the images on the boundary the *projected images* and the images with opposed classification score the *mirror images*.[]{data-label="toyExample"}](toyExample.pdf){width="\textwidth"}
This toy problem is very artificial and the point we make from it might seem little convincing for the moment, but it should not be disputed that there is a priori nothing in the current linear explanation that allows us to predict which classes of images will suffer from the phenomenon of adversarial examples, and which will not. In the following section we consider a more realistic problem: MNIST. We will return to the toy problem in section \[sec: Return to the toy problem\].
Linear classification on MNIST. Are these examples really adversarial? {#sec: Linear classification on MNIST. Are these examples really adversarial?}
----------------------------------------------------------------------
A key argument in favour of the linear explanation of adversarial examples was that logistic regression also suffers from the phenomenon. In contrast, we argue here that what happens with linear classifiers on MNIST is very different from what happens with deep networks on ImageNet.
####
The first difference between the two situations is very clear: the adversarial perturbations have a much higher magnitude and are very perceptible by human observers in the case of linear classifiers on MNIST (see figure \[exampleAE\]). Importantly, the image resolution cannot account for this difference: increasing the size of the MNIST images does not influence the perceptual magnitude of the adversarial perturbations (as shown in section \[sec: An unconvincing argument\]). Not only does the linear explanation unreliably predict whether the phenomenon of adversarial examples will occur on a specific dataset (as shown in section \[sec: Linear behaviour is not sufficient to cause adversarial examples\]), it also cannot predict the magnitude of the adversarial perturbations necessary to make the classifier change its predictions when the phenomenon *does occur*.
####
Another important difference between the adversarial examples shown in [@goodfellow2014explaining] for GoogLeNet on ImageNet and the ones shown for logistic regression on MNIST concerns the appearance of the adversarial perturbations. With GoogLeNet on ImageNet, the perturbation is dominated by high-frequency structure which cannot be meaningfully interpreted; with logistic regression on MNIST, the perturbation is low-frequency dominated and although [@goodfellow2014explaining] argue that it is “not readily recognizable to a human observer as having anything to do with the relationship between 3s and 7s”, we believe that it can be meaningfully interpreted: the weight vector found by logistic regression points in a direction that is close to passing through the mean images of the two classes, thus defining a decision boundary similar to the one of a nearest centroid classifier (see figure \[nearestCentroid\]).
[0.35]{} ![The weight vectors found by linear models resemble the average 3 of the MNIST training data to which the average 7 has been subtracted.[]{data-label="nearestCentroid"}](mean3mean7.pdf "fig:"){width="\textwidth"}
[0.55]{} ![The weight vectors found by linear models resemble the average 3 of the MNIST training data to which the average 7 has been subtracted.[]{data-label="nearestCentroid"}](Ws.pdf "fig:"){width="\textwidth"}
Simple linear models defined by SVM or logistic regression can be deceived on MNIST by perturbations that are visually perceptible and that look roughly like the weight vector of the nearest centroid classifier. This result is hardly surprising and does not help explain why much more sophisticated models — such as deep networks — can be deceived by imperceptible perturbations which look to human observers like random noise. Clearly, the linear explanation is still incomplete.
The Boundary Tilting Perspective {#sec: The Boundary Tilting Perspective}
================================
Pictorial solution to the adversarial examples paradox
------------------------------------------------------
In previous sections, we rejected the linear explanation of [@goodfellow2014explaining]: high dimension is insufficient to explain the phenomenon of adversarial examples and linear models seem to suffer from a weaker type of adversarial examples than deep networks. Without the linear explanation however, the adversarial examples paradox persists: how can two classes of images be well separated, if every element of each class is close to an element of the other class?
####
In figure \[pocketView\], we present a schematic representation of the solution proposed in [@szegedy2013intriguing]: the classes and are well separated, but every element of each class is very close to an element of the other class because low probability adversarial pockets are densely distributed in image space. In figure \[newPerspective\], we introduce a new solution. First, we observe that the data sampled in the training and test sets only extends in a submanifold of the image space. A class boundary can intersect this submanifold such that the two classes are well separated, but will also extend beyond it. Under certain circumstances, the boundary might be lying very close to the data, such that small perturbations directed towards the boundary might cross it.
[0.48]{} {width="\textwidth"}
[0.48]{} {width="\textwidth"}
Note that in the low dimensional representation of figure \[newPerspective\], randomly perturbed images are likely to cross the class boundary. In higher dimension however, the probability that a random perturbation moves exactly in the direction of the boundary is low, such that images that are close to it (and thus sensitive to adversarial perturbations), are robust to random perturbations, in accordance with the results in [@szegedy2013intriguing].
Adversarial examples in linear classification {#sec: Adversarial examples in linear classification}
---------------------------------------------
The drawing of figure \[newPerspective\] is, of course, a severe oversimplification of the reality — but it is a useful one. As we noticed already, it is a low dimensional impression of a phenomenon happening in much higher dimension. It also misrepresents the complexity of real data distributions and the highly non-linear nature of the class boundary defined by a state-of-the-art classifier. Yet it is useful because it allows us to make important predictions. First, the drawing is compatible with a flat class boundary and no non-linearity is required (contrary to the view relying on the presence of low probability pockets). Hence the phenomenon of adversarial examples should be observable in linear classification. At the same time, linear behaviour is not sufficient for the phenomenon to occur either: the class boundary needs to “be tilted” and lie close to the data. In the following, we propose a mathematical analysis of this boundary tilting explanation in linear classification. We start by giving a strict condition for the non-existence of adversarial examples, from which we deduce a measure of *strength* for the adversarial examples affecting a class of images. We also show that the adversarial strength can be reduced to a simple parameter: the *deviation angle* between the classifier considered and the nearest centroid classifier. Then, we introduce the *boundary tilting mechanism* and show that it can lead to adversarial examples of arbitrary strength without affecting classification performance. Finally, we propose a new taxonomy of adversarial examples.
### Condition for the non-existence of adversarial examples
In the standard procedure, adversarial examples are found by moving along the gradient direction by a magnitude $\epsilon$ chosen such that 99% of the data is misclassified [@goodfellow2014explaining]. The smaller $\epsilon$ is, the more “impressive” the resulting adversarial examples. This approach is meaningful when $\epsilon$ is very small — but as $\epsilon$ grows, when should one stop considering the images obtained as adversarial examples? When they start to actually look like images of the other class? Or when the adversarial perturbation starts to be perceptible to the human eye? Here, we introduce a strict condition for the non-existence of adversarial examples.
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Let $I$ and $J$ be two classes of images, and $\mathcal{C}$ a hyperplane boundary defining a linear classifier in $\mathbb{R}^n$. $\mathcal{C}$ is formally specified by a normal weight vector $\boldsymbol{c}$ (we assume that $\|\boldsymbol{c}\|_2 =1$) and a bias $c_0$. For any image $\boldsymbol{x}$ in $\mathbb{R}^n$, we define:
- The *classification score* of $\boldsymbol{x}$ through $\mathcal{C}$ as: $d(\boldsymbol{x},\mathcal{C}) = \boldsymbol{x} \cdot \boldsymbol{c} + c_0$\
$d(\boldsymbol{x},\mathcal{C})$ is the signed distance between $\boldsymbol{x}$ and $\mathcal{C}$.\
$\boldsymbol{x}$ is classified in $I$ if $d(\boldsymbol{x},\mathcal{C}) \leq 0$ and $\boldsymbol{x}$ is classified in $J$ if $d(\boldsymbol{x},\mathcal{C}) \geq 0$.
- The *projected image* of $\boldsymbol{x}$ on $\mathcal{C}$ as: $\boldsymbol{p}(\boldsymbol{x},\mathcal{C}) = \boldsymbol{x} - d(\boldsymbol{x},\mathcal{C})\,\boldsymbol{c}$\
$\boldsymbol{p}(\boldsymbol{x},\mathcal{C})$ is the nearest image $\boldsymbol{y}$ lying on $\mathcal{C}$ (i.e. such that $d(\boldsymbol{y},\mathcal{C}) = 0$).
- The *mirror image* of $\boldsymbol{x}$ through $\mathcal{C}$ as: $\boldsymbol{m}(\boldsymbol{x},\mathcal{C}) = \boldsymbol{x} - 2\,d(\boldsymbol{x},\mathcal{C})\,\boldsymbol{c}$\
$\boldsymbol{m}(\boldsymbol{x},\mathcal{C})$ is the nearest image $\boldsymbol{y}$ with opposed classification score (i.e. such that ${d(\boldsymbol{y},\mathcal{C})=-d(\boldsymbol{x},\mathcal{C})}$).
- The *mirror class* of $I$ through $\mathcal{C}$ as: $m(I,\mathcal{C}) = \{\boldsymbol{m}(\boldsymbol{x},\mathcal{C}) \enspace | \enspace \forall \boldsymbol{x} \in I\}$
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Suppose that $\mathcal{C}$ *does not* suffer from adversarial examples. Then for every image $\boldsymbol{x}$ in $I$, the projected image $\boldsymbol{p}(\boldsymbol{x},\mathcal{C})$ must lie exactly between the classes $I$ and $J$. Since $\boldsymbol{p}(\boldsymbol{x},\mathcal{C})$ is the midpoint between $\boldsymbol{x}$ and the mirror image $\boldsymbol{m}(\boldsymbol{x},\mathcal{C})$, we can say that $\boldsymbol{p}(\boldsymbol{x},\mathcal{C})$ lies exactly between $I$ and $J$ iff $\boldsymbol{m}(\boldsymbol{x},\mathcal{C})$ belongs to $J$. Hence we can say that the class $I$ does not suffer from adversarial examples iff $m(I,\mathcal{C}) \subset J$. Similarly, we can say that the class $J$ does not suffer from adversarial examples iff $m(J,\mathcal{C}) \subset I$. Since the mirror operation is involutive, we have $m(I,\mathcal{C}) \subset J \Rightarrow I \subset m(J,\mathcal{C})$ and $m(J,\mathcal{C}) \subset I \Rightarrow J \subset m(I,\mathcal{C})$. Hence: $$\boxed{\text{$\mathcal{C}$ \underline{does not} suffer from adversarial examples} \enspace \Leftrightarrow \enspace m(I,\mathcal{C}) = J \text{ and } m(J,\mathcal{C}) = I}$$ The non-existence of adversarial examples is equivalent to the classes $I$ and $J$ being mirror classes of each other through $\mathcal{C}$, or to the mirror operator $\boldsymbol{m}(\cdot,\mathcal{C})$ defining a bijection between $I$ and $J$. Conversely, we say that a classification boundary $\mathcal{C}$ suffers from adversarial examples iff ${m(I,\mathcal{C}) \neq J}$ and ${m(J,\mathcal{C}) \neq I}$. In that case, we call *adversarial examples affecting $I$* the elements of $m(I,\mathcal{C})$ that are not in $J$ and we call *adversarial examples affecting $J$* the elements of $m(J,\mathcal{C})$ that are not in $I$.
### Strength of the adversarial examples affecting a class of images
As discussed before, the magnitude $\epsilon$ of the adversarial perturbations used in the standard procedure is a good measure of how “impressive” or “strong” the adversarial examples are. Unfortunately, this measure is only meaningful for small values. We introduce here a measure of *strength* that is valid on the entire spectrum of the adversarial example phenomenon.
Let us note $\boldsymbol{i}$ and $\boldsymbol{j}$ the mean images of $I$ and $J$ respectively. For an element $\boldsymbol{x}$ in $I$, the “strength” of the adversarial example $\boldsymbol{m}(\boldsymbol{x},\mathcal{C})$ is maximised when the distance ${\left\lVert\boldsymbol{x} - \boldsymbol{m}(\boldsymbol{x},\mathcal{C})\right\rVert}$ tends to 0 (this is equivalent to $\epsilon$ tending to 0 in the standard procedure). Averaging over all the elements of $I$, we can say that *the strength of the adversarial examples affecting $I$ is maximised when the distance ${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert}$ tends to 0* (see figure \[strongAE\]).
Remark that ${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert} = 2\,|d(\boldsymbol{i},\mathcal{C})|$ and consider the projections of the elements in $I$ along the direction $\boldsymbol{c}$: their mean value is $d(\boldsymbol{i},\mathcal{C})$ and we note $\sigma$ their standard deviation. Consider in particular the elements $X$ in $I$ that are more than one standard deviation away from the mean in the direction $\boldsymbol{c}$: for each element $\boldsymbol{x}$ in $X$ we have $d(\boldsymbol{i},\mathcal{C}) + \sigma \leq d(\boldsymbol{x},\mathcal{C})$. If there are no strong outliers in the data, a significant proportion of the elements of $I$ belongs to $X$, and if the classifier $\mathcal{C}$ has a good performance, some of the elements in $X$ must be correctly classified in $I$, i.e. some elements in $X$ must verify $d(\boldsymbol{x},\mathcal{C}) < 0$. Hence we must have $d(\boldsymbol{i},\mathcal{C}) + \sigma < 0$ and $|d(\boldsymbol{i},\mathcal{C})| > \sigma$. We can thus write: ${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert} = 2\,|d(\boldsymbol{i},\mathcal{C})| > 2\,\sigma$. The strength of the adversarial examples affecting $I$ is maximised (${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert} \to 0$) when there is a direction $\boldsymbol{c}$ of very small variance in the data ($\sigma \to 0$) and the boundary $\mathcal{C}$ lies close to the data along this direction ($d(\boldsymbol{i},\mathcal{C}) \to 0$).
We call the hyperplane of the nearest centroid classifier the *bisecting boundary*, and denote it $\mathcal{B}$. By definition, $\mathcal{B}$ is the unique classification boundary verifying ${\boldsymbol{m}(\boldsymbol{i},\mathcal{B}) = \boldsymbol{j}}$ (we assume that ${\boldsymbol{i} \neq \boldsymbol{j}}$ such that $\mathcal{B}$ is well-defined). Remark that we have, for a classification boundary $\mathcal{C}$: $${m(I,\mathcal{C}) = J} \implies {\boldsymbol{m}(\boldsymbol{i},\mathcal{C}) = \boldsymbol{j}} \quad \text{but} \quad {\boldsymbol{m}(\boldsymbol{i},\mathcal{B}) = \boldsymbol{j}} \centernot\implies {m(I,\mathcal{B}) = J}$$ Hence, if there exists a classification boundary $\mathcal{C}$ that does not suffer from adversarial examples on $I$, then it is unique and equal to $\mathcal{B}$; but $\mathcal{B}$ can suffer from adversarial examples. In the following, we consider that $\mathcal{B}$ *minimises* the phenomenon of adversarial examples, even when $\mathcal{B}$ does suffer from adversarial examples (see figure \[weakAE\], left). Then, we can say that *the strength of the adversarial examples affecting $I$ is minimised when the distance ${\left\lVert\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert}$ tends to 0* (see figure \[weakAE\], right).
Based on the previous considerations, and using the arctangent in order to bound the values in the finite interval $[0,\,\pi/2[$, we formally define the *strength* $s(I,\mathcal{C})$ of the adversarial examples affecting $I$ through $\mathcal{C}$ as: $$\boxed{s(I,\mathcal{C}) = \arctan\left(\frac{{\left\lVert\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert}}{{\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert}}\right)}$$ $s(I,\mathcal{C})$ is maximised at $\pi/2$ when ${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert} \to 0$ and minimised at 0 when ${\left\lVert\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert} \to 0$
### The adversarial strength is the deviation angle
In our analysis, the bisecting boundary $\mathcal{B}$ of the nearest centroid classifier plays a special role: it minimises the strength of the adversarial examples affecting $I$ and $J$. We note $\boldsymbol{b}$ its normal weight vector (we assume that ${\left\lVert\boldsymbol{b}\right\rVert}_2 = 1$) and $b_0$ its bias. Given a classifier $\mathcal{C}$ specified by a normal weight vector $\boldsymbol{c}$ and a bias $c_0$, we call *deviation angle* of $\mathcal{C}$ with regards to $\mathcal{B}$ the angle $\delta_c$ between $\boldsymbol{c}$ and $\boldsymbol{b}$. More precisely, we can express $\boldsymbol{c}$ as a function of $\boldsymbol{b}$, a unit vector orthogonal to $\boldsymbol{b}$ that we note $\boldsymbol{b}^\perp_c$, and the deviation angle $\delta_c$ as: $$\boldsymbol{c} = \cos(\delta_c)\,\boldsymbol{b} + \sin(\delta_c)\,\boldsymbol{b}^\perp_c$$ We can then derive (see appendix A) the strengths of the adversarial examples affecting $I$ and $J$ through $\mathcal{C}$ in terms of the deviation angle $\delta_c$ and the ratio $r_c = c_0/{\left\lVert\boldsymbol{i}\right\rVert}$ (with the origin $\boldsymbol{0}$ at the midpoint between $\boldsymbol{i}$ and $\boldsymbol{j}$): $$\boxed{s(I,\mathcal{C}) = \arctan\left(\frac{\sqrt{\sin^2(\delta_c)+r_c^2}}{\cos(\delta_c)+r_c}\right) \quad \text{and} \quad s(J,\mathcal{C}) = \arctan\left(\frac{\sqrt{\sin^2(\delta_c)+r_c^2}}{\cos(\delta_c)-r_c}\right)}$$
\
If we assume that $\mathcal{C}$ separates $\boldsymbol{i}$ and $\boldsymbol{j}$, then we must have $-\cos(\delta_c) < r_c < \cos(\delta_c)$.\
$\displaystyle \text{When } r_c\to-\cos(\delta_c) \text{, we have: } s(I,\mathcal{C}) \to \pi/2 \quad \text{and} \quad s(J,\mathcal{C}) \to \pi/2 - \arctan(2\cos(\delta_c))$.\
$\displaystyle \text{When } r_c\to\cos(\delta_c) \text{, we have: } s(I,\mathcal{C}) \to \pi/2 - \arctan(2\cos(\delta_c)), \quad \text{and} \quad s(J,\mathcal{C}) \to \pi/2$.\
The parameter $r_c$ controls the relative strengths of the adversarial examples affecting $I$ and $J$. It can lead to strong adversarial examples on one class at a time (see figure \[rc\]).
In the following, we assume that $r_c \approx 0$, so that: $$\boxed{s(I,\mathcal{C}) \approx s(J,\mathcal{C}) \approx s(\mathcal{C}) = \arctan\left(\frac{\sqrt{\sin^2(\delta_c)}}{\cos(\delta_c)}\right) = |\delta_c|}$$ In words, when $\mathcal{C}$ passes close to the mean of the classes centroids ($r_c \approx 0$), the strength of the adversarial examples affecting $I$ is approximately equal to the strength of the adversarial examples affecting $J$ and can be reduced to the deviation angle $|\delta_c|$. In that case we can speak of the *adversarial strength* without mentioning the class affected: it is minimised for $\delta_c = 0$ (i.e. $\mathcal{C} \approx \mathcal{B}$) and maximised when $|\delta_c|$ tends to $\pi/2$.
### Boundary tilting and its influence on classification
In previous sections, we defined the notion of adversarial strength and showed that it can be reduced to the deviation angle between the weight vector $\boldsymbol{c}$ of the classifier considered and the weight vector $\boldsymbol{b}$ of the nearest centroid classifier. Here, we evaluate the effect on the classification performance of tilting the weight vector $\boldsymbol{c}$ by an angle $\theta$ along an arbitrary direction.
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Let $\boldsymbol{z}$ be a unit vector that we call the *zenith direction*. We can express $\boldsymbol{c}$ as a function of $\boldsymbol{z}$, a unit vector orthogonal to $\boldsymbol{z}$ that we note $\boldsymbol{z}^\perp_c$ and an angle $\theta_c$ that we call the *inclination angle* of $\mathcal{C}$ along $\boldsymbol{z}$: $$\boldsymbol{c} = \cos(\theta_c)\,\boldsymbol{z}^\perp_c + \sin(\theta_c)\,\boldsymbol{z}$$ We say that we *tilt the boundary* $\mathcal{C}$ along the zenith direction $\boldsymbol{z}$ by an angle $\theta$ when we define a new boundary $\mathcal{C}_\theta$ specified by its normal weight vector $\boldsymbol{c}_\theta$ and its bias $c_{\theta0}$ as follow: $$\boldsymbol{c}_\theta = \cos(\theta_c + \theta)\,\boldsymbol{z}^\perp_c + \sin(\theta_c + \theta)\,\boldsymbol{z}$$ $$c_{\theta0} = c_0\,\cos(\theta_c + \theta)/\cos(\theta_c)$$ Let $S$ be the set of all the images in $I$ and $J$. Abusing the notation, we refer to the sets of all classification scores through $\mathcal{C}$ and $\mathcal{C}_\theta$ by $d(S,\mathcal{C})$ and $d(S,\mathcal{C}_\theta)$. We can show (see appendix B) that: $$\boxed{d(S,\mathcal{C}) = \boldsymbol{u} \cdot P \quad \text{and} \quad d(S,\mathcal{C}_\theta) = \boldsymbol{u}_\theta \cdot P}$$ Where $\boldsymbol{u} = (\cos(\theta_c),\; \sin(\theta_c))$ and $\boldsymbol{u}_\theta = (\cos(\theta_c + \theta),\; \sin(\theta_c + \theta))$ are the unit vectors rotated by the angles $\theta_c$ and $\theta_c + \theta$ relatively to the x-axis and $P = S \cdot (\boldsymbol{z}^\perp_c + c_0/\cos(\theta_c),\; \boldsymbol{z})^\top$ is the projection of $S$ on the plane $(\boldsymbol{z}^\perp_c,\; \boldsymbol{z})$ horizontally translated by $c_0/\cos(\theta_c)$.
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Now we define the *rate of change* between $\mathcal{C}$ and $\mathcal{C}_\theta$ and note $roc(\theta)$ the proportion of elements in $S$ that are classified differently by $\mathcal{C}$ and $\mathcal{C}_\theta$ (i.e. the elements $\boldsymbol{x}$ in $S$ for which $\operatorname{sign}(d(\boldsymbol{x},\mathcal{C})) \neq \operatorname{sign}(d(\boldsymbol{x},\mathcal{C}_\theta))$). In general, we cannot deduce a closed-form expression of $roc(\theta)$. However, we can represent it graphically in the plane $(\boldsymbol{z}^\perp_c,\; \boldsymbol{z})$ and we see that $roc(\theta)$ is small as long as the variance of the data in $S$ along the zenith direction $\boldsymbol{z}$ is small and the angle $\theta_c + \theta$ is not too close to $\pi/2$ (see figure \[roc\]).
Let us note $v_{\boldsymbol{z}}^\perp$ and $v_{\boldsymbol{z}}$ the variances of the data in $S$ along the directions $\boldsymbol{z}^\perp_c$ and $\boldsymbol{z}$ respectively. We present below two situations of interest where $roc(\theta)$ can be expressed in closed-form.
1. When $P$ is flat along the zenith component (i.e. when $v_{\boldsymbol{z}}$ is null), we have: $$d(S,\mathcal{C}) = \cos(\theta_c)\,(S \cdot \boldsymbol{z}^\perp_c + c_0/\cos(\theta_c)) \quad \text{and} \quad d(S,\mathcal{C}_\theta) = \cos(\theta_c + \theta)\,(S \cdot \boldsymbol{z}^\perp_c + c_0/\cos(\theta_c))$$ Hence: $$\boxed{d(S,\mathcal{C}_\theta) = \frac{\cos(\theta_c + \theta)}{\cos(\theta_c)}\,d(S,\mathcal{C})}$$ For all $\theta_c + \theta$ in $]-\pi/2,\,\pi/2[$, the sign of $d(S,\mathcal{C}_\theta)$ is equal to the sign of $d(S,\mathcal{C})$: every element of $S$ is classified in the same way by $\mathcal{C}$ and $\mathcal{C}_\theta$ and $roc(\theta) = 0$.\
*When the variance along the zenith direction is null, the classification of the elements in $S$ is unaffected by the tilting of the boundary.*
2. When $P$ follows a bivariate normal distribution $\mathcal{N}(\boldsymbol{0},\boldsymbol{\Sigma})$ with $\boldsymbol{\Sigma} = \operatorname{\textbf{diag}}(v_{\boldsymbol{z}}^\perp, \, v_{\boldsymbol{z}})$, then we can show (see appendix C) that: $$\boxed{roc(\theta) = \frac{1}{\pi}\left[\arctan\left(\sqrt{\frac{v_{\boldsymbol{z}}}{v_{\boldsymbol{z}}^\perp}}\tan(x)\right)\right]_{\theta_c}^{\theta_c+\theta}}$$
For instance if $v_{\boldsymbol{z}}^\perp = 1$ and $v_{\boldsymbol{z}} = 10^{-6}$, and the boundaries $\mathcal{C}$ and $\mathcal{C}_\theta$ are tilted at $10\%$ and $90\%$ respectively along $\boldsymbol{z}$ ($\theta_c = 0.1\,\pi/2$ and $\theta_c+\theta = 0.9\,\pi/2$)), then we have $roc(\theta) = 0.2\%$.\
*When the variance along the zenith direction is small enough, the classification of the elements in $S$ is very lightly affected by the tilting of the boundary.*
### Boundary tilting at the origin of strong adversarial examples
Finally, we show that the boundary tilting mechanism can lead to the existence of strong adversarial examples, without affecting the classification performance.
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Imagine that we choose the zenith direction $\boldsymbol{z}$ orthogonal to $\boldsymbol{b}$. Then we can express $\boldsymbol{z}^\perp_c$ as a function of $\boldsymbol{b}$, a unit vector orthogonal to $\boldsymbol{b}$ (and $\boldsymbol{z}$) that we note $\boldsymbol{y}_c$ and an angle $\phi_c$ that we call the *azimuth angle* of $\mathcal{C}$ with regards to $\boldsymbol{z}$ and $\boldsymbol{b}$: $$\boldsymbol{c} = \cos(\theta_c)\left[\,\cos(\phi_c)\,\boldsymbol{b} + \sin(\phi_c)\,\boldsymbol{y}_c\,\right] + \sin(\theta_c)\,\boldsymbol{z}$$ Now, imagine that we tilt the boundary $\mathcal{C}$ along the zenith direction $\boldsymbol{z}$ while keeping the azimuth angle $\phi_c$ constant. We can express the weight vector $\boldsymbol{c}_\theta$ of the tilted boundary $\mathcal{C}_\theta$ both as a function of its inclination angle $\theta_c + \theta$ and the azimuth angle $\phi_c$, and as a function of its deviation angle $\delta_c+\delta$: $$\boldsymbol{c}_\theta = \cos(\theta_c+\theta)\left[\,\cos(\phi_c)\,\boldsymbol{b} + \sin(\phi_c)\,\boldsymbol{y}_c\,\right] + \sin(\theta_c+\theta)\,\boldsymbol{z} \quad \text{and} \quad \boldsymbol{c}_\theta = \cos(\delta_c+\delta)\,\boldsymbol{b} + \sin(\delta_c+\delta)\,\boldsymbol{b}^\perp_c$$
We see that the deviation angle $\delta_c+\delta$ of $\mathcal{C}_\theta$ depends on the inclination angle $\theta_c+\theta$ and the azimuth angle $\phi_c$: $$\boxed{\cos(\delta_c+\delta) = \cos(\theta_c+\theta)\,\cos(\phi_c)}$$
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In order for $\mathcal{C}_\theta$ to suffer from strong adversarial examples (i.e. $|\delta_c+\delta| \to \pi/2$), it is sufficient to tilt along a zenith direction $\boldsymbol{z}$ orthogonal to $\boldsymbol{b}$ (i.e. $|\theta_c+\theta| \to \pi/2$). If in addition the direction $\boldsymbol{z}$ is such that the variance $v_{\boldsymbol{z}}$ is small, then the rate of change $roc(\theta)$ will be small and the classification boundaries $\mathcal{C}$ and $\mathcal{C}_\theta$ will perform similarly (when $v_{\boldsymbol{z}} = 0$, $\mathcal{C}$ and $\mathcal{C}_\theta$ perform exactly in the same way: see figure \[tilting\]).
> *For any classification boundary $\mathcal{C}$, there always exist a tilted boundary $\mathcal{C}_\theta$ such that $\mathcal{C}$ and $\mathcal{C}_\theta$ perform in the same way $(v_{\boldsymbol{z}} = 0)$ or almost in the same way $(0 < v_{\boldsymbol{z}} \ll 1)$, and $\mathcal{C}_\theta$ suffers from adversarial examples of arbitrary strength (as long as there are directions of low variance in the data).*
### Taxonomy of adversarial examples
Given a classifier $\mathcal{C}$, we note $\delta(\mathcal{C})$ its deviation angle and $er(\mathcal{C})$ its error rate on $S$. In the following, we analyse the distribution of all linear classifiers in the *deviation angle - error rate diagram*. To start with, we consider the nearest centroid classifier $\mathcal{B}$ as a baseline and discard all classifiers with an error rate superior to $er(\mathcal{B})$ as poorly performing. We also note $er_\text{min}$ the minimum error rate achievable on $S$ (in general, $er_\text{min} < er(\mathcal{B})$). For a given error rate comprised between $er(\mathcal{B})$ and $er_\text{min}$, we say that a classifier is *optimal* if it minimises the deviation angle. In particular, we call *label boundary* and we note $\mathcal{L}$ the optimal classifier verifying $er(\mathcal{L}) = er_\text{min}$. In the deviation angle - error rate diagram, the set of optimal classifiers forms a strictly decreasing curve segment connecting $\mathcal{B}$ (minimising the strength of the adversarial examples) to $\mathcal{L}$ (minimising the error rate). Any classifier with a deviation angle greater than $\delta(\mathcal{L})$ is then necessarily suboptimal: there is always another classifier performing at least as well and suffering from weaker adversarial examples (see figure \[types\]).
Based on these considerations, we propose to define the following taxonomy:
Type 0:
: adversarial examples affecting $\mathcal{B}$. They *minimise* the phenomenon of adversarial examples.
Type 1:
: adversarial examples affecting the classifiers $\mathcal{C}$ such that $0 \leq \delta(\mathcal{C}) \leq \delta(\mathcal{L})$. They affect in particular the *optimal classifiers*. The inconvenience of their existence is balanced by the performance gains allowed.
Type 2:
: adversarial examples affecting the classifiers $\mathcal{C}$ such that $\delta(\mathcal{L}) < \delta(\mathcal{C})$. They only affect *suboptimal classifiers* resulting from the tilting of optimal classifiers along directions of low variance.
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Let us call *training boundary* and note $\mathcal{T}$ the boundary defined by a standard classification method such as SVM or logistic regression. In practice, $I$ and $J$ are unlikely to be mirror classes of each other through $\mathcal{B}$ and hence $\mathcal{T}$ is expected to at least suffer from type 0 adversarial examples. In fact, $\mathcal{B}$ is also unlikely to minimise the error rate on $S$ and if $\mathcal{T}$ performs better than $\mathcal{B}$, then $\mathcal{T}$ is also expected to suffer from type 1 adversarial examples. Note that there is no restriction in theory on $\delta(\mathcal{L})$ and on some problems, type 1 adversarial examples can be very strong. However, $\mathcal{T}$ is a priori not expected to suffer from type 2 adversarial examples: why would SVM or logistic regression define a classifier that is suboptimal in such a way? In the following two sections, we show experimentally with SVM that the regularisation level plays a crucial role in controlling the deviation angle of $\mathcal{T}$. When the regularisation level is very strong (i.e. when the SVM margin contains all the data), $\mathcal{T}$ converges towards $\mathcal{B}$ and the deviation angle is null. When SVM is correctly regularised, $\mathcal{T}$ is allowed to deviate from $\mathcal{B}$ sufficiently to converge towards $\mathcal{L}$: the optimal classifier minimising the error rate. However when the regularisation level is too low, the inclination of $\mathcal{T}$ along directions of low variance ends up overfitting the training data, resulting in the existence of strong type 2 adversarial examples.
Return to the toy problem {#sec: Return to the toy problem}
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In light of the mathematical analysis presented in the previous sections, we now return to the toy problem introduced in section \[sec: Linear behaviour is not sufficient to cause adversarial examples\] (see figure \[toyExample\]). Firstly, we can confirm that the boundary defined by SVM satisfies the condition we gave for the non-existence of adversarial examples: the weight vector $\boldsymbol{w}$ is equal to the weight vector $\boldsymbol{b}$ of the nearest centroid classifier $\mathcal{B}$ (see figure \[centroids\]) and we have $m(I,\mathcal{B}) = J$ and $m(J,\mathcal{B}) = I$. Indeed, mirroring an image that belongs to $I$ through $\mathcal{B}$ changes the colour of its right half image from black to white and results in an image that belongs to $J$ (and conversely).
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Secondly, we can illustrate the effect of the regularisation level used on the deviation angle (and hence on the adversarial strength). To start with, we modify the toy problem such that $er_\text{min} > 0$ (when $er_\text{min} = 0$, overfitting is not likely to happen). We do this by corrupting 5% of the images in $I$ and $J$ into fully randomised images, such that $er_\text{min} = 2.5\%$ (half of the corrupted data is necessarily misclassified). Note that on this problem, $er(\mathcal{B}) = er_\text{min}$, hence $\mathcal{B} = \mathcal{L}$ and $\mathcal{B}$ is the only optimal classifier. When we perform SVM with regularisation (soft-margin), we obtain a weight vector $\boldsymbol{w}_\text{soft}$ approximately equal to $\boldsymbol{b}$ (see figure \[withRegularisation\]). The small deviation can be explained by the fact that the training data has been slightly overfitted (the training error is $2.2\% < er_\text{min}$) and corresponds to very weak adversarial examples. Without regularisation however (hard-margin), the deviation of the weight vector $\boldsymbol{w}_\text{hard}$ is very strong (see figure \[withoutRegularisation\]). In that case, the training data is completely overfitted (the training error is 0%), resulting in the existence of strong type 2 adversarial examples. Interestingly, these adversarial examples possess the same characteristics as the ones observed with GoogLeNet on ImageNet in [@goodfellow2014explaining] — the perturbation is barely perceptible, high-frequency and cannot be meaningfully interpreted — even though the classifier is linear.
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Finally, we can visualise the boundary tilting mechanism by plotting the projections of the data on the plane $(\boldsymbol{b},\boldsymbol{z})$, where $\boldsymbol{z}$ is the zenith direction along which $\boldsymbol{w}_\text{hard}$ is tilted (see figure \[projections\]). We observe in particular how the overfitting of the corrupted data leads to the existence of the strong type 2 adversarial examples: maximising the *minimal* separation of the two classes (the margin) results in a very small *average* separation (making adversarial examples possible). This effect is very reminiscent of the *data piling* phenomenon studied by [@marron2007distance] and [@ahn2010maximal] on high-dimension low-sample size data.
![The weight vector $\boldsymbol{w}$ obtained using SVM in figure \[toyExample\] is equal to the weight vector $\boldsymbol{b}$ of the nearest centroid classifier, obtained by subtracting the mean image $\boldsymbol{i}$ of the class $I$ to the mean image $\boldsymbol{j}$ of the class $J$.[]{data-label="centroids"}](centroids.pdf){width="\textwidth"}
![Left: toy problem where 5% of the data is corrupted to purely random images such that the two classes are not linearly separable ($er_\text{min} = 2.5\%$). With a proper level of regularisation (soft-margin), the training data is only slightly overfitted ($er_\text{train} = 2.2\%$) and the weight vector $\boldsymbol{w}_\text{soft}$ defined by SVM only deviates slightly from $\boldsymbol{b}$ ($\delta(\boldsymbol{w}_\text{soft}) = 0.032\,\pi/2$). Right: as a result, adversarial examples are very weak.[]{data-label="withRegularisation"}](withRegularisation.pdf){width="\textwidth"}
![Left: same toy problem as before. Without regularisation (with hard-margin), the training data is entirely overfitted ($er_\text{train} = 0\%$) and the weight vector $\boldsymbol{w}_\text{hard}$ defined by SVM deviates from $\boldsymbol{b}$ considerably ($\delta(\boldsymbol{w}_\text{hard}) = 0.97\,\pi/2$). Right: as a result, adversarial examples are very strong.[]{data-label="withoutRegularisation"}](withoutRegularisation.pdf){width="\textwidth"}
![Projection of the training data in the plane $(\boldsymbol{b},\boldsymbol{z})$ where $\boldsymbol{z} = \text{normalise}(\boldsymbol{w}_\text{hard} - (\boldsymbol{w}_\text{hard} \cdot \boldsymbol{b})\,\boldsymbol{b})$. The images in $I$ appear on the left, the images in $J$ appear on the right, and the corrupted images appear in the middle. The soft-margin and hard-margin boundaries are drawn as dashed red lines. Note that the hard-margin boundary overfits the training data by finding a direction that separates the corrupted data completely (this separation does not generalise to novel test data). The positions of the original images, projected images and mirror images of the figures \[withRegularisation\] and \[withoutRegularisation\] are also shown: the adversarial examples III and VI of the hard-margin boundary are much closer to their respective original images than the adversarial examples 3 and 6 of the soft-margin boundary.[]{data-label="projections"}](projections.pdf){width="\textwidth"}
Return to MNIST {#sec: Return to MNIST}
---------------
We now revisit the 3s vs 7s MNIST problem. In particular, we study the effect of varying the regularisation level by performing SVM classification with seven different values for the soft-margin parameter: ${\log_{10}(C) = -5, -4, -3, -2, -1, 0}$ and $1$. The first remark we can make is that there is a strong, direct correlation between the deviation angle of the weight vector defined by SVM and the regularisation level used (see figure \[deviationAngleErrorRate\], left). When regularisation is high (i.e. when $C$ is low), the SVM weight vector is very close to the weight vector of the nearest centroid classifier $\boldsymbol{b}$ ($\delta = 0.048\,\pi/2$). Conversely when regularisation is low (i.e. when $C$ is high), the SVM weight vector is almost orthogonal to $\boldsymbol{b}$ ($\delta = 0.92\,\pi/2$). As expected, the error rate on test data is minimised for an intermediate level of regularisation and overfitting happens for low regularisation: for $\log_{10}(C) = -1, 0$ and $1$, the error rate on training data approaches 0% while the error rate on test data increases (see figure \[deviationAngleErrorRate\], right).
####
When we look at the SVM weight vector $\boldsymbol{w}$ for the different levels of regularisation (see figure \[deviationAndAE\], left), we see that it initially resembles the weight vector of the nearest centroid classifier ($\log_{10}(C) = -5$), then deviates away into relatively low frequency directions ($\log_{10}(C) = -4, -3$ and $-2$) before deviating into higher frequency directions, resulting in a “random noise aspect”, when the training data starts to be overfitted ($\log_{10}(C) = -1, 0$ and $1$). Let us consider $B$ the one-dimensional subspace of $\mathbb{R}^{784}$ generated by $\boldsymbol{b}$, and $B^\perp$ the 783-dimensional subspace of $\mathbb{R}^{784}$, orthogonal complement of $B$. We note $X_\text{train}$ and $Y_\text{train}$ the projections of the training set $S_\text{train}$ on $B$ and $B^\perp$ respectively and we perform a principal component analysis of $Y_\text{train}$, resulting in the 783 principal vectors $\boldsymbol{u}_1$, ..., $\boldsymbol{u}_{783}$. Then, we decompose $B^\perp$ into 27 subspaces $U_1, ..., U_{27}$ of 29 dimensions each, such that $U_1$ is generated by $\boldsymbol{u}_1, ..., \boldsymbol{u}_{29}$, $U_2$ is generated by $\boldsymbol{u}_{30}, ..., \boldsymbol{u}_{58}$, ..., and $U_{27}$ is generated by $\boldsymbol{u}_{755}, ..., \boldsymbol{u}_{783}$. For each weight vector $\boldsymbol{w}$, we decompose it into a component $\boldsymbol{x}$ in $B$ and a component $\boldsymbol{y}$ in $B^\perp$ and we project $\boldsymbol{y}$ on each subspace $U_1$, ..., $U_{27}$ (see figure \[deviationAndAE\], middle). The norms of the projections of $\boldsymbol{y}$ are shown as orange bar charts and the square roots of the total variances in each subspace $U_1$, ..., $U_{27}$ are shown as blue curves. We see that for $\log_{10}(C) = -4, -3$ and $-2$, $\boldsymbol{y}$ is dominated by components of high variance, while for $\log_{10}(C) = -1, 0$ and $1$, $\boldsymbol{y}$ starts to be more dominated by components of low variance: this result confirms that overfitting happens by the tilting of the boundary along components of low variance. Note that $\boldsymbol{w}$ never tilts along flat directions of variation (corresponding to the subspaces $U_{23}, ..., U_{27}$) because for overfitting to take place, there needs to be some variance in the tilting direction. Interestingly, optimal classification seems to happen when each direction is used proportionally to the amount of variance it contains: for $\log_{10}(C) = -2$, the bar chart follows the blue curve faithfully. Finally, we can look at the adversarial examples affecting each weight vector (see figure \[deviationAndAE\], right). In particular, we look at the images of 3s in the test set that are at a median distance from each boundary (median images). We see that the mirror images are closer to their respective original images when the regularisation level is low, resulting in stronger adversarial examples. For $\log_{10}(C) = -5$, the deviation angle is almost null and we can say that the corresponding adversarial example is of type 0. For $\log_{10}(C) = -4, -3$ and $-2$, the increase in deviation angle is associated with an increase in performance and we can say that the corresponding adversarial examples are of type 1. However, for $\log_{10}(C) = -1, 0$ and $1$, the increase in deviation angle only results in overfitting, and we can say that the corresponding adversarial examples are of type 2.
####
These type 2 adversarial examples, like those found on the toy problem, have similar characteristics to the ones affecting GoogLeNet on ImageNet (the adversarial perturbation is barely perceptible and high-frequency). Hence we may hypothesize that the adversarial examples affecting deep networks are also of type 2, originating from a non-linear equivalent of boundary-tilting and caused by overfitting. If this hypothesis is correct, then these adversarial examples might also be fixable by using adapted regularisation. Unfortunately, straightforward l2 regularisation only works when the classification method operates on pixel values: as soon as the regularisation term is applied in a feature space that does not directly reflect pixel distance, it does not effectively prevent the existence of type 2 adversarial examples any more. We illustrate this by performing linear SVM with soft-margin regularisation after two different standard preprocessing methods: pixelwise normalisation and PCA whitening. In the two cases, the soft-margin parameter $C$ is chosen such that the performance is maximised, resulting in a slight boost in performance both for pixelwise normalisation ($er_\text{test} = 1.2\%$) and for PCA whitening ($er_\text{test} = 1.5\%$). Since the preprocessing steps are linear transformations, we can then project the weight vectors obtained back into the original pixel space. We get a deviation angle for the weight vector defined after pixelwise normalisation that is stronger than that of any weight vector defined without preprocessing ($\delta = 0.95\,\pi/2$) and a deviation angle for the weight vector defined after PCA whitening that appears orthogonal to $\boldsymbol{b}$ ($\delta = 1.00\,\pi/2$). The two weight vectors (see figure \[PreprocessingDeviationAndAEV\], left) have a very peculiar aspect: both are strongly dominated by a few pixels, in the periphery of the image for the weight vector defined after pixelwise normalisation and in the top right corner for the weight vector defined after PCA whitening. When we look at the magnitudes of the projections of the $\boldsymbol{y}$ components on the subspaces $U_1, ..., U_{27}$, we see that the dominant pixels correspond to the components where the variance of the data is smallest but non-null (see figure \[PreprocessingDeviationAndAEV\], middle). Effectively, the rescaling of the components of very low variance puts a disproportionate weight on them, forcing the boundary to tilt very significantly. The phenomenon is particularly extreme with PCA whitening where due to numerical approximations, some residual variance was found in components that were not supposed to contain any, and ended up strongly dominating the weight vector[^1]. The resulting adversarial examples are unusual (see figure \[PreprocessingDeviationAndAEV\], right). For the pixelwise normalisation preprocessing step, it is possible to change the class of an image by altering the value of pixels that do not affect the digit itself. For the PCA whitening preprocessing step, the perturbation is absolutely non-perceptible: the pixel distance between the original image and the corresponding adversarial example is in the order of $10^{-18}$. With such a small distance, classification is now very sensitive to any perturbation, whether it is adversarial or random (despite this obvious weakness, this classifier performs very well on normal data).
![Left: the deviation angle of the weight vector defined by SVM increases almost linearly with the $\log_{10}$ of the soft-margin parameter $C$. Right: The error rate on training data decreases with $\log_{10}(C)$. The error rate on test data is minimised for an intermediate level of regularisation ($\log_{10}(C) = -2$) and overfitting happens for low levels of regularisation ($\log_{10}(C) = -1, 0$ and $1$).[]{data-label="deviationAngleErrorRate"}](deviationAngleErrorRate.pdf){width="\textwidth"}
![Left: weight vector $\boldsymbol{w}$ defined by SVM for different levels of regularisation (controlled with the soft-margin parameter $C$). Middle: decomposition of $\boldsymbol{w}$ into a component $\boldsymbol{x}$ in $B$ and a component $\boldsymbol{y}$ in $B^\perp$. The orange bar charts represent the magnitudes of the projections of $\boldsymbol{y}$ on the subspaces of decreasing variances $U_1, ..., U_{27}$ and the blue curves represent the square root of the total variance in each subspace. Right: Median 3, its projected image and its mirror image for each regularisation level.[]{data-label="deviationAndAE"}](deviationAndAE.pdf){width="\textwidth"}
![Left: weight vector $\boldsymbol{w}$ defined by SVM with soft-margin after two standard preprocessing methods: pixelwise normalisation and PCA whitening (projected back in pixel space). Middle: decomposition of $\boldsymbol{w}$ into a component $\boldsymbol{x}$ in $B$ and a component $\boldsymbol{y}$ in $B^\perp$. Right: Median 3, its projected image and its mirror image for the two weight vectors.[]{data-label="PreprocessingDeviationAndAEV"}](PreprocessingDeviationAndAE.pdf){width="\textwidth"}
Conclusion
==========
####
This paper contributes to the understanding of the adversarial example phenomenon in several different ways. It introduces in particular:
A new perspective.
: The phenomenon is captured in one intuitive picture: a submanifold of sampled data, intersected by a class boundary lying close to it, suffers from adversarial examples.
A new formalism.
: In linear classification, we proposed a strict condition for the non-existence of adversarial examples. We defined adversarial examples as elements of the mirror class and introduced the notion of adversarial strength. Given a classification boundary $\mathcal{C}$, we showed that the adversarial strength can be measured by the deviation angle between $\mathcal{C}$ and the bisecting boundary $\mathcal{B}$ of the nearest centroid classifier. We also defined the boundary tilting mechanism, and showed that there always exists a tilted boundary $\mathcal{C}_\theta$ such that $\mathcal{C}$ and $\mathcal{C}_\theta$ perform in very similar ways, and $\mathcal{C}_\theta$ suffers from adversarial examples of arbitrary strength (as long as there are directions of low variance in the data).
A new taxonomy.
: These results led us to define the notion of optimal classifier, minimising the deviation angle for a given error rate. $\mathcal{B}$ is the optimal classifier minimising the adversarial strength and we called label boundary $\mathcal{L}$ the optimal classifier minimising the error rate. When $\mathcal{C} = \mathcal{B}$ and the two classes of images are not mirror classes of each other, we say that $\mathcal{C}$ suffers from adversarial examples of type 0. When the error rate of $\mathcal{C}$ is strictly inferior to the error rate of $\mathcal{B}$, the deviation angle of $\mathcal{C}$ is necessarily strictly positive; as long as it stays inferior to the deviation angle of $\mathcal{L}$, we say that $\mathcal{C}$ suffers from adversarial examples of type 1. When the deviation angle of $\mathcal{C}$ is superior to the deviation angle of $\mathcal{L}$, $\mathcal{C}$ is necessarily suboptimal. In that case we say that $\mathcal{C}$ suffers from adversarial examples of type 2.
New experimental results.
: We introduced a toy problem that does not suffer from adversarial examples, and presented a minimal set of conditions to provoke the apparition of strong type 2 adversarial examples on it. We also showed on the 3s vs 7s MNIST problem that in practice, the regularisation level used plays a key role in controlling the deviation angle, and hence the type of adversarial examples obtained. Type 2 adversarial examples in particular, can be avoided by using a proper level of regularisation. However, we showed that l2 regularisation only helps when it is applied directly in pixel space.
####
An important distinction must be drawn between the different types of adversarial examples. On the one hand, type 0 and type 1 adversarial examples originate from a lack of expressiveness of linear models: their adversarial perturbations do not correspond to the true features disentangling the classes of images, but they can be interpreted (as optimal linear features). On the other hand, type 2 adversarial examples originate from overfitting: their adversarial perturbations are high frequency and largely meaningless (with a characteristic “random noise aspect”). Due to their similarity with the type 2 adversarial examples affecting linear classifiers, we hypothesised that the adversarial examples affecting state-of-the-art neural networks are also of type 2, symptomatic of overfitting and resulting from a non-linear equivalent of boundary tilting. Unfortunately, we do not know how to effectively regularise deep networks yet. In fact, we do not know whether it is possible to regularise them at all. Neural networks typically operate in a regime where the number of learnable parameters is higher than the number of training images and one could imagine that such models are fundamentally vulnerable to adversarial examples. Perhaps, the adversarial examples phenomenon is to neural systems what Loschmidt’s paradox is to statistical physics: a theoretical aberration of extremely low probability in practice. When Loschmidt pointed out that it is possible to create a system that contradicts the second law of thermodynamics (stating that the entropy of a closed system must always increase) by taking an existing closed system and reversing the motion direction of all the particles constituting it, Boltzmann is reported to have answered: “Go ahead, reverse them!”. Similarly, one could then reply to those who worry about the possible existence of adversarial examples in humans: “Go ahead, generate them!”.
Appendix {#appendix .unnumbered}
========
### A Expression of the adversarial strength as a function of the deviation angle {#a-expression-of-the-adversarial-strength-as-a-function-of-the-deviation-angle .unnumbered}
By choosing the origin $\boldsymbol{0}$ at the midpoint between $\boldsymbol{i}$ and $\boldsymbol{j}$, we can ensure that $\boldsymbol{b} = -\boldsymbol{i}/{\left\lVert\boldsymbol{i}\right\rVert} = \boldsymbol{j}/{\left\lVert\boldsymbol{j}\right\rVert}$ and $b_0 = 0$. We then have:
[ - (,)]{} & = [ - + 2d(,)]{} &&\
& = 2|d(,)| &&\
& = 2| + c\_0| &&\
& = 2|(\_c)( ) + (\_c)(\^[0]{}) + c\_0| &&\
& = 2|(\_c)( (-/)) + c\_0| &&\
& = 2|-(\_c) + c\_0|
Similarly, we have:\
${\left\lVert\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{j},\mathcal{C})\right\rVert} = 2\,|{\left\lVert\boldsymbol{j}\right\rVert}\cos(\delta_c) + c_0|$
If we assume that $\mathcal{C}$ lies between $\boldsymbol{i}$ and $\boldsymbol{j}$, then we must have $-{\left\lVert\boldsymbol{i}\right\rVert} < c_0/\cos(\delta_c) < {\left\lVert\boldsymbol{j}\right\rVert}$ and:\
${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\right\rVert} = 2\,({\left\lVert\boldsymbol{i}\right\rVert}\cos(\delta_c) - c_0)$\
${\left\lVert\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{j},\mathcal{C})\right\rVert} = 2\,({\left\lVert\boldsymbol{j}\right\rVert}\cos(\delta_c) + c_0)$
By applying the law of cosines in the triangle $\boldsymbol{i}\,\boldsymbol{m}(\boldsymbol{i},\mathcal{C})\,\boldsymbol{j}$, we have:
[ - (,)]{} & = &&\
& = &&\
& = 2 &&\
& = 2 &&\
& = 2
Similarly by applying the law of cosines in the triangle $\boldsymbol{j}\,\boldsymbol{m}(\boldsymbol{j},\mathcal{C})\,\boldsymbol{i}$, we have:\
${\left\lVert\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{j},\mathcal{C})\right\rVert} = 2\,\sqrt{{\left\lVert\boldsymbol{j}\right\rVert}^2\,\sin^2(\delta_c) + c_0^2}$
Finally by posing $r_c = c_0/{\left\lVert\boldsymbol{i}\right\rVert} = c_0/{\left\lVert\boldsymbol{j}\right\rVert} = 2\,c_0/{\left\lVert\boldsymbol{j}-\boldsymbol{i}\right\rVert}$, we can write:\
$\displaystyle s(I,\mathcal{C}) = \arctan\left(\frac{\|\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\|}{\|\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{i},\mathcal{C})\|}\right) = \arctan\left(\frac{\sqrt{\sin^2(\delta_c)+r_c^2}}{\cos(\delta_c)+r_c}\right)\\
\displaystyle s(J,\mathcal{C}) = \arctan\left(\frac{\|\boldsymbol{i} - \boldsymbol{m}(\boldsymbol{j},\mathcal{C})\|}{\|\boldsymbol{j} - \boldsymbol{m}(\boldsymbol{j},\mathcal{C})\|}\right) = \arctan\left(\frac{\sqrt{\sin^2(\delta_c)+r_c^2}}{\cos(\delta_c)-r_c}\right)$
### B Expression of the sets of all classification scores through $\mathcal{C}$ and $\mathcal{C}_\theta$ {#b-expression-of-the-sets-of-all-classification-scores-through-mathcalc-and-mathcalc_theta .unnumbered}
If we regard $S$ as a data matrix, then we can write:
d(S,) & = S + c\_0 &&\
& = S ((\_c)\^\_c + (\_c)) + c\_0 &&\
& = (\_c)(S \^\_c) + (\_c)(S ) + c\_0 &&\
& = (\_c)(S \^\_c + c\_0/(\_c)) + (\_c)(S ) &&\
& = ((\_c), (\_c)) S (\^\_c + c\_0/(\_c), )\^&&\
& = V P
With $V = (\cos(\theta_c),\; \sin(\theta_c))$ and $P = S \cdot (\boldsymbol{z}^\perp_c + c_0/\cos(\theta_c),\; \boldsymbol{z})^\top$.
Similarly we have: $d(S,\mathcal{C}_\theta) = V_\theta \cdot P$
With $V_\theta = (\cos(\theta_c+\theta),\; \sin(\theta_c+\theta))$
### C Expression of $roc(\theta)$ when $P$ follows a bivariate normal distribution {#c-expression-of-roctheta-when-p-follows-a-bivariate-normal-distribution .unnumbered}
{width="66.00000%"} \[roc2\]
$\displaystyle roc(\theta) = roc(\mathcal{C},\mathcal{C}_\theta,\boldsymbol{\Sigma_1}) = roc(\mathcal{Z},\mathcal{C}_\theta,\boldsymbol{\Sigma_1}) - roc(\mathcal{Z},\mathcal{C},\boldsymbol{\Sigma_1}) = \frac{\theta_c + \theta}{\pi} - \frac{\theta_c}{\pi} = \frac{\theta}{\pi}$
{width="66.00000%"} \[\]
We have:
$\displaystyle roc(\mathcal{Z},\mathcal{C}_2,\boldsymbol{\Sigma_2}) = roc(\mathcal{Z},\mathcal{C}_1,\boldsymbol{\Sigma_1}) = \frac{\theta_1}{\pi}$
We also have:
$\displaystyle \tan(\theta_1) = \sqrt{\frac{v_{\boldsymbol{z}}}{v_{\boldsymbol{z}}^\perp}}\,\,\frac{y}{x} = \sqrt{\frac{v_{\boldsymbol{z}}}{v_{\boldsymbol{z}}^\perp}}\,\tan(\theta_2) \quad \Rightarrow \quad \theta_1 = \arctan\left(\sqrt{\frac{v_{\boldsymbol{z}}}{v_{\boldsymbol{z}}^\perp}}\,\tan(\theta_2)\right)$
Hence:
$\displaystyle roc(\mathcal{Z},\mathcal{C}_2,\boldsymbol{\Sigma_2}) = \frac{1}{\pi}\,\arctan\left(\sqrt{\frac{v_{\boldsymbol{z}}}{v_{\boldsymbol{z}}^\perp}}\,\tan(\theta_2)\right)$
And:
roc() & = roc(,\_,) - roc(,,) &&\
& = &&\
& = \_[\_c]{}\^[\_c+]{}
[^1]: This effect could be avoided by putting a threshold on the minimum variance necessary before rescaling, as is sometimes done in practice.
|
---
abstract: |
Let $n \ge 2$ be an integer. In this note, we show that the [*oriented*]{} transition matrices over the field $\mathcal R$ of all real numbers (over the finite field $\mathcal Z_2$ of two elements respectively) of all continuous [*vertex maps*]{} on [*all*]{} oriented trees with $n+1$ vertices are similar to one another over $\mathcal R$ (over $\mathcal Z_2$ respectively) and have characteristic polynomial $\sum_{k=0}^n x^k$. Consequently, the [*unoriented*]{} transition matrices over the field $Z_2$ of all continuous [*vertex maps*]{} on [*all*]{} oriented trees with $n+1$ vertices are similar to one another over $\mathcal Z_2$ and have characteristic polynomial $\sum_{k=0}^n x^k$. Therefore, the coefficients of the characteristic polynomials of these [*unoriented*]{} transition matrices, when considered over the field $\mathcal R$, are all odd integers (and hence nonzero).
[[**Keywords**]{}: Similar matrices, oriented trees, (un)oriented transition matrices, vertex maps]{}
[[**AMS Subject Classification**]{}: 15A33; 15A36; 37E25]{}
author:
- 'Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 10617, Taiwan dubs@math.sinica.edu.tw'
title: 'On the Class of Similar Square $\{ -1, 0, 1 \}$-Matrices Arising from Vertex maps on Trees'
---
Let $n \ge 2$ be an integer and let $T$ be a tree with $n+1$ vertices $V_1, V_2, \cdots, V_{n+1}$. The tree $T$ has $n$ edges, say, $E_1, E_2, \cdots, E_n$. If vertices $V_{j_1}$ and $V_{j_2}$ are endpoints of an edge $E$, then, we let $[V_{j_1} : V_{j_2}]$ denote the edge $E$, i.e., the set of all points in $E$ and, following Bernhardt [**[@b1]**]{}, we denote the [*positively oriented edge*]{} from $V_{j_1}$ to $V_{j_2}$ as $\overrightarrow{[V_{j_1}, V_{j_2}]}$ and call $V_{j_1}$ the first vertex of $\overrightarrow{[V_{j_1}, V_{j_2}]}$ and $V_{j_2}$ the second. We also define $-\overrightarrow{[V_{j_1}, V_{j_2}]}$ by putting $-\overrightarrow{[V_{j_1}, V_{j_2}]} = \overrightarrow{[V_{j_2}, V_{j_1}]}$ and call it the [*negatively oriented edge*]{} from $V_{j_1}$ to $V_{j_2}$. So, the first vertex of $-\overrightarrow{[V_{j_1}, V_{j_2}]}$ is $V_{j_2}$ and the second is $V_{j_1}$. Thus, both $\overrightarrow{[V_{j_1}, V_{j_2}]}$ and $-\overrightarrow{[V_{j_1}, V_{j_2}]}$ represent the same edge $[V_{j_1} : V_{j_2}]$, but with the opposite orientations. In the sequel, we denote these $n$ positively oriented edges of $T$ as ${\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n}$ and call the resulting tree [*oriented tree*]{} and denote it as ${\overrightarrow}{T}$. It is clear that there are exactly $2^n$ distinct such oriented trees $\overrightarrow{T}$. Later, we shall see that different choices of orientations on the edges of $T$ will not affect our main results. When no confusion arises, we shall always use $V_1, V_2, \cdots, V_{n+1}$ and ${\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n}$ to denote respectively the vertices and the oriented edges of [*any*]{} tree with $n+1$ vertices.
Following [**[@b1]**]{}, for any two vertices $V_i$ and $V_j$ in the oriented tree ${\overrightarrow}{T}$, a path from $V_i$ to $V_j$ is a sequence of [*oriented edges*]{} ${\overrightarrow}{e_1}, {\overrightarrow}{e_2}, \cdots, {\overrightarrow}{e_m}$, where ${\overrightarrow}{e_k} \in \{ {\overrightarrow}{E_s}, -{\overrightarrow}{E_s} : 1 \le s \le n \}$ for all $1 \le k \le m$, and the first vertex of ${\overrightarrow}{e_1}$ is $V_i$, the second vertex of ${\overrightarrow}{e_m}$ is $V_j$, and the second vertex of ${\overrightarrow}{e_\ell}$ is equal to the first vertex of ${\overrightarrow}{e_{\ell+1}}$ for all $1 \le \ell \le m-1$. It is clear that, for any two vertices, $V_i$ and $V_j$, there is a [*unique shortest path*]{} from $V_i$ to $V_j$ in ${\overrightarrow}{T}$ which will be denoted as ${\overrightarrow}{[V_i, V_j]}$. We also denote $-{\overrightarrow}{[V_i, V_j]} = {\overrightarrow}{[V_j, V_i]}$ as the shortest path from vertex $V_j$ to vertex $V_i$. From now on, when we write [*the shortest path*]{} ${{\overrightarrow}{[V_i, V_j]}}$, we always mean the [*shortest path*]{} from vertex $V_i$ to vertex $V_j$ in ${\overrightarrow}{T}$. We also let $[V_i : V_j]$ denote the collection of all points in all (oriented) edges in [*the shortest path*]{} ${\overrightarrow}{[V_i, V_j]}$.
Following [**[@b1; @b2]**]{}, let $f : T \longrightarrow T$ be a continuous [*vertex map*]{}, i.e., $f$ is a continuous map such that the $n+1$ vertices of $T$ form a periodic orbit and, for each $1 \le i \le n$, $f$ is [*monotonic*]{} on the [*unoriented edge*]{} $E_i = [V_{i_1} : V_{i_2}]$, meaning that, as the point $x$ moves from vertex $V_{i_1}$ to vertex $V_{i_2}$ [*monotonically*]{} along the edge $E_i$, the point $f(x)$ moves [*monotonically*]{} from vertex $f(V_{i_1})$ to vertex $f(V_{i_2})$ along [*the shortest path*]{} $\overrightarrow{[f(V_{i_1}), f(V_{i_2})]}$ from $f(V_{i_1})$ to $f(V_{i_2})$. We now let $\mathcal R$ denote the field of all real numbers and let $\mathcal Z_2$ denote the finite field $\{ 0, 1 \}$ of two elements and let $\mathcal F$ be any field with unity 1. We define the associated [*oriented transition*]{} $n \times n$ $\{ -1, 0, 1 \}$-matrix $\mathcal A_n(f) = (\alpha_{i,j})$ over $\mathcal F$ by putting the [*positively oriented edge*]{} ${\overrightarrow}{E_i} = {\overrightarrow}{[V_{i_1}, V_{i_2}]}$ and putting $$\alpha_{i,j} = \begin{cases}
1, & \text{if ${\overrightarrow}{E_j}$ appears in {\it the shortest path} ${\overrightarrow}{[f(V_{i_1}), f(V_{i_2})]}$ \small{from vertex} $f(V_{i_1})$ \small{to vertex} $f(V_{i_2})$}, \cr
-1, & \text{if $-{\overrightarrow}{E_j}$ appears in {\it the shortest path} ${\overrightarrow}{[f(V_{i_1}), f(V_{i_2})]}$ \small{from vertex} $f(V_{i_1})$ \small{to vertex} $f(V_{i_2})$}, \cr
0, & \text{otherwise}. \cr
\end{cases}$$ and define the associated [*unoriented transition*]{} $n \times n$ $\{ 0, 1 \}$-matrix $\mathcal B_n(f) = (\beta_{i,j})$ over $\mathcal F$ by putting, for all $1 \le i \le n$ and all $1 \le j \le n$, $\beta_{i,j} = 1$ if $\alpha_{i,j} \ne 0$ and $\beta_{i,j} = 0$ otherwise, or equivalently, $$\beta_{i,j} = \begin{cases}
1, & \text{if the set inclusion $f(E_i) \supset E_j$ holds}, \cr
0, & \text{otherwise}. \cr
\end{cases}$$ There are exactly $2^n$ such oriented transition matrices $\mathcal A_n(f)$ for each $f$ and yet they all have the same unoriented transition matrix $\mathcal B_n(f)$. Later, we shall see that the determinant of $\mathcal A_n(f)$ is $(-1)^n$ and that of $\mathcal B_n(f)$ is an odd integer. In [**[@du1]**]{}, we study the special case when $T$ is a compact interval in the real line and $f$ is a continuous vertex map on $T$. In this note, we generalize the main results in [**[@du1]**]{} for interval maps to vertex maps on trees. Surprisingly, the arguments used there [*almost*]{} work for vertex maps on trees. For completeness, we include the proofs.
Recall that $\mathcal F$ denotes a field with unity 1. Let ${\overrightarrow}{W_{\mathcal F}}^n(\mathcal E) = \bigl\{ \sum_{i=1}^{n} r_i{\overrightarrow}{E_i} : r_i \in \mathcal F, 1 \le i \le n \bigr\}$ denote the $n$-dimensional vector space over $\mathcal F$ with $\mathcal E = \{ {\overrightarrow}{E_j} : 1 \le j \le n \}$ as a basis. In the sequel, when there is no confusion, we shall write ${\overrightarrow}{W_{\mathcal F}}^n$ instead of ${\overrightarrow}{W_{\mathcal F}}^n(\mathcal E)$. So, now we regard each positively oriented edge ${\overrightarrow}{E_j}$ as a basis element of the vector space ${\overrightarrow}{W_{\mathcal F}}^n$ and regard the negatively oriented edge $-{\overrightarrow}{E_j}$ of ${\overrightarrow}{E_j}$ as an element in ${\overrightarrow}{W_{\mathcal F}}^n$ such that ${\overrightarrow}{E_j} + (-{\overrightarrow}{E_j}) = {\bf 0}$. Let $\sum_{i=1}^n r_i{\overrightarrow}{E_i}$ be an element of ${\overrightarrow}{W_{\mathcal F}}^n$ such that $r_i \in \{ -1, 0, 1 \}$ for all $1 \le i \le n$. If there exist two vertices $V_i$ and $V_j$ such that $r_k = 1$ if and only if ${\overrightarrow}{E_k}$ appears in [*the shortest path*]{} ${\overrightarrow}{[V_i, V_j]}$ from veretx $V_i$ to vertex $V_j$ and $r_k = -1$ if and only if $-{\overrightarrow}{E_k}$ appears in [*the shortest path*]{} ${\overrightarrow}{[V_i, V_j]}$ from vertex $V_i$ to vertex $V_j$, then we define ${\overrightarrow}{[V_i, V_j]} = \sum_{i=1}^n r_i{\overrightarrow}{E_i}$ and ${\overrightarrow}{[V_j, V_i]} = -{\overrightarrow}{[V_i, V_j]} = -(\sum_{i=1}^n r_i{\overrightarrow}{E_i})$. In particular, if ${\overrightarrow}{[V_i, V_j]} = {\overrightarrow}{e_1}{\overrightarrow}{e_2} \cdots {\overrightarrow}{e_m}$ is [*the shortest path*]{} in the oriented tree ${\overrightarrow}{T}$ defined as above, then, as elements of ${\overrightarrow}{W_{\mathcal F}}^n$, we have ${\overrightarrow}{[V_i, V_j]} = \sum_{k=1}^m {\overrightarrow}{e_k}$. Therefore, for any two vertices $V_i$ and $V_j$, the notation ${\overrightarrow}{[V_i, V_j]}$ will have two meanings: It represents the [*unique shortest path*]{} from vertex $V_i$ to vertex $V_j$ in the oriented tree ${\overrightarrow}{T}$ on the one hand, and represents the element of ${\overrightarrow}{W_{\mathcal F}}^n$ which is a sum of those oriented (positively or negatively) edges which appear in the [*unique shortest path*]{} $\overrightarrow{[V_i, V_j]}$ from vertex $V_i$ to vertex $V_j$ on the other. There should be no confusion from the texts. With respect to the [*oriented transition*]{} $n \times n$ $\{ -1, 0, 1 \}$-matrices $\mathcal A_n(f) = (\alpha_{i,j})$ of the continuous vertex tree map $f$, we define a linear transformation $\Phi_f$ from ${\overrightarrow}{W_{\mathcal F}}^n$ into itself such that, for each $1 \le i \le n$, $$\Phi_f({\overrightarrow}{E_i}) = \sum_{j=1}^n \alpha_{i,j} {\overrightarrow}{E_j}.$$Therefore, if ${\overrightarrow}{E_i} = {\overrightarrow}{[V_{i_1}, V_{i_2}]}$ is a positively oriented edge of ${\overrightarrow}{T}$ from vertex $V_{i_1}$ to vertex $V_{i_2}$, then, when considered as an element of ${\overrightarrow}{W_{\mathcal F}}^n$, we have, by definition of $\mathcal A_n(f)$ and $\Phi_f$, $\Phi_f({\overrightarrow}{E_i}) = \sum_{j=1}^n \alpha_{ij}{\overrightarrow}{E_j} = {\overrightarrow}{[f(V_{i_1}), f(V_{i_2})]}$ [which also represents the]{} [*unique shortest path*]{} [from vertex]{} $f(V_{i_1})$ [to vertex]{} $f(V_{i_2})$ in ${\overrightarrow}{T}$.
We shall need the following fundamental result.
[**Lemma 1.**]{} [*For any distinct vertices $V_i$ and $V_j$ of the oriented tree ${\overrightarrow}{T}$, we have $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$. That is, if ${\overrightarrow}{[V_i, V_j]}$ is the unique shortest path from vertex $V_i$ to vertex $V_j$ in ${\overrightarrow}{T}$, then $\Phi_f({\overrightarrow}{[V_i, V_j]}) \, (= {\overrightarrow}{[f(V_i), f(V_j)]})$ is the unique shortest path from vertex $f(V_i)$ to vertex $f(V_j)$ in ${\overrightarrow}{T}$. Similarly, if $V_{i_1}, V_{i_2}, \cdots, V_{i_m}$ are vertices of ${\overrightarrow}{T}$, then $\sum_{k=1}^{m-1} {\overrightarrow}{[V_{i_k}, V_{i_{k+1}}]} = {\overrightarrow}{[V_{i_1}, V_{i_m}]}$.*]{}
[*Proof.*]{} Let $V_i, V_k, V_j$ be three distinct vertices of the tree ${\overrightarrow}{T}$. Assume that both ${\overrightarrow}{[V_i, V_k]}$ and ${\overrightarrow}{[V_k, V_j]}$ are positively oriented edges of ${\overrightarrow}{T}$. If the set intersection $f([V_i : V_k]) \cap f([V_k : V_j]) = \{ f(V_k) \}$, then the concatenation of [*the shortest path*]{} ${\overrightarrow}{[f(V_i), f(V_k)]}$ and [*the shortest path*]{} ${\overrightarrow}{[f(V_k), f(V_j)]}$ becomes the [*shortest*]{} path ${\overrightarrow}{[f(V_i), f(V_j)]}$ from vertex $f(V_i)$ to vertex $f(V_j)$. Therefore, we have $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$. On the other hand, if the set intersection $f([V_i : V_k]) \cap f([V_k : V_j]) = [V_\ell : f(V_k)] \ne \{ f(V_k) \}$ for some veterx $V_\ell \ne f(V_k)$, then [*the shortest path*]{} ${\overrightarrow}{[V_\ell, f(V_k)]}$ in [*the shortest path*]{} ${\overrightarrow}{[f(V_i), f(V_k)]}$ and [*the shortest path*]{} ${\overrightarrow}{[f(V_k), V_\ell]}$ in [*the shortest path*]{} ${\overrightarrow}{[f(V_k), f(V_j)]}$ cancel out. So, $\Phi_f({\overrightarrow}{[V_i, V_j]}) = \Phi_f({\overrightarrow}{[V_i, V_k]} + {\overrightarrow}{[V_k, V_j]}) = \Phi_f({\overrightarrow}{[V_i, V_k]}) + \Phi_f({\overrightarrow}{[V_k, V_j]}) = {\overrightarrow}{[f(V_i), f(V_k)]} + {\overrightarrow}{[f(V_k), f(V_j)]} =\bigr({\overrightarrow}{[f(V_i), V_\ell]} + {\overrightarrow}{[V_\ell, f(V_k)]}\bigr) + \bigr({\overrightarrow}{[f(V_k), V_\ell]} + {\overrightarrow}{[V_\ell, f(V_j)]}\bigr) = {\overrightarrow}{[f(V_i), V_\ell]} + {\overrightarrow}{[V_\ell, f(V_j)]} = {\overrightarrow}{[f(V_i), f(V_j)]}$.
Assume that both ${\overrightarrow}{[V_i, V_k]}$ and ${\overrightarrow}{[V_j, V_k]} \,(= -{\overrightarrow}{[V_k, V_j]})$ are positively oriented edges of the oriented tree ${\overrightarrow}{T}$. Then [*the shortest path*]{} ${\overrightarrow}{[V_i, V_j]}$ is the concatenation of the positively oriented edge ${\overrightarrow}{[V_i, V_k]}$ and the negatively oriented edge $(-{\overrightarrow}{[V_j, V_k]})$. Thus, as elements of ${\overrightarrow}{W_{\mathcal F}}^n$, we have ${\overrightarrow}{[V_i, V_j]} = {\overrightarrow}{[V_i, V_k]} - {\overrightarrow}{[V_j, V_k]}$. So, $\Phi_f({\overrightarrow}{[V_i, V_j]}) = \Phi_f({\overrightarrow}{[V_i, V_k]} - {\overrightarrow}{[V_j, V_k]}) = \Phi_f({\overrightarrow}{[V_i, V_k]}) - \Phi_f({\overrightarrow}{[V_j, V_k]}) = {\overrightarrow}{[f(V_i), f(V_k)]} - {\overrightarrow}{[f(V_j), f(V_k)]}$. If the set intersection $f([V_i : V_k]) \cap f([V_j : V_k]) = \{ f(V_k) \}$, then the concatenation of [*the shortest path*]{} ${\overrightarrow}{[f(V_i), f(V_k)]}$ and [*the shortest path*]{} $-{\overrightarrow}{[f(V_j), f(V_k)]} \, (= {\overrightarrow}{[f(V_k), f(V_j)]})$ becomes the [*shortest*]{} path ${\overrightarrow}{[f(V_i), f(V_j)]}$ from vertex $f(V_i)$ to vertex $f(V_j)$. Therefore, we obtain that $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$. On the other hand, if the set intersection $f([V_i : V_k]) \cap f([V_j : V_k]) = [V_\ell : f(V_k)] \ne \{ f(V_k) \}$ for some veterx $V_\ell \ne f(V_k)$, then [*the shortest path*]{} ${\overrightarrow}{[V_\ell, f(V_k)]}$ in [*the shortest path*]{} ${\overrightarrow}{[f(V_i), f(V_k)]}$ and [*the shortest path*]{} $-{\overrightarrow}{[V_\ell, f(V_k)]} \, (={\overrightarrow}{[f(V_k), V_\ell]})$ in [*the shortest path*]{} $-{\overrightarrow}{[f(V_j), f(V_k)]}) \, (={\overrightarrow}{[f(V_k), f(V_j)]}$) cancel out. Therefore, $\Phi_f({\overrightarrow}{[V_i, V_j]}) = \Phi_f({\overrightarrow}{[V_i, V_k]} - {\overrightarrow}{[V_j, V_k]}) = \Phi_f({\overrightarrow}{[V_i, V_k]}) - \Phi_f({\overrightarrow}{[V_j, V_k]}) = {\overrightarrow}{[f(V_i), f(V_k)]} - {\overrightarrow}{[f(V_j), f(V_k)]} =\bigr({\overrightarrow}{[f(V_i), V_\ell]} + {\overrightarrow}{[V_\ell, f(V_k)]}\bigr) - \bigr({\overrightarrow}{[f(V_j), V_\ell]} + {\overrightarrow}{[V_\ell, f(V_k)]}\bigr) = {\overrightarrow}{[f(V_i), V_\ell]} - {\overrightarrow}{[f(V_j), V_\ell]} = {\overrightarrow}{[f(V_i), V_\ell]} + {\overrightarrow}{[V_\ell, f(V_j)]} = {\overrightarrow}{[f(V_i), f(V_j)]}$.
If both $\overrightarrow{[V_k, V_i]} \, (= -\overrightarrow{[V_i, V_k]})$ and $\overrightarrow{[V_k, V_j]}$ or, both $\overrightarrow{[V_k, V_i]} \, (= -\overrightarrow{[V_i, V_k]})$ and $\overrightarrow{[V_j, V_k]} \, (= -\overrightarrow{[V_k, V_j]}$ are positively oriented edges of the tree ${\overrightarrow}{T}$, then, by discussing cases depending on the set intersections $f([V_i : V_k]) \cap f([V_j : V_k])$ as above, we obtain that $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$. We omit the details.
So far, we have shown that $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$ as long as [*the shortest path*]{} ${\overrightarrow}{[V_i, V_j]}$ consists of exactly two oriented edges. Now, if [*the shortest path*]{} ${\overrightarrow}{[V_i, V_j]} = {\overrightarrow}{e_1}{\overrightarrow}{e_2}{\overrightarrow}{e_3}$ consists of exactly three oriented edges ${\overrightarrow}{e_1}, {\overrightarrow}{e_2}, {\overrightarrow}{e_3}$. Let the second vertex of ${\overrightarrow}{e_2}$ be $V_k$. Then ${\overrightarrow}{[V_i, V_k]} = {\overrightarrow}{e_1}{\overrightarrow}{e_2}$. It follows from what we just proved above that $\Phi_f({\overrightarrow}{[V_i, V_k]}) = {\overrightarrow}{[f(V_i), f(V_k)]}$. Therefore, $\Phi_f({\overrightarrow}{[V_i, V_j]}) = \Phi_f({\overrightarrow}{e_1}{\overrightarrow}{e_2}{\overrightarrow}{e_3}) = \Phi_f({\overrightarrow}{e_1}{\overrightarrow}{e_2}+{\overrightarrow}{e_3}) = \Phi_f({\overrightarrow}{e_1}{\overrightarrow}{e_2}) + \Phi_f({\overrightarrow}{e_3}) = \Phi_f({\overrightarrow}{[V_i, V_k]}) + \Phi_f({\overrightarrow}{e_3})$. Depending on ${\overrightarrow}{e_3} = {\overrightarrow}{[V_k, V_j]}$ or ${\overrightarrow}{e_3} = -{\overrightarrow}{[V_k, V_j]}$ and arguing as above, we can easily obtain that $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$ whenever ${\overrightarrow}{[V_i, V_j]} = {\overrightarrow}{e_1}{\overrightarrow}{e_2}{\overrightarrow}{e_3}$ consists of exactly three oriented edges ${\overrightarrow}{e_1}, {\overrightarrow}{e_2}, {\overrightarrow}{e_3}$. The general case when ${\overrightarrow}{[V_i, V_j]}$ consists of more than 3 oriented edges can be proved similarly by induction. Therefore, $\Phi_f({\overrightarrow}{[V_i, V_j]}) = {\overrightarrow}{[f(V_i), f(V_j)]}$ as long as $V_i$ and $V_j$ are any two distinct vertices of ${\overrightarrow}{T}$.
Finally, if $V_{i_1}, V_{i_2}, \cdots, V_{i_m}$ are $m \ge 2$ distinct vertices of ${\overrightarrow}{T}$, then similar arguments show that $\sum_{k=1}^{m-1} {\overrightarrow}{[V_{i_k}, V_{i_{k+1}}]} = {\overrightarrow}{[V_{i_1}, V_{i_m}]}$. This completes the proof.
[**Lemma 2.**]{} [*$\Phi_f$ is an isomorphism from ${\overrightarrow}{W_{\mathcal F}}^n$ onto itself.*]{}
[*Proof.*]{} Let $\hat f$ be any continuous vertex map on the tree $T$ such that the composition $\hat f \circ f$ is the identity map on the vertices of $T$. Then, by Lemma 1, for each positively oriented edge ${\overrightarrow}{E_i} = {\overrightarrow}{[V_{i_1}, V_{i_2}]}$, we have $(\Phi_{\hat f} \circ \Phi_f)({{\overrightarrow}{E_i}}) = \Phi_{\hat f}(\Phi_f({\overrightarrow}{[V_{i_1}, V_{i_2}]})) = \Phi_{\hat f}({\overrightarrow}{[f(V_{i_1}), f(V_{i_2})]}) = {\overrightarrow}{[(\hat f \circ f)(V_{i_1}), (\hat f \circ f)(V_{i_2})]} = {\overrightarrow}{[V_{i_1}, V_{i_2}]}$ $= {\overrightarrow}{E_i}$. Therefore, $\Phi_{\hat f}$ is the inverse of $\Phi_f$.
We shall need the following result which is proved in [**[@du1]**]{}. For completeness, we include its proof.
[**Lemma 3.**]{} [*Let $1 \le j \le n$ be any fixed integer and let $b$ denote the greatest common divisor of $j$ and $n+1$. Let $s = (n+1)/b$. For every integer $1 \le k \le s-1$, let $1 \le m_k \le n$ be the unique integer such that $kj \equiv m_k$ (mod $n+1$). Then the $m_k$’s are all distinct and $\{ m_k : 1 \le k \le s-1 \} = \{ kb : 1 \le k \le s-1 \}$.*]{}
[*Proof.*]{} Let $B = \{ m_k : 1 \le k \le s-1 \}$ and $C = \{ kb : 1 \le k \le s-1 \}$. For every integer $1 \le k \le s-1$, since $j/b$ and $(n+1)/b$ are relatively prime, the congruence equation $(j/b)x \equiv k$ (mod $(n+1)/b$) has an integer solution $x$ such that $1 \le x \le s-1 = [(n+1)/b] -1$. Consequently, for every integer $1 \le k \le s-1$, the congruence equation $(m_x \equiv) \, jx \equiv kb$ (mod $n+1$) has an integer solution $x$ such that $1 \le x \le s-1$. Since $1 \le kb \le (s-1)b \le n$ and $1 \le m_k \le n$ for every integer $1 \le k \le s-1$, we obtain that $C \subset B$. Since both $B$ and $C$ consist of exactly $s-1$ elements, we obtain that $B = C$. That is, $\{ m_k : 1 \le k \le s-1 \} = \{ kb : 1 \le k \le s-1 \}$. This completes the proof.
Let $M_1$ and $M_2$ be two $n \times n$ matrices over the field $\mathcal F$. We say that $M_1$ is similar to $M_2$ through the invertible matrix $G$ if $M_1 \cdot G = G \cdot M_2$. We can now prove our main result.
[**Theorem 1.**]{}
*Let $n \ge 2$ be an integer. Let $T$ be any tree with $n+1$ vertices. Let $f$ be a continuous vertex map on $T$. Let $\mathcal R$, $\mathcal Z_2$, $\mathcal F$, ${\overrightarrow}{W_{\mathcal F}}^n$, ${\overrightarrow}{W_{\mathcal Z_2}}^n$, $\Phi_f$, $\mathcal A_n(f)$ and $\mathcal B_n(f)$ be defined as above. Then the following hold:*
- For each integer $1 \le i \le n$, $\sum_{k=1}^n \Phi_f^k({\overrightarrow}{E_i}) = {\bf 0}$ and so, $\sum_{k=1}^n \Phi_f^k(w) = {\bf 0}$ for all $w$ in ${\overrightarrow}{W_{\mathcal F}}^n$.
- Let $i$ and $j$ be two integers in the interval $[1, n]$ and let ${\overrightarrow}{J}$ denote [*the shortest path*]{} ${\overrightarrow}{[V_i, f^j(V_i)]}$ in ${\overrightarrow}{T}$. If $j$ and $n+1$ are relatively prime, then the set $\mathcal W_f = \{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \}$ is a basis for ${\overrightarrow}{W_{\mathcal Z_2}}^n$ and for ${\overrightarrow}{W_{\mathcal F}}^n$ when $\mathcal F$ is a field with characteristic zero or the determinant of the matrix $\mathcal M_f$ of the set $\mathcal W_f$ with respect to the basis $\mathcal E = \{ {\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n} \}$ is not “divisible” by the finite characteristic of $\mathcal F$. Furthermore, when $T$ is a tree in the real line with $n+1$ vertices and $f$ is a continuous vertex map on $T$, then the constant term of the characteristic polynomial of the matrix $\mathcal M_f$ is $\pm 1$ and hence the set $\mathcal W_f$ is a basis of ${\overrightarrow}{W_{\mathcal F}}^n$ for any field $\mathcal F$ (however, not all coefficients of the characteristic polynomial of the matrix $\mathcal M_f$ are odd integers (see, for example, Figure 1(a) with ${\overrightarrow}{J} = {\overrightarrow}{[1, 2]}$ where the corresponding characteristic polynomial is $x^5-x^4-x+1$).
- Over any field $\mathcal F$ with characteristic zero ($\mathcal Z_2$ respectively), the oriented transition matrix $\mathcal A_n(f)$ and its inverse $[\mathcal A_n(f)]^{-1}$, as $\{ -1, 0, 1 \}$-matrices, are similar to the following companion matrix $$\left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 0 & 1 & \cdots & 0 \\ & \vdots & & & \vdots & \\ -1 & -1 & -1 & -1 & \cdots & -1 \\ \end{array}} \right]$$ of the polynomial $\sum_{k=0}^{n} x^k$ through invertible $\{ -1, 0, 1 \}$-matrices over $\mathcal F$ ($\mathcal Z_2$ respectively) and have the same characteristic polynomial $\sum_{k=0}^{n} x^k$ while the unoriented transition matrices of all continuous vertex maps on $T$ may not be similar to each other over the same field $\mathcal F$ ($\mathcal Z_2$ respectively) (see Figures 1 - 3). Furthermore, if $T$ is a tree in the real line, then the oriented transition matrices, when considered over any field, of all continuous vertex maps on $T$ with $n+1$ vertices and their inverses are similar to one another through invertible $\{ -1, 0, 1 \}$-matrices and have the same characteristic polynomial $\sum_{k=0}^{n} x^k$.
- The coefficients of the characteristic polynomial of the unoriented transition matrix $\mathcal B_n(f)$, when considered as a matrix over $\mathcal R$, are all odd integers (see Figures 1 - 4). Furthermore, the unoriented transition matrices over any field of all continuous vertex maps on all trees with $n+1$ vertices, when considered as matrices over $\mathcal Z_2$, are similar to one another and have characteristic polynomial $\sum_{k=0}^{n} x^k$, but may not be similar to each other when considered over the finite field $\mathcal Z_p = \{ 0, 1, 2, \cdots, p-1 \}$, where $p \ge 3$ is a prime number (see Figures 1 - 3).
[*Proof.*]{} To prove Part (1), recall that $f$ is a continuous vertex map on the tree $T_1$. For any fixed integer $1 \le i \le n$, let ${\overrightarrow}{E_i} = {\overrightarrow}{[V_{i_1}, V_{i_2}]}$ and let $1 \le j \le n$ be the unique integer such that $f^j({\overrightarrow}{V_{i_1}}) = {\overrightarrow}{V_{i_2}}$. So, ${\overrightarrow}{E_i} = {\overrightarrow}{[V_{i_1}, V_{i_2}]} = {\overrightarrow}{[V_{i_1}, f^j(V_{i_1})]}$. Let $b$ be the greatest common divisor of $j$ and $n+1$ and let $s = (n+1)/b$. So, $sj = (j/b)(sb) = (j/b)(n+1)$. For every integer $1 \le k \le s-1$, let $1 \le m_k \le n$ be the unique integer such that $kj \equiv m_k$ (mod $n+1$). Then, by Lemma 3, we obtain that $\{ m_k : 1 \le k \le s-1 \} = \{ kb : 1 \le k \le s-1 \}$. Let $m_0 = 0$. Then $\{ m_k : 0 \le k \le s-1 \} = \{ kb : 0 \le k \le s-1 \}$. Hence, the set $\{ 1, 2, \cdots, n-1, n \}$ is the disjoint union of the sets $\{ m_k + m : 0 \le k \le s-1 \}, 0 \le m \le b-1$. Therefore, by Lemma 1, $\sum_{k=0}^{s-1} \Phi_f^{m_k}({\overrightarrow}{E_i}) = \sum_{k=0}^{s-1} \Phi_f^{kj}({\overrightarrow}{E_i})$ (since $kj \equiv m_k$ (mod $n+1)) = {\overrightarrow}{[V_{i_1}, f^j(V_{i_1})]} + {\overrightarrow}{[f^j(V_{i_1}), f^{2j}(V_{i_1})]} + {\overrightarrow}{[f^{2j}(V_{i_1}), f^{3j}(V_{i_1})]} + \cdots + {\overrightarrow}{[f^{(s-2)j}(V_{i_1}), f^{(s-1)j}(V_{i_1})]} + {\overrightarrow}{[f^{(s-1)j}(V_{i_1}), V_{i_1}]} = {\bf 0}$. Thus, $\sum_{\ell=0}^n \Phi_f^\ell({\overrightarrow}{E_i}) = \sum_{m=0}^{b-1} \Phi_f^m\bigr(\sum_{k=0}^{s-1} \Phi_f^{m_k}({\overrightarrow}{E_i})\bigr) = {\bf 0}$. Therefore, $\sum_{k=1}^n \Phi_f^k(w) = {\bf 0}$ for all vectors $w$ in ${\overrightarrow}{W_{\mathcal F}}^n$. This establishes Part (1).
For the proof of Part (2), we first consider those $\Phi_f$ over the field $\mathcal Z_2$. Now we want to show that if $\mathcal N$ is a nonempty subset of $\{ 1, 2, \cdots, n-1, n \}$ such that ${\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J}) = \bf 0$, then $\mathcal N = \{ 1, 2, \cdots, n-1, n \}$. Indeed, for every integer $1 \le k \le n$, let $1 \le m_k \le n$ be the unique integer such that $kj \equiv m_k$ (mod $n+1$). Assume that $(j =) \,\, m_1 \notin \mathcal N$. Then, for any $m \in \mathcal N$, $m \ne j$. So, $f^m(V_i) \ne f^j(V_i)$. If $(f^m(f^j(V_i)) =) \,\,f^{m+j}(V_i) = f^j(V_i)$, then the least period of $f^j(V_i)$ under $f$ divides $m$ ($< n+1$) which contradicts the fact that its least period under $f$ is $n+1$. Therefore, [*the shortest path*]{} $\Phi_f^m({\overrightarrow}{J}) = \Phi_f^m({\overrightarrow}{[V_i, f^j(V_i)]}) = {\overrightarrow}{[f^m(V_i), f^{m+j}(V_i)]}$ either contains the vertex $f^j(V_i)$ in its “interior” or does not contain it. So, in the expression of the element $\Phi_f^m({\overrightarrow}{J}) = \Phi_f^m({\overrightarrow}{[V_i, f^j(V_i)]}) = {\overrightarrow}{[f^m(V_i), f^{m+j}(V_i)]}$ as a sum of the basis elements ${\overrightarrow}{E_k}$’s, the number of the basis elements ${\overrightarrow}{E_k}$ which contain the vertex $f^j(V_i)$ as an endpoint is either 0 or 2. Since ${\overrightarrow}{J} = {\overrightarrow}{[V_i, f^j(V_i)]}$ contains exactly one baisis element ${\overrightarrow}{E_k}$ which has the vertex $f^j(V_i)$ as an endpoint, there are an [*odd*]{} number of basis elements ${\overrightarrow}{E_k}$’s which has the vertex $f^j(V_i)$ as an endpoint in the expression of the element ${\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J})$ as a sum of the basis elements ${\overrightarrow}{E_k}$’s. Consequently, ${\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J}) \ne {\bf 0}$. This is a contradiction. So, $(j =) \,\, m_1 \in \mathcal N$ and $${\bf 0} = {\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J}) = {\overrightarrow}{J} + \Phi_f^{m_1}({\overrightarrow}{J}) + \sum_{k \in \mathcal N \setminus \{ m_1 \}} \Phi_f^k({\overrightarrow}{J})$$ $$\,\, = {\overrightarrow}{[V_i, f^j(V_i)]} + {\overrightarrow}{[f^j(V_i), f^{2j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1 \}} \Phi_f^k(J)$$ $$= {\overrightarrow}{[V_i, f^{2j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1 \}} \Phi_f^k({\overrightarrow}{J}).\qquad \qquad\qquad\quad$$ Now assume that $m_2 \notin \mathcal N \setminus \{ m_1 \}$. Then, for any $m \in \mathcal N \setminus \{ m_1 \}$, $m \notin \{ m_1, m_2 \} = \{ j, m_2 \} \subset \{ 1, 2, \cdots, n \}$. If $f^m(V_i) = f^{2j}(V_i) \, (= f^{m_2}(V_i))$, then $m \equiv m_2$ (mod $n+1$). Since both $m$ and $m_2$ are integers in the set $\{ 1, 2, \cdots, n \}$ such that $m \equiv m_2$ (mod $n+1$), we have $m = m_2$. This is a contradiction. If $f^{m+j}(V_i) = f^{2j}(V_i)$, then $m+j \equiv 2j$ (mod $n+1$) and so, $m \equiv j \, (\equiv m_1)$ (mod $n+1$). Since both $m$ and $m_1$ are integers in the set $\{ 1, 2, \cdots, n \}$ such that $m \equiv m_1$ (mod $n+1$), we have $m = m_1$. This is again a contradiction. Therefore, in the expression of the element $\Phi_f^m({\overrightarrow}{J}) = \Phi_f({\overrightarrow}{[V_i, f^j(V_i)]}) = {\overrightarrow}{[f^m(V_i), f^{m+j}(V_i)]}$ as a sum of the basis elements ${\overrightarrow}{E_k}$’s, the number of the basis elements $E_k$ which contain the vertex $f^{2j}(V_i)$ as an endpoint is either 0 or 2. Since ${\overrightarrow}{[V_i, f^{2j}(V_i)]}$ contains exactly one baisis element ${\overrightarrow}{E_k}$ which has the vertex $f^{2j}(V_i)$ as an endpoint, there are an [*odd*]{} number of basis elements ${\overrightarrow}{E_k}$’s which has the vertex $f^{2j}(V_i)$ as an endpoint in the expression of the element ${\overrightarrow}{[V_i, f^{2j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1 \}} \Phi_f^k({\overrightarrow}{J})$ as a sum of the basis elements ${\overrightarrow}{E_k}$’s. Consequently, ${\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J}) = {\overrightarrow}{[V_i, f^{2j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1 \}} \Phi_f^k({\overrightarrow}{J})\ne {\bf 0}$. This is a contradiction. So, $m_2 \in \mathcal N \setminus \{ m_1 \}$ and $${\bf 0} = {\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J}) = {\overrightarrow}{[V_i, f^{2j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1 \}} \Phi_f^k({\overrightarrow}{J}).\qquad\qquad\qquad\quad\qquad\qquad\qquad\quad$$ $$={\overrightarrow}{[V_i, f^{2j}(V_i)]} + \Phi_f^{m_2}({\overrightarrow}{J}) + \sum_{k \in \mathcal N \setminus \{ m_1, m_2 \}} \Phi_f^k({\overrightarrow}{J})\,\,$$ $$\qquad\quad\,\, = {\overrightarrow}{[V_i, f^{2j}(V_i)]} + {\overrightarrow}{[f^{2j}(V_i), f^{3j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1, m_2 \}} \Phi_f^k({\overrightarrow}{J})$$ $$\quad\quad\,\,= {\overrightarrow}{[V_i, f^{3j}(V_i)]} + \sum_{k \in \mathcal N \setminus \{ m_1, m_2 \}} \Phi_f^k({\overrightarrow}{J}).\qquad\qquad\qquad\quad$$ Proceeding in this manner finitely many times, we obtain that $\{ m_1, m_2, \cdots, m_{n-1}, m_n \} \subset \mathcal N$. Since $j$ and $n+1$ are relatively prime, we see that, by Lemma 3, $\{ m_1, m_2, \cdots, m_n \} = \{ 1, 2, \cdots, n-1, n \}$. Since $\{ m_1, m_2, \cdots, m_n \} \subset \mathcal N \subset \{ 1, 2, \cdots, n-1, n \}$, we obtain that $\mathcal N = \{ 1, 2, \cdots, n-1, n \}$. This proves our assertion.
Now assume that $\sum_{k=0}^{n-1} r_k\Phi_f^k({\overrightarrow}{J}) = {\bf 0}$, where $r_k = 0$ or 1 in $\mathcal Z_2$, for all $0 \le k \le n-1$. If $r_0 = 0$ and $r_\ell \ne 0$ for some integer $1 \le \ell \le n-1$, we may assume that $\ell$ is the smallest such integer. Since $\Phi_f$ is invertible on ${\overrightarrow}{W_{\mathcal Z_2}}^n$, we obtain that ${\overrightarrow}{J} + \sum_{k=1}^{n-1-\ell} r_k\Phi_f^k({\overrightarrow}{J}) = {\bf 0}$. So, without loss of generality, we may assume that $r_0 \ne 0$. That is, we may assume that ${\overrightarrow}{J} + \sum_{k=1}^{n-1} r_k\Phi_f^k({\overrightarrow}{J}) = {\bf 0}$. Let $\mathcal N = \{ k : 1 \le k \le n-1$ and $r_k \ne 0 \}$. Then we have ${\overrightarrow}{J} + \sum_{k \in \mathcal N} \Phi_f^k({\overrightarrow}{J}) = {\bf 0}$. However, it follows from what we just proved above that $\mathcal N = \{ 1, 2, \cdots, n-1, n \}$. This contradicts the assumption that $\mathcal N \subset \{ 1, 2, \cdots, n-1 \}$. Therefore, the set $\{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \}$ is linearly independent in the $n$-dimensional vector space ${\overrightarrow}{W_{\mathcal Z_2}}^n$ and hence is a basis for ${\overrightarrow}{W_{\mathcal Z_2}}^n$. Consequently, the matrix of the basis $\{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \}$ over $\mathcal Z_2$ with respect to the basis $\{ {\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n} \}$ of ${\overrightarrow}{W_{\mathcal Z_2}}^n$, denoted as $$\bigl [{\overrightarrow}{J}, \Phi_f({\overrightarrow}{J}), \Phi_f^2({\overrightarrow}{J}), \cdots, \Phi_f^{n-1}({\overrightarrow}{J}) : {\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n} \bigr ] \,\,\, (\text{over} \,\,\, \mathcal Z_2),$$ has nonzero determinant and hence equals 1. This implies that, over the general field $\mathcal F$ with unity 1, the $\{ -1, 0, 1 \}$-matrix $$\mathcal M_f = \bigl [{\overrightarrow}{J}, \Phi_f({\overrightarrow}{J}), \Phi_f^2({\overrightarrow}{J}), \cdots, \Phi_f^{n-1}({\overrightarrow}{J}) : {\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n} \bigr ] \,\,\, (\text{over} \,\,\, \mathcal F)$$ of the set $\{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \}$ with respect to the basis $\{ {\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n} \}$ of ${\overrightarrow}{W_{\mathcal F}}^n$ also has nonzero determinant if the characteristic of $\mathcal F$ is zero or the determinant of $\mathcal M_f$ is not [*divisible*]{} by the finite characteristic of $\mathcal F$. In particular, if $\mathcal F = \mathcal R$ or $\mathcal F = \mathcal Z_2$, then the set $\{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \}$ is a basis for ${\overrightarrow}{W_{\mathcal F}}^n$ [**[@her]**]{}. Note that when $\mathcal F = \mathcal R$, the determinant of $\mathcal M_f$ is an odd integer. We do not know if it is always equal to $\pm 1$. However, when $T$ is a tree in the real line, by choosing all orientations on the edges [*same direction*]{}, the matrix $\mathcal M_f$ over $\mathcal R$ is a Petrie matrix, i.e., in any row, nonzero entries are [*consecutive*]{} and are all equal to 1 or to $-1$. It follows from easy induction [**[@gor]**]{} that the determinant of any Petrie matrix is 0 or $\pm 1$. Since the determinant of $\mathcal M_f$ over $\mathcal R$ is nonzero, we obtain that the determinant of $\mathcal M_f$ is $\pm 1$. The rest is easy and omitted. This confirms Part (2).
Now, for the general field $\mathcal F$ with unity 1, let ${\overrightarrow}{T}$ be the oriented tree on the interval $[1, n+1]$ in the real line with $n+1$ vertices $\hat V_i = i, 1 \le i \le n+1$ and $n$ positively oriented edges ${\overrightarrow}{D_j} = {\overrightarrow}{[j, j+1]}, 1 \le j \le n$. Let $h$ be the continuous vertex map on ${\overrightarrow}{T}$ such that $h(x) = x+1$ for all $1 \le x \le n$ and $h(x) = -nx + n^2 + n + 1$ for all $n \le x \le n+1$. Then $\Phi_h({\overrightarrow}{D_k}) = {\overrightarrow}{D_{k+1}} = \Phi_h^{k}({\overrightarrow}{D_1})$ for all $1 \le k \le n-1$ and $\Phi_h({\overrightarrow}{D_n}) = \Phi_h({\overrightarrow}{[\hat V_n, \hat V_{n+1}]}) = {\overrightarrow}{[\hat V_{n+1}, \hat V_1]} = -{\overrightarrow}{[\hat V_1, \hat V_{n+1}]} = -\sum_{k=1}^{n} {\overrightarrow}{D_k}$. By definition, the set $\mathcal D = \{ \Phi_h^k({\overrightarrow}{D_1}) : 0 \le k \le n-1 \} = \{ {\overrightarrow}{D_1}, {\overrightarrow}{D_2}, \cdots, {\overrightarrow}{D_n} \}$ is a basis for ${\overrightarrow}{W_{\mathcal F}}^n(\mathcal D)$. Let $1 \le i \le n$ be a fixed integer and choose a fixed integer $1 \le j \le n$ such that $j$ and $n+1$ are relatively prime and let ${\overrightarrow}{J} = {\overrightarrow}{[V_i, f^j(V_i)]}$. $$\text{Suppose the set} \,\,\, \{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \} \,\,\, \text{is a basis for} \,\,\, {\overrightarrow}{W_{\mathcal F}}^n(\mathcal E). \qquad\qquad\qquad\qquad (*)$$ Let $\phi : {\overrightarrow}{W_{\mathcal F}}^n(\mathcal D) \longrightarrow {\overrightarrow}{W_{\mathcal F}}^n(\mathcal E)$ be the linear transformation defined by $$\phi({\overrightarrow}{D_k}) = \Phi_f^{k-1}({\overrightarrow}{J}) \,\, \text{for all} \,\, 1 \le k \le n.$$Then $\phi$ is an isomorphism and the matrix of the basis $\{ \Phi_f^k({\overrightarrow}{J}) : 0 \le k \le n-1 \}$ with respect to the basis $\mathcal E = \{ {\overrightarrow}{E_1}, {\overrightarrow}{E_2}, \cdots, {\overrightarrow}{E_n} \}$ of ${\overrightarrow}{W_{\mathcal F}}^n(\mathcal E)$ is an $n \times n$ $\{ -1, 0, 1 \}$-matrix. Furthermore, $$(\phi \circ \Phi_h)({\overrightarrow}{D_n}) = \phi(\Phi_h({\overrightarrow}{D_n})) = \phi(-\sum_{k=1}^{n} {\overrightarrow}{D_k}) = -\sum_{k=1}^{n} \phi({\overrightarrow}{D_k}) = -\sum_{k=1}^{n} \Phi_f^{k-1}({\overrightarrow}{J}) = \Phi_f^n({\overrightarrow}{J}) \,\, \text{(by Part (1))}$$ $$\,\,\,\,= \Phi_f(\Phi_f^{n-1}({\overrightarrow}{J})) = \Phi_f(\phi({\overrightarrow}{D_n})) = (\Phi_f \circ \phi)({\overrightarrow}{D_n})$$ and, for every integer $1 \le k \le n-1$, $$(\phi \circ \Phi_h)({\overrightarrow}{D_k}) = \phi(\Phi_h({\overrightarrow}{D_k})) = \phi({\overrightarrow}{D_{k+1}}) = \Phi_f^k({\overrightarrow}{J}) = \Phi_f(\Phi_f^{k-1}({\overrightarrow}{J})) = \Phi_f(\phi({\overrightarrow}{D_k})) = (\Phi_f \circ \phi)({\overrightarrow}{D_k}).$$ Therefore, $\Phi_f$ is similar to $\Phi_h$ through $\phi$. Similarly, $\Phi_{\hat f}$ is similar to $\Phi_h$, where $\hat f$ is any continuous [*vertex map*]{} on the oriented tree ${\overrightarrow}{T}$ such that the composition $\hat f \circ f$ on the vertices of ${\overrightarrow}{T}$ is the identity map. So, the matrices $\mathcal A_n(f)$ and $\mathcal A_n(\hat f) = [\mathcal A_n(f)]^{-1}$ are similar to $\mathcal A_n(h)$ over $\mathcal F$. Similarly, the matrices $\mathcal A_n(g)$ and $[\mathcal A_n(g)]^{-1}$ are similar to $\mathcal A_n(h)$ over $\mathcal F$. Consequently, we obtain that the matrices $\mathcal A_n(f)$, $[\mathcal A_n(f)]^{-1}$, $\mathcal A_n(g)$, and $[\mathcal A_n(g)]^{-1}$ are similar to one another over $\mathcal F$. By Part (2), the above (\*) holds for $\mathcal F = \mathcal Z_2$ and for any field $\mathcal F$ with char($\mathcal F) = 0$. Therefore, the matrices $\mathcal A_n(f)$, $[\mathcal A_n(f)]^{-1}$, $\mathcal A_n(g)$, and $[\mathcal A_n(g)]^{-1}$ are similar to one another over $\mathcal Z_2$ and over any field $\mathcal F$ with char($\mathcal F) = 0$. On the other hand, let $P_n(x) = x^n + \cdots$ denote the characteristic polynomial of $\mathcal A_n(f)$ over $\mathcal Z_2$ or over a field $\mathcal F$ with char($\mathcal F) = 0$. By Part (2), the degree of the [*minimal*]{} polynomial of the element ${\overrightarrow}{[V_i, f(V_i)]}$ is at least $n$. It follows from Part (1) that the polynomial $\sum_{k=0}^{n} x^k$ is the minimal polynomial of ${\overrightarrow}{[V_i, f(V_i)]}$. By the well-known Cayley-Hamilton theorem on matrices, we see that the element ${\overrightarrow}{[V_i, f(V_i)]}$ also satisfies the polynomial $P_n(x) - \sum_{k=0}^{n} x^k$ whose degree is at most $n-1 \, (< n)$. Therefore, $P_n(x) - \sum_{k=0}^{n} x^k = 0$, i.e., the characteristic polynomial of $\mathcal A_n(f)$ is $\sum_{k=0}^{n} x^k$. This proves Part (3).
Let $\mathcal A_n(f)$ and $\mathcal B_n(f)$ be the oriented and unoriented transition matrices of $\Phi_f$ over $\mathcal R$ respectively. Then it follows from Part (3) that the characteristic polynomial of $\mathcal A_n(f)$ is $\sum_{k=0}^n x^k$. When we consider $\mathcal A_n(f)$ as a matrix over $\mathcal Z_2$, we obtain that $\mathcal B_n(f) = \mathcal A_n(f)$ and the characteristic polynomial of $\mathcal B_n(f)$ is $\sum_{k=0}^n x^k$ over $\mathcal Z_2$. Consequently, the coefficients of the characteristic polynomial of $\mathcal B_n(f)$ over $\mathcal R$ are all odd integers (see Figures 1 - 4). Furthermore, we see that $\mathcal B_n(f) = \mathcal A_n(f)$ over $\mathcal Z_2$. So, it follows from Part (3) that $\mathcal B_n(f)$ and $\mathcal B_n(g)$, when considered as matrices over $\mathcal Z_2$, are similar to each other. This proves Part (4) and completes the proof of the theorem.
[**Remark.**]{} Let $f$ be a continuous vertex map on the tree $T \,\, (= T_1)$ with $n+1 \ge 3$ vertices. For $\mathcal F = \mathcal R$ and any choices of orientations on the edges of $T$, it follows from Theorem 1(1) that the determinant of the corresponding oriented transition matrix $\mathcal A_n(f)$ is $(-1)^n$ while, by Theorem 1(4), that of the corresponding unoriented transition matrix $\mathcal B_n(f)$ is an odd integer which is not necessarily equal to $\pm 1$. See Figure 2 for some examples. In the following, we present two sufficient conditions which guarantee that the determinant of the corresponding unoriented transition matrix $\mathcal B_n(f)$ is $\pm 1$. For other related problems regarding the unoriented transition matrices $\mathcal B_n(f)$, we refer to [**[@du2]**]{} where (new) notions of one-sided and two-sided similarities or weak similarities of square $\{ 0, 1 \}$-matrices are introduced and examples are presented. It is clear that notions of various similarities of the unoriented transition matrices similar to those considered in [**[@du2]**]{} can be generalized from trees in the real line (i.e., compact intervals) to arbitrary trees.
[**Proposition 1.**]{}
*Let $f$ be a continuous vertex map on the tree $T \,\, (= T_1)$ with $n+1 \ge 3$ vertices. For each integer $1 \le i \le n$, let ${\overrightarrow}{E_i} = {\overrightarrow}{[V_{i_1}, V_{i_2}]}$ be a positively oriented edge of the oriented tree ${\overrightarrow}{T}$. Let the field $\mathcal F = \mathcal R$. Then, by Lemma 1 and the definition of the map $\Phi_f$ on the vector space ${\overrightarrow}{W_{\mathcal R}}^n$, we obtain that $\Phi_f({\overrightarrow}{E_i}) = {\overrightarrow}{[f(V_{i_1}), f(V_{i_2})]}$. Since $f$ is a continuous vertex map on the connected edge $E_i$ of the tree $T$, we can write $\Phi_f({\overrightarrow}{E_i}) = {\overrightarrow}{[f(V_{i_1}), f(V_{i_2})]} = \sum_{j=1}^{m_i} r_{i,j} {\overrightarrow}{[V_{\ell_{i,j}}, V_{\ell_{i,j+1}}]}$, where, for each $1 \le j \le m_i$, $r_{i,j} = \pm 1$, $V_{\ell_{i,1}} = f(V_{i_1})$, $V_{\ell_{i,m_i+1}} = f(V_{i_2})$ and ${\overrightarrow}{[V_{\ell_{i,j}}, V_{\ell_{i,j+1}}]}$ is a positively oriented edge of the oriented tree ${\overrightarrow}{T}$. Then the following hold:*
- If, for each $1 \le i \le n$, $\Phi_f({\overrightarrow}{E_i})$ has only one sign, i.e., $r_{i,1} = r_{i,2} = \cdots = r_{i, m_i}$ (this includes the cases when $T$ is a compact interval in the real line), then the corresponding oriented transition matrix $\mathcal A_n(f)$ of $f$ can be obtained from that of the corresponding unoriented transition matrix $\mathcal B_n(f)$ of $f$ by performing the following row operation: Multiplying one row by $-1$. Consequently, the determinant of the matrix $\mathcal B_n(f)$ is equal to $\pm 1$ times that of the matrix $\mathcal A_n(f)$ which is $\pm 1$ (see [**[@du1]**]{}).
- If, for each $1 \le i \le n$ such that $\Phi_f({\overrightarrow}{E_i})$ does not have one sign, there exists an integer $1 \le k_i < m_i$ such that $r_{i,1} = r_{i,2} = \cdots = r_{i, k_i} \ne r_{i, k_i+1} = r_{i, k_i+2} = \cdots = r_{i, m_i}$ and $|r_{i, 1}| = |r_{i, m_i}| = 1$, let $\hat V_{\ell_{i,k_i}}$ be the unique vertex of $T$ such that $f(\hat V_{\ell_{i,k_i}}) = V_{\ell_{i,k_i}}$ and let ${\overrightarrow}{e_1} {\overrightarrow}{e_2} \cdots {\overrightarrow}{e_s}$ be the [shortest]{} path from either $V_{i_1}$ or $V_{i_2}$ to $\hat V_{\ell_{i,k_i}}$ which passes through the edge $E_i = [V_{i_1}, V_{i_2}]$ (and so the second vertex of ${\overrightarrow}{e_s}$ is $\hat V_{\ell_{i,k_i}}$). If each one of $\Phi_f({\overrightarrow}{e_2}), \Phi_f({\overrightarrow}{e_3}), \cdots, \Phi_f({\overrightarrow}{e_{s-1}})$ and $\Phi_f({\overrightarrow}{e_s})$ has only one sign, then the corresponding oriented transition matrix $\mathcal A_n(f)$ of $f$ can be obtained from that of the corresponding unoriented transition matrix $\mathcal B_n(f)$ of $f$ by performing the following two row operations: (i) Multiplying one row by $-1$ and (ii) Multiplying one row by $\pm 2$ and adding to another row. Consequently, the determinant of the matrix $\mathcal B_n(f)$ is equal to $\pm 1$ times that of the matrix $\mathcal A_n(f)$ which is $\pm 1$ (see Figures 1 $\&$ 3).
[**Remark.**]{} Figure 4 demonstrates a case which is not covered by Proposition 1, yet has the same conclusion. We note that although, for a continuous vertex map $f$ on a tree $T$ with $n+1$ vertices, there are $2^n$ distinct oriented transition matrices, they all have one and the same unoriented transition matrix. Therefore, if we can find an orientation for the tree $T$ so that Proposition 1 applies, then we obtain that the determinant of the unoriented transition matrix is $\pm 1$. Figure 4 is such an example.
![The above 6 matrices are part of oriented transition matrices of continuous vertex maps on the oriented trees with 6 vertices right above them. They are all similar to one another over the field $\mathcal F$. The characteristic polynomials of their corresponding [*unoriented transition matrices*]{} are (a) $x^5-3x^4+x^3+x^2-3x+1$; (b) $x^5-x^4-3x^3-3x^2+x-1$; (c) $x^5-x^4-3x^3+x^2-x+1$; (d) $x^5-x^4-3x^3+x^2+x+1$; (e) $x^5-3x^4+x^3-3x^2-x-1$; and (f) $x^5-x^4-x^3-x^2-x-1$ respectively.](sample1){width="1.6in" height="1.5in"}
![The above 6 matrices are part of oriented transition matrices of continuous vertex maps on the oriented trees with 6 vertices right above them. They are all similar to one another over the field $\mathcal F$. The characteristic polynomials of their corresponding [*unoriented transition matrices*]{} are (a) $x^5-3x^4+x^3+x^2-3x+1$; (b) $x^5-x^4-3x^3-3x^2+x-1$; (c) $x^5-x^4-3x^3+x^2-x+1$; (d) $x^5-x^4-3x^3+x^2+x+1$; (e) $x^5-3x^4+x^3-3x^2-x-1$; and (f) $x^5-x^4-x^3-x^2-x-1$ respectively.](sample2){width="1.6in" height="1.5in"}
![The above 6 matrices are part of oriented transition matrices of continuous vertex maps on the oriented trees with 6 vertices right above them. They are all similar to one another over the field $\mathcal F$. The characteristic polynomials of their corresponding [*unoriented transition matrices*]{} are (a) $x^5-3x^4+x^3+x^2-3x+1$; (b) $x^5-x^4-3x^3-3x^2+x-1$; (c) $x^5-x^4-3x^3+x^2-x+1$; (d) $x^5-x^4-3x^3+x^2+x+1$; (e) $x^5-3x^4+x^3-3x^2-x-1$; and (f) $x^5-x^4-x^3-x^2-x-1$ respectively.](sample3){width="1.6in" height="1.5in"}
$$\left[ {\begin{array}{*{20}c} 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & -1 \\ 0 & 1 & 1 & 1 & 0 \\ -1 & 0 & -1 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ \end{array}} \right], \qquad\quad\, \left[ {\begin{array}{*{20}c}
-1 & 0 & -1 & -1 & 0 \\ -1 & 0 & -1 & -1 & -1 \\ 1 & -1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ \end{array}} \right], \qquad\quad\, \left[ {\begin{array}{*{20}c} 0 & -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & -1 & -1 \\ 0 & 1 & 1 & -1 & 0 \\ 0 & -1 & -1 & 0 & 0 \\ \end{array}} \right]$$
![The above 6 matrices are part of oriented transition matrices of continuous vertex maps on the oriented trees with 6 vertices right above them. They are all similar to one another over the field $\mathcal F$. The characteristic polynomials of their corresponding [*unoriented transition matrices*]{} are (a) $x^5-3x^4+x^3+x^2-3x+1$; (b) $x^5-x^4-3x^3-3x^2+x-1$; (c) $x^5-x^4-3x^3+x^2-x+1$; (d) $x^5-x^4-3x^3+x^2+x+1$; (e) $x^5-3x^4+x^3-3x^2-x-1$; and (f) $x^5-x^4-x^3-x^2-x-1$ respectively.](sample4){width="1.6in" height="1.5in"}
![The above 6 matrices are part of oriented transition matrices of continuous vertex maps on the oriented trees with 6 vertices right above them. They are all similar to one another over the field $\mathcal F$. The characteristic polynomials of their corresponding [*unoriented transition matrices*]{} are (a) $x^5-3x^4+x^3+x^2-3x+1$; (b) $x^5-x^4-3x^3-3x^2+x-1$; (c) $x^5-x^4-3x^3+x^2-x+1$; (d) $x^5-x^4-3x^3+x^2+x+1$; (e) $x^5-3x^4+x^3-3x^2-x-1$; and (f) $x^5-x^4-x^3-x^2-x-1$ respectively.](sample5){width="1.6in" height="1.5in"}
![The above 6 matrices are part of oriented transition matrices of continuous vertex maps on the oriented trees with 6 vertices right above them. They are all similar to one another over the field $\mathcal F$. The characteristic polynomials of their corresponding [*unoriented transition matrices*]{} are (a) $x^5-3x^4+x^3+x^2-3x+1$; (b) $x^5-x^4-3x^3-3x^2+x-1$; (c) $x^5-x^4-3x^3+x^2-x+1$; (d) $x^5-x^4-3x^3+x^2+x+1$; (e) $x^5-3x^4+x^3-3x^2-x-1$; and (f) $x^5-x^4-x^3-x^2-x-1$ respectively.](sample6){width="1.6in" height="1.5in"}
$$\left[ {\begin{array}{*{20}c} -1 & 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & -1 & -1 \\ -1 & 0 & 0 & -1 & 0 \\ 1 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}} \right], \qquad\qquad\, \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & -1 & 0 \\ -1 & -1 & -1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & -1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}} \right], \qquad\qquad \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 1 & 0 & 0 & -1 \\ \end{array}} \right]$$
![The characteristic polynomials of the above corresponding [*unoriented transition matrices*]{} are $x^{11}-x^{10}-7x^9+7x^8+7x^7-7x^6+3x^5-7x^4-x^3-x^2-3x+3$ and $x^{11}-x^{10}-9x^9+5x^8+25x^7-x^6-25x^5-11x^4+9x^3+9x^2-x-3$ respectively.](figure1 "fig:"){width="3.0in" height="1.5in"} ![The characteristic polynomials of the above corresponding [*unoriented transition matrices*]{} are $x^{11}-x^{10}-7x^9+7x^8+7x^7-7x^6+3x^5-7x^4-x^3-x^2-3x+3$ and $x^{11}-x^{10}-9x^9+5x^8+25x^7-x^6-25x^5-11x^4+9x^3+9x^2-x-3$ respectively.](figure2 "fig:"){width="3.0in" height="1.5in"}
![The characteristic polynomials of the above corresponding [*unoriented transition matrices*]{} are $x^{11}-x^{10}-7x^9+7x^8+13x^7-13x^6-7x^5+5x^4+x^3-x^2-x+1$ and $x^{11}-x^{10}-7x^9+3x^8+11x^7+5x^6+x^5-5x^4-5x^3-3x^2+x+1$ respectively.](figure3 "fig:"){width="3.0in" height="1.5in"} ![The characteristic polynomials of the above corresponding [*unoriented transition matrices*]{} are $x^{11}-x^{10}-7x^9+7x^8+13x^7-13x^6-7x^5+5x^4+x^3-x^2-x+1$ and $x^{11}-x^{10}-7x^9+3x^8+11x^7+5x^6+x^5-5x^4-5x^3-3x^2+x+1$ respectively.](figure4 "fig:"){width="3.0in" height="1.5in"}
![The characteristic polynomial of the above corresponding [*unoriented transition matrices*]{} is $x^5-x^4-3x^3-x^2+3x+1$.](figure5 "fig:"){width="2.6in" height="1.5in"} $$\left[ {\begin{array}{*{20}c} 1 & 0 & -2 & 0 & 2 \\ 2 & -1 & -2 & 0 & 2 \\ 0 & 0 & -1 & 0 & 2 \\ -2 & 0 & 2 & 1 & -2 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array}} \right] \left[ {\begin{array}{*{20}c} 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}} \right] = \left[ {\begin{array}{*{20}c} -1 & 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ -1 & 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array}} \right]$$
[**Acknowledgement**]{}\
This work was partially supported by the Ministry of Science and Technology of Taiwan.
[99]{} C. Bernhardt, Vertex maps for trees: Algebra and periods of periodic orbits, *Disc. Cont. Dyn. Sys. [**14**]{}(2006), 399-408.*
C. Bernhardt, A Sharkovsky theorem for vertex maps on trees, *J. Difference Equ. Appl. [**17**]{} (2011), 103-113.*
L. Block and W. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, New York, 1992.
B.-S, Du, On the class of square Petrie matrices induced by cyclic permutations, *Internat. J. Math. Math. Sci. [**31**]{} (2004), 1617-1622.*
B.-S. Du, On the one-sided and two-sided similarities or weak similarities of permutations, arXiv: 0904.3979v1 (2009).
M. Gordon and E. M. Wilkinson, Determinants of Petrie matrices, *Pacific J. Math. [**51**]{} (1974), 451¡V45.*
I. N. Herstein, Topics in Algebra, Blaisdell Publishing, New York, 1964.
|
---
abstract: 'The formation of singularity and breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction are considered. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$ and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In addition, the initial density can even have compact support. The logarithm-type estimate for the Lam[é]{} system and some weighted estimates play a crucial role in the proof.'
author:
- 'Xin Zhong[^1]'
date:
title: '[Singularity formation to the 2D Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction]{} [^2] '
---
Keywords: full compressible Navier-Stokes equations; 2D Cauchy problem; blow-up criterion.
Math Subject Classification: 35Q30; 35B65
Introduction
============
Let $\Omega\subset\mathbb{R}^2$ be a domain, the motion of a viscous, compressible, and heat conducting Navier-Stokes flow in $\Omega$ can be described by the full compressible Navier-Stokes equations $$\begin{aligned}
\label{1.1}
\begin{cases}
\rho_{t}+\operatorname{div}(\rho\mathbf{u})=0,\\
(\rho\mathbf{u})_{t}+\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{u})
-\mu\Delta\mathbf{u}
-(\lambda+\mu)\nabla\operatorname{div}\mathbf{u}+\nabla P=
\mathbf{0},\\
c_{\nu}[(\rho\theta)_{t}+\operatorname{div}(\rho\mathbf{u}\theta)]
+P\operatorname{div}\mathbf{u}-\kappa\Delta\theta
=2\mu|\mathfrak{D}(\mathbf{u})|^2+\lambda(\operatorname{div}\mathbf{u})^2.
\end{cases}\end{aligned}$$ Here, $t\geq0$ is the time, $x\in\Omega$ is the spatial coordinate, and the unknown functions $\rho, \mathbf{u}, P=R\rho\theta\ (R>0), \theta$ are the fluid density, velocity, pressure, and the absolute temperature respectively; $\mathfrak{D}(\mathbf{u})$ denotes the deformation tensor given by $$\mathfrak{D}(\mathbf{u})=\frac{1}{2}(\nabla\mathbf{u}+(\nabla\mathbf{u})^{tr}).$$ The constant viscosity coefficients $\mu$ and $\lambda$ satisfy the physical restrictions $$\label{1.2}
\mu>0,\ \mu+\lambda\geq0.$$ Positive constants $c_\nu$ and $\kappa$ are respectively the heat capacity, the ratio of the heat conductivity coefficient over the heat capacity.
There is huge literature on the studies about the theory of well-posedness of solutions to the Cauchy problem and the initial boundary value problem (IBVP) for the compressible Navier-Stokes system due to the physical importance, complexity, rich phenomena and mathematical challenges, refer to [@CK2006; @CK20062; @H1995; @H1997; @HLX2012; @HL2013; @L1998; @F2004; @NS2004; @LX2013; @LL2014; @YZ2017; @WZ2017; @W2017; @MN1980; @MN1983] and references therein. In particular, non-vacuum small perturbations of a uniform non-vacuum constant state have been shown existing globally in time and remain smooth in any space dimensions [@MN1980; @MN1983], while for general data which may contain vacuum states, only weak solutions are shown to exist for the compressible Navier-Stokes system in multi-dimension with special equation of state as in [@F2004; @FNP2001; @L1998], yet the uniqueness and regularity of these weak solutions remain unknown. Despite the surprising results on global well-posedness of the strong (or classical) solution to the multi-dimensional compressible Navier-Stokes system for initial data with small total energy but possible large oscillations and containing vacuum states [@HLX2012; @HL2013; @LX2013; @YZ2017; @WZ2017], it is an outstanding challenging open problem to investigate the global well-posedness for general large strong solutions with vacuum.
Therefore, it is important to study the mechanism of blow-up and structure of possible singularities of strong (or classical) solutions to the compressible Navier-Stokes equations. The pioneering work can be traced to Serrin’s criterion [@S1962] on the Leray-Hopf weak solutions to the three-dimensional incompressible Navier-Stokes equations, which can be stated that if a weak solution $\mathbf{u}$ satisfies $$\label{1.3}
\mathbf{u}\in L^s(0,T;L^r),\ \text{for}\ \frac2s+\frac3r=1,\ 3<r\leq\infty,$$ then it is regular.
Recently, there are several results on the blow-up criteria of strong (or classical) solutions to the compressible Navier-Stokes equations. Precisely, let $0<T^*<+\infty$ be the maximum time of existence of strong solutions. For the 3D isentropic flows, Huang–Li-Xin [@HLX20112] obtained the following Serrin type criterion $$\label{1.4}
\lim_{T\rightarrow T^{*}}\left(\|\operatorname{div}\mathbf{u}\|_{L^1(0,T;L^\infty)}+\|\mathbf{u}\|_{L^s(0,T;L^r)}\right)=\infty,$$ where $r$ and $s$ as in . In [@HLX20111], they also proved a Beale-Kato-Majda type criterion as follows $$\label{1.04}
\lim_{T\rightarrow T^{*}}\|\mathfrak{D}(\mathbf{u})\|_{L^1(0,T;L^\infty)}=\infty.$$ For the IBVP of 3D isentropic flows, Sun-Wang-Zhang [@SWZ20111] established $$\lim_{T\rightarrow T^{*}}\|\rho\|_{L^\infty(0,T;L^\infty)}=\infty$$ provided that $$\label{1.6}
7\mu>\lambda.$$ For the 3D full compressible Navier-Stokes equations, under the condition , Fan-Jiang-Ou [@FJO2010] showed that $$\label{1.5}
\lim_{T\rightarrow T^{*}}\left(\|\nabla\mathbf{u}\|_{L^1(0,T;L^\infty)}+\|\theta\|_{L^\infty(0,T;L^\infty)}\right)
=\infty.$$ Under just the physical condition $$\label{1.06}
\mu>0,\ 2\mu+3\lambda\geq0,$$ Huang-Li-Wang established the criterion for the 3D barotropic case [@HLX20112] still holds for the full Navier-Stokes system. For the Cauchy problem and the IBVP of 3D full compressible Navier-Stokes system, Huang-Li [@HL2013] proved that $$\label{1.7}
\lim_{T\rightarrow T^{*}}\left(\|\rho\|_{L^\infty(0,T;L^\infty)}+\|\mathbf{u}\|_{L^s(0,T;L^r)}\right)
=\infty,\ \text{for}\ \frac2s+\frac3r\leq1,\ 3<r\leq\infty.$$ However, for the IBVP of 2D full Navier-Stokes equations, Wang [@W2014] showed the formation of singularity must be caused by losing the bound of $\operatorname{div}\mathbf{u}$. More precisely, she obtained $$\label{1.08}
\lim_{T\rightarrow T^{*}}\|\operatorname{div}\mathbf{u}\|_{L^1(0,T;L^\infty)}=\infty.$$ For more information on the blow-up criteria of compressible flows, we refer to [@CJ2006; @R2008; @SWZ20112; @WZ2013; @X1998; @XY2013; @HX2016] and the references therein.
It is worth noting that one would not expect better regularities of the solutions of in general because of Xin’s result [@X1998], where the author proved that there is no global smooth solution to the Cauchy problem of if the initial density is nontrivial compactly supported. Very recently, Liang-Shi [@LS2015] obtained the local existence of strong (or classical) solutions for the non-isentropic compressible Navier-Stokes equations without heat-conductivity. These motivate us to find some possible blow-up criterion of regular solutions to the system with zero heat conduction, especially of strong solutions. In fact, this is the main aim of this paper.
When $\kappa=0$, and without loss of generality, take $c_\nu=R=1$, the system can be written as $$\begin{aligned}
\label{1.10}
\begin{cases}
\rho_{t}+\operatorname{div}(\rho\mathbf{u})=0,\\
(\rho\mathbf{u})_{t}+\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{u})
-\mu\Delta\mathbf{u}
-(\lambda+\mu)\nabla\operatorname{div}\mathbf{u}+\nabla P=
\mathbf{0},\\
P_{t}+\operatorname{div}(P\mathbf{u})
+P\operatorname{div}\mathbf{u}
=2\mu|\mathfrak{D}(\mathbf{u})|^2+\lambda(\operatorname{div}\mathbf{u})^2.
\end{cases}\end{aligned}$$ The present paper is aimed at giving a blow-up criterion of strong solutions to the Cauchy problem of the system with the initial condition $$\label{1.11}
(\rho,\mathbf{u},P)(x,0)=(\rho_0,\mathbf{u}_0,P_0)(x),\ \ x\in\mathbb{R}^2,$$ and the far field behavior $$\label{1.12}
(\rho,\mathbf{u},P)(x,t)\rightarrow(0, \mathbf{0},0),\ \text{as}\ |x|\rightarrow+\infty,\ t>0.$$
Before stating our main result, we first explain the notations and conventions used throughout this paper. For $r>0$, set $$B_r \triangleq\left.\left\{x\in\mathbb{R}^2\right|\,|x|<r \right\},
\quad \int \cdot dx\triangleq\int_{\mathbb{R}^2}\cdot dx.$$ For $1\leq p\leq\infty$ and integer $k\geq0$, the standard Sobolev spaces are denoted by: $$L^p=L^p(\mathbb{R}^2),\ W^{k,p}=W^{k,p}(\mathbb{R}^2), \ H^{k}=H^{k,2}(\mathbb{R}^2),
\ D^{k,p}=\{u\in L_{\operatorname{loc}}^1|\nabla^k u\in L^p\}.$$ Now we define precisely what we mean by strong solutions to the problem –.
\[def1\] $(\rho,\mathbf{u},P)$ is called a strong solution to – in $\mathbb{R}^2\times(0,T)$, if for some $q_0>2$, $$\begin{split}
\begin{cases}
\rho\geq0,\ \rho\in C([0,T];L^1\cap H^1\cap W^{1,q_0}),\ \rho_t\in C([0,T];L^{q_0}),\\
\mathbf{u}\in C([0,T];D^{1,2}\cap D^{2,2})\cap L^{2}(0,T;D^{2,q_0}), \\
\mathbf{u}_t\in L^{2}(0,T;D^{1,2}),\
\sqrt{\rho}\mathbf{u}_{t}\in L^{\infty}(0,T;L^{2}), \\
P\geq0,\ P\in C([0,T];L^1\cap H^1\cap W^{1,q_0}),\ P_t\in C([0,T];L^{q_0}), \\
\end{cases}
\end{split}$$ and $(\rho,\mathbf{u},P)$ satisfies both almost everywhere in $\mathbb{R}^2\times(0,T)$ and almost everywhere in $\mathbb{R}^2$.
Without loss of generality, we assume that the initial density $\rho_0$ satisfies $$\label{1.8}
\int_{\mathbb{R}^2} \rho_0dx=1,$$ which implies that there exists a positive constant $N_0$ such that $$\label{1.9}
\int_{B_{N_0}} \rho_0 dx\ge \frac12\int\rho_0dx=\frac12.$$
Our main result reads as follows:
\[thm1.1\] Let $\eta_0$ be a positive number and $$\label{2.01}
\bar{x}\triangleq(e+|x|^2)^{1/2}\log^{1+\eta_0}(e+|x|^2).$$ In addition to and , assume that the initial data $(\rho_0\geq0, \mathbf{u}_0,P_0\geq0)$ satisfies for any given numbers $a>1$ and $q>2$, $$\label{2.02}
\rho_{0}\bar{x}^{a}\in L^{1}\cap H^{1}\cap W^{1,q},\ \nabla\mathbf{u}_{0}\in H^1,\ \sqrt{\rho_0}\mathbf{u}_0\in L^2, \ P_0\in L^{1}\cap H^{1}\cap W^{1,q},$$ and the compatibility conditions $$\label{A2}
-\mu\Delta\mathbf{u}_0-(\lambda+\mu)\nabla\operatorname{div}\mathbf{u}_0+\nabla P_0=\sqrt{\rho_0}\mathbf{g}$$ for some $\mathbf{g}\in L^2(\Omega)$. Let $(\rho,\mathbf{u},P)$ be a strong solution to the problem –. If $T^{*}<\infty$ is the maximal time of existence for that solution, then we have $$\begin{aligned}
\label{B}
\lim_{T\rightarrow T^{*}}\left(\|\rho\|_{L^{\infty}(0,T;L^\infty)}
+\|P\|_{L^{\infty}(0,T;L^\infty)}\right)=\infty.\end{aligned}$$
Several remarks are in order.
\[re1.1\] The local existence of a strong solution with initial data as in Theorem \[thm1.1\] was established in [@LS2015; @LH2016]. Hence, the maximal time $T^{*}$ is well-defined.
\[re1.2\] According to , the upper bound of the temperature $\theta$ is not the key point to make sure that the solution $(\rho,\mathbf{u},P)$ is a global one, and it may go to infinity in the vacuum region within the life span of our strong solution.
\[re1.3\] Compared with [@HX2016], where the authors investigated blow-up criteria for the 3D Cauchy problem and the IBVP of non-isentropic Navier-Stokes equations with zero heat conduction, there is no need to impose additional restrictions on the viscosity coefficients $\mu$ and $\lambda$ except the physical restrictions .
We now make some comments on the analysis of this paper. We mainly make use of continuation argument to prove Theorem \[thm1.1\]. That is, suppose that were false, i.e., $$\lim_{T\rightarrow T^*}\left(\|\rho\|_{L^{\infty}(0,T;L^\infty)}
+\|P\|_{L^{\infty}(0,T;L^\infty)}\right)\leq M_0<\infty.$$ We want to show that $$\sup_{0\leq t\leq T^*}\left(\|(\rho,P)\|_{H^1\cap W^{1,q}}
+\|\rho\bar{x}^{a}\|_{L^{1}\cap H^{1}\cap W^{1,q}}
+\|\nabla\mathbf{u}\|_{H^1}\right) \leq C<+\infty.$$
It should be pointed out that the crucial techniques of proofs in [@W2014] cannot be adapted directly to the situation treated here, since their arguments depend crucially on the boundedness of the domains and $\kappa>0$. Moreover, technically, it is hard to modify the three-dimensional analysis of [@HX2016] to the two-dimensional case with initial density containing vacuum since the analysis of [@HX2016] depends crucially on the a priori $L^6$-bound on the velocity, while in two dimensions it seems difficult to bound the $L^p(\mathbb{R}^2)$-norm of $\mathbf{u}$ just in terms of $\|\sqrt{\rho}\mathbf{u}\|_{L^{2}(\mathbb{R}^2)}$ and $\|\nabla\mathbf{u}\|_{L^{2}(\mathbb{R}^2)}$ for any $p\geq1$. To overcome these difficulties mentioned above, some new ideas are needed. Inspired by [@LX2013; @LSZ2015], we first observe that if the initial density decays not too slow at infinity, i.e., $\rho_0\bar{x}^a\in L^1(\mathbb{R}^2)$ for some positive constant $a>1$ (see ), then for any $\eta\in(0,1]$, we can show that (see ) $$\label{1.20}
\mathbf{u}\bar{x}^{-\eta}\in L^{p_0}(\mathbb{R}^2),\ \text{for some}\ p_0>1.$$ Then, motivated by [@H1995; @HX2016], in order to get the $L_t^\infty L_x^{2}$-norm of $\sqrt{\rho}\dot{\mathbf{u}}$, we first show the desired a priori estimates of the $L_t^\infty L_x^2$-norm of $\nabla\mathbf{u}$, which is the second key observation in this paper (see Lemma \[lem35\]). Next, the a priori estimates on the $L_t^\infty L_x^{q}$-norm of $(\nabla\rho,\nabla P)$ and the $L_t^1 L_x^{\infty}$-norm of the velocity gradient can be obtained (see Lemma \[lem37\]) simultaneously by solving a logarithm Gronwall inequality based on a logarithm estimate for the Lam[é]{} system (see Lemma \[lem25\]). Finally, with the help of , we can get the spatial weighted estimate of the density (see Lemma \[lem38\]).
The rest of this paper is organized as follows. In Section \[sec2\], we collect some elementary facts and inequalities that will be used later. Section \[sec3\] is devoted to the proof of Theorem \[thm1.1\].
Preliminaries {#sec2}
=============
In this section, we will recall some known facts and elementary inequalities that will be used frequently later.
We begin with the following Gronwall’s inequality (see [@T2006 pp. 12–13]), which plays a central role in proving a priori estimates on strong solutions $(\rho,\mathbf{u},P)$.
\[lem21\] Suppose that $h$ and $r$ are integrable on $(a, b)$ and nonnegative a.e. in $(a, b)$. Further assume that $y\in C[a, b], y'\in L^1(a, b)$, and $$y'(t)\leq h(t)+r(t)y(t)\ \ \text{for}\ a.e\ t\in(a,b).$$ Then $$y(t)\leq \left[y(a)+\int_{a}^{t}h(s)\exp\left(-\int_{a}^{s}r(\tau)d\tau\right)ds\right]
\exp\left(\int_{a}^{t}r(s)ds\right),\ \ t\in[a,b].$$
Next, the following Gagliardo-Nirenberg inequality (see [@N1959]) will be used later.
\[lem22\] For $p\in[2,\infty), r\in(2,\infty)$, and $s\in(1,\infty)$, there exists some generic constant $C>0$ which may depend on $p,$ $r$, and $s$ such that for $f\in H^{1}(\mathbb{R}^2)$ and $g\in L^{s}(\mathbb{R}^2)\cap D^{1,r}(\mathbb{R}^2)$, we have $$\begin{aligned}
& & \|f\|_{L^p(\mathbb{R}^2)}^{p}\leq C\|f\|_{L^2(\mathbb{R}^2)}^{2}\|\nabla f\|_{L^2(\mathbb{R}^2)}^{p-2}, \\
& & \|g\|_{C(\overline{\mathbb{R}^2})}\leq C\|g\|_{L^s(\mathbb{R}^2)}^{s(r-2)/(2r+s(r-2))}\|\nabla g\|_{L^r(\mathbb{R}^2)}^{2r/(2r+s(r-2))}.\end{aligned}$$
The following weighted $L^m$ bounds for elements of the Hilbert space $\tilde{D}^{1,2}(\mathbb{R}^2)\triangleq\{ v \in H_{\operatorname{loc}}^{1}(\mathbb{R}^2)|\nabla v \in L^{2}(\mathbb{R}^2)\}$ can be found in [@L1996 Theorem B.1].
\[1leo\] For $m\in [2,\infty)$ and $\theta\in (1+m/2,\infty),$ there exists a positive constant $C$ such that for all $v\in \tilde{D}^{1,2}(\mathbb{R}^2),$ $$\label{3h}
\left(\int_{\mathbb{R}^2} \frac{|v|^m}{e+|x|^2}\left(\log \left(e+|x|^2\right)\right)^{-\theta}dx \right)^{1/m}
\leq C\|v\|_{L^2(B_1)}+C\|\nabla v\|_{L^2(\mathbb{R}^2) }.$$
The combination of Lemma \[1leo\] and the Poincaré inequality yields the following useful results on weighted bounds, whose proof can be found in [@LX2013 Lemma 2.4].
\[lem26\] Let $\bar x$ be as in . Assume that $\rho \in L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ is a non-negative function such that $$\label{2.i2}
\|\rho\|_{L^1(B_{N_1})} \ge M_1, \quad \|\rho\|_{L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)}\le M_2,$$ for positive constants $M_1, M_2$, and $ N_1\ge 1$. Then for $\varepsilon> 0$ and $\eta>0,$ there is a positive constant $C$ depending only on $\varepsilon,\eta, M_1,M_2$, and $ N_1$, such that every $v\in \tilde{D}^{1,2}(\mathbb{R}^2)$ satisfies $$\label{22}
\|v\bar x^{-\eta}\|_{L^{(2+\varepsilon)/\tilde{\eta}}(\mathbb{R}^2)}
\le C \|{\sqrt\rho}v\|_{L^2(\mathbb{R}^2)}+C \|\nabla v\|_{L^2(\mathbb{R}^2)},$$ with $\tilde{\eta}=\min\{1,\eta\}$.
Next, the following Beale-Kato-Majda type inequality (see [@HLX20112 Lemma 2.3]) will be used to estimate $\|\nabla\mathbf{u}\|_{L^\infty}$.
\[lem25\] For $q\in(2,\infty)$, there is a constant $C(q)>0$ such that for all $\nabla\mathbf{v}\in L^2\cap D^{1,q}$, it holds that $$\label{2.2}
\|\nabla\mathbf{v}\|_{L^\infty}\leq C\left(\|\operatorname{div}\mathbf{v}\|_{L^\infty}
+\|\operatorname{curl}\mathbf{v}\|_{L^\infty}\right)\log(e+\|\nabla^2\mathbf{v}\|_{L^q})
+C\|\nabla\mathbf{v}\|_{L^2}+C.$$
Finally, for $\nabla^{\bot}\triangleq(-\partial_2,\partial_1)$, denoting the material derivative of $f$ by $\dot{f}\triangleq f_t+\mathbf{u}\cdot\nabla f$, then we have the following $L^p$-estimate (see [@LX2013 Lemma 2.5]) for the elliptic system derived from the momentum equations $_2$: $$\label{3.014}
\Delta F=\operatorname{div}(\rho\dot{\mathbf{u}}),\
\mu\Delta\omega=\nabla^{\bot}\cdot(\rho\dot{\mathbf{u}}),$$ where $F$ is the effective viscous flux, $\omega$ is vorticity given by $$\label{3.14}
F=(\lambda+2\mu)\operatorname{div}\mathbf{u}-P,\
\omega=\partial_1u_2-\partial_2u_1.$$
\[lem23\] Let $(\rho, \mathbf{u}, P)$ be a smooth solution of . Then for $p\geq2$ there exists a positive constant $C$ depending only on $p, \mu$ and $\lambda$ such that $$\begin{aligned}
&\|\nabla F\|_{L^p}+\|\nabla\omega\|_{L^p}
\leq C\|\rho\dot{\mathbf{u}}\|_{L^p}, \label{2.3} \\
&\|F\|_{L^p}+\|\omega\|_{L^p}
\leq C\|\rho\dot{\mathbf{u}}\|_{L^2}^{1-\frac2p}
\left(\|\nabla\mathbf{u}\|_{L^2}+\|P\|_{L^2}\right)^{\frac2p}, \label{2.4} \\
&\|\nabla\mathbf{u}\|_{L^p}
\leq C\|\rho\dot{\mathbf{u}}\|_{L^2}^{1-\frac2p}
\left(\|\nabla\mathbf{u}\|_{L^2}+\|P\|_{L^2}\right)^{\frac2p}
+C\|P\|_{L^p}. \label{2.5}\end{aligned}$$
Proof of Theorem \[thm1.1\] {#sec3}
===========================
Let $(\rho,\mathbf{u},P)$ be a strong solution described in Theorem \[thm1.1\]. Suppose that were false, that is, there exists a constant $M_0>0$ such that $$\label{3.1}
\lim_{T\rightarrow T^*}\left(\|\rho\|_{L^{\infty}(0,T;L^\infty)}
+\|P\|_{L^{\infty}(0,T;L^\infty)}\right)\leq M_0<\infty.$$
First, the estimate on the $L^\infty(0,T;L^p)$-norm of the density could be deduced directly from $_1$ and .
\[lem31\] Under the condition , it holds that for any $T\in[0,T^*)$, $$\label{3.2}
\sup_{0\leq t\leq T}\|\rho\|_{L^1\cap L^\infty}\leq C,$$ where and in what follows, $C,C_1,C_2$ stand for generic positive constants depending only on $M_0,\lambda,\mu,T^{*}$, and the initial data.
Next, we have the following estimate which is similar to the energy estimate.
\[lem32\] Under the condition , it holds that for any $T\in[0,T^*)$, $$\label{3.3}
\sup_{0\leq t\leq T}\left(\|\sqrt{\rho}\mathbf{u}\|_{L^2}^2+\|P\|_{L^1\cap L^\infty}
\right)+\int_{0}^T\|\nabla\mathbf{u}\|_{L^2}^2 dt \leq C.$$
[*Proof.*]{} It follows from $_3$ that $$\label{7.01}
P_t+\mathbf{u}\cdot\nabla P+2P\operatorname{div}\mathbf{u}
=F\triangleq2\mu|\mathfrak{D}(\mathbf{u})|^2
+\lambda(\operatorname{div}\mathbf{u})^2\geq0.$$ Define particle path before blowup time $$\begin{aligned}
\begin{cases}
\frac{d}{dt}\mathbf{X}(x,t)
=\mathbf{u}(\mathbf{X}(x,t),t),\\
\mathbf{X}(x,0)=x.
\end{cases}\end{aligned}$$ Thus, along particle path, we obtain from that $$\begin{aligned}
\frac{d}{dt}P(\mathbf{X}(x,t),t)
=-2P\operatorname{div}\mathbf{u}+F,\end{aligned}$$ which implies $$\label{3.03}
P(\mathbf{X}(x,t),t)=\exp\left(-2\int_{0}^t\operatorname{div}\mathbf{u}ds\right)
\left[P_0+\int_{0}^t\exp\left(2\int_{0}^s\operatorname{div}\mathbf{u}d\tau\right)Fds\right]\geq0.$$
Next, multiplying $_2$ by $\mathbf{u}$ and integrating over $\mathbb{R}^2$, we obtain after integrating by parts that $$\begin{aligned}
\label{3.4}
\frac{1}{2}\frac{d}{dt}\int\rho|\mathbf{u}|^2dx
+\int\left[\mu|\nabla\mathbf{u}|^2+(\lambda+\mu)(\operatorname{div}\mathbf{u})^2
\right]dx=\int P\operatorname{div}\mathbf{u}dx.\end{aligned}$$ Integrating $_3$ with respect to $x$ and then adding the resulting equality to give rise to $$\begin{aligned}
\label{3.004}
\frac{d}{dt}\int\left(\frac12\rho|\mathbf{u}|^2+P\right)dx=0,\end{aligned}$$ which combined with , , and leads to $$\begin{aligned}
\label{3.04}
\sup_{0\leq t\leq T}\left(\|\sqrt{\rho}\mathbf{u}\|_{L^2}^2+\|P\|_{L^1\cap L^\infty}
\right)\leq C.\end{aligned}$$ This together with and Cauchy-Schwarz inequality yields $$\begin{aligned}
\label{3.003}
\frac{d}{dt}\|\sqrt{\rho}\mathbf{u}\|_{L^2}^2
+\mu\|\nabla\mathbf{u}\|_{L^2}^2
\leq C.\end{aligned}$$ So the desired follows from and integrated with respect to $t$. This completes the proof of Lemma \[lem32\]. $\Box$
The following lemma gives the estimate on the spatial gradients of the velocity, which is crucial for deriving the higher order estimates of the solution.
\[lem35\] Under the condition , it holds that for any $T\in[0,T^*)$, $$\label{5.1}
\sup_{0\leq t\leq T}\|\nabla\mathbf{u}\|_{L^2}^{2}
+\int_{0}^{T}\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}^{2}
dt \leq C.$$
[*Proof.*]{} Multiplying $_2$ by $\dot{\mathbf{u}}$ and integrating the resulting equation over $\mathbb{R}^2$ give rise to $$\begin{aligned}
\label{5.2}
\int\rho|\dot{\mathbf{u}}|^2dx
= -\int\dot{\mathbf{u}}\cdot\nabla Pdx+\mu\int\dot{\mathbf{u}}\cdot\Delta\mathbf{u}dx
+(\lambda+\mu)\int\dot{\mathbf{u}}\cdot\nabla\operatorname{div}\mathbf{u}dx.\end{aligned}$$ By $_3$ and integrating by parts, we derive from that $$\begin{aligned}
\label{5.3}
-\int\dot{\mathbf{u}}\cdot\nabla Pdx
& = \int \left[(\operatorname{div}\mathbf{u})_tP
-(\mathbf{u}\cdot\nabla\mathbf{u})\cdot\nabla P\right]dx \nonumber\\
&=\frac{d}{dt}\int\operatorname{div}\mathbf{u}Pdx+
\int\left[P(\operatorname{div}\mathbf{u})^2-2\mu\operatorname{div}\mathbf{u}|\mathfrak{D}\mathbf{u}|^2
-\lambda(\operatorname{div}\mathbf{u})^3+P\partial_ju_i\partial_iu_j\right]dx \nonumber\\
&\leq \frac{d}{dt}\int\operatorname{div}\mathbf{u}Pdx+
C\|\nabla\mathbf{u}\|_{L^2}^2+C\|\nabla\mathbf{u}\|_{L^3}^3.\end{aligned}$$ It follows from integration by parts that $$\begin{aligned}
\label{5.4}
\mu\int\dot{\mathbf{u}}\cdot\Delta\mathbf{u}dx
& =\mu\int(\mathbf{u}_{t} +\mathbf{u}\cdot\nabla\mathbf{u})\cdot\Delta\mathbf{u}dx \notag \\
& = -\frac{\mu}{2}\frac{d}{dt}\|\nabla\mathbf{u}\|_{L^2}^{2}
-\mu\int\partial_{i}u_{j}\partial_{i}(u_{k}\partial_{k}u_{j})dx \notag \\
& \leq-\frac{\mu}{2}\frac{d}{dt}\|\nabla\mathbf{u}\|_{L^2}^{2}
+C\|\nabla\mathbf{u} \|_{L^3}^{3}.\end{aligned}$$ Similarly, one gets $$\begin{aligned}
\label{5.5}
(\lambda+\mu)\int\dot{\mathbf{u}}\cdot\nabla\operatorname{div}\mathbf{u}dx
& = -\frac{\lambda+\mu}{2}\frac{d}{dt}\|\operatorname{div}\mathbf{u}\|_{L^2}^{2}
-(\lambda+\mu)\int\operatorname{div}\mathbf{u}\operatorname{div}(\mathbf{u}\cdot\nabla\mathbf{u})dx \notag \\
& \leq-\frac{\lambda+\mu}{2}\frac{d}{dt}\|\operatorname{div}\mathbf{u}\|_{L^2}^{2}
+C\|\nabla\mathbf{u} \|_{L^3}^{3}.\end{aligned}$$ Putting – into , we obtain from and that $$\begin{aligned}
\label{5.6}
\Psi'(t)+\int\rho|\dot{\mathbf{u}}|^2dx
& \leq C\|\nabla\mathbf{u}\|_{L^2}^2+C\|\nabla\mathbf{u}\|_{L^3}^3 \notag \\
& \leq C\|\nabla\mathbf{u}\|_{L^2}^2+C\|\nabla\mathbf{u}\|_{L^3}^3
\notag \\
& \leq C\|\nabla\mathbf{u}\|_{L^2}^2
+C\|\rho\dot{\mathbf{u}}\|_{L^2}\left(\|\nabla\mathbf{u}\|_{L^2}+1\right)^2
\notag \\
& \leq \frac12\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}^2
+C(1+\|\nabla\mathbf{u}\|_{L^2}^2)\|\nabla\mathbf{u}\|_{L^2}^2+C,\end{aligned}$$ where $$\Psi(t)\triangleq\frac{\mu}{2}\|\nabla\mathbf{u}\|_{L^2}^{2}
+\frac{\lambda+\mu}{2}\|\operatorname{div}\mathbf{u}\|_{L^2}^{2}
-\int\operatorname{div}\mathbf{u}Pdx$$ satisfies $$\label{5.7}
\frac{\mu}{2}\|\nabla\mathbf{u}\|_{L^2}^2-C
\leq \Psi(t)\leq\mu\|\nabla\mathbf{u}\|_{L^2}^2+C$$ due to . Thus the desired follows from , , , and Gronwall’s inequality. This completes the proof of Lemma \[lem35\]. $\Box$
Next, motivated by [@H1995], we have the following estimates on the material derivatives of the velocity which are important for the higher order estimates of strong solutions.
\[lem36\] Under the condition , it holds that for any $T\in[0,T^*)$, $$\label{6.1}
\sup_{0\leq t\leq T}\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}^2
+\int_0^T\|\nabla\dot{\mathbf{u}}\|_{L^2}^2dt \leq C.$$
[*Proof.*]{} By the definition of $\dot{\mathbf{u}}$, we can rewrite $\eqref{1.10}_2$ as follows: $$\label{6.2}
\rho\dot{\mathbf{u}}+\nabla P
=\mu\Delta\mathbf{u}+(\lambda+\mu)\nabla\operatorname{div}\mathbf{u}.$$ Differentiating with respect to $t$ and using $_1$, we have $$\begin{aligned}
\label{6.3}
\rho\dot{\mathbf{u}}_t+\rho\mathbf{u}\cdot\nabla\dot{\mathbf{u}}
+\nabla P_t&=\mu\Delta\dot{\mathbf{u}}+(\lambda+\mu)\operatorname{div}\dot{\mathbf{u}}
-\mu\Delta(\mathbf{u}\cdot\nabla\mathbf{u}) \nonumber \\
& \quad -(\lambda+\mu)\operatorname{div}(\mathbf{u}\cdot\nabla\mathbf{u})
+\operatorname{div}(\rho\dot{\mathbf{u}}\otimes\mathbf{u}).\end{aligned}$$ Multiplying by $\dot{\mathbf{u}}$ and integrating by parts over $\mathbb{R}^3$, we get $$\begin{aligned}
\label{6.4}
&\frac12\frac{d}{dt}\int \rho|\dot{\mathbf{u}}|^2\mbox{d}x+\mu\int|\nabla\dot{\mathbf{u}}|^2dx
+(\lambda+\mu)\int|\operatorname{div}\dot{\mathbf{u}}|^2dx \nonumber\\
&=\int \left(P_t\operatorname{div}\dot{\mathbf{u}}+(\nabla P\otimes\mathbf{u}):\nabla\dot{\mathbf{u}}\right)dx
+\mu\int [\operatorname{div}(\Delta\mathbf{u}\otimes\mathbf{u})-\Delta(\mathbf{u}\cdot\nabla\mathbf{u})]\cdot \dot{\mathbf{u}}dx \nonumber\\
& \quad+(\lambda+\mu)\int [(\nabla\operatorname{div}\mathbf{u})\otimes\mathbf{u}
-\nabla\operatorname{div}(\mathbf{u}\cdot\nabla\mathbf{u})]\cdot \dot{\mathbf{u}}dx\triangleq\sum_{i=1}^{3}J_i,\end{aligned}$$ where $J_i$ can be bounded as follows.
It follows from $\eqref{1.10}_3$ that $$\begin{aligned}
\label{6.5}
J_1 & =\int \left(-\operatorname{div}(P\mathbf{u})\operatorname{div}\dot{\mathbf{u}}
-P\operatorname{div}\mathbf{u}\operatorname{div}\dot{\mathbf{u}}
+\mathcal{T}(\mathbf{u}):\nabla\mathbf{u}\operatorname{div}\dot{\mathbf{u}}
+(\nabla P\otimes\mathbf{u}):\nabla\dot{\mathbf{u}}\right)dx \nonumber\\
&=\int \left(P\mathbf{u}\nabla\operatorname{div}\dot{\mathbf{u}}-P\operatorname{div}\mathbf{u}\operatorname{div}\dot{\mathbf{u}}
+\mathcal{T}(\mathbf{u}):\nabla\mathbf{u}\operatorname{div}\dot{\mathbf{u}}
-P\nabla\mathbf{u}^\top:\nabla\dot{\mathbf{u}}
-P\mathbf{u}\nabla\operatorname{div}\dot{\mathbf{u}}\right)dx \nonumber\\
&=\int \left(-P\operatorname{div}\mathbf{u}\operatorname{div}\dot{\mathbf{u}}
+\mathcal{T}(\mathbf{u}):\nabla\mathbf{u}\operatorname{div}\dot{\mathbf{u}}
-P\nabla\mathbf{u}^\top:\nabla\dot{\mathbf{u}}\right)dx\nonumber\\
&\leq C\int \left(|\nabla\mathbf{u}||\nabla\dot{\mathbf{u}}|
+|\nabla\mathbf{u}|^2|\nabla\dot{\mathbf{u}}|\right)dx \nonumber\\
&\leq C\left(\|\nabla\mathbf{u}\|_{L^2}+\|\nabla\mathbf{u}\|_{L^4}^2\right)
\|\nabla\dot{\mathbf{u}}\|_{L^2},\end{aligned}$$ where $\mathcal{T}(\mathbf{u})=2\mu\mathfrak{D}(\mathbf{u})+\lambda\operatorname{div}\mathbf{u}\mathbb{I}_3$.
For $J_2$ and $J_3$, notice that for all $1\leq i,j,k\leq 3,$ one has $$\begin{aligned}
\partial_j(\partial_{kk}u_iu_j)-\partial_{kk}(u_j\partial_j u_i)&=\partial_k(\partial_ju_j\partial_ku_i)-\partial_k(\partial_ku_j\partial_ju_i)
-\partial_j(\partial_ku_j\partial_ku_i),\nonumber\\
\partial_j(\partial_{ik}u_ku_j)-\partial_{ij}(u_k\partial_ku_j)
&=\partial_i(\partial_ju_j\partial_ku_k)-\partial_i(\partial_ju_k\partial_ku_j)
-\partial_k(\partial_iu_k\partial_ju_j).\nonumber\end{aligned}$$ So integrating by parts gives $$\begin{aligned}
J_2&=\mu\int [\partial_k(\partial_ju_j\partial_ku_i)-\partial_k(\partial_ku_j\partial_ju_i)
-\partial_j(\partial_ku_j\partial_ku_i)]\dot{u_i}\mbox{d}x\nonumber\\
&\leq C\|\nabla\mathbf{u}\|_{L^4}^2\|\nabla\dot{\mathbf{u}}\|_{L^2},\label{6.7}\\
J_3&=(\lambda+\mu)\int [\partial_i(\partial_ju_j\partial_ku_k)-\partial_i(\partial_ju_k\partial_ku_j)
-\partial_k(\partial_iu_k\partial_ju_j)]\dot{u_i}\mbox{d}x\nonumber\\
&\leq C\|\nabla\mathbf{u}\|_{L^4}^2\|\nabla\dot{\mathbf{u}}\|_{L^2}.\label{6.8}\end{aligned}$$ Inserting – into and applying lead to $$\begin{aligned}
\label{6.9}
& \frac12\frac{d}{dt}
\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}^2+\mu\|\nabla\dot{\mathbf{u}}\|_{L^2}^2
+(\lambda+\mu)\|\operatorname{div}\dot{\mathbf{u}}\|_{L^2}^2 \nonumber\\
& \leq C(\|\nabla\mathbf{u}\|_{L^2}+\|\nabla\mathbf{u}\|_{L^4}^2)
\|\nabla\dot{\mathbf{u}}\|_{L^2}\nonumber\\
&\leq \frac{\mu}{2}\|\nabla\dot{\mathbf{u}}\|_{L^2}^2 +C(\mu)\left(\|\nabla\mathbf{u}\|_{L^4}^4+1\right),\end{aligned}$$ which implies $$\begin{aligned}
\label{6.16}
\frac{d}{dt}\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}^2
+\mu\|\nabla\dot{\mathbf{u}}\|_{L^2}^2
\leq C\|\nabla\mathbf{u}\|_{L^4}^4+C.\end{aligned}$$ By virtue of , , , and , one has $$\label{6.17}
\|\nabla\mathbf{u}\|_{L^4}^4
\leq C\|\rho\dot{\mathbf{u}}\|_{L^2}^2
\left(\|\nabla\mathbf{u}\|_{L^2}+\|P\|_{L^2}\right)^2
+C\|P\|_{L^4}^4
\leq C\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}^2
+C.$$ Consequently, we obtain the desired from , , and Gronwall’s inequality. This completes the proof of Lemma \[lem36\]. $\Box$
Inspired by [@LX2013; @LSZ2015], we have the following spatial weighted estimate on the density, which plays an important role in deriving the bounds on the higher order derivatives of the solutions $(\rho,\mathbf{u},P)$.
\[lem36\] Under the condition , it holds that for any $T\in[0,T^*)$, $$\label{06.1}
\sup_{0\leq t\leq T}\|\rho\bar{x}^{a}\|_{L^{1}}\leq C.$$
[*Proof.*]{} First, for $N>1,$ let $\varphi_N\in C^\infty_0(\mathbb{R}^2)$ satisfy $$\label{vp1}
0\le \varphi_N \le 1, \quad \varphi_N(x)
=\begin{cases} 1,~~~~ |x|\le N/2,\\
0,~~~~ |x|\ge N,\end{cases}
\quad |\nabla \varphi_N|\le C N^{-1}.$$ It follows from $_1$ that $$\begin{aligned}
\label{oo0}
\frac{d}{dt}\int \rho \varphi_{N} dx &=\int \rho\mathbf{u}
\cdot\nabla \varphi_{N} dx \notag \\
&\ge - C N^{-1}\left(\int\rho dx\right)^{1/2}
\left(\int\rho |u|^2dx\right)^{1/2}\ge - \tilde{C} N^{-1},\end{aligned}$$ where in the last inequality one has used and . Integrating and choosing $N=N_1\triangleq2N_0+4\tilde CT$, we obtain after using that $$\begin{aligned}
\label{p1}
\inf\limits_{0\le t\le T}\int_{B_{N_1}} \rho dx&\ge \inf\limits_{0\le t\le T}\int \rho \varphi_{N_1} dx \notag \\
&\ge \int \rho_0 \varphi_{N_1} dx-\tilde{C}N_1^{-1}T \notag \\
&\ge \int_{B_{N_0}} \rho_0 dx-\frac{\tilde{C}T}{2N_0+4\tilde{C} T} \notag \\
&\ge 1/4.\end{aligned}$$ Hence, it follows from , , , , and that for any $\eta\in(0,1]$ and any $s>2$, $$\label{06.2}
\|\mathbf{u}\bar{x}^{-\eta}\|_{L^{s/\eta}}\leq C\left(\|\sqrt{\rho}\mathbf{u}\|_{L^2}+\|\nabla\mathbf{u}\|_{L^2}\right)\le C.$$ Multiplying $_1$ by $\bar{x}^{a}$ and integrating the resulting equality by parts over $\mathbb{R}^2$ yield that $$\begin{split}
\frac{d}{dt}\int\rho\bar{x}^{a}dx & \leq C\int\rho|\mathbf{u} |\bar{x}^{a-1}\log^{2}(e+|x|^2)dx\\
& \leq C\|\rho\bar{x}^{a-1+\frac{8}{8+a}}\|_{L^{\frac{8+a}{7+a}}}\|\mathbf{u} \bar{x}^{-\frac{4}{8+a}}\|_{L^{8+a}} \\
& \leq C\int\rho\bar{x}^{a}dx+C,
\end{split}$$ which along with Gronwall’s inequality gives and finishes the proof of Lemma \[lem36\]. $\Box$
The following lemma will treat the higher order derivatives of the solutions which are needed to guarantee the extension of local strong solution to be a global one.
\[lem37\] Under the condition , and let $q>2$ be as in Theorem \[thm1.1\], then it holds that for any $T\in[0,T^*)$, $$\label{7.1}
\sup_{0\leq t\leq T}\left(\|(\rho,P)\|_{H^1\cap W^{1,q}}+\|\nabla\mathbf{u}\|_{H^1}\right)
+\int_{0}^T\|\nabla^2\mathbf{u}\|_{L^q}^2dt\leq C.$$
[*Proof.*]{} First, it follows from the mass equation $_1$ that $\nabla\rho$ satisfies for any $r\in[2,q]$, $$\begin{aligned}
\label{7.2}
\frac{d}{dt}\|\nabla\rho\|_{L^r}
& \leq C(r)(1+\|\nabla\mathbf{u} \|_{L^\infty})\|\nabla\rho\|_{L^r}
+C(r)\|\nabla^2\mathbf{u}\|_{L^r} \nonumber \\
& \leq C(1+\|\nabla\mathbf{u} \|_{L^\infty})\|\nabla\rho\|_{L^r}
+C(\|\rho\dot{\mathbf{u}}\|_{L^r}+\|\nabla P\|_{L^r})\end{aligned}$$ due to $$\label{7.3}
\|\nabla^2\mathbf{u}\|_{L^r}
\leq C(\|\rho\dot{\mathbf{u}}\|_{L^r}+\|\nabla P\|_{L^r}),$$ which follows from the standard $L^r$-estimate for the following elliptic system $$\begin{aligned}
\begin{cases}
\mu\Delta\mathbf{u}+(\lambda+\mu)\nabla\operatorname{div}\mathbf{u}=\rho\dot{\mathbf{u}}+\nabla P,\ \ x\in\mathbb{R}^2,\\
\mathbf{u}\rightarrow\mathbf{0},\ \text{as}\ |x|\rightarrow\infty.
\end{cases}\end{aligned}$$ Similarly, one deduces from $_3$ that $\nabla P$ satisfies for any $r\in[2,q]$, $$\begin{aligned}
\label{7.4}
\frac{d}{dt}\|\nabla P\|_{L^r} & \leq C(r)(1+\|\nabla\mathbf{u}\|_{L^{\infty}})(\|\nabla P\|_{L^r}+\|\nabla^2\mathbf{u}\|_{L^r}) \notag \\
& \leq C(1+\|\nabla\mathbf{u}\|_{L^{\infty}})(\|\rho\dot{\mathbf{u}}\|_{L^r}+\|\nabla P\|_{L^r}).\end{aligned}$$
Next, one gets from , Gagliardo-Nirenberg inequality, , , , and that $$\begin{aligned}
\label{7.5}
\|\operatorname{div}\mathbf{u}\|_{L^\infty}+\|\omega\|_{L^\infty}
& \leq C\|P\|_{L^\infty}+C\|F\|_{L^\infty}+\|\omega\|_{L^\infty} \nonumber \\
& \leq C(q)+C(q)\|\nabla F\|_{L^2}^{\frac{q-2}{2(q-1)}}\|\nabla F\|_{L^q}^{\frac{q}{2(q-1)}}
+C(q)\|\nabla\omega\|_{L^2}^{\frac{q-2}{2(q-1)}}\|\nabla\omega\|_{L^q}^{\frac{q}{2(q-1)}} \nonumber \\
& \leq C+C\|\rho\dot{\mathbf{u}}\|_{L^q}^{\frac{q}{2(q-1)}},\end{aligned}$$ which together with Lemma \[lem25\], , and yields that $$\begin{aligned}
\label{7.6}
\|\nabla\mathbf{u}\|_{L^\infty}
& \leq C\left(\|\operatorname{div}\mathbf{u}\|_{L^\infty}+\|\omega\|_{L^\infty}\right)
\log(e+\|\nabla^2\mathbf{u}\|_{L^q})+C\|\nabla\mathbf{u}\|_{L^2}+C \nonumber \\
& \leq C\left(1+\|\rho\dot{\mathbf{u}}\|_{L^q}^{\frac{q}{2(q-1)}}\right)
\log\left(e+\|\rho\dot{\mathbf{u}}\|_{L^q}+\|\nabla P\|_{L^q}\right)+C.\end{aligned}$$ It follows from , , , and that for any $\eta\in(0,1]$ and any $s>2$, $$\begin{aligned}
\label{7.7}
\|\rho^\eta v\|_{L^{\frac{s}{\eta}}}
& \leq C\|\rho^\eta\bar x^{\frac{3\eta a}{4s}}\|_{L^{\frac{4s}{3\eta}}}
\|v\bar x^{-\frac{3\eta a}{4s}}\|_{L^{\frac{4s}{\eta}}} \notag \\
& \leq C\|\rho\|_{L^\infty}^{\frac{(4s-3)\eta}{4s}}\|\rho\bar x^a\|_{L^1}^{\frac{3\eta}{4s}}\left( \|\sqrt{\rho} v\|_{L^2}+\|\nabla v\|_{L^2}\right) \notag \\
& \leq C\left(\|\sqrt{\rho}v\|_{L^2}+\|\nabla v\|_{L^2}\right),\end{aligned}$$ which along with H[ö]{}lder’s inequality, , and shows that $$\begin{aligned}
\label{7.8}
\|\rho\dot{\mathbf{u}}\|_{L^q}
& \leq C\|\rho\dot{\mathbf{u}}\|_{L^2}^{\frac{2(q-1)}{q^2-2}}
\|\rho\dot{\mathbf{u}}\|_{L^{q^2}}^{\frac{q(q-2)}{q^2-2}} \notag \\
& \leq C\|\rho\dot{\mathbf{u}}\|_{L^2}^{\frac{2(q-1)}{q^2-2}}
\left(\|\sqrt{\rho}\dot{\mathbf{u}}\|_{L^2}+\|\nabla\dot{\mathbf{u} }\|_{L^2}\right)^{\frac{q(q-2)}{q^2-2}} \notag \\
& \leq C\left(1+\|\nabla\dot{\mathbf{u}}\|_{L^2}^{\frac{q(q-2)}{q^2-2}}\right).\end{aligned}$$ Then we derive from and that $$\begin{aligned}
\label{7.9}
\|\nabla\mathbf{u}\|_{L^\infty}
\leq C\left(1+\|\nabla\dot{\mathbf{u}}\|_{L^2}\right)
\log\left(e+\|\nabla\dot{\mathbf{u}}\|_{L^2}+\|\nabla P\|_{L^q}\right)+C\end{aligned}$$ due to $\frac{q(q^2-2q)}{(2q-2)(q^2-2)},\ \frac{q^2-2q}{q^2-2}\in(0,1)$. Consequently, substituting and into and , we get after choosing $r=q$ that $$\label{7.11}
f'(t)\leq Cg(t)f(t)\log{f(t)}+Cg(t)f(t)+Cg(t),$$ where $$\begin{aligned}
f(t)&\triangleq e+\|\nabla\rho\|_{L^{ q}}+\|\nabla P\|_{L^{ q}},\\
g(t)&\triangleq (1+\|\nabla\dot{\mathbf{u}}\|_{L^2})
\log(e+\|\nabla\dot{\mathbf{u}}\|_{L^2}).\end{aligned}$$ This yields $$\label{7.12}
(\log f(t))'\leq Cg(t)+Cg(t)\log f(t)$$ due to $f(t)>1.$ Thus it follows from , , and Gronwall’s inequality that $$\label{7.13}
\sup_{0\leq t\leq T}\|(\nabla\rho,\nabla P)\|_{L^{q}}\leq C,$$ which, combined with , , and gives that $$\label{7.14}
\int_{0}^{T}\|\nabla\mathbf{u}\|^2_{L^\infty}dt\leq C.$$ Taking $r=2$ in and , one gets from , , , and Gronwall’s inequality that $$\label{7.15}
\sup_{0\leq t\leq T}\|(\nabla\rho,\nabla P)\|_{L^2}\leq C,$$ which together with , , and yields that $$\label{7.16}
\sup_{0\leq t\leq T}\|\nabla^2\mathbf{u}\|_{L^2}\leq C.$$ Taking $r=q$ in and using and show that $$\label{7.17}
\int_{0}^T\|\nabla^2\mathbf{u}\|_{L^q}^2dt\leq C.$$ Thus the desired follows from , , , , , , and . The proof of Lemma \[lem37\] is finished. $\Box$
\[lem38\] Under the condition , it holds that for any $T\in[0,T^*)$, $$\label{8.1}
\sup_{0\leq t\leq T}\|\rho\bar{x}^{a}\|_{L^1\cap H^{1}\cap W^{1,q}}\leq C.$$
[*Proof.*]{} One derives from $_1$ that $\rho\bar{x}^a$ satisfies $$\label{8.2}
\partial_{t}(\rho\bar{x}^a)+ \mathbf{u}\cdot\nabla(\rho\bar{x}^a)
-a\rho\bar{x}^{a}\mathbf{u}\cdot\nabla\log\bar{x}+\rho\bar{x}^a\operatorname{div}\mathbf{u}=0.$$ Taking the $x_i$-derivative on the both side of gives $$\begin{aligned}
\label{8.3}
0= & \partial_{t}\partial_{i}(\rho\bar{x}^a)+\mathbf{u} \cdot\nabla\partial_{i}(\rho\bar{x}^a)
+\partial_{i}\mathbf{u}\cdot\nabla(\rho\bar{x}^a)
-a\partial_{i}(\rho\bar{x}^{a})\mathbf{u}\cdot\nabla\log\bar{x} \notag \\
& -a\rho\bar{x}^{a}\partial_{i}\mathbf{u}\cdot\nabla\log\bar{x}
-a\rho\bar{x}^{a} \mathbf{u}\cdot\partial_{i}\nabla\log\bar{x}
+\partial_{i}(\rho\bar{x}^a\operatorname{div}\mathbf{u}).\end{aligned}$$ For any $r\in[2,q]$, multiplying by $|\nabla (\rho\bar{x}^a)|^{r-2}\partial_{i}(\rho\bar{x}^a)$ and integrating the resulting equality over $\mathbb{R}^2$, we obtain from integrating by parts, , , , , and that $$\begin{aligned}
\label{8.4}
\frac{d}{dt}\|\nabla(\rho\bar{x}^a)\|_{L^r} \leq &
C\left(1+\|\nabla\mathbf{u}\|_{L^\infty}+\|\mathbf{u}
\cdot\nabla\log\bar{x}\|_{L^\infty}\right)
\|\nabla(\rho\bar{x}^a)\|_{L^r} \notag \\
& + C\|\rho\bar{x}^a\|_{L^\infty}\left(\||\nabla\mathbf{u} ||\nabla\log\bar{x}|\|_{L^r}
+\||\mathbf{u}||\nabla^{2}\log\bar{x}|\|_{L^r}+\|\nabla^2\mathbf{u}\|_{L^r}\right) \notag \\
\leq & C\left(1+\|\nabla\mathbf{u}\|_{W^{1,q}}\right)\|\nabla(\rho\bar{x}^a)\|_{L^r} \notag \\
& +C\|\rho\bar{x}^a\|_{L^\infty}\left(\|\nabla\mathbf{u}\|_{L^r}
+\|\mathbf{u}\bar{x}^{-\frac{2}{5}}\|_{L^{4r}}\|\bar{x}^{-\frac{3}{2}}
\|_{L^{\frac{4r}{3}}}+\|\nabla^2\mathbf{u}\|_{L^r}\right) \notag \\
\leq & C\left(1+\|\nabla^2\mathbf{u}\|_{L^r}+\|\nabla\mathbf{u}\|_{W^{1,q}}\right)
\left(1+\|\nabla(\rho\bar{x}^a)\|_{L^r}+\|\nabla(\rho\bar{x}^a)\|_{L^q}\right)
\notag \\
\leq & C\left(1+\|\nabla^2\mathbf{u}\|_{L^r}+\|\nabla^2\mathbf{u}\|_{L^q}\right)
\left(1+\|\nabla(\rho\bar{x}^a)\|_{L^r}+\|\nabla(\rho\bar{x}^a)\|_{L^q}\right).\end{aligned}$$ Choosing $r=q$ in , together with and Gronwall’s inequality indicates that $$\label{8.5}
\sup_{t\in[0,T]}\|\nabla(\rho\bar{x}^a)\|_{L^q}\leq C.$$ Setting $r=2$ in , we deduce from and that $$\sup_{t\in[0,T]}\|\nabla(\rho\bar{x}^a)\|_{L^2}\leq C.$$ This combined with and gives and completes the proof of Lemma \[lem38\]. $\Box$
With Lemmas \[lem31\]–\[lem38\] at hand, we are now in a position to prove Theorem \[thm1.1\].
**Proof of Theorem \[thm1.1\].** We argue by contradiction. Suppose that were false, that is, holds. Note that the general constant $C$ in Lemmas \[lem31\]–\[lem38\] is independent of $t<T^{*}$, that is, all the a priori estimates obtained in Lemmas \[lem31\]–\[lem38\] are uniformly bounded for any $t<T^{*}$. Hence, the function $$(\rho,\mathbf{u},P)(x,T^{*})
\triangleq\lim_{t\rightarrow T^{*}}(\rho,\mathbf{u},P)(x,t)$$ satisfy the initial condition at $t=T^{*}$.
Furthermore, standard arguments yield that $\rho\dot{\mathbf{u}}\in C([0,T];L^2)$, which implies $$\rho\dot{\mathbf{u}}(x,T^\ast)=\lim_{t\rightarrow
T^\ast}\rho\dot{\mathbf{u}}\in L^2.$$ Hence, $$-\mu\Delta{\mathbf{u}}-(\lambda+\mu)\nabla\mbox{div}\mathbf{u}+\nabla P
|_{t=T^\ast}=\sqrt{\rho}(x,T^\ast)g(x)$$ with $$g(x)\triangleq
\begin{cases}
\rho^{-1/2}(x,T^\ast)(\rho\dot{\mathbf{u}})(x,T^\ast),&
\mbox{for}~~x\in\{x|\rho(x,T^\ast)>0\},\\
0,&\mbox{for}~~x\in\{x|\rho(x,T^\ast)=0\},
\end{cases}$$ satisfying $g\in L^2$ due to . Therefore, one can take $(\rho,\mathbf{u},P)(x,T^\ast)$ as the initial data and extend the local strong solution beyond $T^\ast$. This contradicts the assumption on $T^{\ast}$.
Thus we finish the proof of Theorem \[thm1.1\]. $\Box$
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[^1]: School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China ([xzhong1014@amss.ac.cn]{}).
[^2]: Supported by China Postdoctoral Science Foundation (No. 2017M610579), Fundamental Research Funds for the Central Universities (No. XDJK2017C050) and the Doctoral Fund of Southwest University (No. SWU116033).
|
---
abstract: 'A search for colour reconnection effects in hadronic decays of W pairs is performed with the L3 detector at centre-of-mass energies between 189 and 209 . The analysis is based on the study of the particle flow between jets associated to the same W boson and between two different W bosons in $\qq\qq$ events. The ratio of particle yields in the different interjet regions is found to be sensitive to colour reconnection effects implemented in some hadronisation models. The data are compared to different models with and without such effects. An extreme scenario of colour reconnection is ruled out.'
author:
- The L3 Collaboration
date: 'March 13, 2003'
title: |
Search for Colour Reconnection Effects in\
$\ee \rightarrow \WW \rightarrow \mathrm{hadrons}$ through Particle-Flow Studies at LEP
---
Introduction
============
According to the string model of hadronisation, the particles produced in the process $\ee\rightarrow\WW\rightarrow hadrons$ originate, in the absence of colour reconnection, from the fragmentation of two colour singlet strings each of which is stretched between the two quarks from a W boson. In this case the hadrons are uniquely associated to a particular W and there is a direct correspondence between the jets formed by these hadrons and the primary quarks from the W boson decays. Energy-momentum is separately conserved for each of the W systems. However, it has been suggested that interactions may occur between the decay products of the two W bosons [@gusta; @sjost; @gh; @yellow]. The main justification for this “cross-talk” is the relatively short distance separating the decay vertices of the W bosons produced in $\ee$ annihilation ($\approx$ 0.1 fm) compared to the typical hadronic scale (1 fm), which implies a large space-time overlap of the two hadronising systems.
The main consequence of these interactions, called Colour Reconnection (CR) effects, is a modification of the distribution in phase space of hadrons. CR effects are thought to be suppressed in the hard perturbative phase, but may be more important in the soft gluon emission regime [@sjost]. While hard gluons, with energy greater than the W width, are radiated independently from different colour singlets, soft gluons could in principle be affected by the colour strings of both decaying W’s. Such CR would affect the number of soft particles in specific phase space regions, especially outside the jet cores.
The study of CR is interesting not only for probing QCD dynamics but also for determining a possible bias in the W mass measurement in the four-quark channel. CR could affect the invariant masses of jet pairs originating from W decays. Therefore the precision with which the W mass may be determined using the four-quark channel depends strongly on the understanding of CR effects. Events where only one W decays hadronically are unaffected by CR.
Previous LEP studies of CR, performed at centre-of-mass energy $\rs \le$ 183 , were based on charged particle multiplicity and momentum distributions [@lepnchcr].
The analysis presented in this paper uses the method suggested in Reference based on energy and particle flow to probe the string topology of four-quark events to search for particular effects of particle depletion and enhancement. The results are based on 627 pb$^{-1}$ of data collected with the L3 detector [@l3detect] at $\rs$=189$-$209 . Comparisons with various models are made at detector level and the compatibility with the existence of CR effects in various models is investigated.
Colour Reconnection Models
==========================
Several phenomenological models have been proposed [@sjost; @gh; @geiger; @arcr; @hwcr; @rathsman] to describe CR effects in $\ee\rightarrow\WW\rightarrow hadrons$ events. The analysis presented in this paper is performed with some of those CR models, which are implemented in the PYTHIA [@pythia], ARIADNE [@ariadne] and HERWIG [@herwig] Monte Carlo (MC) programs.
We investigate two models by Sjöstrand and Khoze [@sjost] implemented in PYTHIA. They are based on rearrangement of the string configuration during the fragmentation process. They follow the space-time evolution of the strings and allow local reconnections if the strings overlap or cross, depending on the string definition (elongated bags or vortex lines).
In the type I model (SKI) the strings are associated with colour flux tubes having a significant transverse extension. The reconnection occurs when these tubes overlap and only one reconnection is allowed, the one with the largest overlap volume. The reconnection probability depends on this volume of overlap and is controlled by one free parameter, $k_{\rm{I}}$, which can be varied in the model to generate event samples with different fractions of reconnected events. The relation with the event reconnection probability ($P_{\rm{reco}}$) is given by the following formula: $$\begin{aligned}
P_{\rm{reco}} = 1-{\rm{exp}}(-f k_{\rm{I}})\end{aligned}$$ where $f$ is a function of the overlap volume of the two strings, which depends on W-pair kinematics varying with $\rs$. The default value of $k_{\rm{I}}$ is 0.6 [@sjost], which corresponds to a reconnection probability of about 30% at $\rs$=189 . This analysis is performed with three different values of $k_{\rm{I}}$: 0.6, 3 and 1000, corresponding to reconnection probabilities at $\rs$=189 of about 30%, 66% and nearly 100%, respectively.
In the type II model (SKII) the strings have no lateral extent and the reconnection occurs, with unit probability, when they cross. The fraction of reconnected events in this model is of the order of 30% at $\rs$=189 .
The CR model implemented in ARIADNE is based on reconnection of coloured dipoles before the string fragmentation takes place [@arcr]. In the AR2 scheme, which is investigated here, reconnections are allowed if they reduce the string length. While reconnections within a W are allowed at all scales, those between W’s are only allowed after the parton showers have evolved down to gluon energies less than 2 . At $\rs=$189 they affect about 55% of the events.
The CR scheme implemented in HERWIG is, as for the string fragmentation, a local phenomenon since the cluster fragmentation process follows the space-time development. In this model [@hwcr] the clusters are rearranged if their space-time extension is reduced. This rearrangement occurs with a probability equal to 1/$N^{2}_{\rm{colour}}$, with default value $N_{\rm{colour}}$ = 3, giving about 23% of reconnected events.
All probabilities discussed above are derived as fraction of events where at least one reconnection occurs either within the same W or between two W’s.
Event Selection
===============
The energy measured in the electromagnetic and hadronic calorimeters and in the tracking chamber is used to select $\ee\rightarrow\WW\rightarrow hadrons$ events. The total visible energy (${E_{\mathrm{vis}}}$) and the energy imbalance parallel (${E_{\parallel}}$) and perpendicular (${E_{\perp}}$) to the beam direction are measured. The number of clusters, defined as objects obtained from a non-linear combination of charged tracks with a transverse momentum greater than 100 and calorimetric clusters with a minimum energy of 100 , is denoted by ${N_{\mathrm{cluster}}}$. The selection criteria are:
${E_{\mathrm{vis}}}/ \rs > 0.7$; ${E_{\perp} / E_{\mathrm{vis}}} < 0.2$; ${|E_{\parallel}| / E_{\mathrm{vis}}} < 0.2$; ${N_{\mathrm{cluster}}} \geq 40 $.
In addition the events must have 4 jets reconstructed with the Durham algorithm [@kt] with $y_{\rm{cut}}$ = 0.01. To reduce the contamination from semileptonic W decays, events with energetic $\mu$ or $\rm{e}$ are rejected. Events with hard initial state radiation (ISR) are rejected as described in Reference . Additional criteria select events with nearly perfect quark-jet association, necessary for the study of particle and energy flow between jets. The two largest interjet angles are required to be between $100^{\circ}$ and $140^{\circ}$ and not adjacent. The two other interjet angles must be less than $100^{\circ}$. This selection guarantees similar sharing of energy between the four primary partons with the two strings evolving back-to-back and similar interjet regions between the two W’s. The above cuts are optimized by studying MC $\WW$ events at $\rs$=189 using the KORALW [@koralw] MC generator interfaced with the PYTHIA fragmentation model without CR. Relaxing the angular criteria increases the efficiency but gives lower probability to have correct W-jet pairing due to the more complicated event topology.
The number of selected events, the number of expected events, the selection efficiency and the percentage of correct pairing are given in Table \[tab:selflow\]. After applying all the cuts the full sample contains 666 events with an average efficiency of 12% and a purity of about 85% for $\ee\rightarrow\WW\rightarrow hadrons$. The average probability to have the correct pairing between the W bosons and their associated jets is estimated to be 91%.
The background is composed of $\qq (\gamma)$ events and Z-pair production events, in similar amounts. Background from semileptonic W pair decays is found to be negligible (less than 0.3%). The $\qq (\gamma)$ process is modeled with the KK2F MC program [@kk2f], interfaced with JETSET [@jetset] routines to describe the QCD processes, and the background from Z-pair production is simulated with PYTHIA. For CR studies W-pair events are simulated with PYTHIA. All MC samples are passed through a realistic detector simulation [@l3-simul] which takes into account time dependent detector effects and inefficiencies.
Particle- and Energy-Flow Distributions
=======================================
The algorithm to build the particle- and energy-flow distributions [@crdom] (Figure \[fig:plane\]) starts by defining the plane spanned by the most energetic jet (jet 1) and the closest jet making an angle with jet 1 greater than $100^{\circ}$ which is most likely associated to the same W (jet 2). For each event, the momentum vector direction of each particle is then projected on to this plane. The particle and energy flows are measured as a function of the angle, $\phi$, between jet 1 and the projected momentum vector for the particles located between jets 1 and 2. In order to take into account the fact that the W-pair events are not planar a new plane is defined for each remaining pair of adjacent jets. In this four-plane configuration the angle $\phi$ is defined as increasing from jet 1 toward jet 2, then to the closest jet from the other W (jet 3) toward the remaining jet (jet 4) and back to jet 1. The angle $\phi_{j,i}$ of a particle $i$ having a projected momentum vector located between jets $j$ and $j+1$ is calculated in the plane spanned by these two jets. A particle $i$ making an angle $\phi_{i}$ with respect to jet 1 adds an entry equal to 1 in the particle-flow distribution and adds an entry equal to its energy, normalised to the total event energy, in the energy-flow distribution for the corresponding $\phi$ bin.
The distributions are calculated using, for the particle definition, the clusters defined in the previous section.
Figure \[fig:flow2\] shows the particle- and energy-flow distributions obtained for the data and the MC predictions at detector level by using only the first plane for projecting all the particles. The data and MC distributions agree over the full angular range in both cases.
In order to compare the interjet regions the angles in the planes are rescaled by the angle between the two closest jets. For a particle $i$ located between jets $j$ and $j+1$ the rescaled angle is $$\begin{aligned}
\phi_{i}^{\mathrm{resc}} = j - 1 +\frac{\phi_{j,i}}{\psi_{j,j+1}}\end{aligned}$$ where $\phi_{j,i}$ is the angle between jet $j$ and particle $i$ and $\psi_{j,j+1}$ is the angle between jets $j$ and $j+1$. With this definition the four jets have fixed rescaled angle values equal to 0, 1, 2 and 3.
Figure \[fig:flow3\]a shows the rescaled particle-flow distribution normalised to the number of events after a bin-by-bin background subtraction for the data and MC predictions without CR and for the SKI model with $k_{\rm{I}}$=1000, later referred to as SKI 100%. As expected, the latter shows some depletion in the number of particles in the intra-W regions spanned by the two W bosons (regions A and B) and some particle enhancement in the two inter-W regions (regions C and D) when compared to the model without CR (no-CR).
To improve the sensitivity to CR effects the particle flows in regions A and B are averaged as are the particle flows in regions C and D. The results are shown in Figures \[fig:flow3\]b and \[fig:flow3\]c where the angle is redefined to be in the range \[0,1\]. MC studies at particle level with particles having a momentum greater than 100 show that the CR effects are consistent with the detector level results and have similar magnitudes.
The ratio of the particle flow between the quarks from the same W to that between quarks from different W’s is found to be a sensitive observable to cross-talk effects as predicted by the SKI model. These ratios, computed from the particle- and energy-flow distributions at detector level, are shown in Figure \[fig:flowrat\] for the data, the PYTHIA prediction without CR, the SKI model with $k_{\rm{I}}$=3 and SKI 100%.
The differences between the models with and without CR are larger in the middle of the interjet regions. Therefore, in order to quantify the CR effects the ratio $R$ is computed in an interval, 0.2 $< \phi_{\mathrm{resc}} <$ 0.8, optimized with respect to the sensitivity to SKI 100%. The corresponding variables for particle and energy flow are defined as follows:
$$\begin{aligned}
R_{\rm{N}}= \int_{0.2}^{0.8}{f_{\rm{N}}^{\rm{A+B}}}
\hspace*{0.3cm}{\rm{d}}\phi \left/\ \int_{0.2}^{0.8}{f_{\rm{N}}^{\rm{C+D}}}
\hspace*{0.3cm}{\rm{d}}\phi
\hspace*{0.3cm}\mathrm{ and}\hspace*{0.3cm}
R_{\rm{E}}= \int_{0.2}^{0.8}{f_{\rm{E}}^{\rm{A+B}}}
\hspace*{0.3cm}{\rm{d}}\phi \right/\ \int_{0.2}^{0.8}{f_{\rm{E}}^{\rm{C+D}}}
\hspace*{0.3cm}{\rm{d}}\phi \end{aligned}$$
where, in a region $i$, $$\begin{aligned}
f_{\rm{N}}^{i}= \frac{1}{N_{\rm{evt}}}\frac{{\rm{d}}n}{{\rm{d}}\phi}
\hspace*{0.3cm}\mathrm{ and}\hspace*{0.3cm}
f_{\rm{E}}^{i} = \frac{1}{E}\frac{{\rm{d}}E}{{\rm{d}}\phi}\end{aligned}$$
The measured values of $R_{\mathrm{N}}$ and $R_{\mathrm{E}}$ obtained at each centre-of-mass energy are summarised in Table \[tab:result\]. Correlations in the particle rates between the four interjet regions are taken into account by constructing the full covariance matrix. This results in an increase of about 20% of the statistical uncertainty. The values obtained with the complete data sample are: $$\begin{aligned}
R_{\rm{N}} = 0.911 \pm 0.023\hspace*{0.1cm}(\rm{stat.})\\
R_{\rm{E}} = 0.719 \pm 0.035\hspace*{0.1cm}(\rm{stat.}) \end{aligned}$$ An estimate of the sensitivity to the SKI 100% model, shows that $R_{\rm{N}}$ is 2.6 times more sensitive than $R_{\rm{E}}$. Accordingly, the following results and discussion are only based on $R_{\rm{N}}$.
Figure \[fig:ratio\] shows the measured $R_{\rm{N}}$ as a function of $\rs$ together with PYTHIA no-CR and SKI model predictions. The energy dependence originating from the different pairing purities and jet configurations is in agreement with the model predictions. For the PYTHIA SKI predictions, the ratio decreases with the reconnection probability over the whole energy range with similar magnitude. The data indicate little or no CR.
Semileptonic Decays
===================
To verify the quality of the MC simulation of the $\mathrm{W}\rightarrow\qq$ fragmentation process and the possible biases which may arise when determining the particle yields between reconstructed jets in the detector, the particle- and energy-flow distributions are investigated in $\ee\rightarrow\WW\rightarrow\qq l\nu$ where $l=\mathrm{e},\mu$. For this analysis events are selected with high multiplicity, large missing momentum and a high energy electron or muon. The missing momentum is considered as a fictitious particle in order to apply the Durham jet algorithm to select 4-jet events with $y_{\rm{cut}}$=0.01.
The same angular criteria on the four interjet angles as applied in the fully hadronic channel are used here. The purity obtained after selection is about 96% and the efficiency is about 12%. The number of selected semileptonic events is 315 with an expectation of 314.5 events. Particle- and energy-flow distributions are built in a similar way as in the fully hadronic channel with the additional requirement that the charged lepton should be in jet 3 or 4. Figure \[fig:flowlept\]a shows the corresponding particle-flow distribution projected on to the plane of jets 1 and 2 for the data and the KORALW MC prediction. There is good agreement between data and MC over the whole distribution. Figure \[fig:flowlept\]b shows the rescaled particle-flow distribution where the structure of the two different W’s is clearly visible. The region between jet 1 and jet 2 corresponds to the hadronically decaying W ($W_{1}$) and the region between jet 3 and jet 4 corresponds to the W decaying semileptonically ($W_{2}$). The activity in the $W_{2}$ region is mainly due to low energy fragments from the hadronic decay of the first W. A comparison of data and MC for the particle flow obtained by summing the regions $W_{1}$ and $W_{2}$ is shown in Figure \[fig:flowlept2\]a. The ratio between the data and the MC distributions is shown in Figure \[fig:flowlept2\]b. This ratio is consistent with unity over the whole range. This result gives additional confidence in the correctness of the modelling of the fragmentation process of quark pairs according to the fragmentation parameters used in KORALW and PYTHIA as well as the particle flow definition and reconstruction.
In the absence of CR effects, the activity found in regions A+B of a fully hadronic event should be equivalent to twice the particle activity in the regions $W_{1}$+$W_{2}$ of the distribution for a semileptonic event. Figure \[fig:flowlept2\]c shows the ratio of the particle flow in four-quark events divided by twice the particle flow in semileptonic events for the data and the predictions from no-CR PYTHIA MC and the SKI 100% model. The CR model shows the expected deficit in the hadronic channel compared to the semileptonic one. The data are consistent with the no-CR scheme but the large statistical uncertainty prevents a quantitative statement based on this model-independent comparison.
Systematic Uncertainties
========================
Several sources of systematic uncertainties are investigated. The first important test is whether the result depends on the definition of the particles. The analysis is repeated using calorimetric clusters only. Half the difference between the two analyses is assigned as the uncertainty due to this effect. This is found to be the dominant systematic uncertainty.
The second source of systematic uncertainty is the limited knowledge of quark fragmentation modelling. The systematic effect in the $\qq (\gamma)$ background is estimated by comparing results using the JETSET and HERWIG MC programs. The corresponding uncertainty is assigned to be half the difference between the two models.
The systematic uncertainty from quark fragmentation modelling in W-pair events is estimated by comparing results using PYTHIA, HERWIG and ARIADNE MC samples without CR. The uncertainty is assigned as the RMS between the $R_{\mathrm{N}}$ values obtained with the three fragmentation models. Such comparisons between different models test also possible effects of different fragmentation schemes which are not taken into account when varying only fragmentation parameters within one particular model.
Another source associated with fragmentation modelling is the effect of Bose-Einstein correlations (BEC) in hadronic W decays. This effect is estimated by repeating the analysis using a MC sample with BEC only between particles originating from the same W. An uncertainty is assigned equal to half the difference with the default MC which includes full BEC simulation (BE32 option) [@be32] in W pairs. The sensitivity of the $R_{\rm{N}}$ variable to BEC is found to be small.
The third main source of systematic uncertainty is the background estimation. The $\qq (\gamma)$ background which is subtracted corresponds mainly to QCD four-jet events for which the rate is not well modelled by parton shower programs. PYTHIA underestimates, by about 10%, the four-jet rate in the selected phase space region [@4jetlep]. A systematic uncertainty is estimated by varying the $\qq (\gamma)$ cross section by $\pm$ 5% after correcting the corresponding background by +5%. This correction increases the value of $R_{\rm{N}}$ by 0.004.
A last and small systematic uncertainty is associated with Z-pair production. It is estimated by varying the corresponding cross section by $\pm$ 10%. This variation takes into account all possible uncertainties pertaining to the hadronic channel, from final state interaction effects to the theoretical knowledge of the hadronic cross section.
A summary of the different contributions to the systematic uncertainty is given in Table \[tab:syst\].
The ratio obtained by taking into account the systematic uncertainties is then: $$\begin{aligned}
R_{\rm{N}} = 0.915 \pm 0.023 \hspace*{0.1cm} (\rm{stat.}) \pm 0.021 \hspace*{0.1cm}(\rm{syst.})\end{aligned}$$
Comparison with Models
======================
The $R_{\rm{N}}$ values predicted by the PYTHIA no-CR, SKI, SKII, ARIADNE no-CR, AR2, HERWIG no-CR and HERWIG CR models are given in Table \[tab:model\].
The data disfavour extreme scenarios of CR. A comparison with ARIADNE and HERWIG shows that the CR schemes implemented in these two models do not modify significantly the interjet particle activity in the hadronic W-pair decay events. Thus it is not possible to constrain either of these models in the present analysis.
The dependence of $R_{\rm{N}}$ on the reconnection probability is investigated with the SKI model. For this, four MC samples are used: the no-CR sample and those with $k_{\rm{I}}$=0.6, 3 and 1000. In the SKI model the fraction of reconnected events is controlled by the $k_{\rm{I}}$ parameter and the dependence of $R_{\rm{N}}$ on $k_{\rm{I}}$ is parametrized as $R_{\rm{N}}(k_{\rm{I}}) = p_{1} (1-{\rm{exp}}(-p_{2} k_{\rm{I}}))+p_{3}$ where $p_{i}$ are free parameters. A $\chi^{2}$ fit to the data is performed. The $\chi^{2}$ minimum is at $k_{\rm{I}}=0.08$. This value corresponds to about 6% reconnection probability at $\rs$=189 . Within the large uncertainty the result is also consistent with no CR effect.
The upper limits on $k_{\rm{I}}$ at 68% and 95% confidence level are derived as 1.1 and 2.1 respectively. The corresponding reconnection probabilities at $\rs$ = 189 are 45% and 64%. The extreme SKI scenario, in which CR occurs in essentially all events, is disfavoured by 4.9 $\sigma$.
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namelist266.tex
$\langle\rs\rangle$ (GeV) $ {\cal{L}} (\rm{pb}^{-1})$ $N_{\rm{events}}$ $N_{\rm{MC}}$ $\epsilon$ $\pi$
--------------------------- ----------------------------- ------------------- --------------- ------------ -------
188.6 176.7 208 226.0 14.2% 88%
191.6 29.7 38 37.9 14.3% 90%
195.5 83.7 104 101.0 13.4% 92%
199.5 84.3 97 91.9 12.2% 93%
201.7 35.5 36 37.2 11.3% 93%
205.1 77.8 75 74.8 10.3% 93%
206.6 138.9 108 120.8 8.9% 91%
198.2 626.6 666 689.6 12.0% 91%
: Average centre-of-mass energies, integrated luminosities (${\cal{L}}$), number of selected events ($N_{\rm{events}}$), number of expected events ($N_{\rm{MC}}$), selection efficiency ($\epsilon$) and percentage of correct jet pairing ($\pi$) for the particle flow analysis. The combined figures are given in the last row.[]{data-label="tab:selflow"}
$\langle\rs\rangle$ (GeV) $ R_{\rm{N}}$ $R_{\rm{E}}$
--------------------------- ------------------- -------------------
188.6 0.820 $\pm$ 0.037 0.610 $\pm$ 0.047
191.6 0.929 $\pm$ 0.093 0.822 $\pm$ 0.133
195.5 0.948 $\pm$ 0.059 0.774 $\pm$ 0.077
199.5 1.004 $\pm$ 0.067 0.871 $\pm$ 0.095
201.7 0.770 $\pm$ 0.086 0.626 $\pm$ 0.130
205.1 1.033 $\pm$ 0.083 0.756 $\pm$ 0.111
206.6 0.958 $\pm$ 0.068 0.781 $\pm$ 0.096
: Measured $R_{\rm{N}}$ and $R_{\rm{E}}$ values as a function of energy with their statistical uncertainties.[]{data-label="tab:result"}
Source $\sigma_{R_{\rm{N}}}$
----------------------- -----------------------
Energy flow objects 0.016
$\qq$ fragmentation 0.009
WW fragmentation 0.008
BEC 0.003
4-jet background rate 0.004
ZZ background 0.002
Total 0.021
: Contributions to the systematic uncertainties on $R_{\rm{N}}$.[]{data-label="tab:syst"}
$R_{\rm{N}}$
------------------------ -------------------------------
Data 0.915 $\pm$ 0.023 $\pm$ 0.021
PYTHIA no-CR 0.918 $\pm$ 0.003
SKI ($k_{\rm{I}}$=0.6) 0.896 $\pm$ 0.003
SKI ($k_{\rm{I}}$=3.0) 0.843 $\pm$ 0.003
SKI 100% 0.762 $\pm$ 0.003
SKII 0.916 $\pm$ 0.003
ARIADNE no-CR 0.929 $\pm$ 0.003
AR2 0.919 $\pm$ 0.003
HERWIG no-CR 0.948 $\pm$ 0.005
HERWIG CR 0.946 $\pm$ 0.005
: Measured value of $R_{\rm{N}}$ and model predictions.[]{data-label="tab:model"}
![Determination of the $\phi_{i}$ angle for the particle $i$.[]{data-label="fig:plane"}](fig1_paper.eps){width="14cm"}
![a) Particle- and b) energy-flow distributions at $\sqrt{s} = 189-209 \GeV$ for data and MC predictions.[]{data-label="fig:flow2"}](fig2a_paper.eps "fig:"){width="8.2cm"} ![a) Particle- and b) energy-flow distributions at $\sqrt{s} = 189-209 \GeV$ for data and MC predictions.[]{data-label="fig:flow2"}](fig2b_paper.eps "fig:"){width="8.2cm"}
![a) Particle-flow distribution as a function of the rescaled angle for data and for PYTHIA MC predictions without CR, and with the SKI 100% model. Distributions of b) combined intra-W particle flow and c) combined inter-W particle flow.[]{data-label="fig:flow3"}](fig3_paper.eps){width="12.5cm"}
![ Ratio of a) particle- and b) energy-flow distributions (Equation 4) in regions A+B to that in regions C+D. Statistical uncertainties are shown. []{data-label="fig:flowrat"}](fig4a_paper.eps "fig:"){width="8.cm"} ![ Ratio of a) particle- and b) energy-flow distributions (Equation 4) in regions A+B to that in regions C+D. Statistical uncertainties are shown. []{data-label="fig:flowrat"}](fig4b_paper.eps "fig:"){width="8.cm"}
![The ratio $R_{\rm{N}}$ as a function of $\rs$ at detector level for data and PYTHIA no-CR and SKI model predictions. The parametrisation of the energy dependence is obtained by fitting a second order polynomial function to the predicted MC dependence. The parametrisation obtained with PYTHIA no-CR gives $R_{\rm{N}}(\rs)/R_{\rm{N}}$(189 [[GeV]{}]{}) = $-3.07 \times 10^{-4} s + 0.1297 \rs -12.56$. The dependence obtained with the SKI model ($k_{\rm{I}}$= 3) leads to a 2.3% change in the average rescaled $R_{\rm{N}}$ value at 189 . []{data-label="fig:ratio"}](fig5_paper.eps){width="12.cm"}
![Particle-flow distributions a) before and b) after angle rescaling for the semileptonic W decays for data and KORALW prediction.[]{data-label="fig:flowlept"}](fig6a_paper.eps "fig:"){width="8.2cm"} ![Particle-flow distributions a) before and b) after angle rescaling for the semileptonic W decays for data and KORALW prediction.[]{data-label="fig:flowlept"}](fig6b_paper.eps "fig:"){width="8.2cm"}
![a) Particle-flow distributions as a function of the rescaled angle for the semileptonic W decays for data and the KORALW prediction. b) Ratio of data and MC as a function of the rescaled angle. c) Ratio $R$ of the particle flow in hadronic events divided by twice the particle flow in semileptonic events.[]{data-label="fig:flowlept2"}](fig7_paper.eps){width="8.cm"}
|
---
abstract: 'Here we analyze the effects of an electron-phonon interaction on the one-electron spectral weight $A(k, \omega)$ of a $d_{x^2-y^2}$ superconductor. We study the case of an Einstein phonon mode with various momentum-dependent electron-phonon couplings and compare the structure produced in $A(k, \omega)$ with that obtained from coupling to the magnetic $\pi$-resonant mode. We find that if the strength of the interactions are adjusted to give the same renormalization at the nodal point, the differences in $A(k, \omega)$ are generally small but possibly observable near $k=(\pi,0)$.'
author:
- 'A. W. Sandvik'
- 'D. J. Scalapino'
- 'N.E. Bickers'
title: |
Effect of an Electron-phonon Interaction on the One-electron\
Spectral Weight of a d-wave Superconductor
---
Introduction
============
The role of the electron-phonon coupling in the high $T_c$ cuprates remains a puzzle. The initial finding of the absence of a phonon signature in the temperature dependence of the resistivity[@Mar90] and the small size of the isotope effect in the optimally doped cuprates[@Fra94] suggested that the electron-phonon interaction played a relatively unimportant role in these strongly-correlated materials. However, large isotope effects away from optimal doping,[@Fra94; @Cra90] significant phonon renormalization induced in the superconducting state,[@BHI00; @MRS92; @Mcq01; @Pyk93] and recent interpretations of ARPES data[@Lan01; @SLIN02] continue to raise questions regarding the nature and role of the electron-phonon interaction in the high $T_c$ cuprates.
One point of view is that the effects of the strong Coulomb interaction act to suppress the electron-phonon interaction and that while the electron-lattice interaction enters the problem, it does so on a secondary level coming along as it were for the ride. For example, in this view the large isotope effects observed in some of the cuprates away from optimal doping arises from the influence of the lattice on stripe fluctuations, acting to stabilize these and thus suppressing superconductivity.[@CEKO02] Similarly, the superconductivity-induced phonon renormalization and the possible Englesberg-Schrieffer[@ES63] signature in the ARPES data could be interpreted as naturally occurring in an interacting system but having little effect on the underlying superconducting pairing mechanism. Alternatively one might interpret the isotope effect and the phonon renormalization as supporting the existence of a significant electron-phonon coupling. Furthermore, ARPES measurements have been specifically interpreted in terms of phonon modes that could drive $d_{x^2-y^2}$ pairing.[@SLIN02] Here, we analyze a simple model of an electron-phonon interaction with the goal of obtaining insight into what one expects to see in the ARPES data of a $d_{x^2-y^2}$ superconductor with electron-phonon interactions.
Continuing technological advances along with improved sample quality have allowed angle-resolved photoemission spectroscopy (ARPES) to probe details of the energy and momentum structure of the one-electron excitations in the cuprate materials.[@DHS03] Although simplified, the sudden approximation leads to a useful picture in which the ARPES intensity is equal to the square of a matrix element which depends upon the photon energy, polarization, and the sample geometry times a product of the single particle spectral weight $$A(k, \omega) = \frac{1}{\pi}\ {\rm Im} \lbrace G(k, \omega)\rbrace
\label{one}$$ and a Fermi factor $f(\omega)$. Here $G(k, \omega)$ is the one-electron Green’s function. Thus, the idea is that from the $k$ and $\omega$ dependence of the ARPES data, one can extract information about the spectral weight $A(k, \omega)$. Then, from this, one seeks to learn about the electron self-energy $\Sigma(k, \omega)$ and the structure of the effective interaction. In particular, the role of spin fluctuations and the $\pi$-resonance on the superconducting state spectral function have been studied.[@Kam01; @EN00; @EN03] With the recent suggestions[@Lan01] from ARPES measurements that there may be a significant coupling of the electrons to a phonon with an energy of order 40meV, one would like to understand how this would effect the ARPES spectrum.
From the number of atoms in a unit cell, it is clear that there are a large number of phonon modes in the cuprates. Here we will focus on several of the modes associated with the motion of the O ions. We will treat these as Einstein phonons. Then for a Hubbard-like model in which the Cu sites form the Hubbard lattice, the effective electron-electron interaction is $$V(q, \omega) = \frac{2|g(q)|^2\Omega_0}{\omega^2 - \Omega^2_0 + i\delta}.
\label{two}$$ If $|g(q)|^2 = |g|^2$ is independent of the momentum transfer, $V(q,
\omega)$ does not couple to the $d_{x^2-y^2}$-pairing channel. This could model the coupling to the $c$-axis vibration of the apical oxygen. Alternatively, if the electron-phonon matrix element is momentum dependent, the interaction given by Eq. can couple to the $d_{x^2-y^2}$-pairing channel.
The possibility that an electron-phonon interaction could give rise to $d$-wave pairing has been discussed by various authors.[@SLIN02; @ZK96; @SA95; @Sca95; @ND96; @DMFT96; @BS96] In one approach, the $d$-wave pairing interaction occurs as the result of the interplay of the O half-breathing mode and the exchange interaction.[@SLIN02] Other approaches suggest that the Coulomb interaction can lead to a peaking of the electron-phonon coupling at small momentum transfers which favors $d_{x^2-y^2}$ pairing.[@ZK96; @Hua03] This type of momentum dependence also occurs directly for certain phonon modes. For example, for the Cu-O-Cu buckling-like mode[@SA95; @Sca95; @ND96; @DMFT96; @BS96] the square of the electron-phonon coupling constant is $$|g(q)|^2 = |g|^2\, \left(\cos^2 \left(\frac{q_x}{2}\right)
+ \cos^2 \left(\frac{q_y}{2}\right)\right).
\label{three}$$ Setting $q=k-k^\prime$, the momentum-dependent part of this coupling factors into a sum of separable terms $$\begin{aligned}
&&|g (k-k^\prime)|^2 = \nonumber\\
&&|g|^2 \Bigl(1+\frac{1}{4} (\cos k_x-\cos k_y)
(\cos k^\prime_x - \cos k^\prime_y) + \cdots\Bigr),~~~~~
\label{four}\end{aligned}$$ including additional $(\cos k_x + \cos k_y)$ and $(\sin k_x \pm\ \sin k_y)$ factors. The plus sign in front of the $d$-wave term implies that this type of phonon exchange provides an attractive channel for $d$-wave pairing. The key point is that if the electron-phonon coupling $|g (k, k^\prime)|$ falls off at large $|k-k^\prime|$ momentum transfers, then such a phonon exchange can mediate $d$-wave pairing.
Alternatively, an in-plane O breathing-like mode has $$|g(q)|^2 = |g|^2 \left(\sin^2 \left(\frac{q_x}{2}\right) +
\sin^2 \left(\frac{q_y}{2}\right)\right).
\label{five}$$ This increases at large momentum transfers giving rise to a repulsive interaction in the $d_{x^2-y^2}$-channel. Setting $q=k-k^\prime$ in Eq. one finds that $$\begin{aligned}
&&|g(k-k^\prime)|^2 = \nonumber\\
&&|g|^2\Bigl(1-\frac{1}{4}\ (\cos k_x - \cos k_y)
(\cos k^\prime_x - \cos k^\prime_y) + \cdots\Bigr),~~~~~
\label{six}\end{aligned}$$ and the minus sign in the second term implies that this phonon suppresses $d$-wave pairing.[@SA95; @Sca95; @ND96; @DMFT96; @BS96]
In Section II we discuss the simplified case of a cylindrical Fermi surface and a separable phonon mediated interaction. This provides insight into the differences between the $s$-wave and $d$-wave cases and establishes the structure of the singularities in the self-energy that are reflected in $A(k, \omega)$ for an Einstein mode. While in the actual materials, the singularities are broadened by the dispersion of the phonon mode, quasiparticle lifetime effects due to other interactions and impurities, as well as finite temperature effects, these results provide a useful framework for understanding the structure that appears in $A(k, \omega)$.
In Section III, we include the effects of a $t-t^\prime$ band structure and the momentum dependence of the coupling. We consider the three different electron-phonon coupling constants discussed above and compare these with the response to the $\pi$-resonance spin fluctuation mode. The analysis of the $\pi$-resonance mode has been extensively discussed in Refs . Various estimates for the strength of the $\pi$-resonance-electron coupling have been made.[@KKA02; @ACENS02] Here, we chose this coupling so that the renormalization of the nodal Fermi velocity is comparable with that obtained for the phonon coupling. Then we compare and discuss the spectral weights for the various modes. Section IV contains a summary of the results and our conclusions.
A Cylindrical Fermi Surface and an Einstein Phonon
==================================================
In this section we consider the case of a cylindrical Fermi surface and an interaction arising from the exchange of an Einstein phonon of frequency $\Omega_0$ $$V(\theta, \theta^\prime, \omega) = \frac{2|g(\theta, \theta^\prime)|^2
\Omega_0}{\omega^2 - \Omega^2_0 + i\, \delta}.
\label{seven}$$ Here, $\theta$ and $\theta^\prime$ denote different $k$ vectors on the cylindrical Fermi surface. With Eq. in mind, we will take $|g(\theta, \theta^\prime)|^2$ to have the separable form $$|g(\theta, \theta^\prime)|^2 = |g_z|^2 + |g_\phi|^2 \cos 2\theta \cos
2\theta^\prime.
\label{eight}$$ The one-electron Green’s function can be written as $$G(k, \omega) = \frac{Z(\omega)\, \omega + \epsilon_k}{(Z (\omega)\,
\omega)^2 - \epsilon^2_k - \phi^2(\theta, \omega)},
\label{nine}$$ with $\epsilon_k = k^2/2m-\mu$, $Z(\omega)$ the renormalization parameter and $\phi\, (\theta, \omega)
= \phi\, (\omega) \cos (2\theta)$ the gap parameter. The Eliashberg equations for $Z(\omega)$ and $\phi\, (\omega)$ are
$$\begin{aligned}
&(1-Z(\omega))\, \omega = \lambda_z \int^\infty_0 d\omega^\prime \int
\frac{d\theta}{2\pi}~~~~~\nonumber\\
&{\rm Re}\left\lbrace \left(\frac{Z(\omega^\prime)
\, \omega^\prime}{\left[(Z(\omega^\prime)\, \omega^\prime)^2 - \phi^2
(\omega^\prime) \cos^2 2\theta\right]^{\frac{1}{2}}}\right)\right. \times
\nonumber\\
& \left.\left(\frac{1}{\omega^\prime + \omega + \Omega_0 - i\, \delta}
- \frac{1}{\omega^\prime - \omega + \Omega_0 - i\, \delta}\right)\right\rbrace
\label{ten}\end{aligned}$$
$$\begin{aligned}
& \phi\, (\omega) = \lambda_\phi \int^\infty_0 d\omega^\prime \int
\frac{d\theta}{2\pi}~~~~~~~~\nonumber\\
&{\rm Re}\left\lbrace\left(\frac{\phi\, (\omega^\prime)
\cos^2 2\theta}{\left[(Z(\omega^\prime) \omega^\prime)^2 -
\phi^2(\omega^\prime) \cos^2
2\theta\right]^{\frac{1}{2}}}\right)\right. \times \nonumber\\
& \left.\left(\frac{1}{\omega^\prime + \omega + \Omega_0 - i\, \delta} +
\frac{1}{\omega^\prime - \omega + \Omega_0 - i\, \delta}\right)\right\rbrace,
\label{eleven}\end{aligned}$$
with $\lambda_z = 2|g_q|^2 N(0)/\Omega_0$ and $\lambda_\phi =
2|g_\phi |^2 N(0)/\Omega_0$. Here $N(0)$ is the one-electron density of states at the Fermi surface.
In order to determine the effect of the phonon on $Z(\omega)$ and $\phi\, (\omega)$, we will adapt an approximation used in the early studies of the role of phonons on the superconducting I(V) characteristic.[@SA64] From the form of eqs and , one sees that there will be structure in $Z(\omega)$ and $\phi\, (\omega)$ when $\omega\simeq \pm\, (\Omega_0 + \Delta (\theta))$. In this case, $\omega^\prime$ will be of order the gap $\Delta(\theta)$ at the gap edge $$\Delta(\theta)=\frac{\phi(\theta, \omega = \Delta (\theta))}{Z(\theta,
\omega=\Delta (\theta))}.
\label{twelve}$$ Therefore, if the low-energy response in the superconducting state is well described in terms of BCS $d$-wave quasiparticles, one can replace $Z(\omega^\prime)$ and $\phi(\theta, \omega^\prime)/Z(\omega^\prime)$ inside the integrals by $Z(0)$ and $\Delta(\theta)=\Delta_0 \cos 2\theta$. Then, taking the imaginary parts of eqs and we have for $\omega>0$
$$\begin{aligned}
\omega Z_2(\omega) = 4 \lambda_z \int^{\frac{\pi}{4}}_{\theta_c}
d\theta \frac{(\omega-\Omega_0)}{\left[(\omega-\Omega_0)^2 - \Delta^2_0 \cos^2
2\theta\right]^{\frac{1}{2}}}\label{thirteen}\\
\phi_2(\omega) = 4 \lambda_\Delta \int^{\frac{\pi}{4}}_{\theta_c}
d\theta \frac{\Delta_0 \cos^2
2\theta}{\left[(\omega - \Omega_0)^2 - \Delta^2_0 \cos^2
2\theta\right]^{\frac{1}{2}}}
\label{fourteen}\end{aligned}$$
Here $\theta_c$ is such that $\Delta (\theta_c) =
\omega-\Omega_0$ and $\phi_2(\omega)$ and $Z_2 (\omega)$ are even functions of $\omega$ for a time-ordered zero temperature Green’s function.
Results for $\omega Z_2(\omega)$ and $\phi_2(\omega)$ are shown in the top panel of Fig. 1 for both a $d_{x^2-y^2}$-wave and an $s$-wave gap with $\Omega_0 = 1.5 \Delta_0$. For an $s$-wave gap, $\cos 2\theta$ is set to 1 and $\theta_c=0$ in eqs and . For the $s$-wave case, the imaginary parts of $Z(\omega)$ and $\phi(\omega)$ onset when $\omega$ exceeds $\pm
\ (\Omega_0 + \Delta_0)$ and exhibit a square root singularity. For a $d_{x^2-y^2}$-gap, these functions onset linearly at $\omega=\pm\ \Omega_0$ because of the gap nodes and there is a log singularity at $\pm\ (\Omega_0 +
\Delta_0)$. The real parts of $Z(\omega)$ and $\phi\, (\omega)$ are obtained from the usual dispersion relations, and results for $Z_1 (\omega)$ and $\phi_1(\omega)$ are shown in the lower panel of Fig. 1. For the $s$-wave case, $\phi_1$ and $Z_1$ exhibit square root singularities as $\omega$ approaches $\pm\ (\Omega_0 + \Delta_0)$. This is just the expected Kramers-Kronig transform of the square root singularity in $\phi_2$ and $Z_2$. Similarly, the results for $\phi_1$ and $Z_1$ for the $d_{x^2-y^2}$ case exhibit step discontinuities at $\omega=\pm\ (\Omega_0
+ \Delta_0)$ arising from the log singularities in $\phi_2$ and $Z_2$. Naturally in real materials, phonon dispersion, impurity scattering, and finite temperature effects broaden these features. Nevertheless, they provide a simple framework for analyzing the ARPES data.
An intensity plot of $A(k, \omega)$ for the case of an $s$-wave gap is shown in Fig. 2. Here, $A(k, \omega)$ is obtained from the imaginary part of $G(k, \omega)$, using the $s$-wave results for $Z(\omega)$ and $\phi(\omega)$ shown in Fig. 1 with $\lambda = 0.5$. The real part of the gap function is supplemented by an additional contribution from an underlying pairing interaction so that the magnitude of the gap at the gap edge is equal to $\Delta_0$. Results for both the ARPES accessible region $\omega \leq 0$ and the inverse photoemission region $\omega > 0$ are shown. The shift of spectral weight due to the quasiparticle coherence factors $\frac{1}{2}(1+\frac{\epsilon_k}{E_k})$ is clearly seen as is the Englesberg-Schrieffer signature showing the asymptotic approach of a peak in the spectral function to $\pm
\ (\Omega_0 + \Delta_0)$. Because of the square root singularity in $Z$ and $\phi$, the asymptotic approach of this peak to $\pm\ (\Omega_0 + \Delta_0)$ varies as $(\lambda\Omega_0/\epsilon_k)^2$. In addition, [@note1] the Fermi velocity is renormalized by $Z_1(\Delta_0)\cong 1.3$ so that the dispersion of the peak for $\omega$ near $\Delta_0$ varies as $\sqrt{(\epsilon_k/Z_1(\Delta_0))^2 + \Delta^2_0}$ while for $\omega$ large compared to $\Omega_0$, a broadened quasiparticle peak disperses as $\epsilon_k$. Energy distribution curves (EDC) showing $A(k, \omega)$ versus $\omega$ for various values of $\epsilon_k$ are shown in Fig. 3 for $\omega \leq 0$. This is the type of EDC that one would expect to see for a traditional $s$-wave electron-phonon superconductor with a single dominant Einstein mode.[@Sca69] More generally, one would have multiple phonon modes and their dispersion along with possible finite temperature effects would lead to a richer response.
Intensity plots of $A(k, \omega)$ for the case of a $d_{x^2-y^2}$ gap are shown in Fig. 4. Just as for the $s$-wave case, $\phi_1(\theta, \omega)$ is supplemented so that the gap at the gap edge is $\Delta_0 \cos 2\theta$. Fig. 4(a) shows $A(k, \omega)$ for a cut along the antinodal direction in $k$-space $(\theta=0)$, while Fig. 4(b) shows the results for a cut along the nodal direction $(\theta = \pi/4)$. The antinodal cut resembles the $s$-wave case in the transfer of spectral weight as $\epsilon_k$ passes through the Fermi energy and the renormalization of the quasiparticle dispersion. However, the Englesberg-Schrieffer signature no longer asymptotically approaches $\pm
\ (\Omega_0 + \Delta_0)$, but rather appears to be broadened and cut off. In the $s$-wave case, the broadening due to the electron-phonon interaction did not set in until $|\omega|$ exceeded $\Omega_0 + \Delta_0$ leading to the long sweep of the peak which occurs for $|\omega|$ just below $(\Omega_0 + \Delta_0)$. However, the nodal regions associated with a $d_{x^2-y^2}$ gap lead to a finite broadening when $|\omega|$ exceeds $\Omega_0$. The onset of this broadening is seen by the faint horizontal line in Fig. 4 where the intensity changes from black to blue at larger values of $\epsilon_k$. As we will discuss, termination of this peak is a reflection of the fact that for a $d_{x^2-y^2}$-gap, $Z_1$ and $\phi_1$ have step discontinuities at $\pm\ (\Omega_0 + \Delta_0)$ rather than the square root singularities associated with an $s$-wave gap.
The nodal cut, shown in Fig. 4(b), appears on first glance to be similar to what one would expect for the normal state. That is, a renormalized $\epsilon_k/Z_1(k, 0)$ dispersion for $\omega \ll \Omega_0$ with the dispersion returning to its band value $\epsilon_k$ for $\omega \gg
\Omega_0$. However, the cut-off Englesberg-Schrieffer signature still occurs for $|\omega|=\Omega_0 + \Delta_0$. Thus, the full antinodal gap $\Delta_0$ enters as the characteristic kink energy for all momentum slices. This simply reflects the $|\omega|=\Omega_0+\Delta_0$ singularities in $Z$ and $\phi$ shown in Fig. 1. Again, the broadening of the Englesberg-Schrieffer peak when $|\omega|$ exceeds $\Omega_0$ is clearly seen in Fig. 4(b). In Fig. 5, various EDC slices of $A(k, \omega)$ are shown for the $d_{x^2-y^2}$ case. Comparing these with the $s$-wave case, one sees the broadening and truncation of the Englesberg-Schrieffer lower peak.
The difference in the structure of the Englesberg-Schrieffer signature between the $s$- and the $d_{x^2-y^2}$-cases can be understood from the plots of $$\epsilon_k = - \sqrt{(Z_1(\omega) \omega)^2 - \phi^2_1 (\omega)}
\label{fifteen}$$ shown in Fig. 6. One can see that as one probes $\epsilon_k$ states which are further below the Fermi energy, two solutions of Eq. develops. For the $s$-wave case shown in the upper panel of Fig. 6, an undamped lower energy branch asymptotically approaches $\omega = - (\Omega_0 +
\Delta_0)$, and a second quasiparticle branch at $\omega\simeq -
\epsilon_k$ evolves which is damped by the imaginary parts of $Z$ and $\phi$. As we have seen, these branches are reflected in the structure of $A(k, \omega)$ and the lower energy branch represents the characteristic Englesberg-Schrieffer signature for an $s$-wave superconductor. Similar plots for the $d_{x^2-y^2}$-case with $\theta=0$ and $\theta = \pi/4$ are shown in the lower panel of Fig. 6. Here, unlike the $s$-wave case, the low energy branch is terminated, reflecting the fact that the singularities in $Z_1$ and $\phi_1$ for the $d$-wave case are simply step discontinuities at $\pm\ (\Omega_0 +
\Delta_0)$. The onset of damping processes for the $d_{x^2-y^2}$ case when $\omega < -\Omega_0$ give rise to the discontinuity in slope seen at $\omega= -\Omega_0$.
Band Structure and the Effect of a Momentum-Dependent Coupling
==============================================================
We turn next to the effects of the band structure and to the momentum dependence of the electron-phonon coupling. For the band structure, consider a square lattice with a near-neighbor hopping $t$ and a next-near-neighbor hopping $t^\prime$. In this case $$\epsilon_k = - 2t (\cos k_x + \cos k_y) - 4t^\prime \cos k_x \cos k_y -
\mu.
\label{sixteen}$$ For $t^\prime/t = -0.3$ and $\mu/t = -1$, one has the typical Fermi surface shown in Fig. 7 and the single spin electron density of states shown in the inset. We take the gap to be $$\Delta_k = \Delta_0 (\cos k_x - \cos k_y)/2.
\label{seventeen}$$ In this case, the one-electron Green’s function can be written in the form $$G(k, \omega) = \frac{Z(k, \omega)\omega + (\epsilon_k + X(k,
\omega))}{(Z(k, \omega)\omega)^2 - (\epsilon_k + X(k, \omega))^2 - \phi^2
(k, \omega)}.
\label{eighteen}$$ Adopting the same approximation as before, the phonon-induced contributions to the imaginary parts of the renormalization, energy shift, and gap parameters are given by
$$\begin{aligned}
\omega Z_2 (k, \omega) & = & \frac{\pi}{2N}
\sum\limits_{k^\prime} |g(k-k^\prime)|^2\times~~~~~\nonumber\\
\Bigl(\delta(E_{k^\prime}& + & \Omega_0 - \omega) - \delta
(E_{k^\prime} + \Omega_0 + \omega)\Bigr),
\label{nineteen}\end{aligned}$$
$$\begin{aligned}
X_2 (k, \omega) = - \frac{\pi}{N}\ \sum\limits_{k^\prime}
|g(k-k^\prime)|^2\ \frac{\epsilon_{k^\prime}}{2E_{k^\prime}}\times~~~~~
\nonumber\\
\Bigl(\delta\left(E_{k^\prime} + \Omega_0 - \omega\right) + \delta
\left(E_{k^\prime} + \Omega_0 + \omega\right)\Bigr),
\label{twenty}\end{aligned}$$
$$\begin{aligned}
\phi_2 (k, \omega) = \frac{\pi}{N}\ \sum\limits_{k^\prime}
|g(k-k^\prime)|^2\ \frac{\Delta_{k^\prime}}{2E_{k^\prime}}\times~~~~~
\nonumber\\
\Bigl(\delta\left(E_{k^\prime} + \Omega_0 - \omega\right) + \delta
\left(E_{k^\prime} + \Omega_0 + \omega\right)\Bigr).
\label{twentyone}\end{aligned}$$
Here, as before, we assume that an underlying pairing interaction, most likely spin-fluctuations, gives rise to a zero temperature $d_{x^2-y^2}$ superconducting state. At low energies this state is characterized by a renormalized band structure Eq. and chemical potential, a renormalized coupling constant $g(q)$, and a gap given by Eq. . These parameters have been used in the Eliashberg equations to describe the state which enters when an excitation at energy $\omega >\Omega_0$ decays to a lower energy $E_{k^\prime}$ state (or when $\omega < - \Omega_0$ decays to $-E_{k^\prime}$). The real parts of $Z(k,
\omega)$, $\phi(k, \omega)$, and $X(k, \omega)$ are again found from the Kramer-Kronig dispersion relation. The spectral weight $A(k, \omega)$ is then obtained from Eq. with the chemical potential shift removed from $X_1(k, \omega)$ and a contribution added to the real part of the gap so that the real part of the gap at the gap edge remains equal to $\Delta_k$, Eq. . Note that the contributions of the underlying pairing interaction to $Z$ and $X$, as well as the higher energy part of $\phi$, have not been included. Thus, there are additional renormalization and damping effects which do not appear. We basically are seeking to understand the leading contribution of the electron-phonon interaction which is superimposed on top of the other many-body interactions. This approach rests on the idea that in the superconducting state the low-lying electronic states are well described by BCS $d_{x^2-y^2}$ excitations [@Hof02] with renormalized band parameters $t$, $t^\prime$, and $\mu$, a $d_{x^2-y^2}$-wave gap $\Delta_k$, and renormalized electron-phonon coupling constants. Note, that here we are not taking into account the possible change in $q$-dependence of the electron-phonon couplings produced for example by the Hubbard $U$.[@ZK96; @Hua03]
We begin by looking at the self-energy terms for the case of the buckling mode with $|g(q)|^2$ given by Eq. and $|g|^2 = 0.5$ in units of $t^{-2}$. Results for $\phi(\omega, k)$, $Z(\omega, k)$, and $X(\omega,
k)$ are shown in Fig. 8 for $k$ at point $A$ shown in Fig. 7. The imaginary parts of $Z$ and $\phi$ exhibit the expected log singularity at $\Delta_0 + \Omega_0$ that we previously saw for the case of a circular Fermi surface. In addition, there is a second log singularity at $E(0, \pi) + \Omega_0$ with $E(0, \pi) = \sqrt{\epsilon^2 (0, \pi) +
\Delta^2_0}$ which comes from the Van Hove singularity[@EN00] at $k =(0, \pi)$. These log singularities in $Z_2$ and $\phi_2$ manifest themselves via the Kramers-Kronig dispersion relation as step-down discontinuities in $Z_1$ and $\phi_1$, as seen in Fig. 8. The energy shift parameter $X$ has only the Van Hove singularity. Naturally, the dispersion of the phonon mode as well as finite temperature and lifetime effects will broaden these features in the actual system. The energy distribution of the spectral weight $A(k, \omega)$ for the buckling mode at momentum $k_A$ is plotted in Fig. 9. It shows the quasiparticle peak at the gap edge $\Delta_{k_A}$ as well as structure associated with the buckling phonon at $\Omega_0 + \Delta_0$ and $\Omega_0 + E(0, \pi)$.
As noted in the introduction, one would like to determine whether the structure observed in the ARPES data is due to phonons or the $\pi$-resonance spin fluctuation mode. Eschrig and Norman[@EN00; @EN03] have analyzed the effect of the $\pi$-resonance using a detailed tight binding fit of the band energy $\epsilon_k$ and a coupling to the $\pi$-resonant mode of frequency $\Omega_0$ given by $$|g(q)|^2 = g^2_{SF}\ \frac{w_Q}{1+ 4\xi^2 [\cos^2(q_x/2) +
\cos^2(q_y/2)]}.
\label{twentytwo}$$ Here, we will use the $t-t^\prime$ band structure of Eq. with $t^\prime/t = -0.3$ and $\mu=-1$, set $w_Q=1$, $\xi=2$, and set $g^2_{\rm SF} = 5$ which corresponds to having $\frac{3}{4}(\frac{\bar U}{t})^2 = 5$ in an effective Hubbard RPA interaction. In addition, with this choice for $g^2_{\rm SF}$ we will find that $Z_1(k_F, 0)$ at the nodal point $C$ is comparable with $Z_1(k_F, 0)$ for the phonons. This makes it convenient for addressing the question of whether there are significant spectral differences due simply to the structure of the momentum-dependent couplings that would allow one to determine the nature of the mode from the ARPES data. [@note2] Note that for the spin-fluctuation interaction with $|g(q)|^2$ given by Eq. (\[twentytwo\]), there is a minus sign on the righthand side of Eq. (\[twentyone\]) for the gap parameter. For the three types of phonon couplings we take $|g|^2=0.5$ in units of $t^{-2}$. This gives $Z_1(k_F, 0)
\simeq 1.3$ corresponding to an effective [@ASJL96; @JADS98] $\lambda
\sim 0.3$. For the $\pi$-resonance mode coupling, setting $g_{SF}^2=5$ gives $Z_1(k_F, 0) \simeq 1.3$.
Intensity plots of $A(k, \omega)$ for the constant Holstein coupling, the buckling mode coupling Eq. , the breathing mode coupling Eq. , and the $\pi$-resonance mode coupling Eq. , are shown in Fig. 10 for the momentum cut $A$. Similar intensity plots for the momentum cuts $B$ and $C$ are shown in Figs 11 and 12. In Fig. 10, one sees a high intensity quasiparticle peak and weaker structures onsetting at $\omega=-(\Omega_0 + \Delta_0)$ and $-(\Omega_0 + E(0, \pi)$ due to the coupling to the phonon or magnetic resonance modes. For the $B$ momentum cut shown in Fig. 11, one can now move deep enough inside the Fermi sea that the Englesberg-Schrieffer lower energy peak (the upper bright curve in the figures) is broadened when $\omega$ becomes less than $-\Omega_0$ and terminated at a finite value of $k_x$ as $\omega$ approaches $-(\Omega_0 + \Delta_0)$. At still higher energies ($\omega$ more negative), a damped quasiparticle branch is seen. The nodal $C$ cut is shown in Fig. 12. Here, one clearly sees the Englesberg-Schrieffer signature with a quasiparticle peak which varies as $\epsilon_k/Z_1 (k_F, 0)$ near the Fermi surface, then disperses and bends as $\omega$ approaches $-(\Omega_0 +
\Delta_0)$. This peak is then terminated as a broadened high energy quasiparticle branch appears at more negative values of $\omega$.
The difference of $A(k, \omega)$ for the various modes is in fact subtle since all four have an Einstein spectrum with $\Omega_0 = 0.3t$, a $d_{x^2-y^2}$ gap with $\Delta_0 = 0.2t$, and a band structure with $t^\prime/t = - 0.3$ and $\mu=-1$. Thus, the characteristic energies $\Delta_0$, $\Omega_0 + \Delta_0$, and $\Omega_0 + E(0, \pi)$ are the same. In addition as discussed, we have chosen the coupling constants so that $|g(q)|^2$ averaged over the Brilloin zone is the same for all four cases. Thus, the basic difference is the momentum structure of the different couplings shown in Fig. 13 for $q_x=q_y$. Here, we see that the spin-fluctuation resonant mode is clearly most strongly peaked at large momentum, followed by the breathing mode phonon, the uniform Holstein coupling, and lastly the buckling mode phonon which has $|g(\pi, \pi)|^2=0$. One consequence of the strong peak in the magnetic reasonance-mode coupling is seen in Fig. 10 for the $A$ cut. Here, the increase of the intensity of the spectral weight $A(k, \omega)$ which occurs when $\omega$ decreases below $-(\Omega_0 + E(\pi, 0))$ is greatest for the spin-fluctuation $\pi$-resonance and smallest for the buckling mode.
In Fig. 14 we show the energy distribution curves for the four modes for momentum $k=(0, \pi)$. $A(k, \omega)$ for all of the modes shows a strong peak at $\Delta_0$. For the $\pi$-mode, this is followed by a dip and then a secondary peak which develops as $\omega$ decreases below the Van Hove threshold at $-(\Omega_0 + E(0, \pi))$. It is this peak-dip-hump structure, for the case in which the effects of the bilayer splitting can be eliminated, that has been identified as a ‘fingerprint’ of the $\pi$-resonance.[@EN00; @EN03; @Bor02; @Kim03] Here, we see that indeed this structure is most pronounced for the $\pi$-mode and smallest for the buckling mode. However, this is a quantitative effect rather than a qualitative one and if the phonon coupling increases at large momentum transfers, such as in the case of the breathing mode, this feature returns although not as strongly as for the $\pi$-mode.
Conclusions
===========
The Englesberg-Schrieffer-like structure in the ARPES data of BISCO is consistent with the existence of an Einstein-like mode with $\Omega_0 \sim
40$meV coupled to the electrons as suggested by various authors. [@SS97; @Nor97; @ND98] However, it seems that it will be difficult to determine the origin of the mode based solely upon the $(q_x, q_y)$ momentum dependence of its coupling. One might have thought that the $q$-dependence of the coupling or in the case of the $\pi$ mode, the $q$-dependence of the resonance that has been parameterized as a $q$-dependent coupling, would give rise to clearly identifiable structure in $A(k, \omega)$. However, all of the modes show quite similar characteristic features at energies $\Omega_0 + \Delta_0$ and $\Omega_0 + E(0, \pi)$, which appear throughout the zone.
It would appear that the best place to look for a feature that could distinguish between, for example, the buckling phonon mode and the $\pi$-resonant mode is near the $k=(0, \pi)$ point. Here, the strong coupling of the $\pi$ mode to the electrons for $q$ near $(\pi, \pi)$ leads to a secondary peak onsetting at an energy $\omega= -(E(0, \pi) + \Omega_0)$. For the buckling phonon mode, the coupling at $q=(\pi, \pi)$ vanishes and there is only a relatively weak response in this same frequency range. However, as we have seen, there is a secondary peak for the breathing mode which has nearly the same strength as that for the $\pi$-mode. Thus, the observed peak-dip-hump structure could also be consistent with a coupling to the oxygen-breathing mode. Recently, it has been suggested that the $q_z$ dependence for a bilayer system may identify the mode as having $q_z=\pi$, which would provide support for the $\pi$ resonance.[@EN02] However, further work on the odd and even bilayer phonon coupling is needed for comparison.
While the coupling to the $\pi$-resonance mode along with a higher energy continuum spin-fluctuation spectrum provides an attractive unified framework, our results leave open the possibility that an oxygen phonon mode could also play a role. As we have seen, even with a relatively modest coupling constant $\lambda \sim
0.3$, one would expect to see evidence of some oxygen phonon modes. If they are not seen, then this suggests that the strong Coulomb many-body effects act to suppress the electron-phonon coupling. Alternatively, if it can be shown that the $\pi$-mode is not viable, oxygen phonon modes could provide a source for the resonant mode features. The continuum spin fluctuations would, of course, also contribute in this mixed scenerio. Here we should note that even if the mode were identified as the buckling mode, we find that its contribution to the magnitude of the $d_{x^2-y^2}$ gap is negligible because the increase in $Z_1$ more than offsets the increase in $\phi_1$ (in Eq. the $d$-wave coupling term is only 1/4 of the uniform coupling). This is in agreement with the results of Eliashberg-like $T_c$ calculations [@NSB99] which find that, while the buckling phonons can provide an attraction in the $d_{x^2-y^2}$-channel, its contribution to $Z$ leads to an overall reduction in $T_c$.
To conclude, from the results that we have found, it seems likely that the identification of the excitation responsible for the structure in the ARPES data will be decided on grounds other than the momentum dependence of the effective coupling. One aspect that remains under discussion is the strength of the various couplings. For the O phonon modes, LDA calculations [@ASJL96; @JADS98] find intermediate coupling strengths with $\lambda
\sim 0.3$ to 0.5. From our calculations it would appear that at this strength, one should in fact see structure in $A(k, \omega)$. If this is not seen, it raises the question of why is the electron-phonon coupling weakened in strongly-correlated materials? [@ZK96; @Hua03] The coupling to the $\pi$-resonance mode would appear to raise the opposite problem. That is, if the $\pi$-resonance mode is responsible for the Englesberg-Schrieffer-like signature in the ARPES spectrum, how can it be coupled so strongly? [@EN03; @KKA02; @ACENS02]
We would like to thank Z.-X. Shen for discussing his data with us, for his physical insights, and his enthusiasm for this project. DJS would also like to also acknowledge very useful discussions with S.V. Borisenko, W. Hanke, and S.A. Kivelson. AWS would like to acknowledge support from the Academy of Finland under project No. 26175. DJS acknowledges support from the National Science Foundation under grant No. DMR02-11166.
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---
abstract: 'We investigate possible realizations of exotic SU($N$) symmetry-protected topological (SPT) phases with alkaline-earth cold fermionic atoms loaded into one-dimensional optical lattices. A thorough study of two-orbital generalizations of the standard SU($N$) Fermi-Hubbard model, directly relevant to recent experiments, is performed. Using state-of-the-art analytical and numerical techniques, we map out the zero-temperature phase diagrams at half-filling and identify several Mott-insulating phases. While some of them are rather conventional (non-degenerate, charge-density-wave or spin-Peierls like), we also identify, for even-$N$, two distinct types of SPT phases: an orbital-Haldane phase, analogous to a spin-$N/2$ Haldane phase, and a topological SU($N$) phase, which we fully characterize by its entanglement properties. We also propose sets of non-local order parameters that characterize the SU($N$) topological phases found here.'
author:
- 'V. Bois'
- 'S. Capponi'
- 'P. Lecheminant'
- 'M. Moliner'
- 'K. Totsuka'
title: 'Phase diagrams of one-dimensional half-filled two-orbital SU($N$) cold fermions systems'
---
Introduction
============
High continuous symmetry based on the SU($N$) unitary group with $N>2$ plays a fundamental role in the standard model of particle physics. The description of hadrons stems from an approximate SU($N$) symmetry where $N$ is the number of species of quarks, or flavors. In contrast, the SU($N$) symmetry was originally introduced in condensed matter physics as a mathematical convenience to investigate the phases of strongly correlated systems. For instance, we enlarge the physically relevant spin-SU(2) symmetry to SU($N$) and use the $N$ as a control parameter that makes various mean-field descriptions possible in the large-$N$ limit. We then carry out the systematic $1/N$-expansion to recover the original $N=2$ case.[@Auerbach; @Sachdev]
Extended continuous symmetries have been also used to unify several seemingly different competing orders in such a way that the corresponding order parameters can be transformed to each other under the symmetries. [@Zhang-97; @HermeleSF05] A paradigmatic example is the SO(5) theory[@Zhang-97; @DemlerHZ04] for the competition between $d$-wave superconductivity and antiferromagnetism, where the underlying order parameters are combined to form a unified order parameter quintet. The high continuous symmetry often emerges from a quantum critical point unless it is simply introduced phenomenologically. In this respect, for instance, the consideration of SU(4) symmetry might be a good starting point to study strongly correlated electrons with orbital degeneracy. [@Li-M-S-Z-98; @Yamashita-S-U-98; @Pati-S-K-98; @Frischmuth-M-T-99; @Azaria-G-L-N-99]
At the experimental level, realizations in condensed matter systems of enhanced continuous symmetry (in stark contrast to the SU(2) case) are very rare since they usually require substantial fine-tuning of parameters. Semiconductor quantum dots technology provides a notable exception as it enables the realization of an SU(4) Kondo effect resulting from the interplay between spin and orbital degrees of freedom. [@Keller14]
Due to their exceptional control over experimental parameters, ultracold fermions loaded into optical lattices might be ideal systems to investigate strongly correlated electrons with a high symmetry. While ultracold atomic gases with alkali atoms can, in principle, explore the physics with SO(5) and SU(3) symmetries, [@Wu2003; @HonerkampH04; @Lecheminant-B-A-05; @Wu06; @RappZHH07; @AzariaCL09] alkaline-earth atoms are likely to be the best candidates for experimental realizations of exotic SU($N$) many-body physics. [@Gorshkov-et-al-10; @Cazalilla-H-U-09; @Cazalilla-R-14] These atoms and related ones, like ytterbium atoms, have a peculiar energy spectrum associated with the two-valence outer electrons. The ground state (“$g$” state) is a long-lived singlet state $^1S_0$ and the spectrum exhibits a metastable triplet excited state (“$e$” state) $^3P_0$. Due to the existence of an ultranarrow optical transition $^1S_0$-$^3P_0$ between these states, alkaline-earth-like atoms appear to be excellent candidates for atomic clocks and quantum simulation applications. [@daley11] Moreover, the $g$ and $e$ states have zero electronic angular momentum, so that the nuclear spin $I$ is almost decoupled from the electronic spin. The nuclear spin-dependent variation of the scattering lengths is expected to be smaller than $\sim 10^{-9}$ for the $g$ state and $\sim 10^{-3}$ for the $e$ state. [@Gorshkov-et-al-10] This decoupling of the electronic spin from the nuclear one in atomic collisions paves the way to the experimental realization of fermions with an SU($N$) symmetry where $N= 2 I +1$ ($I$ being the nuclear spin) is the number of nuclear states.
The cooling of fermionic isotopes of these atoms below the quantum degeneracy has been achieved for strontium atoms $^{87}$Sr with $I= 9/2$ [@DeSalvo-Y-M-M-K-10; @Tey-S-G-S-10] and ytterbium atoms $^{171}$Yb, $^{173}$Yb with $I= 1/2,5/2$. [@Fukuhara-T-K-T-07; @Taie-etal-PRL-10] These atoms enable the experimental exploration of the physics of fermions with an emergent SU($N$) symmetry where $N$ can be as large as 10. In this respect, experiments on $^{173}$Yb atoms loaded into a three-dimensional (3D) optical lattice have stabilized an SU(6) Mott insulator [@Taie-Y-S-T-12] while the one-dimensional (1D) regime has also been investigated. [@Pagano-et-al-14] Very recent experiments on $^{87}$Sr (respectively $^{173}$Yb) atoms in a two-dimensional (2D) (respectively 3D) optical lattice have directly observed the existence of the SU($N$) symmetry and determined the specific form of the interactions between the $g$ and $e$ states. [@Zhang-et-al-14; @Scazza-et-al-14] All these results and future experiments might lead to the investigation of the rich exotic physics of SU($N$) fermions as for instance the realization of a chiral spin liquid phase with non-Abelian statistics. [@Hermele-G-11; @Cazalilla-R-14]
The simplest effective Hamiltonian to describe an $N$-component Fermi gas with an SU($N$) symmetry loaded into an 1D optical lattice is the SU($N$) generalization of the famous Fermi-Hubbard model: $${\cal H}_{\text{SU($N$)}} = - t \sum_{i, \alpha} \left(c_{\alpha,\,i}^\dag c_{\alpha,\,i+1} + \text{H.c.}\right)
+ \frac{U}{2} \sum_{i} n_{i}^2 ,
\label{HubbardSUN}$$ $c_{\alpha,\,i}^\dag$ being the fermionic creation operator for site $i$ and nuclear spin states $\alpha = 1, \ldots, N$, and $n_{i} = \sum_{ \alpha} c_{\alpha,\,i}^\dag c_{\alpha,\,i}$ is the density operator. All parameters in model (\[HubbardSUN\]) are independent from the nuclear states which express the existence of an global SU($N$) symmetry: $c_{\alpha,\,i} \mapsto \sum_{\beta} U_{\alpha \beta} c_{\beta,\,i}$, $U$ being an SU($N$) matrix. Model (\[HubbardSUN\]) describes alkaline-earth atoms in the $g$ state loaded into the lowest band of the optical lattice. The interacting coupling constant $U$ is directly related to the scattering length associated with the collision between two atoms in the $g$ state. In stark contrast to the $N=2$ case, the SU($N$) Hubbard model (\[HubbardSUN\]) is not integrable by means of the Bethe ansatz approach. However, most of its physical properties are well understood thanks to field theoretical and numerical approaches. For a commensurate filling of one atom per site, which best avoids issues of three-body loss, a Mott-transition occurs for a repulsive interaction when $N>2$ between a multicomponent Luttinger phase and a Mott-insulating phase with $N-1$ gapless degrees of freedom. [@Assaraf-A-C-L-99; @Manmana-H-C-F-R-11] In addition, the fully gapped Mott-insulating phases of model (\[HubbardSUN\]) are known to be spatially nonuniform for commensurate fillings.[@Szirmai-L-S-2008]
The search for exotic 1D Mott-insulating phases with SU($N$) symmetry requires thus to go beyond the simple SU($N$) Fermi-Hubbard model (\[HubbardSUN\]). One possible generalization is to exploit the existence of the $e$ state in the spectrum of alkaline-earth atoms and to consider a two-orbital extension of the SU($N$) Fermi-Hubbard model which is directly relevant to recent experiments. [@Zhang-et-al-14; @Scazza-et-al-14] The interplay between orbital and SU($N$) nuclear spin degrees of freedom is then expected to give rise to several interesting phases, including symmetry-protected topological (SPT) phases. [@Gu-W-09; @Chen-G-W-10] The latter refer to non-degenerate fully gapped phases which do not break any symmetry and cannot be characterized by local order parameters. Since any gapful phases in one dimension have short-range entanglement, the presence of a symmetry is necessary to protect the properties of that 1D topological phase, in particular the existence of non-trivial edge states. [@Chen-G-W-10; @Chen-G-L-W-12]
In this paper, we will map out the zero-temperature phase diagrams of several two-orbital SU($N$) lattice models at half-filling by means of complementary use of analytical and numerical approaches. A special emphasis will be laid on the description of SU($N$) SPT phases which can be stabilized in these systems. In this respect, as it will be shown here, several distinct SPT phases will be found. In the particular $N=2$ case, i.e. atoms with nuclear spin $I=1/2$, the paradigmatic example of 1D SPT phase, i.e. the spin-1 Haldane phase [@Haldane-PLA-83; @Haldane-PRL-83], will be found for charge, orbital, and nuclear spin degrees of freedom. This phase is a non-degenerate gapful phase with spin-1/2 edges states which are protected by the presence of at least one of the three discrete symmetries: the dihedral group of $\pi$ rotations along the $x,y,z$ axes, time-reversal, and inversion symmetries.[@Pollmann-B-T-O-12] In the general $N$ case, we will show that the spin-$N/2$ Haldane phase emerges only for the orbital degrees of freedom in the phase diagram of the two-orbital SU($N$) model. The resulting phase will be called orbital Haldane (OH) phase and is an SPT phase when $N/2$ is an odd integer. On top of these phases, new 1D SPT phases will be found which stem from the higher SU($N$) continuous symmetry of these alkaline-earth atoms. These phases are the generalization of the Haldane phase for SU($N$) degrees of freedom with $N>2$. As will be argued in the following, these topological phases for general $N$ are protected by the presence of PSU($N$) $=$ SU($N$)/$\mathbb{Z}_N$ symmetry. Even in the absence of the latter symmetry, SU($N$) topological phases may remain topological in the presence of other symmetries. For instance, with the (link-)inversion symmetry present, our SU($N$) topological phase when $N/2$ is odd (i.e., $I=1/2,5/2,9/2,\ldots$ which is directly relevant to ytterbium and strontium atoms) crosses over to the topological Haldane phase. A brief summary of these results has already been given in a recent paper [@Nonne-M-C-L-T-13] where we have found these SU($N$) topological phases for a particular 1D two-orbital SU($N$) model.
The rest of the paper is organized as follows. In Sec. \[sec:models-strong-coupling\], we introduce two different lattice models of two-orbital SU($N$) fermions and discuss their symmetries. Then, strong-coupling analysis is performed which gives some clues about the possible Mott-insulating phases and the global phase structure. We also establish the notations and terminologies used in the following sections, and characterize the main phases that are summarized in Table \[tab:abbreviation\].
The basic properties of the SU($N$) SPT phase identified in the previous section are then discussed in detail in Sec. \[sec:SUN-topological-phase\] paying particular attention to the entanglement properties. The use of non-local (string) order parameters to detect the SU($N$) SPT phases will be discussed, too. In Sec. \[sec:weak-coupling\], a low-energy approach of the two-orbital SU($N$) lattice models is developed to explore the weak-coupling regime of the lattice models. The main results of this section are summarized in the phase diagrams in Sec. \[sec:RG-phase-diag\]. As this section is rather technical, those who are not familiar with field-theory techniques may skip Secs. \[sec:continuum\_description\] and \[sec:RG\_analysis\] for the first reading.
In order to complement the low-energy and the strong-coupling analyses, we present, in Sec. \[sec:DMRG\], our numerical results for $N=2$ and $4$ obtained by the density matrix renormalization group (DMRG) simulations. [@White-92] Readers who want to quickly know the ground-state phase structure may read Sec. \[sec:models-strong-coupling\] first and then proceed to Sec. \[sec:DMRG\]. Finally, our concluding remarks are given in Sec. \[sec:conclusion\] and the paper is supplied with four appendices which provide some technical details and additional information.
Models and their strong-coupling limits {#sec:models-strong-coupling}
=======================================
In this section, we present the lattice models related to the physics of the 1D two-orbital SU($N$) model that we will investigate in this paper. In addition, the different strong-coupling limits of the models will be discussed to reveal the existence of SPT phases in their phase diagrams.
Alkaline-earth Hamiltonian {#sec:Gorshkov-Hamiltonian}
--------------------------
Let us first consider alkaline-earth cold atoms where the atoms can occupy the ground state $g$ and excited metastable state $e$. In this case, four different elastic scattering lengths can be defined due to the two-body collisions between two atoms in the $g$ state ($a_{gg}$), in the $e$ state ($a_{ee}$), and finally between the $g$ and $e$ states ($a^{\pm}_{ge}$). [@Gorshkov-et-al-10] On general grounds, four different interacting coupling constants are then expected from these scattering properties and a rich physics might emerge from this complexity. The model Hamiltonian, derived by Gorshkov [*et al*]{}. [@Gorshkov-et-al-10], which governs the low-energy properties of these atoms loaded into a 1D optical reads as follows ([*$g$-$e$ model*]{}): $$\begin{split}
& \mathcal{H}_{g\text{-}e} =
- \sum_{m=g,e} t_{m} \sum_{i} \sum_{\alpha=1}^{N}
\left(c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1} + \text{H.c.}\right) \\
& -\sum_{m=g,e}\mu^{(m)} \sum_i n_{m,i}
+ \sum_{m=g,e} \frac{U_{mm}}{2} \sum_{i} n_{m,\,i}(n_{m,\,i}-1) \\
& +V \sum_i n_{g,\,i} n_{e,\,i}
+ V_{\text{ex}}^{g\text{-}e} \sum_{i,\alpha \beta}
c_{g\alpha,\,i}^\dag c_{e\beta,\,i}^\dag
c_{g\beta ,\,i} c_{e\alpha,\,i} ,
\end{split}
\label{eqn:Gorshkov-Ham}$$ where the index $\alpha$ labels the nuclear-spin multiplet ($I^{z}=-I,\ldots,+I$, $N=2I+1$, $\alpha=1,\ldots,N$) and the orbital indices $m=g$ and $e$ label the two atomic states ${}^{1}S_{0}$ and ${}^{3}P_{0}$, respectively. The fermionic creation operator with quantum numbers $m,\alpha$ on the site $i$ is denoted by $c_{m\alpha,\,i}^\dag$. The local fermion numbers of the species $m=g,e$ are defined by $$n_{m,i} = \sum_{\alpha=1}^{N}c^{\dagger}_{m\alpha,i}c_{m\alpha,i}
= \sum_{\alpha=1}^{N} n_{m\alpha,i} \; .$$ We also introduce the total fermion number at the site $i$: $$n_{i} = \sum_{m=g,e} n_{m,i} \; .$$
In order to understand the processes contained in this Hamiltonian, it is helpful to represent it as two coupled (single-band) SU($N$) Hubbard chains (see Fig. \[fig:alkaline-2leg\]). On each chain, we have the standard hopping $t$ along each chain (which may be different for $g$ and $e$) and the Hubbard-type interaction $U$, and the two are coupled to each other by the $g$-$e$ contact interaction $V$ and the $g$-$e$ exchange process $V_{\text{ex}}^{g\text{-}e}$. Model (\[eqn:Gorshkov-Ham\]) is invariant under continuous U(1)$_{\text{c}}$ and SU($N$) symmetries: $$c_{m\alpha,\,i} \mapsto {\mathrm{e}}^{i \theta} c_{m\alpha,\,i} \, , \;
c_{m\alpha,\,i} \mapsto \sum_{\beta} \mathcal{U}_{\alpha \beta} c_{m\beta,\,i} ,
\label{eqn:U(N)}$$ with $\mathcal{U}$ being an SU($N$) matrix. The two transformations (\[eqn:U(N)\]) respectively refer to the conservation of the total number of atoms and the SU($N$) symmetry in the nuclear-spin sector. On top of these obvious symmetries, the Hamiltonian is also invariant under $$c_{g\alpha,\,i} \mapsto {\mathrm{e}}^{i \theta_{\text{o}}} c_{g\alpha,\,i} \, , \;
c_{e\alpha,\,i} \mapsto {\mathrm{e}}^{-i \theta_{\text{o}}} c_{e\alpha,\,i} \; .
\label{eqn:orbital-U1}$$ This is a consequence of the fact that the total fermion numbers for $g$ and $e$ are conserved [*separately*]{}.[^1]
![(Color online) The two-leg ladder representation of the $g\text{-}e$ model . Two single-band SU($N$) Hubbard chains are coupled to each other only by the inter-chain particle exchange ($V_{\text{ex}}^{g\text{-}e}$) and the interchain density-density interaction ($V$). Note that splitting of a single physical chain into two is fictitious. \[fig:alkaline-2leg\]](alkaline-2leg)
In the case of SU(2), it is well-known that the orbital ($g$, $e$) exchange process can be written in the form of the Hund coupling. Let us write down such expressions in two ways. First, we introduce the second-quantized SU($N$) generators of each orbital $$\hat{S}_{m,i}^{A} = c^{\dagger}_{m\alpha,i}(S^{A})_{\alpha\beta}c_{m\beta,i} \quad (m=g,e, \;
A=1,\ldots, N^{2}-1),
\label{eqn:second-quant-SUN}$$ as well as the [*orbital pseudo spin*]{} $T^{a}_{i}$ ($a=x,y,z$): $$T_i^a = \frac{1}{2} c_{m \alpha,\,i}^\dag \sigma^a_{m n} c_{n \alpha,\,i}
= \sum_{\alpha=1}^{N}T_{\alpha,i}^{a}
\quad (m,n=g,e) \; ,
\label{eqn:1pseudospinoperator}$$ where a summation over repeated indices is implied in the following and $\sigma^a$ denotes the Pauli matrices. If we normalize the SU($N$) generators $S^{A}$ as[^2] $$\text{Tr}\,(S^{A}S^{B}) = \delta_{AB} \; ,$$ the generators $S^{A}$ satisfy the following identity: $$\sum_{A=1}^{N^{2}-1} (S^A)_{\alpha\beta} (S^A)_{\gamma \delta} = \left(
\delta_{\alpha \delta} \delta_{\beta \gamma} - \frac{1}{N}
\delta_{\alpha \beta} \delta_{\gamma \delta} \right) \; .
\label{eqn:SASA-tensor}$$ The above U(1)$_{\text{o}}$ transformation amounts to the rotation along the $z$-axis: $$\begin{split}
& T_i^{\pm} \mapsto {\mathrm{e}}^{\mp2i \theta_{\text{o}}} T_i^{\pm} \\
& T_i^{z} \mapsto T_i^{z}
\end{split}$$ generated by $$T_i^z = \frac{1}{2}(n_{g,\,i}-n_{e,\,i}) \; .$$
Then, it is straightforward to show that the orbital-exchange ($g \leftrightarrow e$) can be written as the Hund coupling for the SU($N$) ‘spins’ or that for the orbital pseudo spins: $$\begin{split}
& \sum_{i}
c_{g\alpha,\,i}^\dag c_{e\beta,\,i}^\dag
c_{g\beta ,\,i} c_{e\alpha,\,i} \\
& = - \sum_{i}
\hat{S}_{g,i}^{A}\hat{S}_{e,i}^{A}
- \frac{1}{N} \sum_{i}n_{g,i}n_{e,i} \\
& = \sum_{i}( {\mathrm{\bf T}}_{i})^{2}
- \frac{1}{4}\sum_{i} n_{m,i}(n_{m,i}-1)
\\
& \phantom{=}
-\frac{3}{4} \sum_{i} n_{i}
+ \frac{1}{2}\sum_{i}n_{g,i}n_{e,i} \; .
\end{split}
\label{eqn:exchange-to-Hund}$$ The fermionic anti-commutation is crucial in obtaining the two opposite signs in front of the Hund couplings. The above expression enables us to rewrite the original alkaline-earth Hamiltonian in two different ways
$$\begin{split}
\mathcal{H}_{g\text{-}e} =&
- \sum_{i}\sum_{m=g,e} t_{m}
\left(c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1} + \text{H.c.}\right)
- \sum_i \sum_{m=g,e} \mu^{(m)} n_{m,i} \\
& \quad +\sum_{i}\sum_{m=g,e} \frac{U_{mm}}{2} n_{m,\,i}(n_{m,\,i}-1)
+\left(V - \frac{1}{N}V_{\text{ex}}^{g\text{-}e}\right) \sum_i n_{g,\,i} n_{e,\,i}
- V_{\text{ex}}^{g\text{-}e}
\sum_{i}
\hat{S}_{g,i}^{A}\hat{S}_{e,i}^{A}
\\
=&
- \sum_{i}\sum_{m=g,e} t_{m}
\left(c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1} + \text{H.c.}\right)
- \sum_i \sum_{m=g,e}\left(\mu^{(m)} + \frac{3}{4} V_{\text{ex}}^{g\text{-}e}\right) n_{m,i} \\
& \quad
+\sum_{i}\sum_{m=g,e} \frac{U_{m m} - V_{\text{ex}}^{g\text{-}e}/2}{2} n_{m,\,i}(n_{m,\,i}-1)
+\left(V + V_{\text{ex}}^{g\text{-}e}/2 \right) \sum_i n_{g,\,i} n_{e,\,i}
+ V_{\text{ex}}^{g\text{-}e} \sum_{i} ( {\mathrm{\bf T}}_{i})^{2}
\; .
\end{split}
\label{eqn:Gorshkov-Ham-Hund}$$
From this, one readily sees that positive (negative) $V_{\text{ex}}^{g\text{-}e}$ tends to quench (maximize) orbital pseudo spin ${\mathrm{\bf T}}$ and maximize (quench) the SU($N$) spin. This dual nature of the orbital and SU($N$) is the key to understand the global structure of the phase diagram.
Using the orbital pseudo spin $T^{a}$, we can rewrite the original $g$-$e$ Hamiltonian as $$\begin{split}
\mathcal{H}_{g\text{-}e} = & - \, \sum_{i} \sum_{m=g,e}
t_{m} \left( c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1}+ \text{H.c.} \right) \\
& -\frac{1}{2}\left(\mu _e + \mu _g\right) \sum_{i} n_{i} - \left(\mu _g -\mu _e\right)\sum_{i} T^{z}_{i} \\
& +\frac{U}{2} \sum_i n_i^2 + U_{\text{diff}} \sum_i T^{z}_{i} n_{i} \\
& +J \sum_i \left\{ (T_i^x)^2 + (T_i^y)^2\right\} + J_z \sum_i (T_i^z)^2 ,
\end{split}
\label{alkamodel-2b}$$ with $$\begin{split}
& U=\frac{1}{4} (U_{g g}+ U_{ee}
+2 V), \;\;
U_{\text{diff}} = \frac{1}{2}(U_{gg} - U_{ee}) , \\
& J=V^{g\text{-}e}_{\text{ex}} , \;\;
J_{z} = \frac{1}{2}(U_{e e} + U_{g g} -2 V), \\
& \mu_{g} = \frac{1}{2} (2 \mu^{(g)}+ U_{g g} + V^{g\text{-}e}_{\text{ex}}), \\
& \mu_{e} = \frac{1}{2} (2 \mu^{(e)}+ U_{e e} + V^{g\text{-}e}_{\text{ex}})
\; .
\end{split}
\label{eqn:HG-parameters}$$ The site-local part of the above Hamiltonian gives the starting point for the strong-coupling expansion: $$\begin{split}
\mathcal{H}_{\text{atomic}} = &
-\frac{1}{2}\left(\mu _e + \mu _g\right) \sum_{i} n_{i} - \left(\mu _g -\mu _e\right)\sum_{i} T^{z}_{i} \\
& +\frac{U}{2} \sum_i n_i^2
+ U_{\text{diff}} \sum_i T^{z}_{i} n_{i} \\
& +J \sum_i \left\{ (T_i^x)^2 + (T_i^y)^2\right\} + J_z \sum_i (T_i^z)^2
\; .
\end{split}
\label{eqn:atomic-limit-Ham}$$ Since the model contains many coupling constants, it is highly desirable to consider a simpler effective Hamiltonian which encodes the most interesting quantum phases of the problem. In this respect, for the DMRG calculations of Sec. \[sec:DMRG\], we will set $t_g = t_e=t$, $U_{g g} = U_{ee} =U_{mm}$, and $\mu_{g}=\mu_{e}$ to get the following Hamiltonian ([*generalized Hund model*]{})[@Nonne-B-C-L-10]: $$\begin{split}
\mathcal{H}_{\text{Hund}} = & -t \, \sum_{i}
\left( c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1}+ \text{H.c.} \right) \\
& -\mu\sum_i n_i
+\frac{U}{2} \sum_i n_i^2 \\
& +J \sum_i \left\{ (T_i^x)^2 + (T_i^y)^2\right\} + J_z \sum_i (T_i^z)^2 \; .
\end{split}
\label{alkaourmodel}$$ Now, the equivalence mapping between the models and reads as $$\begin{split}
&J=V_{\text{ex}}^{g\text{-}e}, \quad J_z=U_{mm} - V , \\
&U=\frac{U_{mm}+V}{2}, \quad
\mu=\frac{U_{mm}+V^{g\text{-}e}_{\text{ex}}}{2}+\mu_{g} \; .
\end{split}
\label{eqn:Gorshkov-to-Hund}$$ It is obvious that the first three terms in Eq. are U($2N$)-invariant and the remaining orbital part ($J$ and $J_{z}$) breaks it down to $$\begin{split}
\text{U($2N$)} =
\text{U(1)}_{\text{c}}{\times}\text{SU($2N$)} &\xrightarrow{J= J_{z}(\neq 0)}
\text{U(1)}_{\text{c}}{\times}\text{SU($N$)}_{\text{s}}{\times}\text{SU(2)}_{\text{o}} \\
& \xrightarrow{J\neq J_{z}} \text{U(1)}_{\text{c}}{\times}\text{SU($N$)}_{\text{s}}{\times}\text{U(1)}_{\text{o}} \;.
\end{split}
\label{eqn:symmetry-change}$$ Therefore, the generic continuous symmetry of this model is $\text{U}(1)_{\text{c}}\times\text{SU}(N)_{\text{s}}\times \text{U}(1)_{\text{o}}$. Physically, the orbital-$\text{U}(1)_{\text{o}}$ symmetry of $\mathcal{H}_{g\text{-}e}$ may be traced back to the vanishingly weak $g \leftrightarrow e$ transition.[@Gorshkov-et-al-10]
p-band Hamiltonian {#sec:p-band-definition}
------------------
There is yet another way to realize the two orbitals using a simple setting. Let us consider a one-dimensional optical lattice (running in the $z$-direction) with moderate strength of (harmonic) confining potential $V_{\perp}(x,y)=\frac{1}{2}m\omega_{xy}^{2}(x^{2}+y^{2})$ in the direction (i.e. $xy$) perpendicular to the chain. Then, the single-particle part of the Hamiltonian reads as $$\begin{split}
\mathcal{H}_{0} &= \left\{
- \frac{\hbar^{2}}{2m}\partial_{z}^{2} + V_{\text{per}}(z)
\right\} +
\left\{
- \frac{\hbar^{2}}{2m}\left( \partial_{x}^{2} + \partial_{y}^{2} \right)
+ V_{\perp}(x,y)
\right\} \\
& \equiv \mathcal{H}_{\perp}(x,y) + \mathcal{H}_{/\!/}(z)
\; ,
\end{split}
\label{eqn:single-particle-Ham}$$ where $V_{\text{per}}(z)$ is a periodic potential that introduces a lattice structure in the chain (i.e. $z$) direction. If the chain is infinite in the $z$-direction, we can assume the Bloch function in the following form: $$\psi^{(n)}_{n_x,n_y,k_z}(x,y,z) = \phi_{n_x,n_y}(x,y) \varphi^{(n)}_{k_z}(z) \; .$$ The two functions $\varphi^{(n)}_{k_z}(z)$ and $\phi_{n_x,n_y}(x,y)$ respectively satisfy
$$\mathcal{H}_{/\!/}(z) \varphi^{(n)}_{k_z}(z) = \epsilon^{(n)}(k_z) \varphi^{(n)}_{k_z}(z)$$
and $$\mathcal{H}_{\perp}(x,y) \phi_{n_x,n_y}(x,y) = \epsilon_{n_x,n_y} \phi_{n_x,n_y}(x,y) \; .
\label{eqn:Schroedinger-transverse-part}$$
Since the second equation is the Schrödinger equation of the two-dimensional harmonic oscillator, the eigenvalues $\epsilon_{n_x,n_y}$ are given by $$\epsilon_{n_x,n_y} = \left( n_x + n_y + 1 \right) \hbar \omega_{xy} \quad
(n_x, n_y = 0,1,2, \ldots ) \; .$$ The full spectrum of $\mathcal{H}_{0}$ is given by $$E^{(n)}_{n_x,n_y}(k_z) = \epsilon^{(n)}(k_z) + \epsilon_{n_x,n_y}$$ and each Bloch band specified by $n$ splits into the sub-bands labeled by $(n_x,n_y)$. We call the subbands with $(n_x,n_y)=(0,0)$, $(1,0)$, and $(0,1)$ as ‘$s$’, ‘$p_x$’ and ‘$p_y$’, respectively. The shape of the bands depends only on the band index $n$ and the set of integers $(n_x,n_y)$ determines the $k_z$-independent splitting of the sub-bands.
Now let us consider the situation where only the $n=0$ bands are occupied, and, among them, the lowest one (the $s$-band) is completely filled. Then, it is legitimate to keep only the next two bands $p_x$ and $p_y$ in the effective Hamiltonian.[@Kobayashi-O-O-Y-M-12; @Kobayashi-O-O-Y-M-14] To derive a Hubbard-type Hamiltonian, we follow the standard strategy[@Jaksch-Z-05] and move from the Bloch basis $\psi^{(n)}_{n_x,n_y,k_z}(x,y,z)$ to the Wannier basis $$W^{(n)}_{n_x,n_y;R}(x,y,z) \equiv
\frac{1}{\sqrt{N_{\text{cell}}}} \phi_{n_x,n_y}(x,y) \sum_{k_z}
{\mathrm{e}}^{- i k_z R} \varphi^{(n)}_{k_z}(z)
\label{eqn:Wannier-pxpy}$$ ($R$ labels the center of the Wannier function and $N_{\text{cell}}$ is then number of unit cells). Expanding the creation/annihilation operators in terms of the Wannier basis and keeping only the terms with $n=0$ and $(n_x,n_y)=(1,0)$ or $(0,1)$, we obtain the following Hamiltonian (see Appendix \[sec:p-band-hamiltonian\]) $$\begin{split}
& \mathcal{H}_{p\text{-band}} \\
& = - t \sum_{i}
( c_{m\alpha,i}^{\dagger}c_{m\alpha,i+1} + \text{H.c.} ) \\
& + \sum_{i} \sum_{m=p_x,p_y}
(\epsilon_{m} - t_{0}) \, n_{m,i} \\
&+ \frac{1}{2}U_{1} \sum_{i}
n_{m,i}(n_{m,i}-1) + U_{2}\, \sum_{i}n_{p_x,i}n_{p_y,i} \\
& + U_{2} \sum_{i}
c_{p_x\alpha,i}^{\dagger} c_{p_y\beta,i}^{\dagger}c_{p_x\beta,i}c_{p_y\alpha,i} \\
& + U_{2} \sum_{i}\left\{ (T^{x}_{i})^{2} - (T^{y}_{i})^{2} \right\} \; .
\end{split}
\label{eqn:p-band}$$ In the above, we have introduced a short-hand notation $m=p_{x},p_{y}$ with $p_{x}=(n_x,n_y)=(1,0)$ and $p_{y}=(n_x,n_y)=(0,1)$. The last term comes from the pair-hopping between the two orbitals (see Appendix \[sec:p-band-hamiltonian\]) and breaks U(1)$_{\text{o}}$-symmetry in general. Since the Wannier functions are real and the two orbitals $W_{p_x/p_y;R}^{(0)}({\mathrm{\bf r}})$ are related by $C_4$-symmetry, there are only two independent couplings $U_1$ and $U_2$ \[see Eq. \]. In fact, due to the axial symmetry of the potential $V_{\perp}(x,y)$, even the ratio $U_{1}=3 U_{2}$ is fixed and we are left with a single coupling constant.
Except for the last term, $\mathcal{H}_{p\text{-band}}$ coincides with the Hamiltonian after the identification $$\begin{split}
& U=\frac{1}{2}(U_1+U_2) \, , \; U_{\text{diff}}=0 \, , \; J=U_2 \, , \; J_{z}=U_1 - U_2 \\
& \mu = -(\epsilon_{m}-t_{0})+ \frac{1}{2}(U_1 + U_2) \; .
\end{split}$$ Incorporating the last term, we obtain the following (orbital) anisotropic model $$\begin{split}
\mathcal{H}_{p\text{-band}} = & -t \, \sum_{i}
\left( c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1}+ \text{H.c.} \right) \\
& -\mu\sum_i n_i
+\frac{1}{4}(U_1 + U_2) \sum_i n_i^2 \\
& +\sum_i \left\{ 2U_2 (T_i^x)^2 + (U_1 - U_2)(T_i^z)^2\right\} \; .
\end{split}
\label{eqn:p-band-simple}$$ One may think that the last term breaks $\text{U(1)}_{\text{o}}$. However, as $U_{1}=3 U_{2}$ for [*any*]{} axially-symmetric $V_{\perp}(x,y)$, it has in fact a [*hidden*]{} U(1)$_{\text{o}}$-symmetry: $2U_2\left\{ (T_j^x)^2 +(T_j^z)^2\right\}$ and $\mathcal{H}_{p\text{-band}}$ reduces to $\mathcal{H}_{\text{Hund}}$ \[Eq. \] after the due redefinition of $\mathrm{\bf T}$.[^3] Higher continuous symmetries may also appear in model (\[eqn:p-band-simple\]) when $U_2 =0$ since it decouples into two independent U($N$) Hubbard chains, as it can be easily seen from Eq. (\[eqn:p-band\]). Moreover, along the line $U_1 =U_2$, the $p$-band model (\[eqn:p-band-simple\]) is equivalent to the $U_2 =0$ case after a redefinition of $\mathrm{\bf T}$. Finally, as we will see in the next section, the $p$-band model for $N=2$ at half-filling enjoys an enlarged SU(2) $\times$ SU(2) $\sim$ SO(4) symmetry for all $U_1,U_2$ which stems from an additional SU(2) symmetry for the charge degrees of freedom at half-filling.[@Kobayashi-O-O-Y-M-14]
![(Color online) The two-leg ladder representation of the $p$-band model . On top of the interactions included already in the $g$-$e$ model, pair-hopping processes between the two orbitals are allowed. \[fig:p-band-2leg\]](p-band-2leg)
The $p$-band model is convenient since the axial symmetry guarantees that the parameters are fully symmetric for the two orbitals $p_x$ and $p_y$. However, the same symmetry locks the ratio $U_{1}/U_{2}(=3)$ and we cannot control it as far as $V_{\perp}(x,y)$ is axially symmetric. One simplest way of changing the ratio is to break the axial symmetry and consider the following anharmonic potential: $$V_{\perp}(x,y) = \frac{1}{2}m\omega_{xy}^{2} (x^{2}+y^{2})
+ \frac{1}{2}\beta (x^{4}+y^{4}) \quad (\beta \geq 0) \; .
\label{eqn:anharmonic-potential}$$ In Fig. \[fig:U1overU2\], we plot the ratio $U_{1}/U_{2}$ as a function of anharmonicity $\beta$. Clearly, the ratio calculated using Eqs. and deviates from 3 with increasing $\beta$. In that case ($U_1 < 3 U_2$), the original anisotropic model should be considered.
![The ratio $U_{1}/U_{2}$ for anharmonic potential obtained by solving the Schrödinger equation numerically. \[fig:U1overU2\]](U1overU2)
Symmetries {#sec:symmetries}
----------
The different models that we have introduced in the previous section enjoys generically an $\text{U}(1)_{\text{c}}\times\text{SU}(N)_{\text{s}}\times \text{U}(1)_{\text{o}}$ continuous symmetry or an $\text{U}(1)_{\text{c}}\times\text{SU}(N)_{\text{s}}$ symmetry for the $p$-band model . On top of these continuous symmetries, the models display hidden discrete symmetries which are very useful to map out their global zero-temperature phase diagrams.
### Spin-charge interchange {#sec:spin-charge-interchange}
The first transformation is a direct generalization of the Shiba transformation[@Shiba-72; @Emery-76] for the usual Hubbard model and is defined [*only*]{} for $N=2$: $$\begin{split}
& c_{m\uparrow,i} \mapsto \tilde{c}_{m\uparrow,i} \\
& c_{m\downarrow,i} \mapsto (-1)^{i} \tilde{c}^{\dagger}_{m\downarrow,i}
\quad (m=g,e \text{ or } p_x,p_y) \; .
\end{split}
\label{eqn:Shiba-tr-SU2-2band}$$ It is easy to show that it interchanges spin and charge \[see Eq. \]: $$S^{A}_{m,i} \; \leftrightarrow \; K^{A}_{m,i} \quad (A=x,y,z) \; ,
\label{eqn:Shiba-S-to-K}$$ where $K^{A}_{m,i}$ are defined as $$\begin{split}
& K^{+}_{m,i} \equiv (-1)^{i} c_{m\uparrow,i}^{\dagger}c_{m\downarrow,i}^{\dagger} , \quad
K^{-}_{m,i} \equiv (-1)^{i} c_{m\downarrow,i} c_{m\uparrow,i} , \\
& K_{m,i}^{z} \equiv \frac{1}{2}(n_{m\uparrow,i} + n_{m\downarrow,i} -1 )
= \frac{1}{2}(n_{m,i}-1) \; .
\label{chargepseudospin}
\end{split}$$ The latter operator carries charge and is a SU(2) spin-singlet. It generalizes the $\eta$-pairing operator introduced by Yang for the half-filled spin-$\frac{1}{2}$ Hubbard model [@yang89] or by Anderson in his study of the BCS superconductivity [@anderson58].
Now let us consider how the transformation affects the fermion Hamiltonians $\mathcal{H}_{g\text{-}e}$ \[Eq. \] and $\mathcal{H}_{p\text{-band}}$ \[Eq. \]. The first three terms of the alkaline-earth Hamiltonian $\mathcal{H}_{g\text{-}e}$ \[Eq. \] do not change their forms under the transformation , while the last two are asymmetric in ${\mathrm{\bf S}}_{m,i}$ and $\mathbf{K}_{m,i}$. Hence the $g\text{-}e$ Hamiltonian $\mathcal{H}_{g\text{-}e}$ does not preserve its form under ${\mathrm{\bf S}}_{m,i} \leftrightarrow \mathbf{K}_{m,i}$.
On the other hand, the $p$-band Hamiltonian, written in terms of ${\mathrm{\bf S}}_{m,i}$ and $\mathbf{K}_{m,i}$, $$\begin{split}
\mathcal{H}_{p\text{-band}} = &
- t \sum_{i}
( c_{m\alpha,i}^{\dagger}c_{m\alpha,i+1} + \text{H.c.} ) \\
&+ U_{1} \sum_{i}
(n_{m\uparrow,i}-1/2)(n_{m\downarrow,i}-1/2) \\
& - 2 U_{2}\sum_{i} {\mathrm{\bf S}}_{p_x,i}{\cdot} {\mathrm{\bf S}}_{p_y,i}
+ 2U_{2} \sum_{i} \mathbf{K}_{p_x,i}{\cdot} \mathbf{K}_{p_y,i} \; ,
\end{split}
\label{pbandSO4}$$ preserves its form and the Shiba transformation changes the coupling constants as $$(U_{1}, \, U_{2}) \rightarrow (-U_{1}, \, -U_{2}) \; .
\label{eqn:Shiba-tr-SU2-U1U2}$$
The expression (\[pbandSO4\]) reveals the hidden symmetry of the half-filled $p$-band model for $N=2$. On top of the SU(2) symmetry for the nuclear spins, which is generated by $\sum_{i,m} {\mathrm{\bf S}}_{m,i}$, the $p$-band Hamiltonian (\[pbandSO4\]) enjoys a second independent SU(2) symmetry related to the (charge) pseudo spin operator (\[chargepseudospin\]): $$\left[ \mathcal{H}_{p\text{-band}} , \sum_{i,m} \mathbf{K}_{m,i} \right] = 0.$$ The continuous symmetry group of the $N=2$ half-filled $p$-band model is therefore: SU(2) $\times$ SU(2) $\sim$ SO(4) for all $U_1,U_2$, i.e., without any fine-tuning. In this respect, the latter model shares the same continuous symmetry group as the half-filled spin-1/2 Hubbard chain [@yangZ89; @zhang91] but, as we will see later, the physics is strongly different.
### orbital-charge interchange {#sec:orbita-charge-interchange}
For general $N$, we can think of another ‘Shiba’ transformation: $$\begin{split}
& c_{g\alpha,i} \mapsto \tilde{c}_{g\alpha,i} \\
& c_{e\alpha,i} \mapsto (-1)^{i} \tilde{c}^{\dagger}_{e\alpha,i}
\quad (\alpha=1,\ldots, N) \; ,
\end{split}
\label{eqn:Shiba-tr-SUN}$$ which interchanges the orbital pseudo spin $\mathbf{T}_{i}$ and another charge-SU(2) $\boldsymbol{\mathcal{K}}_{i}$. Now the charge-SU(2) is generated by the following orbital-singlet operators $$\begin{split}
& \mathcal{K}_{i}^{+} \equiv (-1)^{i} c_{g\alpha,i}^{\dagger}c_{e\alpha,i}^{\dagger} \; , \quad
\mathcal{K}_{i}^{-} \equiv (-1)^{i} c_{e\alpha,i}c_{g\alpha,i} \\
& \mathcal{K}_{i}^{z} \equiv \frac{1}{2}(n_{g,i}+n_{e,i} -N)
= \frac{1}{2}( n_{i} -N) \; .
\end{split}$$
The transformation changes the $g\text{-}e$ Hamiltonian by flipping the sign of $\left(V - V_{\text{ex}}^{g\text{-}e}/N \right)$ and replacing $S^{A}_{e,i}$ with the generators of the conjugate representation. Therefore, one sees that only when $J(=V^{g\text{-}e}_{\text{ex}})=0$ the $g\text{-}e$ Hamiltonian $\mathcal{H}_{g\text{-}e}$ preserves its form after $$V \mapsto - V \quad
(\text{or } J_{z} \leftrightarrow 2U )\; .
\label{eqn:Gorshkov-VG-flip}$$ We will come back to this point later in Sec. \[sec:N4-Gorshkov\] in the discussion of the numerical phase diagram of the $N=4$ $g\text{-}e$ model.
The case $N=2$ is special since any SU(2) representations are self-conjugate. In fact, when $N=2$, the transformation , supplemented by the $\pi$-rotation along the $y$-axis in the SU(2) space ($c_{e\uparrow,i} \mapsto -c_{e\downarrow,i}$, $c_{e\downarrow,i} \mapsto c_{e\uparrow,i}$), preserves the form of the Hamiltonian after the mapping $$\begin{split}
& V - \frac{1}{2}V_{\text{ex}}^{g\text{-}e} \rightarrow
- \left(V - \frac{1}{2}V_{\text{ex}}^{g\text{-}e}\right) \quad \left(
\text{or } V \rightarrow -V + V_{\text{ex}}^{g\text{-}e} \right) \\
& V_{\text{ex}}^{g\text{-}e} \rightarrow V_{\text{ex}}^{g\text{-}e} \, , \;\;
U_{m m} \rightarrow U_{m m} \; .
\end{split}
\label{eqn:N2-Gorshkov-orbital-charge}$$ Due to the orbital anisotropy $\left\{ (T^{x}_{j})^{2} - (T^{y}_{j})^{2} \right\}$ in $\mathcal{H}_{p\text{-band}}$ \[the last term Eq. \], the $p$-band Hamiltonian in general does not preserve its form under the orbital-charge interchange . When $U_2=0$, the model is U(1)-orbital symmetric and is invariant (self-dual) under . A summary of the effect of the two Shiba transformations on the two models is summarized in Tables \[tab:Gorshkov\] and \[tab:p-band\].
[lc]{} transformation & mapping\
spin-charge \[Eq. \] & not defined\
orbital-charge \[Eq. \] &
------------ -----------------------------------------------------
$N=2$: $V \rightarrow -V + V_{\text{ex}}^{g\text{-}e}$
$N\geq 3$: $V \rightarrow -V$ ($V_{\text{ex}}^{g\text{-}e}=0$)
------------ -----------------------------------------------------
: \[tab:Gorshkov\] Two Shiba transformations and $g$-$e$ Hamiltonian \[Eq. \].
\
[lc]{} transformation & mapping\
spin-charge \[Eq. \] &
-------- ---------------------------------
$N=2$: $U_{1,2} \rightarrow - U_{1,2}$
-------- ---------------------------------
: \[tab:p-band\]Two Shiba transformations and $p$-band Hamiltonian \[Eq. \]. Orbital-charge interchange exists only when $U_2=0$ and then the Hamiltonian is kept invariant.
\
orbital-charge \[Eq. \] &
---------------- ------------------------------
$N$ arbitrary: invariant (only for $U_2=0$)
---------------- ------------------------------
: \[tab:p-band\]Two Shiba transformations and $p$-band Hamiltonian \[Eq. \]. Orbital-charge interchange exists only when $U_2=0$ and then the Hamiltonian is kept invariant.
\
Strong-coupling limits {#sec:strong-coupling}
----------------------
Useful insight into the global structure of the phase diagram may be obtained by investigating the strong-coupling limit where the hopping $t_{(g,e)}$ are very small. Then, the starting point is the atomic-limit Hamiltonian . In the following, we assume that $N=2I +1$ is even since the nuclear spin $I$ is half-odd-integer for alkaline-earth fermions. The dominant phases found in the strong-coupling analysis are summarized in Table \[tab:abbreviation\].
### Positive-$J$ {#sec:strong-coupling-positive-J}
First, we assume that $U$ and the chemical potential $\mu_{g}+\mu_{e}$ \[see Eq. \] are tuned in such a way that the fermion number at each site is $n_{i}=N$. Then, the remaining ${\mathrm{\bf T}}$-dependent terms in determine the optimal orbital and SU($N$) states. From Eq. , we see that for large positive $J(=V_{\text{ex}}^{g\text{-}e})$ the orbital pseudo spin ${\mathrm{\bf T}}$ at each site tends to be quenched thereby maximizing the SU($N$) spin as $$\text{\scriptsize $N/2$} \left\{
\yng(2,2,2)
\right. \quad (N=\text{even}) \; .
\label{selfrep}$$ When considering second-order perturbation, it is convenient to view our system as a two-leg ladder of SU($N$) fermions \[see Fig. \[fig:alkaline-2leg\]\]. The resulting effective Hamiltonian reads then as follows $$\mathcal{H}_{\text{SU(N)}} = J_{\text{s}}
\sum_{A=1}^{N^{2}-1} \mathcal{S}_{i}^{A}\mathcal{S}_{i+1}^{A} + \text{const.} \; ,
\label{eqn:2nd-order-effective-Ham-Gorshkov}$$ where the exchange coupling $J_{\text{s}}$ is $N$-independent $$J_{\text{s}} \equiv
\frac{1}{2} \left\{
\frac{(t^{(g)})^{2}}{U+U_{\text{diff}}+J+\frac{J_z}{2}}
+ \frac{(t^{(e)})^{2}}{U-U_{\text{diff}}+J+\frac{J_z}{2}}
\right\} \; .
\label{eqn:exch-coupling-Gorshkov}$$
In the case of $\mathcal{H}_{\text{$p$-band}}$, $T^{z}$ is no longer conserved and we cannot use the same argument as above. However, we found that when $U_{1}>U_{2}(>0)$, the lowest-energy state has $T=0$ enabling us to follow exactly the same steps and obtain $$\mathcal{H}_{\text{SU(N)}} = \frac{t^{2}}{U_1 +U_2}
\sum_{A=1}^{N^{2}-1} \mathcal{S}_{i}^{A}\mathcal{S}_{i+1}^{A} + \text{const.} \; .
\label{eqn:2nd-order-effective-Ham-p-band}$$
One observes that models and take the form of an SU($N$) spin chain in the self-conjugate representation (\[selfrep\]) at each site and is not solvable. The physical properties of that model are unknown for general $N$. In the special $N=2$ case where the model reduces to the SU(2) spin-1 Heisenberg chain, it is well-known that the Haldane phase[@Haldane-PLA-83; @Haldane-PRL-83] is formed by the nuclear spins. The resulting spin Haldane (SH) phase for $N=2$ is depicted in Fig. \[fig:4-MottPhases-SU2\](a). Using the spin-charge interchange transformation , one concludes, for $N=2$, the existence of a charge Haldane (CH) phase [@Nonne-L-C-R-B-10] in the $p$-band model for $U_2 <0$ which is illustrated in Fig. \[fig:4-MottPhases-SU2\](c). We will come back to this point later in Sec. \[sec:strong-coupling-negative-J\].
When $N>2$, the situation is unclear and a non-degenerate gapful phase is expected from the large-$N$ analysis of Refs. . We will determine the nature of the underlying phase in the next section.
![(Color online) Four translationally invariant Mott states for $N=2$: (a) spin Haldane (SH), (b) orbital Haldane (OH), (c) charge Haldane (CH), and (d) rung-singlet (RS) phases (see also Appendix \[sec:N2-p-band-continuum\]). Singlet bonds formed between spins (orbital pseudo spins) are shown by thick solid (dashed) lines \[singlet bonds are not shown in (c)\]. Dashed ovals (rectangles) denote spin-singlets (triplets). \[fig:4-MottPhases-SU2\]](4-MottPhases-SU2)
### $J=0$ {#sec:strong-coupling-J-0}
Another interesting line is the generalized Hund model (\[alkaourmodel\]) with $J=J_z=0$ which becomes equivalent to the U($2N$) Hubbard model. In the strong-coupling limit with $U>0$, the lowest-energy states correspond to representations of the SU($2N$) group which transform in the antisymmetric self-conjugate representations of SU($2N$), described by a Young diagram with one column of $N$ boxes. The model is then equivalent to an SU($2N$) Heisenberg spin chain where the spin operators belong to the antisymmetric self-conjugate representation of SU($2N$). The latter model is known for all $N$ to have a dimerized or spin-Peierls (SP) twofold-degenerate ground state, where dimers are formed between two neighboring sites [@Affleck-M-88; @*Marston-A-89; @Affleck-88; @OnufrievM99; @Assaraf-A-B-C-L-04; @Nonne-L-C-R-B-11].
In the attractive case ($U<0$), the lowest-energy states are the empty and the fully occupied state, which is an SU($2N$) singlet. At second order of perturbation theory, the effective model reads as follows: [@Zhao-U-W-06; @*Zhao-U-W-07] $${\cal H}_{\textrm{eff}} = \frac{t^2}{N(2N-1)|U|} \sum_i \left(n_i n_{i+1} - N n_i\right),$$ The first term introduces an effective repulsion interaction between nearest neighbor sites. This leads to a two-fold degenerate fully-gapped charge-density wave (CDW) where empty ($n_{i}=0$) and fully occupied ($n_{i}=2N$) states alternate. The resulting CDW phase for $N=2$ is depicted in Fig. \[fig:2-DWPhases-SU2\](a).
![(Color online) Two density-wave states for $N=2$. In-phase and out-of-phase combinations of two density waves in $g$ and $e$ orbitals respectively form (a) CDW and (b) ODW. \[fig:2-DWPhases-SU2\]](2-DWPhases-SU2)
### Negative-$J$ {#sec:strong-coupling-negative-J}
Now let us discuss the case with $J<0$ (and $U>0$). For small enough anisotropies $|J-J_{z}|$, $|\mu_{g}-\mu_{e}|$, the atomic-limit ground states are obtained by applying the lowering operators $T^{-}_{i}$ onto the reference state $$c^{\dagger}_{g1,i}c^{\dagger}_{g2,i}\cdots c^{\dagger}_{gN,i}
|0\rangle \; .$$ To carry out the second-order perturbation, it is convenient to regard the model $\mathcal{H}_{g\text{-}e}$ as the $N$ coupled Hubbard-type chains, along which the $g$ and $e$ fermions move (see Fig. \[fig:alkaline-Nleg\]). Since each “site” of the chains is occupied by exactly one fermion in the ground states, it is clear that the two hopping processes must occur on the same chain. Therefore, the calculation is similar to that in the usual single-band Hubbard chain (except that we have to symmetrize the $N$ resultant $T=1/2$ chains at the last stage) and we finally obtain the pseudo spin $T=N/2$ Hamiltonian $$\begin{split}
\mathcal{H}_{\text{orb}} = &\sum_{i}
\biggl\{
\mathcal{J}_{xy} \left(
T^{x}_{i} T^{x}_{i+1} + T^{y}_{i} T^{y}_{i+1}
\right)
+ \mathcal{J}_{z} T^{z}_{i} T^{z}_{i+1} \\
& -(J- J_z) (T_{i}^z)^2 \biggr\} \\
& + \sum_{i} \left\{
N U_{\text{diff}}
- \left(\mu _g -\mu _e\right)
\right\} T_{i}^{z}
+ \text{const.}
\end{split}
\label{eqn:eff-Ham-orbital-Haldane}$$ with the following exchange couplings
\[eq:eff\_coupling\_Hund\] $$\begin{aligned}
& \mathcal{J}_{xy} \equiv \frac{4 t_{g}t_{e}}{N\left\{U -J \left( N+\frac{1}{2} \right)\right\}} \\
& \mathcal{J}_{z} \equiv
\frac{2 \left\{ t_{g}^{2} + t_{e}^{2}\right\}}{N \left\{U -J \left( N+\frac{1}{2} \right)\right\}}
\quad (\mathcal{J}_{xy} \leq \mathcal{J}_{z}) \; .\end{aligned}$$
Since the atomic-limit ground state where we have started does not depend on $N$, the final effective Hamiltonian is valid for both even-$N$ and odd-$N$. When $g$ and $e$ are symmetric (i.e., $U_{\text{diff}}=0$, $\mu_{g}=\mu_{e}$, $t_{g}=t_{e}$), $\mathcal{J}_{xy} = \mathcal{J}_{z}$ and the above effective Hamiltonian reduces to the usual spin $T=N/2$ Heisenberg model with the single-ion anisotropy, whose phase diagram has been studied extensively (see, e.g. Refs. and references cited therein). It is well-known[@Haldane-PRL-83; @Haldane-PLA-83] that the behavior of the spin-$S(=N/2)$ Heisenberg chain differs dramatically depending on the parity of $N$. Therefore, we may conclude that, when $N$ is even, the gapped “orbital” Haldane (OH) phase[@Nonne-B-C-L-10] appears for large negative $J$ (at least for small anisotropy $J \approx J_{z}$, $t_{g}\approx t_{e}$), while, for odd $N$, the same region is occupied by the gapless Tomonaga-Luttinger-liquid phase. The non-trivial hidden ordering of orbital degrees of freedom in the OH phase is illustrated for $N=2$ in Fig. \[fig:4-MottPhases-SU2\](b).
When we increase $|J-J_{z}|$ ($J<J_{z}$), the OH phase finally gets destabilized and is taken over by a gapful SU($N$)-singlet non-degenerate phase. This is an orbital-analog of the “large-$D$ phase” whose wave function is given essentially by a product of $T_{i}^{z} = 0$ states \[see Fig. \[fig:4-MottPhases-SU2\](d)\]. In the following, we call it “rung-singlet (RS)” as this state reduces in the case of $N=2$ to the well-known rung-singlet state in the spin-$\frac{1}{2}$ two-leg ladder.[@Dagotto-R-96] On the other hand, when $J-J_z$ takes a large positive value (as will be seen in Sec. \[sec:N4-gen-Hund-woSU2-J-pm4\]), the effective Hamiltonian develops easy-axis anisotropy and enters a phase with antiferromagnetic ordering of the orbital pseudo spin $T^{z}$: $-N/2,+N/2,-N/2,+N/2,\cdots$ \[see Fig. \[fig:2-DWPhases-SU2\](b)\]. This phase will be called ‘orbital-density wave (ODW)’ and is depicted in Fig. \[fig:2-DWPhases-SU2\](b) for $N=2$.
![(Color online) The $N$-leg ladder representation of the model . $N$ Hubbard-type chains for “spinful” fermions ($g$ and $e$) are coupled to each other by $U$ (interchain density-density interaction among like fermions), $V$ (that between $g$ and $e$), and the inter-chain Hund couplings ($V_{\text{ex}}^{g\text{-}e}$). \[fig:alkaline-Nleg\]](alkaline-Nleg)
Due to the strong easy-plane anisotropy in the orbital sector, a different conclusion is drawn for the $p$-band model . Now the single-site energy is given as $$-\mu \, n_{i}
+\frac{1}{4}(U_1 + U_2) n_{i}^2
+ \left\{ 2U_2 (T_{i}^x)^2 + (U_1 - U_2)(T_{i}^z)^2\right\} \; .$$ Since $V_{\text{ex}}^{g\text{-}e} = J \leftrightarrow U_{2}$, the condition $J<0$ translates to $U_{2}<0$ in the $p$-band model. Since the condition $U_{2}<0$ in the physical region $U_1 \simeq 3U_2$ implies an attractive interaction $U_1 + U_2 <0$, we have to take into account several different values of $n_{i}$. We follow the same line of argument as in Sec. \[sec:strong-coupling-J-0\] to show that at $\mu=-N |U_1+U_2|$, we have two degenerate SU($N$)-singlet states $n_{i}=0$ ($T=0$) and $n_{i}=2N$ ($T=0$) which feel a repulsive interaction coming from $t^2$-processes. Therefore, $2k_{\text{F}}$-CDW occupies a region around the line $U_1 = 3U_2$ for $N \geq 3$.
The case $N=2$ is exceptional due to the existence of the spin-charge symmetry . In fact, at $\mu= -4|U_{2}|$, the following three spin-singlet states $$\begin{split}
& c^{\dagger}_{p_x\uparrow,i}c^{\dagger}_{p_x\downarrow,i}
c^{\dagger}_{p_y\uparrow,i}c^{\dagger}_{p_y\downarrow,i}|0\rangle
\;\; (n_{i}=4) \\
& \frac{1}{\sqrt{2}}\left(
c^{\dagger}_{p_x\uparrow,i}c^{\dagger}_{p_x\downarrow,i}
+ c^{\dagger}_{p_y\uparrow,i}c^{\dagger}_{p_y\downarrow,i} \right) |0\rangle
\; (\equiv |\text{OLD}_y\rangle, n_{i}=2 ) \\
& |0\rangle \;\; (n_{i}=0) \; ,
\end{split}
\label{eqn:charge-triplet}$$ are degenerate on the U(1)-symmetric line $U_1=3U_2$ and form a triplet of charge-SU(2) at each site.
The effective Hamiltonian for the ground-state manifold spanned by these triplets is readily obtained by applying the transformation to , which is nothing but the spin-1 Heisenberg model. From the known ground state of the effective Hamiltonian, one sees that, instead of CDW for $N \geq 3$, CH appears around the line $U_1=3U_2$ when $N=2$. Note that the existence of the Shiba transformation, which guarantees the symmetry between spin and charge, is crucial for the appearance of the CH phase in the $N=2$ case.
Phases Abbreviation SU($N$) Orbital ($T$)
-------------------------- -------------- --------------- ---------------
Spin-Haldane SH $S=1$ Local singlet
Orbital-Haldane OH Local singlet $N/2$
Charge-Haldane CH Local singlet $-$
Orbital large-$D_{x,y}$ OLD$_{x,y}$ Local singlet $N/2$
Rung-singlet (OLD$_{z}$) RS Local singlet $N/2$
Spin-Peierls SP $-$ $N/2$
Charge-density wave CDW Local singlet Local singlet
Orbital-density wave ODW Local singlet $N/2$
: \[tab:abbreviation\] List of dominant phases and their abbreviations. Local SU($N$)/orbital degrees of freedom are shown, too.
SU(N) topological phase {#sec:SUN-topological-phase}
=======================
In this section, we investigate the nature of the ground state of the SU($N$) Heisenberg spin chain and its main physical properties.
SU(N) valence-bond-solid (VBS) state {#sec:SUN-VBS-state}
------------------------------------
In Sec. \[sec:models-strong-coupling\], we have seen that for positive $J$ (or positive $U_{2}$), we obtain the SU($N$) Heisenberg model or for relatively wide parameter regions. This SU($N$) spin chain has the self-conjugate representation (with $N/2$ rows and 2 columns) at each site and is not solvable. Nevertheless, we can obtain[@Nonne-M-C-L-T-13] a fairly good understanding of the properties of the ground state by constructing a series of model ground states, the VBS states [@Affleck-K-L-T-87; @Affleck-K-L-T-88], whose parent Hamiltonian is close to the original ones and .
We start from a pair of the self-conjugate representations \[characterized by a Young diagram with [*one*]{} column and $N/2$ rows; see \] on each site and create maximally-entangled pairs between adjacent sites \[see Figs. \[fig:VB-construction\](a) and (b)\]. To obtain the physical wave function, we apply the projection \[see Figs. \[fig:VB-construction\](a)$^{\prime}$ and \[fig:VB-construction\](b)$^{\prime}$\] $$\text{\scriptsize $N/2$} \left\{
\yng(1,1,1) \otimes \yng(1,1,1)
\right.
\longrightarrow
\yng(2,2,2)
\label{eqn:two-auxiliary-spaces}$$ onto the tensor-product state obtained above and construct the physical Hilbert space \[i.e., SU($N$) representation with its Young diagram having $N/2$ rows and two columns\] at each site. This procedure may be most conveniently done by using the matrix-product state (MPS)[@Garcia-V-W-C-07] $$\sum_{\{m_{i}\}}A_{1}(m_1)A_{2}(m_2)\cdots A_{i}(m_i)\cdots |m_1,m_2,\ldots,m_i
,\ldots\rangle \; ,
\label{eqn:VBS-state-in-MPS}$$ where $m_{i}$ labels the states of the $d$-dimensional local Hilbert space at the site-$i$ and $A_{i}(m_{i})$ is $D{\times}D$ matrices with $D$ being the bond dimensions. The dimensions of the local Hilbert space are $d=20$ \[SU(4)\], $d=175$ \[SU(6)\], $d=1764$ \[SU(8)\], and so on.
Although it is in principle possible to write the MPS for general $N$, the construction rapidly becomes cumbersome with increasing $N$. Therefore, we focus below only on the $N=4$ case where the ground state is given by the MPS with $D=6$ (the dimensions of ${\tiny \yng(1,1)}$). The parent Hamiltonian bearing the above VBS state as the exact ground state is not unique and, aside from the overall normalization, there are two free (positive) parameters. Among them, the one with lowest order in $(S^{A}_{i} S^{A}_{i+1})$ is given by[@Nonne-M-C-L-T-13] $$\begin{split}
&{\cal H}^{(N=4)}_{\text{VBS}} \\
& = J_{\text{s}}
\sum_{i}\Big\{ S^{A}_{i} S^{A}_{i+1} + \frac{13}{108}(S^{A}_{i} S^{A}_{i+1})^2
+ \frac{1}{216}(S^{A}_{i} S^{A}_{i+1})^3 \Big\},
\end{split}
\label{eqn:SU4-VBSmodel}$$ where $S^{A}_{i}$ ($A=1,\ldots,15$) denote the SU($4$) spin operators in the 20-dimensional representation \[normalized as $\text{Tr}\, (S^{A}S^{B})=16 \delta^{AB}$\] and $J_{\text{s}}$ is the exchange interaction between SU($N$) “spins”.[^4]
The ground state is SU(4)-symmetric and featureless [*in the bulk*]{}, and has the “spin-spin” correlation functions $$\langle S^{A}_{j} S^{A}_{j+n} \rangle
=
\begin{cases}
\frac{12}{5} \left(-\frac{1}{5} \right)^n & n \neq 0 \\
\frac{4}{5} & n=0
\end{cases}$$ that are exponentially decaying with a very short correlation length $1/\ln 5\approx 0.6213$. In spite of the featureless behavior in the bulk, the system exhibits a certain structure near the boundaries. In fact, if one measures ${\langle S^{A}_{i} \rangle}$ (with $S^{A}_{i}$ being any three commuting generators), one can clearly see the structure localized around the two edges. At each edge, there are six different states distinguished by the value of the set of the three generators ${\langle S^{A}_{i} \rangle}$. As in the spin-1 Haldane systems where two spin-$\frac{1}{2}$’s emerge at the edges,[@Affleck-K-L-T-88; @Kennedy-90] one may regard these six edge states as the emergent SU($N$) ‘spin’ ${\tiny \yng(1,1)}$ appearing near the each edge.
![(Color online) SU($N$) VBS states are constructed out of a pair of self-conjugate representations at each site. Dashed lines denote maximally-entangled pairs. (a) SU(4) with 20-dimensional representation and (b) SU(6) with 175-dimensional representation. (a)’ and (b)’ are the corresponding matrix-product states. \[fig:VB-construction\]](SU4-VBS-20rep-v3){width="0.9\columnwidth"}
Symmetry-protected topological phases
-------------------------------------
We observe that the model (\[eqn:SU4-VBSmodel\]) is not very far from the original pure Heisenberg Hamiltonian or obtained by the strong-coupling expansion in Sec. \[sec:strong-coupling\]. This strongly suggests that the SU(4) topological phase realizes in the strong-coupling regime of the SU($N$) fermion system $\mathcal{H}_{g\text{-}e}$ \[Eq. \] or $\mathcal{H}_{p\text{-band}}$ \[Eq. \] with the emergent edge states that belong to the six-dimensional representation of SU(4). In Ref. , it is predicted using the group-cohomology approach[@Chen-G-W-11; @Fidkowski-K-11; @Schuch-G-C-11], that there are $N$ topologically distinct phases (including one trivial phase) protected by PSU($N$) $=$ SU($N$)/$\mathbb{Z}_N$ symmetry, which are characterized by the number of boxes $n_{\text{y}}$ (mod $N$) contained in the Young diagram corresponding to the emergent edge “spin” at the (right) edge. Since the six-dimensional representation ${\tiny \yng(1,1)}$ appears at the edge of the $N=4$ VBS state , one expects that the ground state of the Heisenberg Hamiltonian \[or \] as well as that of the $N=4$ VBS Hamiltonian belongs to the $n_{\text{y}}=2$ member (we call it [*class-2*]{} hereafter) of the four topological classes.
Nevertheless, the observation of the edge-state degeneracy alone may lead to erroneous answers. A firmer evidence may be provided by the entanglement spectrum,[@Li-H-08] which is essentially the logarithm of the eigenvalues of the reduced density matrix. For instance, by tracing the entanglement spectrum, we can distinguish between different topological phases.[@Pollmann-T-B-O-10; @Zang-J-W-Z-10; @Zheng-Z-X-L-11; @Pollmann-B-T-O-12] On general grounds, one may expect that any representations compatible with the group-cohomology classification[@Chen-G-W-11; @Schuch-G-C-11] can appear in the entanglement spectrum.[^5] Quite recently, the entanglement spectrum for the model has been calculated[@Tanimoto-T-14] by using the infinite-time evolving block decimation (iTEBD)[@Vidal-iTEBD-07; @Orus-V-08] method. It has been found that the spectrum indeed consists of several different levels whose degeneracies are all compatible with the dimensions of the SU(4) irreducible representations allowed for the edge states of the class-2 topological phase. Specifically, the lowest-lying entanglement levels consist of ${\tiny \yng(1,1)}$ (6-dimensional), ${\tiny \yng(3,2,1)}$ (64-dimensional), ${\tiny \yng(3,3)}$ (50-dimensional), etc. Moreover, the continuity between the ground state of the model and that of has been demonstrated[@Tanimoto-T-14] by tracing the entanglement spectrum along the path ($0 \leq \lambda \leq 1$): $$\begin{split}
&{\cal H}(\lambda)
= J_{\text{s}} \sum_{i} S^{A}_{i} S^{A}_{i+1} \\
& \quad + \lambda J_{\text{s}} \sum_{i} \Big\{ \frac{13}{108}(S^{A}_{i} S^{A}_{i+1})^2
+ \frac{1}{216}(S^{A}_{i} S^{A}_{i+1})^3 \Big\} \; .
\end{split}
\label{eqn:SU4-VBSmodel-lambda}$$ At $\lambda=0$, ${\cal H}(\lambda)$ reduces to the effective Hamiltonian $\mathcal{H}_{\text{SU(N)}}$ \[Eq. or \] and ${\cal H}(1)$ is the VBS Hamiltonian whose entanglement spectrum consists only of the sixfold-degenerate level. When we move from $\lambda=0$ to 1, the entanglement levels other than the lowest one gradually go up and finally disappear from the spectrum at $\lambda=1$ while preserving the structure of the spectrum.
It is interesting to consider the protecting symmetries other than PSU($N$). The result from group cohomology[@Chen-G-L-W-13] $H^{2}(\text{PSU($N$)},\text{U(1)})=H^{2}(\mathbb{Z}_{N}{\times}\mathbb{Z}_{N},\text{U(1)})=\mathbb{Z}_{N}$ suggests that $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ will do the job. Since it has been recently demonstrated that the even-fold degenerate structure in the entanglement spectrum signals the topological Haldane phase, [@Pollmann-T-B-O-10; @Pollmann-B-T-O-12] one may ask whether there is a relation between our class-2 topological phase and the Haldane phase. However, as we will show in the following, the even-fold degeneracy found in the entanglement spectrum of our SU(4) state comes from the protecting $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$-symmetry that is a subgroup of PSU(4).
The first $\mathbb{Z}_{4}$-generator $Q$ is defined in terms of the two commuting SU(4) generators (Cartan generators) as $$\begin{split}
& Q \equiv {\mathrm{e}}^{i\frac{3\pi}{4}} \exp\left( i \frac{2\pi}{4}G_{Q} \right) , \;\; Q^{4}=1 \\
& G_{Q} \equiv 2H_{1} + H_{2} \; .
\end{split}$$ On the other hand, the second $\mathbb{Z}_{4}$ is generated by $$\begin{split}
& P \equiv {\mathrm{e}}^{i\frac{3\pi}{4}} \exp\left( i \frac{2\pi}{4}G_{P} \right) , \;\; P^{4}=1 \\
& G_{P} \equiv - \frac{1}{2} \sum_{\alpha} E_{\alpha} + \frac{i}{2} \left( \sum_{i=1}^{3}E_{\alpha_{i}}
- E_{\alpha_{1}+\alpha_{2}+\alpha_{3}} \right) \\
& \phantom{G_{P} \equiv}
- \frac{i}{2} \left( \sum_{i=1}^{3}E_{-\alpha_{i}}- E_{-\alpha_{1}-\alpha_{2}-\alpha_{3}} \right) \; .
\end{split}$$ In the above equations, we have used the Cartan-Weyl basis $\{ H_{a},E_{\alpha}\}$ that satisfies $$\begin{split}
& [H_{a},H_{b}]=0 \, , \;\;
[H_{a},E_{\alpha}] = (\alpha)_{a}E_{\alpha} \, , \\
& [E_{\alpha},E_{-\alpha}] = \sum_{a=1}^{3} (\alpha)_{a}H_{a} \, , \;
\text{Tr}\, (H_{a}H_{b})=16 \delta_{ab} ,
\\
&(a,b=1,2,3)
\end{split}$$ with $\alpha$ being the roots of SU(4) normalized as $|\alpha|=\sqrt{2}$ which are generated by the simple roots $\alpha_{i}$ ($i=1,2,3$). The summation $\sum_{\alpha}$ is taken over all 12 roots $\alpha$ of SU(4). Here we do not give the explicit expressions of the generators which depend on a particular choice of the basis, since giving the commutation relations suffices to define $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$. In the actual calculations, one may use, e.g., the generators and the weights given in Sec. 13.1 of Ref. with due modification of the normalization. [^6] It is important to note that the two $\mathbb{Z}_{4}$s defined above commute with each other (i.e., $[Q,P]=0$) [*only*]{} when the number of boxes in the Young diagram is an integer multiple of 4. To put it another way, the two operators $Q$ and $P$ constructed here generate $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ only for PSU(4) as the two $\pi$-rotations along the $x$ and $z$ axes generate $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$ only when the spin quantum number is integer.
Now let us consider the relation between the PSU($4$) topological classes and the above $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ symmetry. To this end, we recall the fundamental property of MPS. If a given MPS generated by the matrices $\{A(m)\}$ is invariant under the $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ symmetry introduced above, there exists a set of unitary matrices $U_{Q}$ and $U_{P}$ satisfying [@Garcia-W-S-V-C-08] $$\begin{split}
& A(m) \xrightarrow{Q}
e^{i\theta_{Q}}{U_{Q}}^{\dagger} A(m) U_{Q} \\
& A(m) \xrightarrow{P}
e^{i\theta_{P}}{U_{P}}^{\dagger} A(m) U_{P} \; .
\end{split}
\label{unitary_matrix}$$ Then, the property $QP=PQ$ mentioned above implies[^7] that they obey the following non-trivial relation[@Duivenvoorden-Q-ZnxZn-13]: $$U_{Q}U_{P} = \omega^{n_{\text{y}}} U_{P}U_{Q} \;\;
(\omega \equiv {\mathrm{e}}^{i \frac{2\pi}{N}})
\label{eqn:commutation-projective}$$ with the same $n_{\text{y}}(=0,1,2,3)$ as above. Reflecting the entanglement structure, $U_{P}$ and $U_{Q}$ are both block-diagonal. By taking the determinant of both sides, one immediately sees that the degree of degeneracy $D_{\xi}$ of each entanglement level $\xi$ (i.e., the size of each block) satisfies $\omega^{D_{\xi} n_{\text{y}}}=1$. In our SU(4) case, $D_{\xi}=4n$ ($n$: positive integer) for class-1 ($n_{\text{y}}=1$) and class-3 ($n_{\text{y}}=3$), while $D_{\xi}=2n$ for class-2 ($n_{\text{y}}=2$). The relation implies that the crucial information on the PSU(4) topological phase is encoded in the exchange property of the [*projective*]{} representations $U_Q$ and $U_P$ of $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$. This is the key to the construction of non-local string order parameters of our PSU($N$) topological phases.
Non local order parameters {#sec:non-local-OP}
--------------------------
By definition, local order parameters are not able to capture the SU($N$) SPT phases. Nevertheless, elaborate choice [@Haegeman-G-C-S-12; @Pollmann-T-12; @Hasebe-T-13] of [*non-local*]{} order parameters could detect hidden topological orders in those phases. We adapt the method[@Duivenvoorden-Q-ZnxZn-13] of constructing non-local order parameters in generic $(\mathbb{Z}_{N}{\times}\mathbb{Z}_{N})$-invariant systems to our SU(4) system. As in the usual spin systems[@Kennedy-T-92-PRB; @Kennedy-T-92-CMP], one can construct the following sets of order parameters in terms of SU(4) generators
$$\begin{aligned}
\begin{split}
& \mathcal{O}_{1}(m,n) \\
&\equiv \lim_{|i-j|\nearrow \infty}
\Biggl\langle \left\{\hat{X}_{P}(i)\right\}^{m} \left\{
\prod_{i\leq k <j} \hat{Q}(k)^{n}
\right\} \left\{ \hat{X}_{P}^{\dagger}(j) \right\}^{m} \Biggr\rangle
\label{eqn:def-stringOP-1b}
\end{split}
\\
\begin{split}
& \mathcal{O}_{2}(m,n) \\
& \equiv \lim_{|i-j|\nearrow \infty}
\Biggl\langle
\left\{ \hat{X}_{Q}(i) \right\}^{m}
\left\{ \prod_{i < k \leq j} \hat{P}(k)^{n}
\right\} \left\{\hat{X}_{Q}^{\dagger}(j) \right\}^{m} \Biggr\rangle \\
& \qquad (0 \leq m,n < N)
\; .
\end{split}
\label{eqn:def-stringOP-2b}\end{aligned}$$
The subscripts 1 and 2 refer to the string order parameters corresponding to the two commuting $\mathbb{Z}_{N}$’s. The operators $\hat{X}_{Q}$ and $\hat{X}_{P}$ appearing in the above can be expressed by the SU(4) generators as $$\begin{split}
& \hat{X}_{Q} = \frac{1}{2}( E_{-\alpha_1}+E_{-\alpha_2}+E_{-\alpha_3} + E_{\alpha_1+\alpha_2+\alpha_3}) \\
& \hat{X}_{P} = \frac{1}{\sqrt{2}}(H_{1} - i H_{3} )
\end{split}$$ and obey the following relations ($\omega={\mathrm{e}}^{i\frac{2\pi}{4}}$) $$\begin{split}
& \hat{Q}^{\dagger}\hat{X}_{Q} \hat{Q} = \omega \hat{X}_{Q} \;\; , \quad
\hat{P}^{\dagger}\hat{X}_{Q} \hat{P} = \hat{X}_{Q} \\
& \hat{Q}^{\dagger}\hat{X}_{P} \hat{Q} = \hat{X}_{P} \;\; , \quad
\hat{P}^{\dagger}\hat{X}_{P} \hat{P} = \omega^{-1}\hat{X}_{P}
\end{split}$$ for [*any*]{} irreducible representations of SU(4).
It is known[@Pollmann-T-12; @Hasebe-T-13] that the boundary terms of $\mathcal{O}_{1,2}(m,n)$ carry crucial information about the projective representation under which the physical edge states transform and hence give a physical way of characterizing the topological phases. By carefully analyzing the phase factors appearing in the boundary terms, one sees that the three sets of non-local string order parameters $\{\mathcal{O}_{1,2}(1,3),\mathcal{O}_{1,2}(2,1),\mathcal{O}_{1,2}(1,1)\}$ can distinguish among the four distinct phases (one trivial and three topological) protected by PSU(4) symmetry (see Table \[tab:3-string-OP\]).[@Tanimoto-T-14] In fact, one can check[@Tanimoto-T-14] numerically that $\mathcal{O}_{1,2}(2,1)$ remains finite all along the interpolating path $\mathcal{H}(\lambda)$, while all the others are zero (at the solvable point $\lambda=1$, $\mathcal{O}_{1,2}(2,1)=1$).
Phases $\mathcal{O}_{1,2}(1,3)$ $\mathcal{O}_{1,2}(2,1)$ $\mathcal{O}_{1,2}(1,1)$
------------------------------- -------------------------- -------------------------- --------------------------
Trivial ($n_{\text{y}}=4n$) 0 0 0
Class-1 ($n_{\text{y}}=4n+1$) Finite 0 0
Class-2 ($n_{\text{y}}=4n+2$) 0 Finite 0
Class-3 ($n_{\text{y}}=4n+3$) 0 0 Finite
: \[tab:3-string-OP\] Three sets of string order parameters characterizing the four distinct phases protected by PSU(4). The entry ‘finite’ means that the corresponding $\mathcal{O}_{1,2}$ in principle can take non-zero values.
The weak-coupling approach {#sec:weak-coupling}
==========================
In this section, we map out the zero-temperature phase diagram of the different lattice models , and related to the physics of the 1D two-orbital SU($N$) cold fermions by means of a low-energy approach. In particular, we will investigate the fate of the different topological Mott-insulating phases, revealed in the strong-coupling approach, in the regime where the hopping term is not small.
Continuum description {#sec:continuum_description}
---------------------
The starting point of the analysis is the continuum description of the lattice fermionic operators $c_{m\alpha,\,i}$ in terms of $2N$ left-right moving Dirac fermions ($m=g,e$ or $m=p_x, p_y$, $\alpha=1,\ldots,N$): [@Gogolin-N-T-book; @Giamarchi-book] $$c_{m \alpha,\,i} \rightarrow \sqrt{a_0} \left(L_{m \alpha}(x)
e^{-i k_\text{F} x} + R_{m \alpha}(x) e^{i k_\text{F} x} \right),
\label{contlimitDirac}$$ where $x= i a_0$ ($a_0$ being the lattice spacing). Here we assume $t_{g}=t_{e}$ and $\mu^{(g)}=\mu^{(e)}$, and hence $k^{(g)}_\text{F} =k^{(e)}_\text{F} =k_\text{F} = \pi/(2 a_0)$ for half-filling. The non-interacting Hamiltonian is equivalent to that of $2N$ left-right moving Dirac fermions: $${\cal H}_0=-i v_\text{F} \left(R_{m \alpha} ^\dag \partial_x R_{m \alpha} ^{\phantom \dag} -
L_{m \alpha}^\dag \partial_x L_{m \alpha}^{\phantom \dag}\right) ,
\label{HamcontDirac}$$ where $v_\text{F} = 2t a_0$ is the Fermi velocity. The non-interacting model enjoys an U(2$N$)$|_\text{L}$ $\otimes$ U(2$N$)$|_\text{R}$ continuous symmetry which results from its invariance under independent unitary transformations on the $2N$ left and right Dirac fermions. It is then very helpful to express the Hamiltonian directly in terms of the currents generated by these continuous symmetries. To this end, we introduce the U(1)$_{\text{c}}$ charge current and the SU(2$N$)$_1$ current which underlie the conformal field theory (CFT) of massless $2N$ Dirac fermions: [@Affleck-NP86; @Affleck-88] $$\begin{split}
& J_{\text{c} \text{L}} = :L_{n\alpha}^\dagger L_{n\alpha}: \quad\textrm{U(1)$_\text{c}$ charge current} \\
& J_\text{L}^{A} = L_{m\alpha}^\dagger {\cal T}^{A}_{m,\alpha;n,\beta} L_{n\beta} \quad
\text{SU($2N$)}_1 \text{ currents},
\end{split}
\label{U(2N)currents}$$ with $m,n = g,e$ (or $m,n = p_x, p_y$ for the $p$-band model), $\alpha,\beta = 1, \ldots, N$, and we have similar definitions for the right currents. In Eq. (\[U(2N)currents\]), the symbol $::$ denotes the normal ordering with respect to the Fermi sea, and ${\cal T}^{A}$ ($A =1, \ldots, 4 N^2 -1$) stand for the generators of SU($2N$) in the fundamental representation normalized such that: $\text{Tr}({\cal T}^A {\cal T}^B)=\delta^{A\,B}/2$. The non-interacting model (\[HamcontDirac\]) can then be written in terms of these currents (the so-called Sugawara construction of the corresponding CFT[@DiFrancesco-M-S-book]): $$\begin{split}
{\cal H}_0 =& \frac{\pi v_\text{F}}{2N}
\left[ : J^2_{c\text{R}} : + : J^2_{c\text{L}} :
\right] \\
&+ \frac{2\pi v_\text{F}}{2N + 1} \left[ : J^A_{\text{R}} J^A_{\text{R}}: + : J^A_{\text{L}} J^A_{\text{L}}:
\right] .
\end{split}
\label{contfreehambis}$$
The non-interacting part is thus described by an U(1)$_{\text{c}}$ $\times$ SU(2$N$)$_1$ CFT. Since the lattice model has a lower SU($N$) symmetry originating from the nuclear spin degrees of freedom, it might be useful to consider the following conformal embedding [@DiFrancesco-M-S-book], which is also relevant to multichannel Kondo problems [@Affleck-L-91]: U(1)$_{\text{c}}$ $\times$ SU(2$N$)$_1$ $\supset$ U(1)$_{\text{c}}$ $\times$ SU($N$)$_2$ $\times$ SU($2$)$_N$. In this respect, let us define the following currents which generate the SU($N$)$_2$ $\times$ SU($2$)$_N$ CFT: $$\begin{split}
& J_\text{L}^a = L_{n\alpha}^\dagger (T^a)_{\alpha,\beta} L_{n\beta} \quad \text{SU($N$)}_2
\text{ (nuclear) spin currents}\\
& j_{\text{L}}^i = L_{m\alpha}^\dagger (\sigma^i/2)_{m,n} L_{n\alpha} \quad
\text{SU(2)}_N \text{ orbital currents} \\
& \mathcal{J}_{\text{L}}^{a,i} = L_{m\alpha}^\dagger T^{a,i}_{m,\alpha;n,\beta} L_{n\beta} \quad
\text{remaining SU($2N$)$_1$ currents},
\end{split}
\label{defalkacurrents}$$ where $T^a$ ($a=1,\ldots,N^{2}-1$) and $\sigma^i$ ($i=x,y,z$) respectively are the SU($N$) generators and the Pauli matrices. The $4N^2-1$ SU(2$N$) generators can be expressed in a unifying manner as a direct product between the SU($N$) and the SU(2) generators: $$\begin{split}
& T^{a,0} = \frac{1}{\sqrt{2}}T^a\otimes I_2 \\
& T^{0,i} = \frac{1}{2\sqrt{N}}I_N\otimes \sigma^i \\
& T^{a,i} = \frac{1}{\sqrt{2}}T^a\otimes \sigma^i ,
\end{split}$$ where all the above generators are normalized in such a way that: $\text{Tr}(T^X T^Y)=\delta^{X\,Y}/2$ ($X,Y=(a,i)$). The current $j_{\text{L}}^i$, being the sum of $N$ SU(2)$_1$ currents, the CFT corresponding to spin-1/2 degrees of freedom [@Gogolin-N-T-book], becomes an SU(2)$_N$ current, that accounts for the critical properties of the orbital degrees of freedom. Similarly, $J_\text{L}^a$ is a sum of two level-1 SU($N$) currents and the low-energy properties of the nuclear spin degrees are governed by an SU($N$)$_2$ CFT which is generated by the $J_{\text{L}}^a$ ($a = 1, \ldots, N^2 -1$) current.
At half-filling, we need to introduce, on top of these currents, additional operators which carry the U(1) charge to describe various umklapp operators in the continuum limit: $$\begin{split}
& A^{\alpha \beta +}_{mn \text{L}} =
\frac{-i}{2} \left( L^{\dagger}_{m \alpha} L^{\dagger}_{n \beta}
- L^{\dagger}_{m \beta} L^{\dagger}_{n \alpha} \right) \\
& S^{\alpha \beta +}_{\text{L}} = \frac{1}{2} \left( L^{\dagger}_{g \alpha} L^{\dagger}_{e \beta}
+ L^{\dagger}_{g \beta} L^{\dagger}_{e \alpha} \right) ,
\end{split}
\label{umklappop}$$ with $m,n = g,e$ (or $m,n = p_x, p_y$ for the $p$-band model), and $\alpha,\beta = 1, \ldots, N$. We introduce a similar set of operators for the right fields as well.
With all these definitions at hand, we are able to derive the continuum limit of two-orbital SU($N$) models of Sec. II. We will neglect all the velocity anisotropies for the sake of simplicity. Performing the continuum limit, we get the following interacting Hamiltonian density: $$\begin{split}
& \mathcal{H}_{\text{int}} \\
&= g_1 J_\text{L}^a J_\text{R}^a +\frac{g_2}{2}\left(\mathcal{J}_\text{L}^{a,+} \mathcal{J}_\text{R}^{a,-}
+\textrm{H.c.}\right) + g_3 \mathcal{J}_\text{L}^{a,3} \mathcal{J}_\text{R}^{a,3} \\
& + \frac{g_4}{2}\left(j_{\text{L}}^+j_{\text{R}}^-+\textrm{H.c.}\right)
+g_5 j_{\text{L}}^zj_{\text{R}}^z + g_6 J_{\text{c L}} J_{\text{c R}} \\
& + g_7 \left(S^{\alpha \beta +}_{\text{L}} S^{\alpha \beta -}_{R} +\textrm{H.c.}\right)
+ \frac{g_8}{2} \sum_{m=g,e} \left(A^{\alpha \beta +}_{mm \text{L}}
A^{\alpha \beta -}_{mm \text{R}} +\textrm{H.c.}\right) \\
&+ g_9 \left(A^{\alpha \beta +}_{g e \text{L}} A^{\alpha \beta -}_{g e \text{R}} +\textrm{H.c.}\right) .
\end{split}
\label{lowenergyham}$$
Although the different lattice models, having the same continuous symmetry, share the same continuum Hamiltonian in common, the sets of initial coupling constants are different. For the generalized Hund model , we find the following identification for the coupling constants: $$\begin{split}
& g_1 = - \left( U + J + \frac{J_z}{2}\right)a_0 \\
& g_2 = \left(-2U+J_z\right)a_0 \\
& g_3 = \left(-2U+2J-J_z\right)a_0 \\
& g_4 = \left(-\frac{2U}{N}+2J+\frac{J_z}{N}\right)a_0 \\
& g_5 = \left( -\frac{2U}{N} +\frac{2J}{N}+\frac{2N-1}{N}J_z \right) a_0 \\
& g_6 = \left(\frac{U(2N-1)}{2N} - \frac{J}{2N} - \frac{J_z}{4N}\right)a_0 \\
& g_7 = - \left(-U + J + \frac{J_z}{2}\right)a_0 \\
& g_8 = \left(U + \frac{J_z}{2}\right)a_0 \\
& g_9 = \left(U + J - \frac{J_z}{2}\right) a_0 ,
\end{split}
\label{couplings}$$ while, for the $g\text{-}e$ model with fine-tuning $U_{gg} = U_{ee} = U_{mm}$, we use Eq. to obtain: $$\begin{split}
& g_1 = - a_0 \left( U_{mm} + V_{\text{ex}}^{g\text{-}e} \right) \\
& g_2 = -2 a_0 V \\
& g_3 = 2 a_0 \left(V_{\text{ex}}^{g\text{-}e} - U_{mm} \right) \\
& g_4 = 2 a_0 \left(V_{\text{ex}}^{g\text{-}e} - \frac{V}{N}\right)\\
& g_5 = 2 a_0 \left( \frac{(N-1)}{N} U_{mm} + \frac{1}{N} V_{\text{ex}}^{g\text{-}e} - V\right)\\
& g_6 = \frac{a_0}{2N} \left( - V_{\text{ex}}^{g\text{-}e} + (N-1) U_{mm} + N V \right)\\
& g_7 = a_0 \left(V - V_{\text{ex}}^{g\text{-}e} \right)\\
& g_8 = a_0 U_{mm} \\
& g_9 = a_0 \left(V + V_{\text{ex}}^{g\text{-}e} \right) .
\end{split}
\label{couplingsGorsh}$$
Since the effective Hamiltonian enjoys an $\text{U}(1)_{\text{c}}\times\text{SU}(N)_{\text{s}}\times \text{U}(1)_{\text{o}}$ continuous symmetry, it governs also the low-energy properties the $p$-band model with an harmonic confinement potential where $U_1 = 3 U_2$ and also along the line $U_1 =U_2$ as discussed in Sec. \[sec:p-band-definition\]. In absence of the U(1)$_{o}$ orbital symmetry, model will be more complicated with 12 independent coupling constants and we will not investigate this case here.
RG analysis {#sec:RG_analysis}
-----------
The interacting part consists of marginal current-current interactions. The one-loop RG calculation enables one to deduce the infrared (IR) properties of that model and thus the nature of the phase diagram of the SU($N$) two-orbital models. After very cumbersome calculations, we find the following one-loop RG equations: $$\begin{split}
\dot{g_1} =& \frac{N}{4\pi} g^2_1 + \frac{N}{8\pi} g^2_2 + \frac{N}{16\pi} g^2_3
+ \frac{N+2}{4\pi} g^2_7 + \frac{N-2}{4\pi} \left( 2 g^2_8 + g^2_9 \right) \\
\dot{g_2} =& \frac{N}{2\pi} g_1 g_2 +
\frac{N^2 -4}{4\pi N} g_2 g_3 + \frac{1}{2\pi} (g_2 g_5+g_3 g_4) + \frac{N}{\pi} g_7 g_8 \\
&+ \frac{N-2}{\pi} g_8 g_9 \\
\dot{g_3} =& \frac{N}{2\pi} g_1 g_3 +
\frac{N^2 -4}{4\pi N} g^2_2 + \frac{1}{\pi} g_2 g_4
+ \frac{N}{\pi} g_7 g_9 + \frac{N-2}{\pi} g^2_8 \\
\dot{g_4} =& \frac{1}{2\pi} g_4 g_5 +
\frac{N^2 - 1}{2\pi N^2} g_2 g_3 + \frac{2(N-1)}{\pi N} g_8 g_9 \\
\dot{g_5} =& \frac{N^2 -1}{2\pi N^2} g^2_2 + \frac{1}{2\pi} g^2_4 + \frac{2(N-1)}{\pi N} g^2_8 \\
\dot{g_6} =& \frac{N +1}{4\pi N} g^2_7 + \frac{N - 1}{2\pi N} g^2_8 + \frac{N-1}{4\pi N} g^2_9 \\
\dot{g_7} =& \frac{(N + 2)(N - 1)}{2\pi N} g_1 g_7 + \frac{2}{\pi} g_6 g_7
+ \frac{N-1}{4\pi} (2 g_2 g_8 + g_3 g_9) \\
\dot{g_8} =& \frac{N + 1 }{4\pi} g_2 g_7 + \frac{2}{\pi} g_6 g_8
+ \frac{1}{2\pi} \left(g_4 g_9 + g_5 g_8 \right) \\
& + \frac{(N-2)(N+1)}{4\pi N} \left(2 g_1 g_8 + g_2 g_9 + g_3 g_8 \right) \\
\dot{g_9} =& \frac{N + 1 }{4\pi} g_3 g_7 + \frac{1}{\pi} (g_4 g_8 + 2 g_6 g_9 ) \\
&+ \frac{(N-2)(N+1)}{2\pi N} \left( g_1 g_9 + g_2 g_8 \right),
\end{split}
\label{RG}$$ where ${\dot g}_i = \partial g_i/ \partial l (i =1,\ldots, 9)$ with $l$ being the RG time. First, we note that the RG flow of these equations is drastically different for $N=2$ and $N>2$ as we observe, from Eqs. (\[RG\]), that some terms vanish in the special $N=2$ case. In the latter case, the RG analysis has been done in detail already in Refs. , where the phase diagram of the generalized Hund and $g\text{-}e$ cold fermions have been mapped out. We thus assume $N>2$ hereafter and, for completeness, we will also determine the phase diagram of the half-filled $p$-band model for $N=2$ (see Appendix \[sec:N2-p-band-continuum\]).
The next step is to solve the RG equations numerically using the Runge-Kutta procedure. For the initial conditions (\[couplings\], \[couplingsGorsh\]) corresponding to the different lattice models of Sec. \[sec:models-strong-coupling\], the numerical analysis reveals the existence of the two very different regimes that we will now investigate carefully below.
### Phases with dynamical symmetry enlargement {#sec:Phases_with_dynamical_symmetry_enlargement}
One striking feature of 1D interacting Dirac fermions is that when the interaction is marginally relevant, a dynamical symmetry enlargement (DSE)[@Lin-B-F-98; @Boulat-A-L-09; @Konik-S-L-02] emerges very often in the far IR. Such DSE corresponds to the situation where the Hamiltonian is attracted under an RG flow to a manifold possessing a symmetry higher than that of the original field theory. Most of DSEs have been discussed within the one-loop RG approach. Among those examples is the emergence of SO(8) symmetry in the low-energy description of the half-filled two-leg Hubbard model [@Lin-B-F-98; @Chen-C-L-C-M-04] and the SU(4) half-filled Hubbard chain model. [@Assaraf-A-B-C-L-04]
It is convenient to introduce the following rescaling of the coupling constants to identify the possible DSEs compatible with the one-loop RG Eqs. (\[RG\]): $$\begin{split}
& f_{1,7,8,9} = \frac{N}{\pi} g_{1,7,8,9} \, , \;\; f_{2,3} = \frac{N}{2 \pi} g_{2,3} \\
& f_{4,5} = \frac{N^2}{2 \pi} g_{4,5} \, , \;\; f_{6} = \frac{2 N^2}{\pi} g_{6} .
\end{split}
\label{rescaling}$$ One then observes that along a special direction of the flow (dubbed ‘ray’[^8]) where $f_i = f$, all the nine one-loop RG equations (\[RG\]) reduces to a single equation: $$\dot{f} = \frac{2N-1}{N} f^2 \; .$$ This signals the emergence of an SO($4N$) symmetry which is the maximal continuous symmetry enjoyed by $2N$ Dirac fermions, i.e., $4N$ Majorana (real) fermions. To see this, one notes that along this special ray, model (\[lowenergyham\]) reduces to the SO($4N$) Gross-Neveu (GN) model: [@Gross-N-74] $$\begin{split}
{\cal H}_{\text{GN}} =& -i v_\text{F}\left(R_{m \alpha} ^\dag \partial_x R_{m \alpha} ^{\phantom \dag} -
L_{m \alpha}^\dag \partial_x L_{m \alpha}^{\phantom \dag}\right) \\
&+ \frac{\pi f}{2N} \left( L_{m \alpha} ^{\dag} R_{m \alpha} - \text{H.c.} \right)^2,
\end{split}
\label{GN}$$ where the SO($4N$) symmetry stems from the decomposition of Dirac fermions into Majorana fermions: $L_{m \alpha} =
\xi_{m \alpha} + i \chi_{m \alpha}$. The GN model (\[GN\]) is a massive integrable field theory when $f>0$ whose mass spectrum is known exactly [@Zamolodchikov-Z-79; @Karowski-T-81].
The numerical integration of RG Eqs. (\[RG\]) revealed that for some set of initial conditions, the coupling constants flow along the highly-symmetric ray where $f_i = f >0$ in the far IR (see Sec. \[sec:RG-phase-diag\]). The model is then equivalent to the SO($4N$) GN model and a non-perturbative spectral gap is generated. The development of this strong-coupling regime in the SO($4N$) GN model signals the formation of a SP phase for all $N \ge 2$ with the order parameter: $${\cal O}_{\rm SP} =
i \left( L_{m \alpha}^\dagger R_{m \alpha} - H.c.\right),
\label{SP}$$ which is the continuum limit of the SP operator on a lattice $${\cal O}_{\rm SP}(i) = (-1)^i\sum_{m \alpha}c^\dagger_{m \alpha,i+1}c_{m \alpha,i} \; .
\label{SP-lattice}$$ Since the interacting part of the GN model (\[GN\]) can be written directly in terms of ${\cal O}_{\rm SP}$: ${\cal H}^{\rm int}_{\text{GN}} = - \pi f {\cal O}_{\rm SP}^2 /(2N)$, we may conclude that $ \langle {\cal O}_{\rm SP} \rangle \ne 0$ in the ground state for $f>0$, i.e., the emergence of a dimerized phase. The latter is two-fold degenerate and breaks spontaneously the one-step translation symmetry: $$T_{a_0} : \quad
L_{m \alpha} \to -i L_{m \alpha}\, , \;\; R_{m \alpha} \to i R_{m \alpha} ,$$ since ${\cal O}_{\text{SP}} \to - {\cal O}_{\text{SP}}$ under $T_{a_0}$. It turns out that the SU$(2N)$ line ($J=J_{z}=0$) with $U >0$ of the generalized Hund model is described by the $f_i = f >0$ manifold with an SO($4N$) DSE. This is in full agreement with the fact that the repulsive SU($2N$) Hubbard model for $N \ge 2$ displays a SP phase at half filling.[@Nonne-L-C-R-B-11]
On top of this phase, we can define other DSE phases with global SO($4N$) symmetry. These phases are described by RG trajectories along the rays $f_i = \epsilon_i f $ ($\epsilon_i = \pm 1$) in the long-distance limit. The physical properties of these phases are related to those of the SO($4N$) GN model up to some duality symmetries on the Dirac fermions. [@Boulat-A-L-09] These duality symmetries can be determined using the symmetries of the RG Eqs. (\[RG\]):
$$\begin{aligned}
& \Omega_1: f_{7,8,9} \rightarrow - f_{7,8,9}
\label{eq:Omega_1} \\
& \Omega_2 : f_{2,4,8} \rightarrow - f_{2,4,8}
\label{eq:Omega_2} \\
& \Omega_3 (= \Omega_1 \Omega_2): f_{2,4,7,9} \rightarrow - f_{2,4,7,9} ,
\label{eq:Omega_3}\end{aligned}$$
which are indeed symmetries of Eqs. (\[RG\]) in the general $N$ case. Using the definitions , , and , one can represent these duality symmetries simply in terms of the Dirac fermions: $$\begin{split}
& \Omega_1:
L_{m \alpha}
\rightarrow i L_{m \alpha} \\
& \Omega_2 :
L_{m \alpha} \rightarrow \left(-1\right)^{m} i L_{m \alpha} \\
& \Omega_3:
L_{m \alpha} \rightarrow \left(-1\right)^{m+1} L_{m \alpha} ,
\end{split}
\label{dualitiesDirac}$$ while the right fermions remain invariant. These transformations are automorphisms of the different current algebra in Eq. (\[defalkacurrents\]). [@Boulat-A-L-09]
Starting from the gapful SP phase found above, one can deduce the three other insulating phases by exploiting the duality symmetries : $$\begin{split}
& {\cal O}_{\rm SP} \xrightarrow{\Omega_1} {\cal O}_{\rm CDW} \equiv
L_{m \alpha}^\dagger R_{m \alpha} + \text{H.c.} \\
& {\cal O}_{\rm SP} \xrightarrow{\Omega_2} {\cal O}_{\rm ODW} \equiv
\sum_m \left(- 1\right)^{m} L_{m \alpha}^\dagger R_{m \alpha} + \text{H.c.} \\
& {\cal O}_{\rm SP} \xrightarrow{\Omega_3} {\cal O}_{\rm SP_{\pi}} \equiv
\sum_m \left(-1 \right)^{m} i \left( L_{m \alpha}^\dagger R_{m \alpha} - \text{H.c.} \right) \; .
\end{split}
\label{dualityorderparameters}$$ Using , one can identify the lattice order parameters corresponding to these operators as: $$\begin{split}
& {\cal O}_{\rm CDW}(i) = (-1)^i n_i
\\
& {\cal O}_{\rm ODW}(i) =
(-1)^i\sum_{m} \left(- 1\right)^{m} c^\dagger_{m \alpha,i}c_{m \alpha,i}
\\
& {\cal O}_{\rm SP_{\pi}}(i) = (-1)^i\sum_{m} \left(- 1\right)^{m} c^\dagger_{m \alpha,i+1}c_{m \alpha,i} ,
\end{split}
\label{latticedualityorderparameters}$$ which describe respectively a CDW, an orbital-density wave (ODW), and an alternating SP phase (SP$_{\pi}$). For instance, by using $\Omega_{1}$, one can immediately conclude that on the SU$(2N)$ line ($J=J_{z}=0$) with $U <0$, the generalized Hund model is in a CDW phase $ \langle {\cal O}_{\rm CDW} \rangle \ne 0$ exhibiting the SO($4N$) DSE. This is fully consistent with the known result that the [*attractive*]{} SU($2N$) Hubbard model for $N \ge 2$ displays a CDW phase at half filling.[@Zhao-U-W-06; @Zhao-U-W-07; @Nonne-L-C-R-B-11]
In summary, in the first regime of the RG flow characterized by DSE, we found four possible Mott-insulating phases which are two-fold degenerate and spontaneously break the one-site translation symmetry. The RG approach developed here tells that each of these four phases is characterized by one of the four SO($4N$)-symmetric DSE rays related to each other by the duality symmetries $\Omega_{1,2,3}$.
### Non-degenerate Mott insulating phases {#sec:Non-degenerate_Mott_insulating_phases}
In the second regime, the RG flow displays no symmetry enlargement, and we can no longer use any duality symmetry to relate the underlying insulating phases to a single phase ([*e.g.*]{} the SP phase in the above). Indeed, in stark contrast, the numerical solution of the one-loop RG equations for $N > 2$ reveals that the coupling constant $g_1$ in the low-energy effective Hamiltonian (\[lowenergyham\]) reaches the strong-coupling regime before the other coupling constants such as $g_{2,4,5,8}$. Since the operator corresponding to $g_1$ depends only on the nuclear spin degrees of freedom, one expects a separation of the energy scales in this second region of the RG flow. Neglecting all the other couplings for the moment, the resulting perturbation corresponds to an SU($N$)$_2$ CFT perturbed by a marginally relevant current-current interaction $g_1 >0$. This model is an integrable massive field theory [@Ahn-B-L-90; @Babichenko-04] and a spin gap $\Delta_{\text{s}}$ thus opens for the SU($N$) (nuclear) spin sector in this regime. The next task is to integrate out these (nuclear) spin degrees of freedom to derive an effective Hamiltonian for the remaining degrees of freedom in the low-energy limit $E \ll \Delta_{\text{s}}$ from which the physical properties of the second regime of the RG approach will be determined.
#### SU(2)$_\text{o}$ symmetric case. {#sec:SU2o_symmetry}
Let us first consider the SU(2)$_{\text{o}}$ symmetric case to derive the low-energy limit $E \ll \Delta_\text{s}$. In this case, the model (\[lowenergyham\]) simplifies as: $$\begin{split}
\mathcal{H}^{\text{SU(2)}_{\text{o}}}_{\text{int}} =&
g_1 J_\text{L}^a J_\text{R}^a +
g_2 \mathcal{J}_\text{L}^{a,i} \mathcal{J}_\text{R}^{a,i}
+ g_4 \, \mathbf{j}_\text{L} \cdot \mathbf{j}_\text{R}
\\
& + g_6 J_{\text{c L}} J_{\text{c R}}
+ g_7 \left(S^{\alpha \beta +}_{\text{L}} S^{\alpha \beta -}_{\text{R}} +\textrm{H.c.}\right)
\\
& + \frac{g_8}{2} \left[ A^{\alpha \beta +}_{mn \text{L}} A^{\alpha \beta -}_{mn \text{R}}
+\textrm{H.c.} \right] ,
\end{split}
\label{lowenergyhamsu2}$$ since $g_2=g_3, g_4= g_5$ and $g_8 = g_9$ as a consequence of the SU(2)$_\text{o}$-symmetry. At this point, we need to express all operators appearing in Eq. (\[lowenergyhamsu2\]) in the U(1)$_\text{c} \times$ SU(2)$_N \times$ SU($N$)$_2$ basis. To this end, we will use the so-called non-Abelian bosonization [@Knizhnik-Z-84; @Affleck-NP86]: $$\begin{split}
& L^{\dagger}_{m \alpha} R_{n \beta} \simeq \exp \left( i \sqrt{2 \pi /N} \Phi_\text{c} \right) g_{nm} G_{\beta \alpha} ,
\\
& R^{\dagger}_{m \alpha} L_{n \beta} \simeq \exp \left( - i \sqrt{2 \pi /N} \Phi_\text{c} \right) g^{\dagger}_{mn}
G^{\dagger}_{\alpha \beta} ,
\end{split}
\label{nonabelboso}$$ where the charge field $\Phi_{\text{c}} $ is a compactified bosonic field with radius $R_\text{c} = \sqrt{N/2\pi} $: $ \Phi_{\text{c}} \sim \Phi_{\text{c}} + \sqrt{2 \pi N}$. This field describes the low-energy properties of the charge degrees of freedom. In Eq. , $g$ (respectively $G$) is the SU(2)$_N$ (respectively SU($N$)$_2$) primary field with spin-1/2 (respectively which transforms in the fundamental representation of SU($N$)). The scaling dimensions of these fields are given as $$\Delta_{g}= \frac{3}{N+2} \, , \;\; \Delta_{G} = \frac{N^{2}-1}{N(N+2)}$$ (see Appendix \[sec:CFT-data\]) so that Eq. (\[nonabelboso\]) is satisfied at the level of the scaling dimension: $1 = 1/2N + 3/(N+2) + (N^2-1)/N(N+2)$.
By the correspondence (\[nonabelboso\]), the different operators of the low-energy effective Hamiltonian can then be expressed in terms of the U(1)$_\text{c} \times$ SU(2)$_N \times$ SU($N$)$_2$ basis. Let us first find the decomposition of $\mathcal{J}_\text{L}^{a,i} \mathcal{J}_\text{R}^{a,i}$ of Eq. (\[lowenergyhamsu2\]). Using the SU($N$) identity $$\sum_a T^a_{\alpha \beta} T^a_{\gamma \rho} =
\frac{1}{2} \left(\delta_{\alpha \rho} \delta_{\beta \gamma}
- \frac{1}{N}\; \delta_{\alpha \beta} \delta_{\gamma \rho} \right),
\label{SUNident}$$ and ${\vec \sigma}_{m n} \cdot {\vec \sigma}_{p q} =
2 \left(\delta_{m q} \delta_{n p}
- \frac{1}{2}\; \delta_{m n} \delta_{p q} \right)$, we obtain: $$\begin{split}
\mathcal{J}_\text{L}^{a,i} \mathcal{J}_\text{R}^{a,i} = &
- \frac{1}{2} L^{\dagger}_{l \alpha} R_{l \alpha} R^{\dagger}_{m \beta} L_{m \beta}
+ \frac{1}{4} L^{\dagger}_{l \alpha} R_{m \alpha} R^{\dagger}_{m \beta} L_{l \beta}
\\
&+ \frac{1}{2N} L^{\dagger}_{l \alpha} R_{l \beta} R^{\dagger}_{m \beta} L_{m\alpha}
- \frac{1}{4N} L^{\dagger}_{l \alpha} R_{m \beta} R^{\dagger}_{m \beta} L_{l \alpha} .
\end{split}
\label{g2term}$$ Using Eq. (\[nonabelboso\]), we get: $$\begin{split}
\mathcal{J}_\text{L}^{a,i} \mathcal{J}_\text{R}^{a,i} = &
- \frac{1}{2} \left[ {\rm Tr} \left( g \right) {\rm Tr} \left( g^{\dagger} \right)
- \frac{1}{2} g_{mn} g^{\dagger}_{mn} \right] \\
& \times \left[
{\rm Tr} \left( G^{\dagger} \right) {\rm Tr} \left( G \right)
- G_{\beta \alpha} G^{\dagger}_{\beta \alpha} /N \right] .
\end{split}
\label{g2termbis}$$
Now we use the expression of the trace of the SU(2)$_N$ primary field which transforms in the spin-1 representation that we have derived in Appendix \[sec:CFT-data\] \[Eq. \] and a similar one for the SU($N$)$_2$ primary field in the adjoint representation of SU($N$): $${\rm Tr} \left( \Phi^{\text{SU($N$)}_2}_{\text{adj}} \right) = {\rm Tr} \left( G^{\dagger} \right) {\rm Tr} \left( G \right)
- \frac{1}{N} G_{\beta \alpha} G^{\dagger}_{\beta \alpha} ,
\label{adjointSUN}$$ so that Eq. (\[g2term\]) simplifies as follows: $$\mathcal{J}_\text{L}^{a,i} \mathcal{J}_\text{R}^{a,i} \sim
- {\rm Tr} \left( \Phi^{\text{SU(2)}_N}_{j=1} \right)
{\rm Tr} \left( \Phi^{\text{SU($N$)}_2}_{\text{adj}} \right) .
\label{g2decomp}$$ The expression of the operator $S^{\alpha \beta +}_{\text{L}} S^{\alpha \beta -}_{\text{R}}$ in Eq. (\[lowenergyhamsu2\]) in the U(1)$_\text{c} \times$ SU(2)$_N \times$ SU($N$)$_2$ basis can be obtained by observing that $S^{\alpha \beta +}_{\text{L}} $ is symmetric with respect to the exchange $ \alpha \leftrightarrow \beta$ and a singlet under the SU(2) orbital. The decomposition will then involve the SU($N$)$_2$ primary field in the symmetric representation of SU($N$) with dimension $N(N+1)/2$: $$S^{\alpha \beta +}_{\text{L}} S^{\alpha \beta -}_{\text{R}} \sim \exp \left(i \sqrt{8 \pi /N} \Phi_\text{c} \right)
{\rm Tr} \left( \Phi^{\text{SU($N$)}_2}_{\text{s}} \right) .
\label{g7decomp}$$ Finally, the last operator in Eq. (\[lowenergyhamsu2\]) is symmetric under the SU(2) orbital symmetry and antisymmetric with respect to the exchange $ \alpha \leftrightarrow \beta$ of SU($N$). Therefore, it will involve the spin 1 operator $\Phi^{\text{SU(2)}_N}_{j=1}$ and SU($N$)$_2$ primary field in the antisymmetric representation of SU($N$) with dimension $N(N-1)/2$: $$\begin{aligned}
A^{\alpha \beta +}_{mn L} A^{\alpha \beta -}_{mn R}
\sim
e^{ i \sqrt{8 \pi /N} \Phi_\text{c} } {\rm Tr} \left( \Phi^{\text{SU(2)}_N}_{j=1} \right)
{\rm Tr} \left( \Phi^{\text{SU($N$)}_2}_{\text{a}} \right).
\label{g8decomp}\end{aligned}$$
In the low-energy limit $E \ll \Delta_\text{s}$, we can average the SU($N$) degrees of freedom in the decompositions , , and to get the effective interacting Hamiltonian which controls the physics in the second region of the RG analysis: $$\begin{split}
\mathcal{H}^{\text{SU(2)}_\text{o}}_{\text{eff}} =&
\lambda_2 \text{Tr} \left( \Phi^{\text{SU(2)}_N}_{j=1} \right)
+ g_4 \, \mathbf{j}_{\text{L}} \cdot \mathbf{j}_{\text{R}} \\
&+ \frac{2N g_6}{\pi} \; \partial_x \Phi_{\text{c L}} \partial_x \Phi_{\text{c R}}
+ \lambda_7 \cos \left(\sqrt{8 \pi /N} \Phi_\text{c} \right) \\
&
+ \lambda_8 {\rm Tr} \left( \Phi^{\text{SU(2)}_N}_{j=1} \right) \cos \left(\sqrt{8 \pi /N} \Phi_\text{c} \right) ,
\end{split}
\label{effhamregion3}$$ where we have used the bosonized description of the chiral charge currents: $J_{\text{c L,R}} = \sqrt{2N/ \pi} \; \partial_x \Phi_{\text{c L,R}}$. In Eq. (\[effhamregion3\]), the coefficients are phenomenological since they involve the form factors of the SU($N$) operators in the integrable model with SU($N$)$_2$ current-current interaction which are not known to the best of our knowledge: $ \lambda_2 \simeq - 2 g_2 \left\langle {\rm Tr} \left( \Phi^{\text{SU($N$)}_2}_{\rm adj} \right)\right\rangle$, $ \lambda_{7,8} \simeq g_{7,8}
\left\langle \text{Tr} \left( \Phi^{\text{SU($N$)}_2}_{\rm S,A} \right) + \text{H.c.} \right\rangle$. We assume, in the following, that the expectation values of the SU($N$)$_2$ operators are positive. We can safely neglect the last term ($\lambda_{8}$) in Eq. (\[effhamregion3\]) which is less relevant than the perturbations with $ \lambda_2$ and $ \lambda_7$ to obtain the following residual interaction for the charge and the orbital sectors: $$\begin{split}
\mathcal{H}^{\text{\text{SU(2)}}_\text{o}}_{\rm eff} =&
\lambda_2 {\rm Tr} \left( \Phi^{\text{SU(2)}_N}_{\rm j=1} \right)
+ g_4 \, \mathbf{j}_{\text{L}} \cdot \mathbf{j}_{\text{R}} \\
&+ \lambda_7 \cos \left(\sqrt{8 \pi K_\text{c} /N} \Phi_\text{c} \right),
\end{split}
\label{effhamregion3fin}$$ where the charge Luttinger parameter $K_\text{c}$ satisfies $$K_\text{c} = \frac{1}{\sqrt{1 + 2 N g_6/\pi v_\text{F}}} < 1,
\label{Luttingerpara}$$ since $g_6 >0$ from the numerical solution of the RG flow in the second region.
Therefore, for the energy scale lower than the gap $\Delta_{\text{s}}$ in the nuclear-spin sector, the effective Hamiltonian for the charge degrees of freedom is the well-known sine-Gordon model at $\beta_{\text{c}}^2 = 8 \pi K_\text{c} /N$. The model is known to develop a charge gap $\Delta_\text{c}$ for all $N$ satisfying $K_\text{c} < N$, which is always the case as far as the weak-coupling expression is valid. The development of the strong-coupling regime of the sine-Gordon model is accompanied by the pinning of the charged field on either of the two minima: $$\langle \Phi_{\text{c}} \rangle = \sqrt{\frac{N\pi}{8 K_\text{c}}} + p \sqrt{N \pi/2K_\text{c}} \quad
(p=0,1) \; ,
\label{pinningcharge}$$ since $\lambda_7 > 0$ in the second region of the RG flow.
For energy smaller than the charge gap $\Delta_\text{c}$, the effective interaction (\[effhamregion3fin\]) governing the fate of the orbital degrees of freedom simplifies as follows: $$\begin{aligned}
\mathcal{H}^{\text{SU(2)}_\text{o}}_{\text{eff}} =
\lambda_2 {\rm Tr} \left( \Phi^{\text{SU(2)}_N}_{\rm j=1} \right)
+ g_4 \, \mathbf{j}_{\text{L}} \cdot \mathbf{j}_{\text{R}},
\label{effhamregion3orbital}\end{aligned}$$ which is nothing but the low-energy theory of the spin-$N/2$ SU(2) Heisenberg chain derived by Affleck and Haldane in Ref. . This is quite natural in view of the strong-coupling effective Hamiltonian obtained in Sec. \[sec:strong-coupling\].
The nature of the ground state of this Hamiltonian can be inferred from a simple semiclassical approach. The operator with the coupling constant $\lambda_{2}$ in Eq. has the scaling dimension $4/(N+2)$ and is strongly relevant. By using Eq. , the minimization of that operator in the second regime of the RG flow with $\lambda_2 >0$ (since $g_2 <0$) gives the condition $\text{Tr} \,g =0$, $g$ being an SU(2) matrix. We have thus $g = i \boldsymbol{\sigma} \cdot {\mathrm{\bf n}}$, with ${\mathrm{\bf n}}$ being an unit vector. From Eq. (\[nonabelboso\]), one may expect that the ‘dimerization’ operator for the orbital pseudo spin $\mathbf{T}_i = c^{\dagger}_{ m \alpha,\,i} \boldsymbol{\sigma}_{m n} c_{ n \alpha,\,i}/2$ would be related, when $E \ll \Delta_{\text{c}}$, to $g$ as $$(-1)^{i} {\mathrm{\bf T}}_{i+1} {\cdot} \mathbf{T}_i \sim \text{Tr}\, g .
\label{orbitaldim}$$ Therefore, the ground state is not dimerized when $\lambda_2 >0$. The nature of the phase can be determined by exploiting the result of Affleck and Haldane in Ref. that model (\[effhamregion3orbital\]) with $g = i \boldsymbol{\sigma} \cdot {\mathrm{\bf n}}$ is the non-linear sigma model with the topological angle $\theta = \pi N$. Since $N$ is even in our cold fermion problem, the topological term is trivial and the resulting model is then equivalent to the non-linear sigma model which is a massive field theory in $(1+1)$-dimensions. [@Zamolodchikov-Z-79] As is well-known, the latter model describes the physics of integer-spin Heisenberg chain in the large-spin limit [@Haldane-PLA-83].
To summarize, in the $\text{SU(2)}_\text{o}$ symmetric case, the second region of the RG flow describes the emergence of a non-degenerate gapful phase with no CDW or SP ordering. Such phase is an Haldane phase for the orbital pseudo spin $\mathbf{T}$, i.e., the OH phase that we found in the strong-coupling investigation for all even $N$ (see Sec. \[sec:strong-coupling\]). The resulting OH phase exhibits an hidden ordering which is revealed by a non-local string order parameter. On top of this hidden ordering, the OH phase has edge state with pseudo spin $T_{\text{edge}}= N/4$. According to Ref. , this is a SPT phase when $N/2$ is odd.
#### U(1)$_\text{o}$ symmetric case. {#sec:U1o_symmetry}
We now investigate the nature of the RG flow in the second regime in the generic case $J \neq J_{z}$ with an U(1)$_\text{o}$ symmetry. For energy $E \ll \Delta_\text{c}$, the interacting part of the effective Hamiltonian for the orbital sector now takes the following anisotropic form: $$\begin{split}
\mathcal{H}^{\text{U(1)}_\text{o}}_{\rm eff} =&
\lambda_{2\|} \left( \Phi^{1}_{1, 1} + \Phi^{1}_{-1, -1} \right) +
\lambda_{2\perp} \Phi^{1}_{0, 0} \\
&+ \frac{g_{4\perp}}{2}\left(j_{\text{L}}^+j_{\text{R}}^-+\textrm{H.c.}\right) +g_{4\|} j_{\text{L}}^zj_{\text{R}}^z,
\end{split}
\label{HamU1}$$ where the SU(2)$_N$ primary operators with spin $j = 0,\ldots, N/2$ are denoted by $\Phi^{j}_{m, \bar m}$ ($-j \le m, \bar m \le j$) with scaling dimension $d_j = 2 j (j+1)/(N+2)$ (see Appendix \[sec:CFT-data\]).
The low-energy properties of model (\[HamU1\]) can then be determined by introducing ${\mathbb{Z}}_N$ parafermion degrees of freedom and relating the fields of the SU(2)$_N$ CFT to those of the U(1)$_\text{o}$ CFT. Such a mapping is realized by the conformal embedding: ${\mathbb{Z}}_N$ $\sim$ SU(2)$_N$ / U(1)$_\text{o}$, which defines the series of the $\mathbb{Z}_N$ parafermionic CFTs with central charge $c= 2 (N -1)/(N+2)$. [@Zamolodchikov-F-JETP-85; @Gepner-Q-87] These CFTs describe the critical properties of two-dimensional ${\mathbb{Z}}_N$ generalizations of the Ising model,[@Zamolodchikov-F-JETP-85] where the lattice spin $\sigma_r$ takes values: $e^{i 2 \pi m /N }, m=0, \ldots, N-1$ and the corresponding generalized Ising lattice Hamiltonian is ${\mathbb{Z}}_N$ invariant. In the scaling limit, the conformal fields $\sigma_k$ with scaling dimensions $\Delta_k = k(N-k)/N(N+2)$ $(k=1, \ldots, N -1)$ describe the long-distance correlations of $\sigma^{k}_r$ at the critical point. [@Zamolodchikov-F-JETP-85] In the context of cold atoms, the ${\mathbb{Z}}_N$ CFT is also very useful to map out the zero-temperature phase diagram of general 1D higher-spin cold fermions. [@Nonne-L-C-R-B-11; @Lecheminant-B-A-05; @Lecheminant-A-B-08]
The orbital SU(2)$_N$ currents can be directly expressed in terms of the first parafermionic current $\Psi_{1 L,R}$ with scaling dimension $1 - 1/N$ and a bosonic field $\Phi_{o}$ which accounts for orbital fluctuations: [@Zamolodchikov-F-JETP-85] $$\begin{aligned}
j_{\text{L,R}}^{\dagger}
&\simeq& \frac{\sqrt{N}}{2\pi}
:\exp \left(\pm i \sqrt{8 \pi/N} \; \Phi_{\text{o L,R}} \right): \Psi_{1 \text{L,R}}
\nonumber \\
j_{\text{L,R}}^{z}
&\simeq& \sqrt{\frac{N}{2\pi}} \partial_x \Phi_{\text{o L,R}} ,
\label{parasu2Ncurrent}\end{aligned}$$ where the orbital bosonic field $\Phi_{\text{o}} = \Phi_{\text{o L}} + \Phi_{\text{o R}}$ is a compactified bosonic field with radius $R_\text{o} = \sqrt{N/2\pi} $: $ \Phi_{\text{o}} \sim \Phi_{\text{o}} + \sqrt{2 \pi N}$. Under the ${\mathbb{Z}}_N$ symmetry, the parafermionic currents $\Psi_{1 \text{L,R}}$ transform as [@Zamolodchikov-F-JETP-85] $$\begin{aligned}
\Psi_{1 \text{L,R}} &\rightarrow& e^{i 2 \pi k /N} \Psi_{1 \text{L,R}} ,
\label{chargepara}\end{aligned}$$ with $k=0,\ldots, N-1$. Using Eq. (\[parasu2Ncurrent\]), we identify the ${\mathbb{Z}}_N$ symmetry of the parafermions directly on the Dirac fermions through: $$L_{g\alpha} \to e^{- i \pi k/N} L_{g\alpha}, \; \; \; L_{e\alpha} \to e^{i \pi k/N} L_{e\alpha},
\label{ZN}$$ with a similar transformation for the right-moving Dirac fermions. It is easy to check that the low-energy description (\[lowenergyham\]) is invariant under this transformation, and thus ${\mathbb{Z}}_N$-symmetric. Using the definition (\[contlimitDirac\]), one can deduce a lattice representation of this ${\mathbb{Z}}_N$ in terms of the original fermions $c_{m \alpha,i}$: $$c_{g\alpha} \to e^{- i \pi k/N} c_{g\alpha}, \; \; \; c_{e \alpha} \to e^{i \pi k/N} c_{e \alpha} ,
\label{ZNlattice}$$ which is indeed a symmetry of all lattice models introduced in Sec. \[sec:models-strong-coupling\].
As described in the Appendix, the SU(2)$_N$ primary operators can be related to that of the ${\mathbb{Z}}_N$ CFT. Using the results (\[identiprimZnApp\]) and (\[spin1primaryparaApp\]) of Appendix \[sec:CFT-data\] and Eq. (\[parasu2Ncurrent\]), the low-energy effective Hamiltonian (\[HamU1\]) can then be expressed in terms of ${\mathbb{Z}}_N$ primary fields as follows: $$\begin{split}
\mathcal{H}^{\text{U(1)}_\text{o}}_{\rm eff} = &
\lambda_{2\|} \left\{ \mu_2 \exp \left(-i \sqrt{8 \pi/N} \; \Phi_{\text{o}} \right)
+ \textrm{H.c.} \right\} -
\lambda_{2\perp} \epsilon_1 \\
&+ \frac{g_{4\perp} N}{2 \pi} \left\{ \Psi_{\text{1 L}} \Psi^{\dagger}_{\text{1 R}}
\exp \left(i \sqrt{8 \pi/N} \; \Phi_{\text{o}} \right) + \textrm{H.c.} \right\} \\
&+ \frac{N g_{4\|}}{2 \pi} \partial_x \Phi_{\text{oL}} \partial_x \Phi_{\text{oR}},
\end{split}
\label{effectiveHamU1para}$$ where $\epsilon_1$ (respectively $\mu_2$) is the thermal (respectively second disorder) operator of the ${\mathbb{Z}}_N$ CFT with scaling dimension $4/(N+2)$ (respectively $(N-2)/N(N+2)$). In our convention, $\langle \epsilon_1 \rangle > 0 $ in a phase where the ${\mathbb{Z}}_N$-symmetry is broken so that the disorder parameters do not condense $\langle \mu_k\rangle=0$ ($k=1,\ldots, N-1$), as they are dual to the order fields $\sigma_k$. Since the second disorder and the thermal operators themselves are known to be ${\mathbb{Z}}_N$-invariant, the model (\[effectiveHamU1para\]) is invariant under the ${\mathbb{Z}}_N$-symmetry as it should be.
The low-energy effective field theory (\[effectiveHamU1para\]) appears in such different contexts as the field theory approach to the Haldane’s conjecture [@Cabra-P-R-98] and the half-filled 1D general spin-$S$ cold fermions.[@Nonne-L-C-R-B-11] It was shown[@Nonne-L-C-R-B-11] that the phase diagram of the latter model strongly depends on the parity of $N$. The numerical solution of the RG flow shows that the operator with the coupling constant $\lambda_{2\perp}$ dominates the strong-coupling regime. Such perturbation describes an integrable deformation of the ${\mathbb{Z}}_N$ CFT[@Fateev-91] which is always a massive field theory for all sign of $\lambda_{2\perp}$; when $\lambda_{2\perp} >0$ (i.e. $g_3 < 0$), we have $\langle \epsilon_1 \rangle > 0 $ and the mass is generated from the spontaneous ${\mathbb{Z}}_N$-symmetry breaking and all the order fields of the ${\mathbb{Z}}_N$ CFT condense: $\langle \sigma_k \rangle \ne 0$, while the disorder one $\langle \mu_k \rangle = 0$ for all $k=1,\ldots, N-1$.
One can immediately see that the nature of the underlying phase can be captured neither by the SP nor by the density-order parameters and since they are all invariant under the ${\mathbb{Z}}_N$ symmetry (\[ZN\]). In fact, by using the identifications (\[identiprimZnApp\]), it is straightforward to check that these order parameters involve the first disorder operator $\mu_1$ and therefore cannot sustain a long-range ordering in the ${\mathbb{Z}}_N$-broken phase. In this respect, the first regime, in which we have DSE, corresponds to a region where the ${\mathbb{Z}}_N$-symmetry is not broken spontaneously.
Since all the parafermionic operators in average to zero in the ${\mathbb{Z}}_N$ broken phase, one has to consider higher orders in perturbation theory to derive an effective theory for the orbital bosonic field $\Phi_\text{o}$. When $N$ is even, one needs the $N/2$-th order of perturbation theory to cancel out the operator $\mu_2$ in Eq. (\[effectiveHamU1para\]). The resulting low-energy Hamiltonian then reads as follows: $$\begin{split}
\mathcal{H}^{\rm even}_{\text{o}} =& \frac{v_\text{o}}{2} \left\{
\frac{1}{K_\text{o}} \left(\partial_x \Phi_\text{o}\right)^{2}
+ K_\text{o} \left(\partial_x \Theta_\text{o}\right)^{2} \right\} \\
&+ g_\text{o} \cos \left(\sqrt{2 \pi N} \; \Phi_{\text{o}} \right) ,
\end{split}
\label{hoNeven}$$ where $v_\text{o}$ and $K_\text{o}$ are the velocity and the Luttinger parameters for the orbital boson $\Phi_\text{o}$: $$K_\text{o} = \frac{1}{\sqrt{1 + N g_{4\|}/(2\pi v_\text{F}})}.
\label{Luttingerparaorbital}$$ A naive estimate of the coupling constant $g_\text{o}$ in higher orders of perturbation theory reads as: $g_\text{o} \sim - (-\lambda_{2\|})^{N/2}$.
The resulting low-energy Hamiltonian (\[hoNeven\]) which governs the physical properties of the orbital sector takes the form of the sine-Gordon model at $\beta_{\text{o}}^2 = 2 \pi N K_\text{o}$ . The latter turns out to be the effective field theory of a spin-$S=N/2$ Heisenberg chain with a single-ion anisotropy as shown by Schulz in Ref. . From the integrability of the quantum sine-Gordon model, we expect that a gap for orbital degrees of freedom opens when $K_\text{o} < 4/N$. As usual, it is very difficult to extract the precise value of the Luttinger parameter $K_\text{o}$ from a perturbative RG analysis. Along the SU(2)$_\text{o}$ line, the exact value $K_\text{o}$ is known by the SU(2)-symmetry, i.e. $K_\text{o} = 1/N$, since the $\beta_{\text{o}}^2 = 2 \pi$ sine-Gordon model is known to display a hidden SU(2) symmetry. [@Affleck-chiral-86] In the vicinity of that line, we thus expect that there is a region where $K_\text{o} < 4/N$ and a Mott-insulating phase emerges. In that situation, the orbital bosonic field is pinned into the following configurations: $$\begin{aligned}
{2}
\langle \Phi_{\text{o}} \rangle &= \sqrt{\frac{\pi}{2N}} + p \sqrt{\frac{2\pi}{N}}\; ,& & \qquad {\rm if}
\; g_\text{o} > 0 \notag \\
\langle \Phi_{\text{o}} \rangle &= p \sqrt{\frac{2\pi}{N}} \; ,& &\qquad {\rm if} \; g_\text{o} < 0 ,
\label{pinningNeven}\end{aligned}$$ where $p=0,\ldots, N-1$. This semiclassical analysis naively gives rise to a ground-state degeneracy. However, there is a gauge-redundancy in the continuum description. On top of the ${\mathbb{Z}}_N$ symmetry (\[ZN\]) of the parafermions CFT, there is an independent discrete symmetry, ${\tilde {\mathbb{Z}}}_N$, such that the parafermionic currents transform as follows: [@Zamolodchikov-F-JETP-85] $$\begin{aligned}
\Psi_{1 \text{L,R}} &\rightarrow& e^{ \pm i 2 \pi m /N} \Psi_{1 \text{L,R}} ,
\label{chargeparadual}\end{aligned}$$ with $m=0,\ldots, N-1$. The two ${\mathbb{Z}}_N$ symmetries are related by a Kramers-Wannier duality transformation. [@Zamolodchikov-F-JETP-85] The thermal operator $\epsilon_1$ is a singlet under the ${\tilde {\mathbb{Z}}}_N$ while the disorder operator $\mu_2$ transforms as: $\mu_{2} \rightarrow e^{i 4 \pi m /N} \mu_{2}$. [@Zamolodchikov-F-JETP-85] The combination of the ${\tilde {\mathbb{Z}}}_N$ (\[chargeparadual\]) and the identification on the orbital bosonic field: $$\Phi_\text{o} \sim \Phi_\text{o} - m \sqrt{\frac{2 \pi}{N}} + p \sqrt{\frac{N \pi}{2}},
\; m=0, \ldots, N -1,
\label{tildeznorbitalbose}$$ becomes a symmetry of model (\[effectiveHamU1para\]), as it can be easily seen. In fact, this symmetry is a gauge redundancy since it corresponds to the identity in terms of the Dirac fermions. Using the redundancy (\[tildeznorbitalbose\]), we thus conclude that the gapful phase of the quantum sine-Gordon model (\[hoNeven\]) is non-degenerate with ground state: $$\begin{aligned}
\langle \Phi_{\text{o}} \rangle &=& \sqrt{\frac{\pi}{2N}}, \; {\rm if} \; g_\text{o} > 0 \nonumber \\
\langle \Phi_{\text{o}} \rangle &=& 0, \; \; \; \; \; \; \; \; \; {\rm if} \; g_\text{o} < 0 .
\label{pinningGSNeven}\end{aligned}$$
The lowest massive excitations are the soliton and the antisoliton of the quantum sine-Gordon model; they carry the orbital pseudo spin: $$T^z = \pm \sqrt{N/ 2\pi} \int dx \; \partial_x \Phi_\text{o} = \pm 1,
\label{chargekinkHaldane}$$ and correspond to massive spin-1 magnon excitations.
At this point, it is worth observing that the duality symmetry $\Omega_2$ of Eq. (\[dualitiesDirac\]) plays a subtle role in the even $N$ case. Indeed, the change of sign of the coupling constants $g_{2,4}$ can be implemented by the shift: $\Phi_{\text{o}} \rightarrow \Phi_{\text{o}} + \sqrt{N/ 8\pi}$ so that the cosine term of Eq. (\[hoNeven\]) transforms as $$\cos \left(\sqrt{2 \pi N} \; \Phi_{\text{o}} \right) \to \left(-1 \right)^{N/2} \cos \left(\sqrt{2 \pi N} \; \Phi_{\text{o}} \right).
\label{dualcosNeven}$$ The latter result calls for a separate analysis depending on the parity of $N/2$.
*.*
When $N/2$ is odd, the cosine term of Eq. (\[hoNeven\]) is odd under the $\Omega_2$ duality transformation and there is thus two distinct fully gapped phases depending on the sign of $g_\text{o}$. The numerical solution of the RG equations shows that $g_2 < 0$, i.e. $\lambda_{2\|} > 0$, in the vicinity of the SU(2)$_\text{o}$ line. We thus expect that $g_\text{o} > 0$ in this region and the ground state of the sine-Gordon model (\[Luttingerparaorbital\]) with $K_\text{o} < 4/N$ is described by the pinning: $\langle \Phi_{\text{o}} \rangle = \sqrt{\pi/2N} $ \[first line of Eq. \]. The corresponding Mott-insulating phase is the continuation of the OH phase that we have found along the SU(2)$_\text{o}$ line. This phase can be described by a string-order ordermeter which takes the form: $$\begin{split}
& \lim_{|i-j| \rightarrow \infty}
\left\langle
{T}^{z}_i
e^{ i \pi \sum_{k=i+1}^{j-1}{T}^{z}_k}
{T}^{z}_j
\right\rangle \simeq \\
& \lim_{|x-y| \rightarrow \infty}
\left\langle{\sin \left( \sqrt{N \pi/2} \; \Phi_\text{o} \left( x \right) \right)
\sin \left( \sqrt{N \pi/2} \; \Phi_\text{o} \left( y \right) \right)} \right\rangle \ne 0 .
\end{split}
\label{stringorbitalNodd}$$ This result is in full agreement with the known properties of the Haldane phase when the orbital pseudo spin $T=N/2$ is odd.
According to Eq. , the duality symmetry $\Omega_2$ changes the sign of the cosine operator in the sine-Gordon model when $N/2$ is odd. Therefore, there exists yet another Mott-insulating phase obtained by the duality $\Omega_{2}$ when $K_\text{o} < 4/N$ which is characterized by the pinning: $\langle \Phi_{\text{o}} \rangle = 0$ \[the second of Eq. \]. In this phase, the string-order parameter vanishes, i.e., we have a new fully gapped non-degenerate phase which is different from the OH phase. A simple non-zero string order parameter in this phase, that we can estimate within our low-energy approach, reads as follows $$\begin{split}
& \lim_{|i-j| \rightarrow \infty}
\left\langle{\cos \left(\pi \sum_{k<i} T^{z}_k \right)
\cos \left(\pi \sum_{k<j} T^{z}_k \right)}\right\rangle \\
& \simeq \lim_{|x-y| \rightarrow \infty}
\left\langle{\cos \left( \sqrt{N \pi/2} \; \Phi_\text{o} \left( x \right) \right)
\cos \left( \sqrt{N \pi/2} \; \Phi_\text{o} \left( y \right) \right)} \right\rangle \\
&\ne 0 .
\end{split}
\label{HaldanestringN4}$$ The latter phase is expected to be the RS phase (i.e., the orbital-analogue of the large-$D$ phase with $T^z =0$) that we have already identified in the strong-coupling analysis of Sec. \[sec:strong-coupling\].
*.*
When $N/2$ is even, the cosine term of Eq. is now even under the $\Omega_2$ duality transformation and there is thus a single fully gapped phase. In this phase, we have $g_\text{o} <0$ and the orbital bosonic field is pinned when $K_\text{o} < 4/N$ into configurations: $\langle \Phi_{\text{o}} \rangle = 0$. The phase is thus characterized by the long-range ordering of the string-order parameter while the standard one vanishes. In this respect, the physics is very similar to the properties of the even-spin Haldane phase. The authors of Ref. have conjectured that there is an adiabatic continuity between the Haldane and large-$D$ phases in the even-spin case. Such continuity has been shown numerically in the spin-2 XXZ Heisenberg chain with a single-ion anisotropy by finding a path where the two phases are connected without any phase transition. [@Tonegawa-O-N-S-N-K-11] The Haldane phase for integer spin is thus equivalent to a topologically trivial insulating phase in this case. In our context, the two non-degenerate Mott insulating OH and RS (the orbital large-$D$) phases belong to the same topologically trivial phase when $N/2$ is even, while they exhibit very different topological properties for odd $N/2$.
*.*
Regardless of the parity of $N/2$, there is a room to have, on top of the Mott-insulating phases, an algebraic (metallic) one since the Luttinger parameter $K_\text{o}$ can be large in the second region of the RG flow. When $K_\text{o} > 4/N$, the interaction of the sine-Gordon model (\[hoNeven\]) becomes irrelevant and a critical Luttinger-liquid phase emerges having one gapless mode in the orbital sector. At low energies $E \ll \Delta_\text{c}$, the staggered part of the orbital-pseudo spin $\mathbf{T}_i$ simplifies as follows using the identifications (\[identiprimZnApp\]): $$\begin{aligned}
T^{+}_{\pi} &\sim& \sigma_1 e^{ i \sqrt{2 \pi/N} \;
\Theta_{\text{o}} } \left( \left\langle e^{ i \sqrt{2 \pi K_\text{c} /N} \Phi_\text{c} } \right\rangle
\left \langle \text{Tr}\, G \right \rangle \;+ \text{c.c.} \right)
\nonumber \\
T^{z}_{\pi} &\sim& \left\langle e^{ i \sqrt{2 \pi K_\text{c} /N} \Phi_\text{c} } \right\rangle \langle {\rm Tr} G \rangle
\left( \mu_1 e^{- i \sqrt{2 \pi/N} \Phi_{\text{o}} } - \mu^{\dagger}_1 e^{ i \sqrt{2 \pi/N}\Phi_{\text{o}} } \right)
\nonumber \\
&+& \text{H.c.}
\label{staggeredorbitalspin}\end{aligned}$$ Since the $\mathbb{Z}_N$-symmetry is broken in the second region of the RG flow, we have $\langle \sigma_1 \rangle \ne 0$ and $\langle \mu_1 \rangle = 0$ so that the $z$-component of $\mathbf{T}_{\pi}$ is thus short-range while the transverse ones are gapless: $T^{+}_{\pi} \sim e^{ i \sqrt{2 \pi/N} \Theta_{\text{o}} }$. Taking into account the uniform part of the $z$-component of the orbital-pseudo spin $\mathbf{T}_i$, i.e. the SU(2)$_N$ current $j^z_{\text{L}} + j^z_{\text{R}} $, we get the following leading asymptotics for the equal-time orbital pseudo spin correlations: $$\begin{aligned}
\langle T^{+} \left(x\right) T^{-} \left(0\right) \rangle
&\sim& \left(-1\right)^{x/a_0} x^{- 1/NK_\text{o}} \nonumber \\
\langle T^{z} \left(x\right) T^{z} \left(0\right) \rangle &\sim&
-\frac{NK_\text{o}}{4\pi^2 x^2}.
\label{correlBCSNeven}\end{aligned}$$ The leading instability is thus the transverse orbital correlation when $K_\text{o} > 4/N$, i.e., the formation of a critical orbital-XY phase, i.e., an orbital Luttinger-liquid phase.
![(Color online) General phase diagrams for the $N=6$ generalized Hund model (\[alkaourmodel\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplings\]). T coupling constants $(J_z, J, U)$ are parametrized by $(\theta, \phi)$ as Eqs. and the meaning of the extra bold lines is discussed in the text. The signs of $J_z$, $J$ and $U$ in each quadrant are indicated. In the region shown as “no-DSE”, RGE flow does not exhibit dynamical symmetry enlargement. For other abbreviations, see TABLE \[tab:abbreviation\]. \[fig:RG\_general\_phase\_diagram\]](RG_g2_N6){width="0.99\columnwidth"}
Phase diagrams {#sec:RG-phase-diag}
--------------
We have determined the possible phases of general 1D two-orbital SU($N$) models in the weak-coupling regime by means of the one-loop RG analysis combined with CFT techniques. We now exploit all these results to map out the zero-temperature phase diagram of the generalized Hund model and the $g$-$e$ model defined in Sec. \[sec:models-strong-coupling\]. The phase diagram of the $N=2$ $p$-band model is presented in Appendix \[sec:N2-p-band-continuum\], together with the study of its low-energy limit. The correspondence between the parameters used in the phase diagrams and the physical interactions is summarized in TABLE \[tab:three-models\].
Before solving numerically the one-loop RG analysis, one immediately observes that our global approach of the phases in the weak-coupling regime does not give any SPT phases when $N>2$ in stark contrast to the strong coupling result of Sec. \[sec:strong-coupling\]. It might suggest that there is no adiabatic continuity between weak and strong coupling regimes and necessarily a quantum phase transition occurs in some intermediate regime which is not reachable by the one-loop RG analysis. In this respect, a two-loop analysis might be helpful but it is well beyond the scope of this work. The possible occurence of a quantum phase transition will be investigated in Sec. \[sec:DMRG\] by means of DMRG calculations to study the extension of the SU(4) SPT phase.
The sets of first-order differential equations obtained with the one-loop RG analysis, $\{ {\dot g}_i \} = \{ \partial g_i/ \partial l\}$, $l$ being the RG time, can be solved numerically with Runge-Kutta methods. The initial conditions $g_{i,0}$ depend on the lattice model and we loop on values of the couplings taken in $[-0.1;0.1]$ to scan the zero-temperature phase diagrams in the weak coupling regime. For each run, the couplings $g_i$ flow to the strong coupling regime as the RG time increases. The procedure is stopped at $l_{\text{max}}$ when one of the couplings, which turns out to be $g_1$ (see Sec. \[sec:Non-degenerate\_Mott\_insulating\_phases\]), reaches an arbitrary large value $g_{\text{max}}$. Typically, we choose $g_{\text{max}} \ge 10^{10}$ so that the directions taken by the RG flow in the far IR appear clearly. For simplicity, we consider renormalized ratios $g_i(l_{\text{max}})/g_1(l_{\text{max}})$. For instance, when the procedure stops in the SP phase, all the couplings have reached a value $g_i(l_{\text{max}})/g_1(l_{\text{max}}) \sim +1$, as a signature of the SO(4$N$) maximal DSE.
As discussed in Sec. \[sec:RG\_analysis\], we distinguish in the weak coupling limit two types of regimes: phases with DSE and non-degenerate Mott insulating phases. On the one hand, the first ones can be readily identified by looking at the ratios $g_i(l_{\text{max}})/g_1(l_{\text{max}})$ that are either $+1$ in the SP phase or $\pm 1$ in the phases obtained by applying the duality symmetries Eqs. (\[eq:Omega\_1\]-\[eq:Omega\_3\]). On the other hand, couplings $g_{2,4,5,8}$ flow very slowly to the strong coupling regime in the non-degenerate phases. Determining the exact nature of the phase is thus more approximative in that case. In particular, as detailed in Sec. \[sec:U1o\_symmetry\], the sign of $g_2$ allows to distinguish between OH and the RS phase only in the $N/2$ odd case. Next, we therefore show results for $N=6$. [^9] In order to have an overview of the phases that appear, we first compute the general phase diagram of the generalized Hund model (\[alkaourmodel\]) for all $J_z$, $J$ and $U$, see Fig. \[fig:RG\_general\_phase\_diagram\]. We solve the RG equations (\[RG\]) using the initial conditions (\[couplings\]) and introduce sphere variables: $$\begin{aligned}
U &=& R \cdot \sin 4\phi \cdot\cos\theta \nonumber \\
J &=& R \cdot \sin 4\phi \cdot\sin\theta \nonumber \\
J_z &=& R \cdot \cos 4\phi ,
\label{eq:sphere_variable} \end{aligned}$$ where $R=0.1$. Eight quadrants are required to get all the possible combinations of signs for $U$, $J$ and $J_z$ ($\theta \in [0,2\pi]$ and $\phi \in [0,\frac{\pi}{4}]$). We directly identify three phases with DSE (SP, CDW and ODW) while the SP$_{\pi}$ phase obtained by applying the duality $\Omega_3$ (\[eq:Omega\_3\]) is not realized. [^10] The SU(2)$_o$ symmetry ($J=J_z$) corresponds to $\theta = \arcsin (\cot 4\phi)$ and is showed with bold dashed lines in Fig. \[fig:RG\_general\_phase\_diagram\]. In the ‘no-DSE’ region, the sign of $g_2$ changes on the blue line and the nature of the phases obtained is discussed next, in special cuts of the phase diagram. The one-loop RG analysis does not allow to confirm if the SU(2)$_{\text{o}}$ line is exactly at the ODW/‘No DSE’ transition but the latter is clearly in its vicinity as seen in Fig. \[fig:RG\_general\_phase\_diagram\].
### Generalized Hund model
![(Color online) Phase diagram for the $N=6$ generalized Hund model (\[alkaourmodel\]) with SU(2)$_o$ symmetry obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplings\]). \[fig:RG\_phasediag\_alka\_N6\_SU2o\]](RG_Phase_diag_Hund_N6_SU2o){width="0.5\columnwidth"}
Let us continue with the generalized Hund model (\[alkaourmodel\]) and take a closer look at special cuts in the general phase diagram Fig. \[fig:RG\_general\_phase\_diagram\].
![(Color online) Phase diagram for the $N=6$ generalized Hund model (\[alkaourmodel\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplings\]). From top to bottom, and from right to left: $J_z/t=-0.03$, $J_z/t=0$ and $J_z/t=0.03$. \[fig:RG\_phasediag\_alka\_N6\_Jz\]](RG_Phase_diag_Hund_N6_Jz_Neg "fig:"){width="0.5\columnwidth"} ![(Color online) Phase diagram for the $N=6$ generalized Hund model (\[alkaourmodel\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplings\]). From top to bottom, and from right to left: $J_z/t=-0.03$, $J_z/t=0$ and $J_z/t=0.03$. \[fig:RG\_phasediag\_alka\_N6\_Jz\]](RG_Phase_diag_Hund_N6_Jz0 "fig:"){width="0.49\columnwidth"}![(Color online) Phase diagram for the $N=6$ generalized Hund model (\[alkaourmodel\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplings\]). From top to bottom, and from right to left: $J_z/t=-0.03$, $J_z/t=0$ and $J_z/t=0.03$. \[fig:RG\_phasediag\_alka\_N6\_Jz\]](RG_Phase_diag_Hund_N6_Jz_Pos "fig:"){width="0.49\columnwidth"}
#### SU(2)$_\text{o}$ symmetric case. {#su2_texto-symmetric-case.}
We first consider the case of SU(2) orbital symmetry ($J=J_z$, along bold dashed lines in Fig. \[fig:RG\_general\_phase\_diagram\]). We focus on $N=6$, although the position of the phases is almost not sensitive to the value of $N$ in this case. In Fig. \[fig:RG\_phasediag\_alka\_N6\_SU2o\], we identify three regions: the SP phase, the degenerate CDW phase obtained by applying the duality symmetry $\Omega_1$ (\[eq:Omega\_1\]) and a region that displays no DSE with $|g_{2,4,5,8}(l_{\text{max}})| \ll g_{\text{max}}$. The latter was identified in Sec. \[sec:SU2o\_symmetry\] as the non-degenerate OH phase for even $N$. It is a SPT phase for $N/2$ odd. Besides, on the particular SU(2$N$) line $J = 0$, for $U>0$ (respectively $U<0$) we recover the SP (respectively CDW) phase expected for the repulsive (respectively attractive) SU(2$N$) Hubbard model at half-filling.
#### U(1)$_\text{o}$ symmetric case. {#u1_texto-symmetric-case.}
We now turn to the phase diagrams of the generic case of U(1) orbital symmetry ($J \ne J_z$) at $N=6$. We chose arbitrary cuts of the general phase diagram Fig. \[fig:RG\_general\_phase\_diagram\] at constant $J_z$: $J_z =-0.03$, $J_z = 0 $ and $J_z = 0.03$ (see Fig. \[fig:RG\_phasediag\_alka\_N6\_Jz\]). As discussed the *$N/2$ odd case* of Sec. \[sec:U1o\_symmetry\], the sign of $g_2$ allows us to determine if the non-degenerate Mott insulating phase (blue ‘no-DSE’ region in Fig. \[fig:RG\_general\_phase\_diagram\]) is either OH or RS. We find that the change of sign takes place at $J_z^* < 0$. The one-loop RG analysis does not allow us to determine the value of the Luttinger parameter $K_\text{o}$ except in the vicinity of the SU(2)$_\text{o}$ symmetric line where $K_\text{o}$ is fixed by symmetry. We cannot thus conclude that the phases, obtained by varying $J_z$, are indeed fully gapped from this analysis. However, the DMRG calculations in this regime of parameters strongly support that $K_\text{o}$ is small enough to get gapful phases. In Fig. \[fig:RG\_phasediag\_alka\_N6\_Jz\], for $J_z=0$ and $J_z>0$, we find thus that the non-degenerate Mott insulating phase is the RS phase, while for $J_z<0$, a transition takes place between RS and OH. At the transition, the line $g_2=0$ (bold dashed line in Fig. \[fig:RG\_phasediag\_alka\_N6\_Jz\], top panel) corresponds to the Luttinger critical line in which the cosine term of Eq. (\[hoNeven\]) is canceled. Interestingly, the phase diagram for $J_z<0$ obtained in the weak coupling regime is in agreement with the prediction from the strong coupling regime, i.e., an OH region followed by a RS region as $|J-J_z|$ increases (see Sec. II D 2).
### $g$-$e$ model
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![(Color online) Phase diagram for the $N=6$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplingsGorsh\]). From top to bottom, and from right to left: SU(2)$_o$ symmetry, $V_{\text{ex}}^{g\text{-}e}/t=-0.06$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=0.02$. \[fig:RG\_phasediag\_Gorshkov\_N6\_Vex\]](RG_Phase_diag_Hae_N6_SU2o "fig:"){width="0.5\columnwidth"} ![(Color online) Phase diagram for the $N=6$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplingsGorsh\]). From top to bottom, and from right to left: SU(2)$_o$ symmetry, $V_{\text{ex}}^{g\text{-}e}/t=-0.06$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=0.02$. \[fig:RG\_phasediag\_Gorshkov\_N6\_Vex\]](RG_Phase_diag_Hae_N6_Vex_Neg "fig:"){width="0.5\columnwidth"}
![(Color online) Phase diagram for the $N=6$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplingsGorsh\]). From top to bottom, and from right to left: SU(2)$_o$ symmetry, $V_{\text{ex}}^{g\text{-}e}/t=-0.06$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=0.02$. \[fig:RG\_phasediag\_Gorshkov\_N6\_Vex\]](RG_Phase_diag_Hae_N6_Vex0 "fig:"){width="0.5\columnwidth"} ![(Color online) Phase diagram for the $N=6$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by solving numerically the one-loop RG equations (\[RG\]) with initial conditions (\[couplingsGorsh\]). From top to bottom, and from right to left: SU(2)$_o$ symmetry, $V_{\text{ex}}^{g\text{-}e}/t=-0.06$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=0.02$. \[fig:RG\_phasediag\_Gorshkov\_N6\_Vex\]](RG_Phase_diag_Hae_N6_Vex_Pos "fig:"){width="0.5\columnwidth"}
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For completeness, we also present the phase diagrams of the $g$-$e$ model (\[eqn:Gorshkov-Ham\]) with coupling constants $U_{gg} = U_{ee} = U_{mm}, V$ and $V_{\text{ex}}^{g\text{-}e}$. The mapping to the couplings $J$, $J_z$ and $U$ is defined in Eqs. (\[eqn:Gorshkov-to-Hund\]), in particular, $V_{\text{ex}}^{g\text{-}e}=J$. As explained in Sec. \[sec:continuum\_description\], the $g$-$e$ model shares the same continuum Hamiltonian with the generalized Hund model. Only the initial conditions differ and we solve the set of equation (\[RG\]) starting from (\[couplingsGorsh\]). In Fig. \[fig:RG\_phasediag\_Gorshkov\_N6\_Vex\], we show the phase diagrams for the SU(2)$_\text{o}$-symmetric (i.e. $V=U_{mm}-V_{\text{ex}}^{g\text{-}e}$) cases, $V_{\text{ex}}^{g\text{-}e}/t=-0.06$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=0.02$. In the presence of the orbital SU(2)$_\text{o}$-symmetry, we recover the SP, CDW and OH phases from Fig. \[fig:RG\_phasediag\_alka\_N6\_SU2o\]. For $V_{\text{ex}}^{g\text{-}e}> 0$ and $V_{\text{ex}}^{g\text{-}e}= 0$, the phase diagram exhibits only regions with DSE (SP, CDW and ODW), in agreement with the phase found for $J \ge 0$ in the preceding section. Their positions are little affected by the value of $V_{\text{ex}}^{g\text{-}e}$. Finally, for $V_{\text{ex}}^{g\text{-}e} < 0$, as for $J <0$, we have a non-degenerate Mott insulating region in which the sign of $g_2$ changes. We thus identify the OH and the RS regions.
DMRG calculations {#sec:DMRG}
=================
We now turn to numerical simulations using DMRG algorithm in order to determine some of the phase diagrams that were discussed in the previous sections (Sections \[sec:strong-coupling\] and \[sec:RG-phase-diag\]), namely the $g$-$e$ model with orbital SU(2)$_{\text{o}}$ symmetry (\[eqn:Gorshkov-Ham\]), the generalized Hund model with or without SU(2)$_\text{o}$ symmetry , and the $p$-band model . As already mentioned in Sec. \[sec:Gorshkov-Hamiltonian\], for concreteness we assume that the two orbitals behave in a similar manner, i.e. we restrict ourselves to the case $$t_g = t_e=t, \; \; U_{g g} = U_{ee} =U_{mm}, \;\; \mu_{g}=\mu_{e}$$ of the $g$-$e$ model or the generalized Hund one . The parametrization used in the three models ($g$-$e$ model, generalized Hund model, and $p$-band model) considered here is summarized in TABLE \[tab:three-models\]. Also the definitions of the abbreviations used in the phase diagrams are given in TABLE. \[tab:abbreviation\].
This numerical investigation is especially needed (i) to check our weak-coupling predictions (Sec. \[sec:RG-phase-diag\]) and (ii) to go beyond this regime and make connection with strong-coupling results (Sec. \[sec:strong-coupling\]). Moreover, it allows us to get precise numerical estimates of the locations of the phases and the transitions among them, which is of fundamental importance to decide whether they could be accessed experimentally. Typically, we used open boundary conditions, keeping between 2000 and 4000 states depending on the model and the parameters in question in order to keep a discarded weight below $10^{-6}$. Note also that for the sake of the efficiency of the simulations, for all models with $N=4$ and for the $p$-band with $N=2$ too, we map the *original* two-orbital SU($N$) models onto the *equivalent* (pseudo)spin-1/2 (where the pseudo spin corresponds to the orbital) fermionic models on some $N$-leg ladder (with generalized rung interactions which are tailored to reproduce the original interactions) shown in Fig. \[fig:alkaline-Nleg\]. As a last remark, let us mention that we worked at half-filling and except for the $p$-band model, we have implemented the abelian U(1) symmetry corresponding to the conservation of particles in each orbital.
In order to map out the phase diagrams, we worked at fixed length $L=36$ (for $N=4$) or $L=64$ (for $N=2$) and measured the local quantities (densities, pseudospin densities, kinetic energies, etc.) as well as the presence/absence of edge states. One may wonder why we do not use the string order parameters introduced in Sec. \[sec:non-local-OP\] in determining (a part of) the phase diagram. In fact, for purely bosonic models, the string order parameters combined with, e.g., the Binder-parameter analysis may yield a reasonably good results [@Totsuka-N-H-S-95]. However, the string order parameters are defined for [*fixed*]{} SU($N$) ‘spins’ which are meaningful only deep inside the Mott phases [@Manmana-H-C-F-R-11]. When the charge fluctuations are not negligible, entanglement spectrum necessarily contains the contribution from the fermionic sector[@Hasebe-T-13], for which the relation between the SPTs and the string order parameters mentioned in Sec. \[sec:non-local-OP\] is not very clear. For this reason, in order for the search in the full parameter space, more conventional methods seem robust. We refer the interested reader to Refs. which contain more details on our procedure.
N=2 g-e model {#sec:N2-Gorshkov}
-------------
For completeness, we present, in Figs. \[fig:phasediag\_Gorshkov\_N2\_Vex\], some phase diagrams of the $g$-$e$ model (\[eqn:Gorshkov-Ham\]) with $N=2$ which exhibit a large variety of phases: (i) charge density wave (CDW), (ii) orbital density wave (ODW), (iii) spin-Peierls (SP), (iv) charge Haldane (CH), (v) orbital Haldane (OH), (vi) spin Haldane (SH), and (vii) rung singlet (RS) (see the previous sections and TABLE. \[tab:abbreviation\] for the definitions). These very rich phase diagrams are in rather good agreement with the low-energy predictions, and they were already discussed in Ref. . In Figs. \[fig:phasediag\_Gorshkov\_N2\_Vex\], one notes that the phases concerning the charge sector (CDW and CH) and those concerning the orbital sector (ODW and OH) appear in a very symmetric manner. In fact, this is quite natural since the $N=2$ $g$-$e$ model possesses the symmetry discussed in Sec. \[sec:orbita-charge-interchange\]: $$\begin{split}
& V \rightarrow -V + V_{\text{ex}}^{g\text{-}e} \\
& V_{\text{ex}}^{g\text{-}e} \rightarrow V_{\text{ex}}^{g\text{-}e} \, , \;\;
U_{mm} \rightarrow U_{mm} \; ,
\end{split}$$ that swaps a phase related to charge and the corresponding orbital phase.
models parameters hopping intra-orbital inter-orbital Hund pair-hop.
------------------------ ------------------------------------------- --------- --------------- --------------- ------------------------------ -----------
$g$-$e$ model $(t,U_{mm},V,V_{\text{ex}}^{g\text{-}e})$ $t$ $U_{mm}$ $V$ $V_{\text{ex}}^{g\text{-}e}$ –
generalized Hund model $(t,U,J,J_{z})$ $t$ $U+J_z/2$ $U-J_z/2$ $J$ –
$p$-band model $(t,U_1,U_2)$ $t$ $U_1$ $U_2$ $U_2$ $U_2$
: \[tab:three-models\] Three models considered in Sec. \[sec:weak-coupling\], \[sec:DMRG\] and their parametrization. See Figs. \[fig:alkaline-2leg\] and \[fig:p-band-2leg\] for the physical process to which each parameter corresponds. In the first two models, pair-hopping does not exist.
![(Color online) Phase diagram for $N=2$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by DMRG. Top and bottom panels correspond respectively to $V_{\text{ex}}^{g\text{-}e}/t=-1$ and $V_{\text{ex}}^{g\text{-}e}/t=1$. Due to the symmetry Eq. (which exists only for $N=2$), CH and OH, as well as CDW and ODW, appear in a symmetrical way with respect to the symmetry axis $V=V_{\text{ex}}^{g\text{-}e}/2$ indicated with a dashed line. \[fig:phasediag\_Gorshkov\_N2\_Vex\]](phasediag_Gorshkov_N2_Vex_-1 "fig:") ![(Color online) Phase diagram for $N=2$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by DMRG. Top and bottom panels correspond respectively to $V_{\text{ex}}^{g\text{-}e}/t=-1$ and $V_{\text{ex}}^{g\text{-}e}/t=1$. Due to the symmetry Eq. (which exists only for $N=2$), CH and OH, as well as CDW and ODW, appear in a symmetrical way with respect to the symmetry axis $V=V_{\text{ex}}^{g\text{-}e}/2$ indicated with a dashed line. \[fig:phasediag\_Gorshkov\_N2\_Vex\]](phasediag_Gorshkov_N2_Vex_1 "fig:")
N=2 [*p*]{}-band model {#sec:DMRG_N2pband}
----------------------
We now map out the phases of the $N=2$ $p$-band model as a function of $(U_1/t,U_2/t)$. While the physical realization with a harmonic trap imposes $U_1=3U_2$, we think that it is worth investigating the full phase diagram which could be accessible using other trapping schemes for instance (see Sec. \[sec:p-band-definition\]). Note also that Kobayashi [*et al.*]{} have recently reported in Ref. the presence of the spin Haldane (SH) phase in the same model at a slightly different ratio $U_2/U_1$.
The phase diagram (Fig. \[fig:phasediag\_Gorshkov\_N2\_Vex\]) obtained exhibits a remarkable symmetry with respect to the origin. In fact, as has been discussed in Sec. \[sec:spin-charge-interchange\], the $p$-band model possesses the symmetry under the Shiba transformation under which spin and charge are interchanged by the mapping: $(U_1,U_2) \mapsto (-U_1,-U_2)$. Consequently, the SH and CH phases appear in a symmetric manner in Fig. \[fig:phasediag\_pband\_N2\]. The remaining areas of the phase diagram are filled respectively with the trivial RS phase (with $T^{z}=0$) and its symmetry partner, the orbital large-$D$ (OLD$_{x,}$) one. We have not investigated in details the transition between these phases, but their locations are in excellent agreement with the weak coupling predictions (i.e. $U_2=0$ and $U_1=U_2$). Moreover, using block entanglement entropy scaling at the transition, one can obtain an estimate of the central charge [@Calabrese-C-04; @Capponi-L-M-13], estimated to be 1.8 (on $L=64$ chain with $U_1=U_2=-8t$ for instance, data not shown), rather close to the expected $c=2$ behavior discussed in Appendix D.
![(Color online) Phase diagram for $N=2$ $p$-band model obtained by DMRG. Note the mapping $(U_1,U_2) \rightarrow (-U_1,-U_2)$ which interchanges spin and charge. The line $U_1=3U_2$ corresponds to the axially symmetric trapping scheme. The two other lines denote the transitions and are compatible with the expected $c=2$ Luttinger liquid behavior. \[fig:phasediag\_pband\_N2\]](phasediag_pband_N2_n2){width="\columnwidth"}
N=4 g-e model {#sec:N4-Gorshkov}
-------------
Here we consider again model $\mathcal{H}_{g\text{-}e}$ (\[eqn:Gorshkov-Ham\]) as in Sec. \[sec:N2-Gorshkov\], but in the $N=4$ case. In the low-energy analysis of Sec. \[sec:weak-coupling\], it was argued that, in comparison with the rich $N=2$ case, there were no more (symmetry-protected) topological phases for the nuclear spin degrees of freedom, but only degenerate ones (CDW, ODW or SP) and the non-degenerate OH and RS phases. Our numerical simulations do confirm these predictions at weak-coupling as shown in Fig. \[fig:phasediag\_Gorshkov\_N4\_Vex\] for fixed $V_{\text{ex}}^{g\text{-}e}/t=-1$, $0$ and $1$, although the one-loop RG results from Sec. \[sec:RG-phase-diag\] were obtained at much smaller $V_{\text{ex}}^{g\text{-}e}/t$ values. The phase diagram for $V_{\text{ex}}^{g\text{-}e}/t=0$ clearly shows symmetry with respect to $V=0$ (see the middle panel of Fig. \[fig:phasediag\_Gorshkov\_N4\_Vex\]). Actually, this is a natural consequence of the orbital-charge interchange symmetry discussed in Sec. \[sec:orbita-charge-interchange\]; the transformation $V \rightarrow -V$ maps the CDW phase on the $V>0$ side to the ODW one on the $V<0$ side (see Table \[tab:Gorshkov\]).
Moreover, both CDW and ODW are rather insensitive to the value of $V_{\text{ex}}^{g\text{-}e}$. On the contrary, as was emphasised in the previous sections, the sign of $V_{\text{ex}}^{g\text{-}e}$ plays a major role in the positive $U_{mm}$ region. For $V_{\text{ex}}^{g\text{-}e}<0$, the SP phase gives way to the trivial RS phase. For $V_{\text{ex}}^{g\text{-}e}>0$, on the other hand, the SP phase remains stable at weak and intermediate coupling as found using RG. There is however a crucial difference for $V_{\text{ex}}^{g\text{-}e}/t=1$ at strong coupling since we also find a large region of the topological SU(4) phase discussed in Sec. \[sec:SUN-topological-phase\] (see the lower panel of Fig. \[fig:phasediag\_Gorshkov\_N4\_Vex\]).
![(Color online) Phase diagrams for $N=4$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by DMRG. From top to bottom, panels correspond respectively to $V_{\text{ex}}^{g\text{-}e}/t=-1$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=1$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. \[fig:phasediag\_Gorshkov\_N4\_Vex\]](phasediag_Gorshkov_N4_Vex_m1 "fig:"){width="\columnwidth"} ![(Color online) Phase diagrams for $N=4$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by DMRG. From top to bottom, panels correspond respectively to $V_{\text{ex}}^{g\text{-}e}/t=-1$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=1$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. \[fig:phasediag\_Gorshkov\_N4\_Vex\]](phasediag_Gorshkov_N4_Vex_0 "fig:"){width="\columnwidth"} ![(Color online) Phase diagrams for $N=4$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) obtained by DMRG. From top to bottom, panels correspond respectively to $V_{\text{ex}}^{g\text{-}e}/t=-1$, $V_{\text{ex}}^{g\text{-}e}/t=0$ and $V_{\text{ex}}^{g\text{-}e}/t=1$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. \[fig:phasediag\_Gorshkov\_N4\_Vex\]](phasediag_Gorshkov_N4_Vex_1 "fig:"){width="\columnwidth"}
Clear signatures of the topological SU(4) phase are given by the existence of 6-fold degenerate edge states [@Nonne-M-C-L-T-13] (see Fig. \[fig:edge\_DMRG\](a)), or by the 6-fold degeneracy of the dominant eigenvalue in the entanglement spectrum of half a system [@Tanimoto-T-14] (data not shown). While the edge states should not occur in the true ground-state, which is highly entangled but exponentially close in energy to the other low-lying states (similarly to the spin-1 Haldane non-magnetic ground-state which lies very close to the so-called Kennedy triplets), it is known that DMRG will target a minimally entangled state [@Jiang-W-B-12] and thus for a large enough system size (at a fixed number of states $m$), the DMRG algorithm will ultimately lead to one of the quasi-degenerate ground-states with some edge states configurations, as is observed in Fig. \[fig:edge\_DMRG\]. For $N=4$, a simple physical interpretation of the 6-fold degeneracy is given by the number of ways of choosing two colors among four. Using the VBS wave function obtained in Sec. \[sec:SUN-VBS-state\], one can explicitly compute the local fermion densities $n_{\alpha,i}$ ($\alpha=1,2,3,4$). Near the left edge of a sufficiently large system, two of the four $\{n_{\alpha,i}\}$ decay as $1+3 (-1/5)^r$ and the other two as $1-3 (-1/5)^r$ ($r$ being the distance from the left edge). The existence of the two different kinds of color-pairing on the left and right edges is clearly seen in Fig.\[fig:edge\_DMRG\](a) and gives another support for the SPT nature of the SU(4) phase found here.
![(Color online) DMRG results for the local fermion densities and kinetic energies for each flavor $\alpha=1,\ldots,4$ in the $N=4$ case. Panel (a) corresponds to the $g$-$e$ model with $U/t=8$, $V=0$ and $V_{\text{ex}}/t=1$ on $L=54$ chain. Panel (b) corresponds to the $p$-band model with $U_1/t=12$ and $U_2/t=4$ on $L=54$ chain. The presence of localized edge states is clearly visible in both cases. \[fig:edge\_DMRG\]](edge_N4){width="\columnwidth"}
N=4 SU(2)$_\text{o}$ g-e model
------------------------------
We now consider the same $N=4$ model but imposing SU(2)$_\text{o}$ symmetry, i.e. $U_{mm}-V=V_{\text{ex}}^{g\text{-}e}$ \[$J=J_{z}$; see Eq. \]. The phase diagram as a function of $(U_{mm},V)$ is shown in Fig. \[fig:phasediag\_Gorshkov\_N4\_SU2\] together with the one-loop RG result. We observe that the agreement is excellent at weak-coupling, and rather good at all couplings for the phase boundaries CDW/SP, CDW/OH and OH/SP. Still, we emphasize that the RG results shown as dashed lines are mostly guide to the eyes for these transitions. Moreover, as expected from our strong-coupling analysis, we do confirm the presence of the SU(4) topological phase along the special line $V=U_{mm}/5$ at strong $V>0$.[^11] In fact, this topological phase occupies a large fraction of the phase diagram, which in our opinion makes its potential observation quite promising. A quantum phase transition necessarily takes place between the SP and the SU(4) topological phase. A precise numerical determination of its nature is beyond the scope of this paper.
![(Color online) Phase diagram for the $N=4$ $g$-$e$ model (\[eqn:Gorshkov-Ham\]) with SU(2)$_\text{o}$ symmetry, i.e. $U_{mm}-V=V_{\text{ex}}^{g\text{-}e}$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. We also plot the special line $V=U_{mm}/5$ (see text). \[fig:phasediag\_Gorshkov\_N4\_SU2\]](phasediag_Gorshkov_N4_SU2_orb){width="\columnwidth"}
N=4 SU(2)$_\text{o}$ generalized Hund model
-------------------------------------------
As discussed in Sec. \[sec:Gorshkov-Hamiltonian\], the SU(2)$_\text{o}$ model can also be parametrized as a function of $(U,J)$ in the generalized Hund model (\[alkaourmodel\]). This means that we can simply take the data of the previous paragraph and replot them accordingly in Fig. \[fig:phasediag\_alka\_N4\]. Obviously, we obtain the same set of phases, and the same extent of agreement with the one-loop RG numerical result as far as the structure in the weak-coupling region and the locations of the phase transitions are concerned. As already noted in Ref. , the topological SU(4) phase is stable along the special line $J=4U/3$ at strong coupling $J>0$, but our numerical results prove that it has an unexpectedly large extent in the first quadrant $U,J>0$.
![(Color online) Phase diagram for the $N=4$ generalized Hund model (\[alkaourmodel\]) with SU(2)$_{\text{o}}$ symmetry. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. We also plot the special line $J=4U/3$ (see text). \[fig:phasediag\_alka\_N4\]](phasediag_alka_N_4){width="\columnwidth"}
N=4 generalized Hund model without SU(2)$_\text{o}$ symmetry {#sec:N4-gen-Hund-wo-SU2}
------------------------------------------------------------
### $J_z=0$
We can also investigate parameter region *without* SU(2)$_\text{o}$ symmetry ($J\neq J_z$) for the generalized Hund model in order to check the robustness of the observations made for the (fine-tuned) SU(2)-symmetric model. In Fig. \[fig:phasediag\_alka\_N4\_Jz\_0\], we present our numerical results for $J_z=0$ together with the RG phase boundaries. Again, we obtained remarkable agreement at weak coupling as well as the semi-quantitative results concerning the phase transitions. The main difference from the SU(2)$_\text{o}$ case consists in the disappearance of the OH which is replaced by the trivial singlet phase RS. In the strong-coupling picture, this result is obvious since the model maps onto a large-$D$ spin-2 chain \[see eq.\]. However, the topological SU(4) phase is scarcely affected by the breaking of SU(2)$_\text{o}$ and it still occupies a large fraction of the $U,J>0$ region. Finally, we have indicated in Fig. \[fig:phasediag\_alka\_N4\_Jz\_0\] the $J=U$ line which can be mapped onto the special line $U_1= 3U_2$ of the $N=4$ $p$-band model upon the identification $J=U=2U_2$. We will use this property later in Sec \[sec:N4-p-band\].
![(Color online) Phase diagram for the $N=4$ generalized Hund model (\[alkaourmodel\]) with $J_z=0$. Symbols correspond to the numerical data obtained by DMRG with $L=36$ while colored regions and dashed lines indicate the one-loop numerical RG results. We also plot the special line $J=U$ where the model can be mapped onto the $N=4$ $p$-band model. \[fig:phasediag\_alka\_N4\_Jz\_0\]](phasediag_alka_N4_Jz_0){width="\columnwidth"}
### $J_z/t=\pm 4$ {#sec:N4-gen-Hund-woSU2-J-pm4}
![(Color online) Phase diagram for the $N=4$ generalized Hund model (\[alkaourmodel\]) with $J_z/t=4$ \[(a)\] and $J_z/t=-4$ \[(b)\]. Symbols correspond to the numerical data obtained by DMRG with $L=36$. Dash-dotted lines correspond to SU(2)$_\text{o}$ lines when $J=J_z$. The critical region for $J_z/t=-4$ is almost invisible on this scale (see text). \[fig:phasediag\_alka\_N4\_Jz\_4\]](phasediag_alka_N4_Jz_4 "fig:"){width="\columnwidth"} ![(Color online) Phase diagram for the $N=4$ generalized Hund model (\[alkaourmodel\]) with $J_z/t=4$ \[(a)\] and $J_z/t=-4$ \[(b)\]. Symbols correspond to the numerical data obtained by DMRG with $L=36$. Dash-dotted lines correspond to SU(2)$_\text{o}$ lines when $J=J_z$. The critical region for $J_z/t=-4$ is almost invisible on this scale (see text). \[fig:phasediag\_alka\_N4\_Jz\_4\]](phasediag_alka_N4_Jz_m4 "fig:"){width="\columnwidth"}
For fixed finite $J_z$, generically there is no SU(2)$_\text{o}$ symmetry. Nevertheless, one can understand part of the phase diagram starting from the line $J=J_z$.
For $J_z/t=4$, as is seen in Fig. \[fig:phasediag\_alka\_N4\], if we fix $J=J_z$ then the system will evolve from the CDW phase to the topological SU(4) one through the SP region with increasing $U$. Since these are all gapped phases, they must have a finite extension in the phase diagram. Our numerical results in Fig. \[fig:phasediag\_alka\_N4\_Jz\_4\](a) confirm this expectation and moreover prove that these phases occupy a large fraction of the phase diagram. The remaining part of it contains the RS phase in agreement with strong-coupling picture.
Considering now a fixed $J_z/t=-4$ and our previous results in Fig. \[fig:phasediag\_alka\_N4\] for $J=J_z$, we expect the appearance of CDW and OH by varying $U$. Our numerical phase diagram in Fig. \[fig:phasediag\_alka\_N4\_Jz\_4\](b) recovers, of course, this result, but there is a crucial difference from the previous case. Indeed, our strong-coupling analysis reveals that starting from the OH phase, deviations from $J=J_z$ will induce an effective $D (T^z)^2$ term with $D=J_z-J$ \[see Eq. (\[eqn:eff-Ham-orbital-Haldane\])\]. This on-site anisotropy is well-known for spin-2 chain [@Schollwock-G-J-96], and it drives the OH phase either to the Ising ODW phase for large $D<0$, or to a non-degenerate singlet phase for $D>0$ (the so-called large-$D$ phase, which is equivalent to RS here) through an intermediate extended gapless $c=1$ phase lying in the interval $0.04 \lesssim D/{\cal J} \lesssim 3.0$, where ${\cal J}={\cal J}_{xy}={\cal J}_z$ is the effective spin exchange . [^12] This scenario away from the OH region is confirmed by our numerical phase diagram, although the extent of the intermediate critical region is rather small in Fig. \[fig:phasediag\_alka\_N4\] due to the smallness of ${\cal J}$. For the same reason, we have not investigated here whether the intermediate-$D$ phase, which has been proposed long-time ago by Oshikawa [@Oshikawa1992] and only recently observed numerically in anisotropic spin-2 chains [@Tonegawa-O-N-S-N-K-11; @Tu2011; @Tzeng2012; @Kjall2013], could appear in our phase diagram.
The existence of the critical region may be further evidenced by the measurement of the pseudo spin correlation functions. Using the low-energy predictions for $N=4$, and taking into account that we are measuring correlation from the middle of a chain with OBC, we use the appropriate functional form for the distance [@Hikihara2004; @Cazalilla-04; @Roux-C-L-A-09]: $$\langle T^+(L/2) T^-(L/2+x) \rangle \sim (-1)^x \left(d_c(x)\right)^{-1/4K_o}$$ where $$d_c(x)=\frac{d(x+L/2|2(L+1)) d(x-L/2|2(L+1))}{\sqrt{d(2x|2(L+1))d(L|2(L+1))}}$$ with $d(x|L)=L |\sin(\pi x/L)|/\pi$ is the conformal distance. Thus fitting, we get an excellent agreement (see Fig. \[fig:corr\_critical\]) with the data and a Luttinger parameter $K_\text{o}=1.09$ indeed larger than 1 as expected. An identical value was obtained when fitting the longitudinal correlations, too. This critical phase is thus described by the orbital Luttinger liquid .
![(Color online) Absolute values of the transverse and longitudinal pseudo spin correlation functions measured from the middle of the chain on a $L=72$ system with parameters $U=0$, $J/t=-4.1$ and $J_z/t=-4$. Both can be fitted with a similar Luttinger parameter $K_\text{o} \simeq 1.09>1$ and appropriate functional forms (see text). \[fig:corr\_critical\]](corr_L72_U0_Jm4_1_Jz_m4){width="\columnwidth"}
Another difference from the weak-coupling results lies in the large $J/t>0$ region where we have found surprisingly the reentrance of the SP and topological SU(4) phases that were found in other parts of the full three-dimensional parameters phase diagram. This confirms again that, contrary to the OH phase whose stability is limited to the proximity of the SU(2)$_\text{o}$-symmetric points, the SU(4) SPT phase could be stabilized for a large variety of parameters and thus could potentially be realized experimentally.
Varying $J_z$ in the N=4 g-e model
----------------------------------
As was shown before, if one starts from the OH phase in the SU(2)$_\text{o}$ case and then increases $J_z$, the OH phase will ultimately be replaced by the trivial RS phase. However, in the strong-coupling, we have an effective spin-$N/2$(=2) chain with some on-site anisotropy $D$ term. For such a system, we know that the transition from the Haldane phase to the trivial large-$D$ phase goes through an *extended* gapless region [@Schollwock-G-J-96] with central charge $c=1$. In Fig. \[fig:entropy\_N4\_L72\], we present measurements of the von Neumann entropy $S_{\text{vN}}$ vs conformal distance $d(x|L)=(L/\pi) \sin (\pi x/L)$ for various parameters ($U=0$, $J/t=-4$ and $J_z>J$) obtained on $L=72$ chains. It is known [@Calabrese-C-04] that this quantity will saturate in a gapped phase, and will scale as $S_{\text{vN}}=(c/6) \log d(x|L) + \mathrm{Cst}$ in a critical phase with central charge $c$. As is expected from our strong-coupling results, our numerical data do confirm the presence of an extended critical phase compatible with $c=1$.
If one uses the expressions from the strong-coupling (\[eq:eff\_coupling\_Hund\]) for our choice of parameters, we are thus starting from an SU(2) spin-2 chain with exchange ${\cal J}=1/18$ (using $t=1$ as the unit of energy). As recalled in the previous subsection, an on-site anisotropy $D=J_z-J$ will induce a critical phase when $0.04 \lesssim D/{\cal J} \lesssim 3$, or assuming that ${\cal J}$ is not changed, $ -3.998 \lesssim J_z \lesssim < -3.83$ in good agreement with our numerical data too.
![(Color online) Von Neumann entanglement entropy $S_{\text{vN}}$ of a block of $x$ sites (starting from the left open edge) vs conformal distance $d(x|L)=(L/\pi) \sin (\pi x/L)$ for $U=0$ and $J/t=-4$ with varying parameters $J_z$ from the SU(2)$_\text{o}$ point $J_z=J$ with OH phase to the $J_z=0$ RS singlet phase. In the intermediate region, there is an extended critical gapless phase compatible with $c=1$ central charge. \[fig:entropy\_N4\_L72\]](entropy_N4_L72){width="\columnwidth"}
N=4 p-band model {#sec:N4-p-band}
----------------
Lastly, we investigate the $N=4$ $p$-band model (\[eqn:p-band\]) which we believe to be quite relevant experimentally. Its phase diagram as a function of $(U_1/t,U_2/t)$ is depicted in Fig. \[fig:phasediag\_pband\_N4\]. While the physical realization with an axially symmetric trap imposes $U_1=3U_2$, we have already discussed that other trapping schemes could remove this constraint.
![(Color online) Phase diagram for half-filled $N=4$ $p$-band model (\[eqn:p-band\]) obtained by DMRG on $L=32$. Dashed line corresponds to the condition $U_1=3U_2$ satisfied for an axially symmetric trap. $U_2=0$ correspond to two decoupled SU(4) Hubbard chains (see text). \[fig:phasediag\_pband\_N4\]](phasediag_pband_N4_n4){width="\columnwidth"}
Starting from this special line and using the equivalence to the generalized Hund model with $J_z=0$ (see Sec. \[sec:N4-gen-Hund-wo-SU2\]), we obtain identical results as in Fig. \[fig:phasediag\_alka\_N4\_Jz\_0\], i.e. when increasing $U_1(=3U_2)$, we find respectively the CDW phase (when $U_1<0$), the SP phase (for small $U_1>0$, as found in weak-coupling), and our topological SU(4) phase (for large $U_1>0)$. Since these are gapped phases, they do have a finite extension in the phase diagram. Again, the topological SU(4) phase occupies a rather large portion which makes it a good candidate for being realized experimentally. As was done for the $g$-$e$ model in the above, this topological SU(4) phase can be easily identified numerically thanks to the existence of characteristic edge-states in the DMRG simulations, see Fig. \[fig:edge\_DMRG\](b) and related comments in Sec. \[sec:N4-Gorshkov\].
The rest of the phase diagram is dominated by trivial singlet phases. However, contrary to the $N=2$ $p$-band model (see Fig. \[fig:phasediag\_pband\_N2\]) where two trivial phases could be distinguished with respect to the symmetry $p_x \leftrightarrow p_y$, here we do not have a full picture. For instance, for $U_1=0$ and large $U_2/t \gg 1$, the ground-state is a complicated superposition of different $T^z$ eigenstates (that are neither $T^x$ nor $T^y$ eigenstates) which has thus no special features concerning the orbital degrees of freedom.
Before concluding this section, we have to comment about the special line $U_2=0$ where the model decouples into two identical (single-band) SU(4) Hubbard chains (one for each orbital). Such a chain is known to be either in a CDW (for $U_1<0$) or in a SP (for $U_1>0$) phase, each of which is two-fold degenerate. As a consequence, for $U_1>0$ we have four-fold degenerate SP phase depending on how dimerization patterns on the two chains are combined; for $U_1<0$, the CDW on each chain can be in-phase or out-of-phase, which in our terminology translates respectively into CDW or ODW for the whole system (see Fig. \[fig:2-DWPhases-SU2\]), again giving four-fold degeneracy. Any small finite $U_2$ splits these four degenerate states into two pairs of degenerate states, thereby stabilizing either CDW or ODW depending on its sign.[^13]
Concluding remarks {#sec:conclusion}
==================
The possibility to realize SU($N$)-symmetric models in alkaline-earth cold atoms experiments has revived the interest in determining what kind of electronic phases, possibly exotic, can be stabilized in these systems and more generally in establishing their phase diagrams. While this remains a challenging problem in general, we have presented a fairly complete study relevant for alkaline-earth fermionic atoms loaded into a 1D optical lattice at half-filling. The two models we considered take into account two orbitals as well as the SU($N$) internal degrees of freedom and we focused on the $N$-even case, which seems to harbor various interesting topological phases.
Working in 1D allows us to use rather powerful analytical and unbiased numerical tools in order to complete this program. Moreover, this strategy has often been used in the past even to gain insight on possible phases in higher dimensions. Last but not least, 1D optical lattices are easily created experimentally so that the exotic phases proposed could be investigated in future experiments.
Our choice of working at half-filling aims at investigating Mott phases, which are presumably simpler in the sense that some (charge) degrees of freedom will be frozen, but may still exhibit a variety of properties as exemplified in our phase diagrams where several exotic SPT phases have been found and characterized thanks to their nontrivial edge states, for instance. Let us remind that precisely in an SPT phase, edge states are protected (and thus cannot be removed without closing a gap) as long as some particular (protecting) symmetry is present.
The addition of the orbital degree of freedom is the key ingredient in our study. Indeed, without it, there are no SPT phases for 1D (singe-band) SU($N$) Hubbard models. This additional degree of freedom may be provided either by a metastable $e$ state (on top of the ground-state $g$) or by populating the two degenerate first-excited $p_x$ and $p_y$ states forming the $p$-bands of the optical lattice. Now, if one considers contact interactions only, the resulting minimal models are respectively the $g$-$e$ model \[see Eqs. (\[eqn:Gorshkov-Ham\]) or (\[alkaourmodel\])\] and the $p$-band Hamiltonian (\[eqn:p-band-simple\]). Depending on their parameters, we have first clarified their symmetries as well as their strong-coupling limits, which provided a firm ground for the subsequent analyses and allowed a physical interpretation of some of their phases.
Combining the strong-coupling approach, a low-energy field-theory and a large-scale unbiased numerical (DMRG) simulations, we have obtained a large number of phase diagrams of the two models depending on the value of $N$ (specifically, $N=2$ and $4$) and its parameters. Our main conclusion is that the interplay between the orbital and the SU($N$) nuclear-spin degrees of freedom gives rise to several interesting phases: in particular, we presented microscopic models whose ground-states realize two different kinds of SPT phases (see Sec. \[sec:strong-coupling-positive-J\] and \[sec:strong-coupling-negative-J\]).
One of these SPT phases concerns the orbital pseudo spins $\mathbf{T}$ and can be described by an effective (pseudo spin) $T=N/2$ Heisenberg chain, possibly with some single-ion anisotropy. If the original model we consider possesses the orbital SU(2)$_{\text{o}}$ symmetry (which may require some fine-tuning), then there is no such anisotropy so that the physical properties are identical to those of the spin-$N/2$ Heisenberg chain (see Figs. \[fig:phasediag\_Gorshkov\_N4\_SU2\] and \[fig:phasediag\_alka\_N4\]). Recent studies have shown[@Pollmann-B-T-O-12] that this gapped phase, when $N/2$ odd, is topologically protected by *any* one of the following symmetries: (i) $\pi$ rotations around two of the three spin axis; (ii) time-reversal; (iii) bond inversion. Away from the SU($2$)$_{\text{o}}$ regime, the phase diagram is dominated by the trivial rung-singlet (RS) phase corresponding to the so-called large-$D$ phase in the spin-chain language, so that the observation of the SPT phase remains challenging. Quite interestingly too, in the case of intermediate values of $D$, there is an extended critical phase for the integer $N/2$ strictly larger than 1, that we have been able to characterise as the Luttinger liquid of this orbital pseudo spin degree of freedom.
Our main observation is the appearance in a much wider region of parameter space of another SU($N$) topological phase, corresponding in the strong-coupling limit to an SU($N$) Heisenberg chain with a self-conjugate representation (Young diagram with $N/2$ rows and 2 columns) at each site. Thanks to the VBS approach, we have been able to show: (i) this is a featureless gapped phase in the bulk, (ii) with open boundary conditions, there exist edge states (corresponding to self-conjugate representation with $N/2$ rows and 1 column), (iii) this is an SPT phase protected by $\text{PSU}(N) \simeq \text{SU}(N)/\mathbb{Z}_{N}$ (this is the case in the SU($N$) phase of our systems) or $\mathbb{Z}_N \times \mathbb{Z}_N$ symmetry for any $N$. Therefore, this provides a microscopic realization of one (among $N$) possible SPT phases for SU($N$) models [@Duivenvoorden-Q-13], characterised by the number of boxes modulo $N$ in the Young diagram describing the edge state (here $N/2$). Note also that even if the SU($N$) symmetry is broken but there remains some bond inversion symmetry, then this topological phase remains protected iff $N/2$ is odd as the Haldane one.
Both our strong-coupling approach and our numerical simulations have confirmed the existence of this phase in a large regime of parameters, which make its potential observation more realistic. Nevertheless, the detection of our topological phases is still a real challenge given that the edge states may be substantially suppressed or even absent if one takes into accout a harmonic trap [@Kobayashi-O-O-Y-M-12; @Kobayashi-O-O-Y-M-14] and it appears difficult, though not hopeless[@Endres-etal-stringOP-11], at the moment to directly measure the rather involved non-local order parameters. An exciting possibility would be to use a box trapping scheme [@Gaunt2013] where presumably edge states should be more visible.
Quite remarkably, this topological SU($N$) phase is not found in the weak-coupling regime, both in the low-energy approach as well as in the numerical simulations, but instead is replaced by the spin-Peierls-like ground state with bond-strength modulations. As discussed in Ref. , we expect that the quantum phase transition between the topological SU($N$) phase and the dimerized one is described by a SU($N$)$_2$ CFT with central charge $c= 2(N^2 -1)/(N+2)$. Since this prediction is independent on the microscopic model, we are looking forward to checking it using simpler Hamiltonians with less degrees of freedom, which will be easier from the numerical point of view.
In this paper, we did not consider the case of odd $N$, which can also be realized in the systems of alkaline-earth fermions by trapping only a subset of $N$(=even) nuclear multiplet. In fact, already in the strong-coupling limit, one can see that the systems with even-$N$ considered here and those with odd-$N$ behave quite differently. For instance, as the orbital pseudo spin can never be quenched even in the Mott region when $N$ odd, one obtains an SU($N$)-orbital-coupled effective Hamiltonian for the region that was described by the SU($N$) Heisenberg model or when $N$ even. Mapping out the phases in the odd-$N$ system would be an interesting future problem.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank H. Nonne for collaboration on this project. Numerical simulations have been performed using HPC resources from GENCI–TGCC, GENCI–IDRIS (Grant 2014050225) and CALMIP. S.C. would like to thank IUF for financial support. K.T. has benefited from stimulating discussions with A. Bolens, K. Penc, and K. Tanimoto on related projects. He was also supported in part by JSPS Grant-in-Aid for Scientific Research(C) No. 24540402.
Decomposition of SU([*2N*]{}) in terms of SU([*N*]{})$\boldsymbol{\times}$SU(2) {#sec:decomp-SUN-SU2}
===============================================================================
As we have seen in Sec. \[sec:models-strong-coupling\], the largest symmetry of the system is U($2N$) since we deal with fermions with two different types of indices: $\alpha=1,\ldots,N$ for SU($N$) and $m=g,e$ for orbitals (or $p_x$ and $p_y$ for the $p$-band model). The Mott state with the fixed number of fermions at each site corresponds to one of the irreducible representations of SU($2N$). In the presence of interactions, the symmetry of the system changes as Eq. . Therefore, it is helpful to know how a given irreducible representation of SU($2N$) decomposes into those of SU($N$) and SU(2) (orbital).
As a warming-up, we begin with the $N=2$ case. Then, we have four species of fermions $c_{g\uparrow}$, $c_{g\downarrow}$, $c_{e\uparrow}$ and $c_{e\downarrow}$ and the largest symmetry is SU(4) \[U(4), precisely\]. Let us consider the Mott-insulating state where we have an integer number ($n$) of fermions at each site. Then, the fermionic property restricts the possible representations at each site to the following four: $$\yng(1) \; (n=1), \;\; \yng(1,1) \; (n=2), \;\; \yng(1,1,1) \; (n=3), \;\; \yng(1,1,1,1) \; (n=4).$$ These on-site states correspond respectively to SU(4) irreducible representations with dimensions 4, 6, 4 and 1.
It is easy to see that the four states in the $n=1$ (${\tiny \yng(1)}$) case are grouped into two
$$\left\{ c_{g\uparrow}^{\dagger}|0\rangle \; , \;
c_{g\downarrow}^{\dagger}|0\rangle \right\} \; , \;\;
\left\{ c_{e\uparrow}^{\dagger}|0\rangle \; , \;
c_{e\downarrow}^{\dagger}|0\rangle \right\} \; ,$$
which span the two independent ($g$ and $e$) sets of the two-dimensional ($S=1/2$) representations of spin-SU(2). Note that the spin operators ${\mathrm{\bf S}}_{g}+{\mathrm{\bf S}}_{e}$ does not see the orbital indices. For the orbital SU(2), we see that another grouping $$\left\{ c_{g\uparrow}^{\dagger}|0\rangle \; , \;
c_{e\uparrow}^{\dagger}|0\rangle \right\} \; , \;\;
\left\{ c_{g\downarrow}^{\dagger}|0\rangle \; , \;
c_{e\downarrow}^{\dagger}|0\rangle \right\}$$ gives the two ($\uparrow$ and $\downarrow$) basis sets for the two-dimensional ($T=1/2$) representations of orbital-SU(2).
We write these results as $$\underbrace{\yng(1)}_{\text{SU}(4)} \sim (\underbrace{\yng(1)}_{\text{SU}(2)_{\text{s}}} ,
\underbrace{\yng(1)}_{\text{SU}(2)_{\text{o}}}) \; .$$
There are six states with two fermions at each site ($n=2$; half-filled) and these six states can be grouped into $$\left\{
c_{g\uparrow}^{\dagger}c_{e\uparrow}^{\dagger}|0\rangle \; , \;\;
\frac{1}{\sqrt{2}}\left(
c_{g\uparrow}^{\dagger}c_{e\downarrow}^{\dagger}|0\rangle
+ c_{g\downarrow}^{\dagger}c_{e\uparrow}^{\dagger}|0\rangle \right) \; , \;\;
c_{g\downarrow}^{\dagger}c_{e\downarrow}^{\dagger}|0\rangle
\right\}$$ and $$\left\{
c_{g\uparrow}^{\dagger}c_{g\downarrow}^{\dagger}|0\rangle \; , \;\;
\frac{1}{\sqrt{2}}\left(
c_{g\uparrow}^{\dagger}c_{e\downarrow}^{\dagger}|0\rangle
- c_{g\downarrow}^{\dagger}c_{e\uparrow}^{\dagger}|0\rangle \right) \; , \;\;
c_{e\uparrow}^{\dagger}c_{e\downarrow}^{\dagger}|0\rangle
\right\} \; .$$ One can easily see that the above two respectively correspond to $$(S=1){\otimes}(T=0) \;\; \text{and} \;\; (S=0){\otimes}(T=1) \; .$$ Therefore, the spin-SU(2) and the orbital-SU(2) are entangled and when the former is in a triplet (singlet), the latter should be in a singlet (triplet). Again, in terms of Young diagrams, this may be written as $$\yng(1,1) \sim
(\underbrace{\yng(2)}_{\text{SU}(2)_{\text{s}}} , \underbrace{\yng(1,1)}_{\text{SU}(2)_{\text{o}}}) \oplus
(\underbrace{\yng(1,1)}_{\text{SU}(2)_{\text{s}}}, \underbrace{\yng(2)}_{\text{SU}(2)_{\text{o}}})
\Rightarrow
(\yng(2) , \bullet) \oplus (\bullet, \yng(2)) \; ,$$ where $\bullet$ denotes the singlet.
For general $N$, we use the rules described in Refs. (chapter 15) and (in particular, Table C of Ref. is quite useful). The decomposition of fermionic states reads, for various local fermion number $n$ ($n_{c}\leq 2N$), as
$$\begin{aligned}
& \yng(1) \sim (\underbrace{\yng(1)}_{\text{SU}(N)},\underbrace{\yng(1)}_{\text{SU(2)}})
\quad (n=1) \\
& \yng(1,1) \sim \left(\yng(2),\bullet \right) \oplus \left(\yng(1,1), \yng(2) \right) \quad (n=2) \\
& \yng(1,1,1) \sim \left( \yng(2,1), \yng(1) \right) \oplus \left(\yng(1,1,1),\yng(3) \right) \quad (n=3)
\end{aligned}$$
$$\begin{split}
& \yng(1,1,1,1) \sim \left( \yng(2,2), \bullet \right) \oplus \left( \yng(2,1,1),\yng(2) \right)
\oplus \left(\yng(1,1,1,1), \yng(4) \right) \\
& \qquad \qquad (n=4)
\end{split}
\label{eqn:decomp-SU2-SUN-4}$$
$$\begin{split}
& \yng(1,1,1,1,1) \sim \left( \yng(2,2,1), \yng(1) \right) \oplus \left( \yng(2,1,1,1),\yng(3) \right) \\
& \qquad \oplus \left(\yng(1,1,1,1,1), \yng(5) \right) \quad (n=5)
\end{split}$$
It is easy to check that the dimensions on the both sides match. Consider the decomposition for $N=4$. Apparently, the dimensions of the left-hand side is $8!/(4!4!)=70$. The sum of the dimensions appearing on the right-hand side is given by $$20{\times}1 + 15{\times}3 + 1{\times}5=70 \; ,$$ which coincides with the one on the left-hand side. From these results, it is obvious that the SU($N$) irreducible representations contained in the fermionic states of the form $\prod c_{m\alpha}^{\dagger}|0\rangle$ are represented by Young diagrams with [*at most two columns*]{}. If we denote the lengths of the two columns by $p$ and $q$ ($p+q=n$, $p \geq q$), $$\text{\scriptsize $p$} \left\{
\yng(2,2,2,1,1)
\right.
\raisebox{2.2ex}{$\left.
\vphantom{\yng(2,2,2)}
\right\} \text{\scriptsize $q$}
$ }$$ the ‘spin’ $T$ of the orbital SU(2) is given by $$\underbrace{\yng(2)}_{p-q} \; , \quad
T = \frac{1}{2}(p-q) \; .$$
$p$-band Hamiltonian {#sec:p-band-hamiltonian}
====================
In this appendix, we sketch the derivation of the $p$-band Hamiltonian . The eigenfunctions of the single-particle part $\mathcal{H}_{0}$ is given by the Bloch function: $$\psi^{(n)}_{n_x,n_y,k_z}(x,y,z) = \phi_{n_x,n_y}(x,y) \varphi^{(n)}_{k_z}(z) \; ,$$ where $\phi_{n_x,n_y}(x,y)$ and $\varphi^{(n)}_{k_z}(z)$ are the eigenfunctions of $\mathcal{H}_{\perp}(x,y)$ and $\mathcal{H}_{/\!/}(z)$, respectively. Since $\mathcal{H}_{\perp}$ is the two-dimensional harmonic oscillator, we can obtain the explicit form of $\phi_{n_x,n_y}(x,y)$. First three (normalized) eigenfunctions are given as (see Fig. \[fig:pxpy-orbitals\])
$$\begin{aligned}
& \phi_{0,0}(x,y) = \frac{1}{\sqrt{\pi } x_{0}}{\mathrm{e}}^{-\frac{x^2+y^2}{2 x_{0}^2}} \\
& \phi_{1,0}(x,y) = \frac{\sqrt{2}}{\sqrt{\pi}x_{0}}
\left(\frac{x}{x_{0}} \right) {\mathrm{e}}^{-\frac{x^2+y^2}{2 x_{0}^2}} \, , \\
& \phi_{0,1}(x,y) = \frac{\sqrt{2}}{\sqrt{\pi}x_{0}}
\left(\frac{y}{x_{0}} \right) {\mathrm{e}}^{-\frac{x^2+y^2}{2 x_{0}^2}}
\label{eqn:pxpy-orbitals}\end{aligned}$$
with $x_{0} = \sqrt{\hbar/(m\omega_{xy})}$. We call the levels with $(n_x,n_y)=(0,0)$, $(1,0)$ and $(0,1)$ ‘$s$’, ‘$p_x$’ and ‘$p_y$’, respectively.
![Contour plots of squared wave functions $| \phi_{n_x,n_y}|^{2}$ for three orbitals $(n_x,n_y)=(0,0)$, $(1,0)$ and $(0,1)$. \[fig:pxpy-orbitals\]](pxpy-orbitals)
To derive an effective Hubbard-like Hamiltonian[@Jaksch-Z-05], it is convenient to move from the Bloch function $\psi^{(n)}_{n_x,n_y,k_z}(x,y,z)$ to the Wannier function defined as $$W^{(n)}_{n_x,n_y;R}(x,y,z) \equiv
\frac{1}{\sqrt{N_{\text{cell}}}} \phi_{n_x,n_y}(x,y) \sum_{k_z}
{\mathrm{e}}^{- i k_z R} \varphi^{(n)}_{k_z}(z)
$$ ($R$ labels the center of the Wannier function and $N_{\text{cell}}$ is then number of unit cells) and introduce the corresponding creation/annihilation operators $$\begin{split}
& c_{a\alpha}({\mathrm{\bf r}})
= \sum_{R}\sum_{n=\text{bands}} W_{a;R}^{(n)}(x,y,z) c_{a\alpha,R}^{(n)} \\
& c^{\dagger}_{a\alpha}({\mathrm{\bf r}})
= \sum_{R}\sum_{n=\text{bands}} W_{a;R}^{(n)\ast}(x,y,z) c_{a\alpha,R}^{(n)\dagger} \\
& \quad (a=p_x,p_y, \; \alpha=1,\ldots, N) \; ,
\end{split}$$ As in Sec. \[sec:models-strong-coupling\], we have used the short-hand notation $a=p_{x},p_{y}$ meaning $p_{x}=(n_x,n_y)=(1,0)$ and $p_{y}=(n_x,n_y)=(0,1)$.
Following the standard procedure[@Jaksch-Z-05], we can derive the Hubbard-type interactions from the original contact interaction $g \delta^{3}({\mathrm{\bf r}})$: $$\begin{split}
& \frac{1}{2} \sum_{R} \sum_{a=p_x,p_y} U_{aaaa}
\hat{V}_{aaaa}(R) \\
& + \frac{1}{2} \sum_{R} \sum_{\substack{a\neq b\\=p_x,p_y}} \Biggl\{
U_{aabb} \hat{V}_{aabb}(R) + U_{abba} \hat{V}_{abba}(R) \\
& \phantom{+ \frac{1}{2} \sum_{R} \sum_{\substack{a\neq b\\=p_x,p_y}} \Biggl\{ }
+ U_{abab} \hat{V}_{abab}(R) \Biggr\}
\end{split}
\label{eqn:contact-int-by-Wannier}$$ where the superscript ‘$(0)$’ for the fermion operators of the lowest Bloch band has been suppressed and $U_{abcd}$ is defined by $$\begin{split}
& U_{abcd} \equiv g
\int\! d{\mathrm{\bf r}}W_{a;R}^{(0)\ast}({\mathrm{\bf r}})W_{b;R}^{(0)\ast}({\mathrm{\bf r}})
W_{c;R}^{(0)}({\mathrm{\bf r}})W_{d;R}^{(0)}({\mathrm{\bf r}}) \\
& \hat{V}_{abcd}(R) \equiv
c_{a\alpha,R}^{\dagger} c_{b\beta,R}^{\dagger}c_{c\beta,R}c_{d\alpha,R} \\
& \quad
(a,b,c,d=p_{x}, p_{y}) \; .
\end{split}
\label{eqn:def-Uabcd-pxpy}$$ Since the Wannier functions are real and the two orbitals $W_{p_x/p_y;R}^{(0)}({\mathrm{\bf r}})$ are symmetry-related ($C_4$), there are only two independent couplings: $$\begin{split}
& U_{1} \equiv U_{p_x p_x p_x p_x} = U_{p_y p_y p_y p_y} \\
& U_{2} \equiv U_{p_x p_x p_y p_y} = U_{p_y p_y p_x p_x}
= U_{p_x p_y p_y p_x} \\
& \phantom{U_{2}} = U_{p_y p_x p_x p_y}
= U_{p_x p_y p_x p_y}= U_{p_y p_x p_y p_x} \; .
\end{split}
\label{eqn:def-U1-U2}$$ Using the explicit forms , one can readily verify that the above two coupling constants $U_1$ and $U_2$ actually are [*not*]{} independent and satisfy $U_1 = 3 U_2$. In fact, this ratio is constant for [*any*]{} axially symmetric potential $V_{\perp}(x,y)$.
Plugging the above into Eq. , we obtain the Hamiltonian of the $p$-band model : $$\begin{split}
\mathcal{H}_{p\text{-band}} =& - t \sum_{i}
( c_{a\alpha,i}^{\dagger}c_{a\alpha,i+1} + \text{H.c.} ) \\
& + \sum_{i} \sum_{a=p_x,p_y}
(\epsilon_{a} - t_{0}) \, n_{a,i} \\
&+ \frac{1}{2}U_{1} \sum_{i}
n_{a,i}(n_{a,i}-1) + U_{2}\, \sum_{i} n_{p_x,i} n_{p_y,i} \\
& + U_{2} \sum_{i}
c_{p_x\alpha,i}^{\dagger} c_{p_y \beta,i}^{\dagger}c_{p_x \beta,i}c_{p_y \alpha,i} \\
& + \frac{1}{2}U_{2}\sum_{i} \sum_{\substack{a\neq b\\=p_x,p_y}}
c_{a,\alpha,i}^{\dagger} c_{a,\beta,i}^{\dagger}c_{b,\beta,i}c_{b,\alpha,i} \; ,
\end{split}
\label{eqn:p-band-in-appendix}$$ where $\epsilon_{p_x}=\epsilon_{p_y}=3\hbar \omega_{xy}/2$ and the hopping amplitude $t$ is defined as $$\begin{split}
& t = t^{(0)}(\pm 1) \\
& \int\! d{\mathrm{\bf r}}\,
W_{a;R_1}^{(n_1)\ast}({\mathrm{\bf r}})\mathcal{H}_{/\!/}(z) W_{b;R_2}^{(n_2)}({\mathrm{\bf r}})
\equiv - \delta_{ab} \delta_{n_1 n_2} t^{(n_1)}(R_1-R_2) \;.
\end{split}$$ When the last term in ([*pair-hopping*]{}) is rewritten in terms of the orbital pseudo spin ${\mathrm{\bf T}}$, eq. is recovered.
Conformal field theory data {#sec:CFT-data}
===========================
In this Appendix, we recall some useful formula of SU($N$)$_k$ CFT which are useful in the low-energy approach of two-orbital SU($N$) models (Sec. \[sec:SUN-topological-phase\]).
Let us first consider the SU(2)$_N$ CFT which is generated by the orbital current $\mathbf{j}_{\text{L,R}}$ in our problem. The left chiral current satisfies the SU(2)$_N$ Kac-Moody algebra which reads as follows within our conventions: $$\begin{aligned}
j^{i}_{\text{L}}\left(z\right) j^{j}_{\text{L}}\left(0\right) \sim \frac{N \delta^{ij}}{8 \pi^2 z^2}
+ \frac{ i \epsilon^{ijk}}{2 \pi z} j^{k}_{\text{L}}\left(0\right),
\label{KacMoodyApp}\end{aligned}$$ with a similar result for the right current. The SU(2)$_N$ primary operators with spin $j = 0,\ldots, N/2$ is an SU(2) $\times$ SU(2) tensor with $(2j+1)^2$ components which are denoted by $\Phi^{j}_{m, \bar m}$ ($|m, \bar m| \le j$). They transform in the spin-j representation of SU(2) and have scaling dimension $d_j = 2 j (j+1)/(N+2)$. [@Knizhnik-Z-84; @DiFrancesco-M-S-book] They are defined through the OPE: [@DiFrancesco-M-S-book] $$\begin{aligned}
j^{i}_{\text{L}}\left(z\right) \Phi_{m, \bar m}^{j}\left(\omega, \bar \omega\right)
&\sim & -\frac{1}{z-\omega}
T^i_{ms} \Phi_{s, \bar m}^{j}\left(\omega, \bar \omega\right)
\nonumber \\
j^{i}_{\text{R}}\left(\bar z\right) \Phi_{m, \bar m}^{j}\left(\omega, \bar \omega\right)
&\sim & \frac{1}{\bar z- \bar \omega}
\Phi_{m, \bar s}^{j}\left(\omega, \bar \omega\right)
T^i_{\bar s \bar m} ,
\label{theopeApp}\end{aligned}$$ where $T^i$ are the usual spin-$j$ matrices. The conjugate of $\Phi^{j}_{m, \bar m}$ is defined by: $$\Phi^{j \dagger}_{m, \bar m} = \left(-1\right)^{2j - m - \bar m} \Phi^{j}_{-m, -\bar m} .
\label{Su2conj}$$
We need also the SU(2)$_N$ fusion rule which describes the product between two primary operators with spin $j_1$ and $j_2$: [@Zamolodchikov-F-86] $$\begin{aligned}
j_1 \otimes j_2 &=& | j_1 - j_2 |, | j_1 - j_2 | +1, \ldots, \nonumber \\
&& {\rm min} \left( j_1 + j_2, N/2 - j_1 - j_2 \right) .
\label{Su2fusion}\end{aligned}$$ Related to this decomposition is the SU(2)$_N$ operator algebra:[@Zamolodchikov-F-86] $$\begin{aligned}
\Phi^{j_1}_{m_1, \bar m_1} \left( z, \bar z\right) && \Phi^{j_2}_{m_2, \bar m_2} \left( 0, 0 \right) \sim
\sum_{j=0}^{N/2} \sum_{m, \bar m=-j}^{j} | z |^{d_j - d_{j_1} - d_{j_2}} \nonumber \\
&& C \left(\begin{array}{cccccc}
j & m & \bar m \\
j_1 & m_1 & \bar m_1 \\
j_2 & m_2 & \bar m_2
\end{array}\right) \Phi^{j}_{m, \bar m} \left( 0, 0 \right) ,
\label{SU2Nfusionop}\end{aligned}$$ where $C$ are the structure constants of the operator algebra which are related to the Wigner $3j$ symbols as: $$\begin{aligned}
&& C \left(\begin{array}{cccccc}
j & m & \bar m \\
j_1 & m_1 & \bar m_1 \\
j_2 & m_2 & \bar m_2
\end{array}\right) = \rho_{ j, j_1, j_2 }
\nonumber \\
&& \left(\begin{array}{cccccc}
j & j_1 & j_2 \\
-m & m_1 & m_2 \\
\end{array}\right)
\left(\begin{array}{cccccc}
j & j_1 & j_2 \\
- \bar m & \bar m_1 & \bar m_2 \\
\end{array}\right),
\label{structconst}\end{aligned}$$ where $ \rho_{ j, j_1, j_2 }$ is a constant which can be found in Ref. and we have the constraints: $m = m_1 + m_2$, $\bar m = \bar m_1 + \bar m_2$ which stem from the properties of $3j$ symbols. The explicit application of the operator algebra (\[SU2Nfusionop\]) for $j_1=j_2 = 1/2$ leads to $$\begin{aligned}
\Phi^{1/2}_{1/2, 1/2} \left( z, \bar z\right) && \Phi^{1/2}_{1/2, 1/2} \left( 0, 0 \right) \sim
\nonumber \\
&& \frac{1}{3} | z |^{1/(N+2)} \rho_{ 1, 1/2, 1/2 } \Phi^{1}_{1, 1} \left( 0, 0 \right)
\nonumber \\
\Phi^{1/2}_{-1/2, -1/2} \left( z, \bar z\right) && \Phi^{1/2}_{-1/2, -1/2} \left( 0, 0 \right) \sim \nonumber \\
&& \frac{1}{3} | z |^{1/(N+2)} \rho_{ 1, 1/2, 1/2 } \Phi^{1}_{-1, -1} \left( 0, 0 \right)
\nonumber \\
\Phi^{1/2}_{1/2, 1/2} \left( z, \bar z\right) && \Phi^{1/2}_{-1/2, -1/2} \left( 0, 0 \right) \sim
\nonumber \\
&& \frac{1}{2} | z |^{-3/(N+2)} \rho_{ 0, 1/2, 1/2 }
\nonumber \\
&+& \frac{1}{6} | z |^{1/(N+2)} \rho_{ 1, 1/2, 1/2 } \Phi^{1}_{0, 0} \left( 0, 0 \right)
\nonumber \\
\Phi^{1/2}_{1/2, -1/2} \left( z, \bar z\right) && \Phi^{1/2}_{-1/2, 1/2} \left( 0, 0 \right) \sim
\nonumber \\
&& - \frac{1}{2} | z |^{-3/(N+2)} \rho_{ 0, 1/2, 1/2 }
\nonumber \\
&+& \frac{1}{6} | z |^{1/(N+2)} \rho_{ 1, 1/2, 1/2 } \Phi^{1}_{0, 0} \left( 0, 0 \right) .
\label{SU2Nfusionopspin1demi}\end{aligned}$$
At this stage, we introduce another parametrization of the spin-1/2 SU(2)$_N$ field which will be used in Sec. \[sec:SUN-topological-phase\]: $g_{pl} \equiv \Phi^{1/2}_{m, \bar m}$ where $p=g,e \; ({\rm or} \; p_x, p_y)
\rightarrow m=1/2,-1/2$ and $l=g,e \; ({\rm or} \; p_x, p_y) \rightarrow \bar m =1/2,-1/2$. With this definition and the OPEs (\[SU2Nfusionopspin1demi\]), we deduce that the trace of the SU(2)$_N$ primary field which transforms in the spin-1 representation, reads as follows: $${\rm Tr} \left( \Phi^{\text{SU(2)}_N}_{\rm j=1} \right) \sim
{\rm Tr} \left( g \right) {\rm Tr} \left( g^{\dagger} \right)
- \frac{1}{2} g_{p l} g^{\dagger}_{p l} .
\label{adjointSU2App}$$
The SU(2)$_N$ primary operators can also be related to that of the ${\mathbb{Z}}_N$ CFT ($f^{2j}_{2m, 2\bar m}$) through the coset construction ${\mathbb{Z}}_N$ $\sim$ SU(2)$_N$ / U(1)$_\text{o}$. [@Zamolodchikov-F-JETP-85; @Gepner-Q-87] In the paper, the U(1)$_\text{o}$ CFT is described by a bosonic field which is the orbital field $\Phi_{o}$ with chiral components $\Phi_{o L,R}$. Within our conventions, the relationship between the primary fields is: $$\Phi^{j}_{m, \bar m} = f^{2j}_{-2m, 2\bar m} \exp \left( - i m \sqrt{8 \pi/N} \; \Phi_{o L}
- i \bar m \sqrt{8 \pi/N} \; \Phi_{o R} \right),
\label{primariesApp}$$ where the ${\mathbb{Z}}_N$ primary operators have scaling dimension $ \Delta^{j}_{m, \bar m} = 2 j (j+1)/(N+2) - (m^2 + {\bar m}^2)/N$. The most important one for our purpose are the ${\mathbb{Z}}_N$ ordered spin operators $\sigma_k \sim
f^{k}_{k, k}$ and the disordered ones $\mu_k \sim f^{k}_{-k, k}$ ($k=1,\ldots, N-1$). The relation (\[primariesApp\]) gives in particular the following identifications: $$\begin{aligned}
\Phi^{1/2}_{1/2, 1/2} &\simeq& \mu_1 \exp \left( - i \sqrt{2 \pi/N} \; \Phi_{o } \right)
\nonumber \\
\Phi^{1/2}_{-1/2, -1/2} &\simeq& \mu^{\dagger}_1 \exp \left( i \sqrt{2 \pi/N} \; \Phi_{o } \right)
\nonumber \\
\Phi^{1/2}_{-1/2, 1/2} &\simeq& \sigma_1 \exp \left( i \sqrt{2 \pi/N} \; \Theta_{o } \right)
\nonumber \\
\Phi^{1}_{1, 1} &\simeq& \mu_2 \exp \left(- i \sqrt{8 \pi/N} \; \Phi_{o} \right)
\nonumber \\
\Phi^{1}_{-1, -1} &\simeq& \mu^{\dagger}_2 \exp \left(i \sqrt{8 \pi/N} \; \Phi_{o} \right),
\label{identiprimZnApp}\end{aligned}$$ where $\Theta_{o }$ is the dual field associated with $\Phi_\text{o}$. The last identification that we need is the ${\mathbb{Z}}_N$ description of $\Phi^{1}_{0, 0}$ which can be determined by the SU(2)$_N$ fusion rule $\Phi^{1/2}_{1/2, 1/2} \Phi^{1/2}_{-1/2, -1/2}$ (see Eq. (\[SU2Nfusionopspin1demi\])). Using the identification (\[identiprimZnApp\]) for $\Phi^{1/2}_{\pm 1/2, \pm 1/2}$ and the following OPE for the ${\mathbb{Z}}_N$ CFT ($C$ being an unimportant positive constant) $$\begin{aligned}
\mu_1 \left(z, \bar z \right) \mu^{\dagger}_1 \left(0, 0 \right) &\sim& |z|^{-\frac{2(N-1)}{N(N+2)}}
\nonumber \\
&- C&
|z|^{\frac{2(N+1)}{N(N+2)}} \epsilon_1 \left(0, 0 \right) ,
\label{ZnOPEApp}\end{aligned}$$ we get: $$\Phi^{1}_{0, 0} \simeq - \epsilon_1 ,
\label{spin1primaryparaApp}$$ where $\epsilon_1$ is the thermal operator of the $Z_N$ CFT with scaling dimension $4/(N+2)$. In our convention, $\langle \epsilon_1 \rangle > 0 $ in a phase where the ${\mathbb{Z}}_N$ is broken so that the disorder parameters cannot condense.
These results generalize in the SU($N$) case. We will only need for our purpose the values of scaling dimensions of SU($N$)$_2$ primary fields. The SU($N$)$_k$ primary field transforms in some representation $R$ of the SU($N$) group and its scaling dimension is given by [@Knizhnik-Z-84]: $$\Delta_{R} = \frac{2 C_{R}}{N+k},
\label{dimensioprimary}$$ where $C_{R}$ is the quadratic Casimir in the representation $R$. Its expression can be obtained from the general formula where $R$ is written as a Young diagram: $$C_{R} = T^aT^a = \frac{1}{2} \left\{
l(N-l/N) +\sum_{i=1}^{n_{\text{row}}} b_i^2 - \sum_{i=1}^{n_{\text{col}}}a_i^2
\right\}
\label{Cas}$$ for Young diagram of $l$ boxes consisting of $n_{\text{row}}$ rows of length $b_i$ each and $n_{\text{col}}$ columns of length $a_i$ each. For instance, we get $C_{R} = (N^2 -1)/2N$ for the fundamental representation, $C_{R} = N$ for the adjoint representation, $C_{R} (k) = k(N+1) (N - k)/2N$ for the $k$-th basic antisymmetric representation made of a Young diagram with a single column and $k$ boxes, and $C_{R} = N - 2/N +1$ for the symmetric representation with dimension $N(N+1)/2$. In particular, in the SU($N$)$_2$ case, i.e. the CFT which describes the nuclear spin degrees of freedom in our paper, the scaling dimensions of various primary operators needed in Sec. \[sec:SUN-topological-phase\] are: $$\begin{aligned}
\Delta_G &=& \frac{N^2-1}{N(N+2)} \nonumber
\\
\Delta_{\rm adj} &=& \frac{2N}{N+2} \nonumber
\\
\Delta_{S} &=& \frac{2(N - 2/N +1)}{N+2} \nonumber
\\
\Delta_{A} &=& \frac{2(N+1)(N-2)}{N(N+2)},
\label{scalingdimSUN2App}\end{aligned}$$ which describes respectively the scaling dimension of the SU($N$)$_2$ primary field which transforms in the fundamental, adjoint, symmetric representation with dimension $N(N+1)/2$, and antisymmetric representation with dimension $N(N-1)/2$ of SU($N$).
Majorana-fermionization of the half-filled N=2 p-band model {#sec:N2-p-band-continuum}
===========================================================
In this Appendix, we investigate the zero-temperature phase diagram of the half-filled $N=2$ $p$-band model in the general case with two different coupling constants $U_{1,2}$ by means of the low-energy approach. As seen in Sec. \[sec:models-strong-coupling\], the U(1)$_\text{o}$ continuous orbital symmetry is explicitly broken when $U_1 \ne 3 U_2$ and the low-energy effective Hamiltonian is no longer parametrized by nine coupling constants as in Eq. (\[lowenergyham\]). In the special $N=2$ case, one can use the standard field-theoretical methods based on bosonization and refermionization techniques as in the two-leg ladders. [@Gogolin-N-T-book] In the context of the $N=2$ generalized Hund model at half-filling, we have described extensively this approach in Ref. .
Using the Abelian bosonization, one can define four chiral bosonic field $\Phi_{m \sigma \text{R,L}}$ ($m= p_x,p_y;\sigma= \uparrow, \downarrow$) from the four left-right moving Dirac fermions of the continuum limit for $N=2$. The next step of the approach is to introduce a bosonic basis which singles out the different degrees of freedom for $N=2$, i.e. charge, (nuclear) spin, orbital, and spin-orbital degrees of freedom: $$\begin{aligned}
&&\Phi_{p_x\uparrow \text{L,R}}=\frac{1}{2}
(\Phi_\text{c}+\Phi_\text{s}+\Phi_\text{o}+\Phi_{\text{so}})_{\text{L,R}} \nonumber\\
&&\Phi_{p_x\downarrow \text{L,R}}=\frac{1}{2}
(\Phi_\text{c}-\Phi_\text{s}+\Phi_\text{o}-\Phi_{\text{so}})_{\text{L,R}} \nonumber\\
&&\Phi_{p_y\uparrow \text{L,R}}=\frac{1}{2}
(\Phi_\text{c}+\Phi_\text{s}-\Phi_\text{o}-\Phi_{\text{so}})_{\text{L,R}} \nonumber\\
&&\Phi_{p_y\downarrow \text{L,R}}=\frac{1}{2}
(\Phi_\text{c}-\Phi_\text{s}-\Phi_\text{o}+\Phi_{\text{so}})_{\text{L,R}}.
\label{basebosons}\end{aligned}$$
From these new bosonic fields, one can now consider a refermionization procedure by introducing eight left and right moving Majorana fermions through: $$\begin{aligned}
&&\xi_{\text{L}}^2+i\xi_{\text{L}}^1=
\frac{\eta_1}{\sqrt{\pi a_0}}
\exp{(-i\sqrt{4\pi}\Phi_{sL})}\nonumber\\
&&\xi_{\text{R}}^2+i\xi_{\text{R}}^1=
\frac{\eta_1}{\sqrt{\pi a_0}}
\exp{(i\sqrt{4\pi}\Phi_{sR})}\nonumber\\
&&\xi_{\text{L}}^4-i\xi_{\text{L}}^5=
\frac{\eta_2}{\sqrt{\pi a_0}}
\exp{(-i\sqrt{4\pi}\Phi_{oL})}\nonumber\\
&&\xi_{\text{R}}^4-i\xi_{\text{R}}^5=
\frac{\eta_2}{\sqrt{\pi a_0}}
\exp{(i\sqrt{4\pi}\Phi_{oR})}\nonumber\\
&&\xi_{\text{L}}^6+i\xi_{\text{L}}^3=
\frac{\eta_3}{\sqrt{\pi a_0}}
\exp{(-i\sqrt{4\pi}\Phi_{soL})}\nonumber\\
&&\xi_{\text{R}}^6+i\xi_{\text{R}}^3=
\frac{\eta_3}{\sqrt{\pi a_0}}
\exp{(i\sqrt{4\pi}\Phi_{soR})} \nonumber\\
&&\xi_{\text{L}}^8+i\xi_{\text{L}}^7=
\frac{\eta_4}{\sqrt{\pi a_0}}
\exp{(- i\sqrt{4\pi}\Phi_{cL})} \nonumber\\
&&\xi_{\text{R}}^8+i\xi_{\text{R}}^7=
\frac{\eta_4}{\sqrt{\pi a_0}}
\exp{(i\sqrt{4\pi}\Phi_{cR})} ,
\label{refer}\end{aligned}$$ where the Klein factors $\eta_{1,2,3,4}$ ensure the anti-commutation rules for the Majorana fermions.
With these definitions, the continuum Hamiltonian of the half-filled $N=2$ $p$-band model can then be expressed in terms of these eight Majorana fermions: $$\begin{aligned}
\mathcal{H}&=& - \frac{i v_\text{F}}{2}\sum_{a=1}^8
(\xi_{\text{R}}^a \partial_x \xi_{\text{R}}^a
- \xi_{\text{L}}^a \partial_x \xi_{\text{L}}^a)\nonumber\\
&+&\frac{g_1}{2}\left(\sum_{a=1}^3
\xi_{\text{R}}^a\xi_{\text{L}}^a\right)^2+g_2\left(\sum_{a=1}^3
\xi_{\text{R}}^a\xi_{\text{L}}^a\right) \xi_{\text{R}}^4\xi_{\text{L}}^4\nonumber\\
&+& \xi_{\text{R}}^6\xi_{\text{L}}^6\left[ g_3 \sum_{a=1}^3
\xi_{\text{R}}^a\xi_{\text{L}}^a + g_4 \xi_{\text{R}}^4 \xi_{\text{L}}^4 \right] +\frac{g_5}{2}\left(
\xi_{\text{R}}^5\xi_{\text{L}}^5 + \sum_{a=7}^8 \xi_{\text{R}}^a\xi_{\text{L}}^a \right)^2
\nonumber\\
&+& \left(\xi_{\text{R}}^5\xi_{\text{L}}^5 + \sum_{a=7}^8 \xi_{\text{R}}^a\xi_{\text{L}}^a \right)
\Biggl( g_6 \sum_{a=1}^3 \xi_{\text{R}}^a\xi_{\text{L}}^a + g_7 \xi_{\text{R}}^4\xi_{\text{L}}^4
\nonumber\\
&+&
g_8 \xi_{\text{R}}^6\xi_{\text{L}}^6 \Biggl),
\label{Hamiltonian_tperp0_majoranapp}\end{aligned}$$ where we have neglected the velocity-anisotropy terms for the sake of simplicity. The different coupling constants of the continuum limit are given by: $$\begin{aligned}
&&g_1= - g_5 = -a_0\left(U_1 + U_2 \right) \nonumber\\
&&g_2= - g_8 = -2a_0 U_2\nonumber\\
&&g_3= -g_7 = a_0\left(U_2 - U_1 \right) \nonumber\\
&&g_4= g_6 = 0 ,
\label{majocouplingsapp}\end{aligned}$$ where we have included the operators with coupling constants $g_{4,6}$ since they will be generated in the one-loop RG calculation.
From Eq. (\[Hamiltonian\_tperp0\_majoranapp\]), one observes that the three Majorana fermions $\xi_{\text{R,L}}^a, (a = 1,2,3)$, which accounts for the physical properties of the (nuclear) spin degrees of freedom, play a symmetric role as the result of the SU(2)$_s$ spin-symmetry of the lattice model. In addition, the two Majorana fermions $\xi_{\text{R,L}}^a, (a = 7,8)$, associated to the charge degrees of freedom, are unified with one Majorana fermion $\xi_{\text{R,L}}^5$ of the orbital sector. This signals the emergence of a new independent SU(2) symmetry for all $U_1$ and $U_2$ that we have revealed on the lattice from the charge pseudo spin operator (\[chargepseudospin\]). The continuous symmetry of model (\[Hamiltonian\_tperp0\_majoranapp\]) is actually SU(2) $\times$ SU(2) $\sim$ SO(4).
The one-loop RG of model (\[Hamiltonian\_tperp0\_majoranapp\]) can be easily determined within the Majorana formalism and we find: $$\begin{aligned}
\dot{g_1} &=& \frac{1}{2\pi} g^2_1 + \frac{1}{2\pi} g^2_2 + \frac{1}{2\pi} g^2_3
+ \frac{3}{2\pi} g^2_6
\nonumber\\
\dot{g_2} &=& \frac{1}{\pi} g_1 g_2 + \frac{1}{2\pi} g_3 g_4 + \frac{3}{2\pi} g_6 g_7
\nonumber\\
\dot{g_3} &=& \frac{1}{\pi} g_1 g_3 + \frac{1}{2\pi} g_2 g_4 + \frac{3}{2\pi} g_6 g_8
\nonumber\\
\dot{g_4} &=& \frac{3}{2\pi} g_2 g_3 + \frac{3}{2\pi} g_7 g_8
\nonumber\\
\dot{g_5} &=& \frac{1}{2\pi} g^2_5 + \frac{3}{2\pi} g^2_6 + \frac{1}{2\pi} g^2_7
+ \frac{1}{2\pi} g^2_8
\nonumber\\
\dot{g_6} &=& \frac{1}{\pi} g_1 g_6 + \frac{1}{2\pi} g_2 g_7 + \frac{1}{2\pi} g_3 g_8
+ \frac{1}{\pi} g_5 g_6
\nonumber\\
\dot{g_7} &=& \frac{3}{2\pi} g_2 g_6 + \frac{1}{2\pi} g_4 g_8 + \frac{1}{\pi} g_5 g_7
\nonumber\\
\dot{g_8} &=& \frac{3}{2\pi} g_3 g_6 + \frac{1}{2\pi} g_4 g_7 + \frac{1}{\pi} g_5 g_8 .
\label{RGN2}\end{aligned}$$
These RG equations enjoy some hidden symmetries: $$\begin{aligned}
\Omega_1 &:& g_{2,3,6} \to - g_{2,3,6} \nonumber \\
\Omega_2 &:& g_{3,4,8} \to - g_{3,4,8} \nonumber \\
\Omega_3 &:& g_{6,7,8} \to - g_{6,7,8} \nonumber \\
\Omega_4 &:& g_{2,4,7} \to - g_{2,4,7},
\label{eq:symmRG2}\end{aligned}$$ which correspond to duality symmetries on the Majorana fermions $$\begin{aligned}
\Omega_1 &:& \xi^{1,2,3}_\text{L} \to -\xi^{1,2,3}_\text{L} \nonumber \\
\Omega_2 &:& ~~ \xi^6_\text{L} ~ \to -\xi^6_\text{L} \nonumber \\
\Omega_3 &:& \xi^{5,7,8}_\text{L} \to -\xi^{5,7,8}_\text{L} \nonumber \\
\Omega_4 &:& ~~ \xi^4_\text{L} ~ \to -\xi^4_\text{L} ,
\label{dualitiespband}\end{aligned}$$ while the right-moving Majorana fermions remain invariant. The four dualities (\[dualitiespband\]), together with the trivial one $\Omega_0$, gives five possible $\mathrm{SO}(8)$-symmetric rays which attract the one-loop RG (\[RGN2\]) flows in the far IR regime. Along these rays, the interacting part of the effective Hamiltonian (\[Hamiltonian\_tperp0\_majoranapp\]) simplifie as follows: $$\begin{aligned}
\Omega_0 &:& \mathcal{H}^{\Omega_0}_{\text{int}} = \frac{g}{2}\left(\sum_{a=1}^8\xi^a_\text{R}\xi^a_\text{L}\right)^2
\nonumber \\
\Omega_1 &:& \mathcal{H}^{\Omega_1}_{\text{int}} = \frac{g}{2}\left(\sum_{a=4}^8\xi^a_\text{R}\xi^a_\text{L}
- \sum_{a=1}^3\xi^a_\text{R}\xi^a_\text{L}\right)^2 \nonumber \\
\Omega_2 &:& \mathcal{H}^{\Omega_2}_{\text{int}}
= \frac{g}{2}\left(\sum_{a\neq6}\xi^a_\text{R}\xi^a_\text{L} - \xi^6_\text{R}\xi^6_\text{L}\right)^2 \nonumber \\
\Omega_3 &:& \mathcal{H}^{\Omega_3}_{\text{int}}
= \frac{g}{2}\left(\sum_{a\neq5,7,8}\xi^a_\text{R}\xi^a_\text{L} -
\sum_{b=5,7,8}\xi^b_\text{R}\xi^b_\text{L}\right)^2 \nonumber \\
\Omega_4 &:& \mathcal{H}^{\Omega_4}_{\text{int}}
= \frac{g}{2}\left(\sum_{a\neq4}\xi^a_\text{R}\xi^a_\text{L} - \xi^4_\text{R}\xi^4_\text{L}\right)^2 .
\label{SO8raysapp}\end{aligned}$$ with $g>0$. The nature of the underlying electronic phase can then be inferred by a straightforward semiclassical approach on the bosonic representation of the different models in Eqs. (\[SO8raysapp\]) by means of the identification (\[refer\]). The following five different fully gapped Mott-insulating phases are found in this analysis.
**Spin-Peierls phase:**
The trivial duality $\Omega_0$ correspond to the SO(8) GN model. As seen in Sec. \[sec:Phases\_with\_dynamical\_symmetry\_enlargement\] in the general SO($4N$) case, the underlying Mott-insulating phase is a SP one with spontaneous dimerization.
**Spin Haldane phase:**
For the first non-trivial duality symmetry $\Omega_1$, the semi-classical approach leads to a non-degenerate phase where the bosonic fields are pinned as follows: $$\langle \Phi_{\text{s}} \rangle = \langle \Theta_{\text{so}} \rangle = \frac{\sqrt{\pi}}{2} ~ ; ~~
\langle \Phi_{\text{c,o}} \rangle = 0 ~~ \mbox{(SH phase)},
\label{HSsemi}$$ where $\Phi_{a} = \Phi_{a\text{L}} + \Phi_{a\text{R}}$ and $\Theta_{a} = \Phi_{a\text{L}} - \Phi_{a\text{R}}$ ($a =\text{c},\text{s},\text{o},\text{so}$) are respectively the total bosonic field and the dual field. The field configurations (\[HSsemi\]) correspond to the SH phase. [@Nonne-B-C-L-10]
**Rung-Singlet phase:**
The duality symmetry $\Omega_2$ leads to a non-degenerate phase with field configurations: $$\langle \Phi_{\text{c,s,o}} \rangle = \langle \Theta_{\text{so}} \rangle = 0 ~~ \mbox{(RS phase)}.
\label{RSso}$$ The physical picture of the corresponding phase is a singlet formed between the orbital and nuclear spins: $${| \mathrm{RS} \rangle} = \prod_{i}\frac{1}{\sqrt{2}}\big(c^\dagger_{p_x\uparrow,i}c^\dagger_{p_y\downarrow,i}-c^\dagger_{p_x\downarrow,i}c^\dagger_{p_y\uparrow,i}\big){| 0 \rangle}.
\label{RSWF}$$ Such phase is similar to the RS phase of the two-leg spin-1/2 ladder where a singlet is formed on each rung of the ladder. [@Gogolin-N-T-book] Since $T^z_i {| \mathrm{RS} \rangle} = 0$, the RS phase can also be interpreted as an orbital large-$D$ (OLD) phase along the $z$-axis.
**Charge Haldane phase:**
For the duality symmetry $\Omega_3$, we obtain again a non-degenerate phase with the following pinning: $$\langle \Phi_{\text{c}} \rangle = \langle \Theta_{\text{o}} \rangle = \frac{\sqrt{\pi}}{2} ~ ; ~~
\langle \Phi_{\text{s,so}} \rangle = 0 ~~ \mbox{(CH phase)}.$$ Such field configurations signal the emergence of a Haldane phase for the charge degrees of freedom, which has been dubbed charge Haldane (CH) phase (or equivalently Haldane charge) in Refs. . The spin degrees of freedom of this phase are described by the pseudo spin operator (\[chargepseudospin\]), which is a spin-singlet that carries charge. This CH phase is deduced from the usual SH phase by the Shiba transformation (\[eqn:Shiba-tr-SU2-2band\]).
**Orbital large-D phase:**
For the last duality symmetry, *i.e.* $\Omega_4$, the semi-classical approach gives the following vacuum expectation values: $$\langle \Phi_{\text{c,s,so}} \rangle = \langle \Theta_{\text{o}} \rangle = 0 .$$ The corresponding Mott insulating phase is non-degenerate and featureless. In the strong-coupling regime, a ground state for that phase is the singlet state: $${| \text{OLD}_x \rangle} = \prod_{i}\frac{1}{\sqrt{2}}\left(
c^\dagger_{p_x\uparrow,i}c^\dagger_{p_x\downarrow,i}
-c^\dagger_{p_y\uparrow,i}c^\dagger_{p_y\downarrow,i}\right){| 0 \rangle} ,
\label{oldx}$$ which is characterized by $T^x_i {| \text{OLD}_x \rangle} = 0$. The resulting spin-singlet phase is an orbital large-$D$ (OLD) phase along the $x$-axis. We can also think of a similar state along the $y$-axis: $${| \text{OLD}_y \rangle} = \prod_{i} \frac{1}{\sqrt{2}}\left(
c^{\dagger}_{p_x\uparrow,i}c^{\dagger}_{p_x\downarrow,i}
+ c^{\dagger}_{p_y\uparrow,i}c^{\dagger}_{p_y\downarrow,i} \right) |0\rangle \; .$$ The latter is different from the RS phase (\[RSWF\]) since (\[oldx\]) (respectively (\[RSWF\])) is antisymmetric (respectively symmetric) under the $p_x \leftrightarrow p_y$ exchange.
**Phase diagram in the weak coupling regime:**
![(Color online) Phase diagram for $N=2$ $p$-band model obtained by solving numerically the one-loop RG Eqs (\[RGN2\]) with initial conditions (\[majocouplingsapp\]). The line $U_1=3U_2$ corresponds to the axially symmetric trapping scheme. \[fig:RG\_phasediag\_pband\_N2\]](RG_Phase_diag_pband_N2_U1_U2_with_lines){width="0.5\columnwidth"}
Following the same procedure as described in Sec. \[sec:RG-phase-diag\], we solve numerically the RG equations (\[RGN2\]) with initial conditions (\[majocouplingsapp\]) to obtain the low-energy phase diagram of the $N=2$ $p$-band model in the ($U_1, U_2$) plane. We identify four out of the five regions discussed above. Indeed, the SP phase is not realized. These fours regions are readily identified as SH, RS, CH and OLD by the flows of the couplings $g_i(l_{\text{max}}) = \pm g_{\text{max}}$ that are in agreement with the symmetries . The phase diagram in the low energy limit Fig. \[fig:RG\_phasediag\_pband\_N2\] is equivalent with the one obtained with the DMRG technique in Fig. \[fig:phasediag\_pband\_N2\] (see discussion in Sec. \[sec:DMRG\_N2pband\]).
From the duality symmetries (\[dualitiespband\]), we can, as well, discuss the nature of the quantum phase transitions that occur in Fig. \[fig:RG\_phasediag\_pband\_N2\] by investigating the self-dual manifolds.
**CH/RS or SH/OLD transition:**
The transition between the CH and RS phases, or between SH and OLD, is governed by the self-dual manifold of the duality $\Omega_2 \Omega_3$ where $\xi^{5,6,7,8}_\text{L} \to -\xi^{5,6,7,8}_\text{L}$. The self-dual manifold is then described by $g_3=g_4=g_6=g_7 =0$. From the initial conditions (\[majocouplingsapp\]), we observed that the line $U_1 =U_2$ of the $p$-band model belongs to that manifold. The interacting part of the effective Hamiltonian (\[Hamiltonian\_tperp0\_majoranapp\]) simplifies as follows along that line: $$\begin{aligned}
\mathcal{H}^{\mathrm{CH/RS}}_{\tiny\mbox{int}}
= \frac{g_1}{2}\left(\sum_{a=1}^4\xi^a_\text{R}\xi^a_\text{L} \right)^2
- \frac{g_1}{2}\left(\sum_{a=5}^8\xi^a_\text{R}\xi^a_\text{L} \right)^2 ,
\label{transHCRS}\end{aligned}$$ which takes the form of two decoupled SO(4) GN models. Due to the particular structure of model (\[transHCRS\]), one of this SO(4) GN displays a critical behavior while the other is massive. We thus conclude that the quantum phase transition CH/RS or SH/OLD belongs to the SO(4)$_1$ universality class with central charge $c=2$.
**SH/RS or CH/OLD transition:**
One can repeat the analysis for the transition between the SH and RS phases, or between CH and OLD. In that case, the relevant duality is $\Omega_1 \Omega_2$ with $\xi^{1,2,3,6}_L \to -\xi^{1,2,3,6}_L$. The resulting self-dual manifold is $g_2=g_4=g_6=g_8 =0$. From the initial conditions (\[majocouplingsapp\]), we observed that the line $U_2 = 0$ of the $p$-band model belongs to that manifold. The interacting part of the effective Hamiltonian (\[Hamiltonian\_tperp0\_majoranapp\]) simplifies as follows along that line: $$\begin{split}
\mathcal{H}^{\mathrm{SH/RS}}_{\text{int}}
=& \frac{g_1}{2}\left(\sum_{a=1,2,3,6}\xi^a_\text{R}\xi^a_\text{L} \right)^2 - \frac{g_1}{2}\left(\sum_{a=4,5,7,8}\xi^a_\text{R}\xi^a_\text{L} \right)^2 ,
\end{split}
\label{transHSRS}$$ which takes also the form of two decoupled SO(4) GN models with an emerging SO(4)$_1$ quantum criticality with $c=2$. This last result can be easily understood since when $U_2 = 0$ the $p$-band model (\[eqn:p-band\]) is equivalent to two decoupled half-filled Hubbard chains and therefore a critical behavior with central charge $c=1+1=2$ occurs.
**SH/CH or RS/OLD transition:**
In this last case, the quantum phase transition is described by the duality $\Omega_1 \Omega_3$ with $\xi^{1,2,3,5,7,8}_\text{L} \to -\xi^{1,2,3,5,7,8}_\text{L}$. The self-dual manifold is $g_2 = g_3 = g_7 = g_8 =0$. Using the initial conditions of the $p$-band model (\[majocouplingsapp\]), the non-interacting point belongs to that manifold and we expect thus that the SH/CH and RS/OLD transitions occur for $U_1 =U_2 =0$.
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Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases
with symmetries, which have gapless or degenerate boundary states as long as
the symmetries are not broken. The SPT phases in free fermion systems, such
as topological insulators, can be classified; however, it is not known what
SPT phases exist in general interacting systems. We present a systematic way
to construct SPT phases in interacting bosonic systems. Just as group theory
allows us to construct 230 crystal structures in three-dimensional space, we
use group cohomology theory to systematically construct different interacting
bosonic SPT phases in any dimension and with any symmetry, leading to the
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[^1]: This breaks down when there is a hopping (transition) between $g$ and $e$.
[^2]: This corresponds to, e.g., using the SU(2) generators $\sigma^{a}/\sqrt{2}$ ($a=x,y,z$) instead of the standard ones $\sigma^{a}/2$.
[^3]: This is in a sense an artifact of the choice of the basis ($p_x$ and $p_y$). In fact, if we choose the angular-momentum (along the $z$-axis) basis, the U(1)-symmetry is obvious.
[^4]: It is evident that one can generalize this strategy to general even-$N$ once we know the Clebsch-Gordan decomposition of the two physical spaces on the adjacent sites.
[^5]: For instance, according to the group-cohomology scheme[@Chen-G-W-11; @Schuch-G-C-11], the topological phases protected by $\text{PSU(2)}\simeq \text{SO(3)}$ are classified by the parity of integer-spin $S$. In the topological Haldane phase corresponding to odd-$S$, the entanglement spectrum consists of even-fold degenerate levels reflecting the even-dimensional edge states emerging at the edge.
[^6]: In order to obtain the generators in the 20-dimensional representation ($\mathbf{20}$), one may start, e.g., from $\mathbf{4}\otimes\mathbf{4}\simeq \mathbf{6}\oplus\mathbf{10}$ and then use $\mathbf{6}\otimes \mathbf{6} \simeq \mathbf{1}\oplus\mathbf{15}\oplus\mathbf{20}$.
[^7]: We use an argument similar to that in Refs. .
[^8]: The simplest example of such rays is the separatrices in the Kosterlitz-Thouless RG flow.
[^9]: Actually, for $N>2$ the position of the different regions obtained by solving numerically the RG equations is almost not affected by $N$. However, the nature of the phases differs. For instance, for $N$ odd, the ‘No DSE’ region in Fig. \[fig:RG\_general\_phase\_diagram\] turns out to be critical.
[^10]: Our global RG approach, based on duality symmetries, give all possible DSE phases compatible with the global symmetry group of the low-energy Hamiltonian. Some phases might not be realized in concrete lattice model with the same continuous symmetry. The SP$_{\pi}$ phase is one example and a more general lattice model is necessary to stabilize such a phase.
[^11]: See also Ref. where this line was investigated in strong coupling and numerically in the equivalent formulation of a generalized Hund model
[^12]: The upper critical value has been estimated more recently [@Tonegawa-O-N-S-N-K-11] to be $D/{\cal J}=2.4$.
[^13]: Near the line $U_2=0$, the impact of the deviation from $U_2=0$ is readily estimated. When $U_2>0$, the two CDWs on different orbitals (‘chains’) repel each other due to the positive $V=U_2$-interaction and favor out-of-phase CDW, i.e., ODW. When $U_2<0$, on the other hand, they attract each other and consequently stabilize usual CDW.
|
---
abstract: 'We present a power efficient clock-less fully asynchronous bit-serial link with event-driven instant wake-up and self-sleep features, optimized for high speed inter-chip communication of asynchronous address-events between neuromorphic chips. The proposed link makes use of the representation and a token-ring architecture to encode and transmit data, avoiding the use of conventional large modules with power-hungry DLL or PLL circuits. We implemented the circuits in a device fabricated with a standard 0.18$\mu$m CMOS process. The total silicon area used for such block is of 0.14mm$^{2}$. We present experimental measurement results to demonstrate that, with a bit rate of 1.5Gbps and an event width of 32-bit, the proposed link can achieve transmission event rates of 35.7MEvents/second with current consumption of 19.3mA and 3.57mA for receiver and transmitter blocks, respectively. Given the clock-less and instant on/off design choices made, the power consumption of the whole link depends linearly on the data transmission rate. We show that the current consumption can go down to sub-$\mu$A for low event rates (e.g., $<$1kEvents/second), with a floor of 80nA for transmitter and 42nA for receiver, determined mainly by static off-leakage currents.'
author:
-
-
title: 'A clock-less ultra-low power bit-serial [LVDS]{} link for Address-Event multi-chip systems'
---
Introduction
============
protocol has been widely used in neuromorphic computing systems to connect multiple cores and chips together [@Moradi_etal17; @Park_etal16; @Merolla_etal14a; @Furber_etal14; @Benjamin_etal14; @Liu_etal14], in single-chip devices for encoding sensory signals [@Liu_Delbruck10] or for implementing spike-based learning mechanisms [@Qiao_etal15; @Giulioni_etal12], and in multi-chip sensory-processing systems [@Neftci_etal13; @Serrano-Gotarredona_etal09; @Chicca_etal07a]. By exploiting the asynchronous principle, the protocol is extremely efficient for event-driven neural system in terms of power consumption and low latency. Bit-parallel AER is the most commonly used implementation, due to its ease of design and configuration. This strategy however is not scalable, as the width of the parallel bus and the power required to transmit these parallel event bits scales with the size of the network. This can become a critical issue for large scale neuromorphic systems, which typically employ multiple copies of buses for routing events to multiple destinations and receiving events from multiple sources [@Moradi_etal17; @Merolla_etal14a; @Furber_etal14]: these systems are normally arranged and tiled in 2D arrays with North-South, East-West, and possibly diagonal links between them. This requires a very large pin-count and can lead to significant leakage and dynamic power consumption. Rather than using the full parallel protocol, some approaches have resorted to employing a “word-serial” protocol, which groups multiple row addresses for a given column address to reduce pin count [@Benjamin_etal14; @Brandli_etal14]. However, it has been argued that one of the most efficient solutions for transmitting data in terms of both speed and power consumption, is to use a bit-serial scheme [@Zamarreno-Ramos_etal13a].
Event rates in neuromorphic systems tend to be sparse, but to have high peak values [@Brandli_etal14]. As the time information is typically important, low latency is an essential requirement. Traditional schemes are designed for continuous data transmission with power consumption that depends on clock frequency, and independent of the input data rate. In some schemes it is possible to send idle comma characters and signal a pause in the data transmission. However, these idle states may cause loss of synchronization between transmitter and receiver, and many (e.g., in the order of hundreds) clock cycles are typically required for these lock recoveries. Therefore, as traditional implementations are likely to cause significant latency for event transmission, they are not suitable for neuromorphic systems. Previous approaches have proposed to optimize the scheme so that the phase lock of transmitter and receiver can be recovered on the fly [@Zamarreno-Ramos_etal13a], but they required additional clock generation and synchronization circuits, such as DLL or PLL circuits, for the which are very expensive in terms of power and area requirements. The event-based nature of data transmission in neuromorphic systems calls for the development of a new fully asynchronous clock-less event-based switchable bit-serial link, that does not need clock recovery circuits. In this paper we propose a new clock-less scheme optimized for neuromorphic systems, and demonstrate its implementation in a prototype chip, fabricated using a standard 0.18$\mu$m CMOS process. We show that the chip designed successfully implements the following features:\
1) Pure asynchronous design without PLL/DLL for .\
2) Instant on ($<$0.5ns) wake-up for new event data and instant off ($< $0.5ns) self-sleep in absence of data, maintaining low latency and low power consumption.\
3) Sub-nW (220nW) static power consumption and event-rate based dynamic power consumption.\
4) Compact layout as a building block for multi-core and multi-chip neuromorphic systems.
The paper is organized as follows: Section \[sec:encod-scheme-arch\] presents the data transmission scheme and link architecture; Section \[sec:circ-impl\] describes the circuits implementation of the proposed bit-serial link; Section \[sec:experiment\] presents the measurements made with the prototype chip and describes the experimental results; Section \[sec:conclusions\] shortly concludes the work.
Encoding Scheme and Architecture {#sec:encod-scheme-arch}
================================
Data encoding
-------------
![Encoding example of . Shaded regions represent the even phase and the non-shaded regions represent the odd phase.[]{data-label="fig:encoding"}](encode){width="28.00000%"}
It is possible to implement a clock-less fully asynchronous event-driven bit-serial link by choosing a proper data encoding scheme that eliminates the need of traditional , which is expensive for asynchronous systems. One scheme that is optimally suited for data is the signaling scheme [@Dean_etal91]. In signaling data bits are encoded using two rails: given a sequence of bits, a data rail is used to represent the bit value and a parity rail is used to represent the parity relative to the encoding phase and data rail. The encoding alternates between an even and an odd phase. In the even phase the parity rail takes the inverted bit value, and in the odd phase the parity rail takes the same bit value of the data rail. Formally the data rail value $D[i]$ and the parity rail value $P[i]$ are: $$\begin{cases}
D[i]=B[i]; \quad P[i]=\overline{B[i]}
& \text{for odd phase }
\\
D[i]=B[i]; \quad P[i]=B[i]
& \text{for even phase }
\end{cases}$$ where B\[i\] represents the encoded bit value of the sequence. Figure \[fig:encoding\] shows an encoding example for an 8-bit data sequence.
The is a protocol: sequential bits can easily be distinguished by checking whether D\[i\]=P\[i\] or not. So, by encoding address event data strings using , it is possible to build fully asynchronous bit-serial links without using a clock generation block or a clock synchronization block for .
According to this encoding scheme, it is possible to implement both asynchronous encoder and decoder. On the encoder side, the data rail should always take the original sequence bit value while the parity rail should take the inverted sequence bit value for the odd phase, and the original sequence bit value for the even phase. On the decoder side, it is sufficient to check if $D[i]=P[i]$ or $P[i]=\overline{D[i]}$ to determine the bit phase, and then to read incoming bits one by one. This scheme leads to a very compact design in terms of hardware resources. Because encoding follows a two-phase handshaking (or ) protocol, it allows a full bit rate and provides a significant bandwidth advantage comparing to alternative schemes based on or methods.
LVDS with token-rings {#sec:token_ring}
---------------------
![A typical 8-bit transceiver based on token-ring architecture. Token-cells are labeled as “TCell”.[]{data-label="fig:token_ring"}](token_ring.png){width="45.00000%"}
{width="65.00000%"}
Token-ring schemes have already been proposed for asynchronous sequential data transmission [@Teifel_Manohar03]. A token-ring comprises a number of mutually exclusive token-cells to transmit their data content one by one. Figure \[fig:token\_ring\] shows a typical 8-bit transceiver based on token-ring architecture [@Teifel_Manohar03]. Token-cells in the “Transmitter block” are activated sequentially to take one bit at a time from a parallel data bus and to write it on a shared interconnection link. Token-cells in the “Receiver block” take bit values sequentially from the shared bus to reconstruct the parallel data.
A token-ring based serializer can be built following the scheme to sequentially encode both data and corresponding parity bits from a parallel bus to a shared serial one. Accordingly, a token-ring based de-serializer can be built to de-serialize and decode the data by taking data bit-by-bit from shared data/parity wires. The block diagram of the asynchronous bit-serial link we propose based on these concepts is shown in Fig. \[fig:arch\]. It comprises the following blocks: “Input Buffer”, “TX Token-Ring”, “RX Token-Ring”, “ Drivers”, “ Receivers”, “Output Buffer” and “Control Queue”.
The “TX Token-Ring” block is implemented to serialize and encode event parallel bits into data and parity rails following the scheme. The “RX Token-Ring” is implemented to de-serialize and reconstruct parallel bits from data and parity rails. [The “ Drivers” convert data and parity rails into low-voltage differential signals for low-power consumption and high-speed inter-chip data transmission. Similarly, the “ Receivers” convert signals back to normal digital signals. In order to minimize power consumption and make it depend only on the event-rate, we propose a novel instant on/off scheme for Drivers and Receivers, described in the following section.]{} Finally, the data transmission is done in a “burst mode”, such that the acknowledge signal is returned once per address event word, rather than bit-by-bit. Address event input and output buffers are included to pipeline the transmission cycle and increase data depth on both sides. A small “Control Queue” block with the same depth of the output buffer is employed to pre-store multiple acknowledges, so that the transmitter can keep on sending events without waiting for their corresponding acknowledge signals to arrive, in order to minimize latency.
Instant On/Off driver and receiver
----------------------------------
Instant on/off drivers and receivers that implement event-driven wake-up and sleep-mode mechanisms are crucial for minimizing consumption in neuromorphic systems that operate with sparse activity and low average event rates. Since the main digital blocks communicate with each other following a four-phase handshaking protocol, no dynamic power is dissipated in idle states. The “ Drivers” can be easily turned on or off by a digital signal, such as $TX.r$ of Fig. \[fig:arch\], as new event data appears on the “Input Buffer” block. In order to turn on/off the “ Drivers” instantly, we exploited the common voltage of pairs. As shown in Fig. \[fig:LVDS\_CM\], during the idle state, when no data is being transmitted, the two pairs of signals are both pulled down to *Gnd*, resulting in a 0V common mode voltage. In this way the “ Receivers” , designed with NMOS input transistors, will be fully tuned off and power consumption will be due only to off-leakage level static power dissipation. As soon as a new event arrives, the common-mode feedback circuit will drive both data pair and parity pair voltage lines back to a *Vref* common mode voltage, which is set to about 1V in this design. Simultaneously, the differential voltages of data pair and parity pair will recover back to their previous bits value with $D=P$. In this way the “ Receivers” with NMOS input transistors will be turned on and will start to convert the signals to standard digital ones. Because the first odd token-cell in the decoder will only take data when $P[i]=\overline{D[i]}$, the receiver will ignore potential spurious repeated LSB bits until a new MSB bit arrives. After transmitting the full word, the common-mode voltage of the pairs will be pull down to *Gnd* again, turning the receiver off. The recovery speed of common-mode voltage is controlled by a common-mode feedback circuit in the driver. In our measurement, the recovery latency of common-mode voltage is less than 0.5ns, which is much shorter than previously reported values (e.g., 6.6ns in [@Zamarreno-Ramos_etal12]).
![Proposed signaling scheme with LVDS for data transmission and common-mode voltage for instant on/off receiver. []{data-label="fig:LVDS_CM"}](LVDS_CM){width="40.00000%"}
Transmission Scheme
-------------------
![Event transmission timing diagram of the bit-serial link. The “Stb” represents the stand-by state.[]{data-label="fig:scheme"}](scheme){width="48.00000%"}
Figure \[fig:scheme\] describes the timing diagram for the transmission of one event in the proposed bit-serial link. A four-phase handshaking protocol is implemented between “Input Buffer” and “TX Token-Ring”. Once the event data $D_{1}<n-1:0>$ appears on the “TX Token-Ring” input bus $TX.in$, the signal $TX.r$ will be set to high by the “Input Buffer”, thus requesting a new data transmission which will trigger the first stage token-cell of “TX Token-Ring” to take the first bit. Meanwhile, this request signal will turn on the drivers to be ready for sending new data. After a tunable delay $t_{wd}$, the first odd token-cell of “Token-Ring” will push new bit value $D=D_{1}<n-1>$ and parity $P=\overline{D_{1}<n-1>}$ to shared data and parity wires $Data$ and $Parity$. It will then enable the following stage, i.e., the first even token-cell to take new bit value. After a cycle delay $t_{d}$, the enabled stage will disable previous stage and push new data/parity $D=D_{1}<n-2>$ and $P=D$ on the shared wires, while enabling its following stage. Mutual exclusion is implemented stage by stage till the end the token-ring. After pushing data/parity of the last bit value to shared wires, the “TX Token-Ring” block will acknowledge the “Input Buffer” by asserting $Enc.a$ to high. Subsequently, $TX.r$ will be reset to low for the successful removal of data $D_{1}<n-1:0>$. Finally, $TX.a$ will be reset to low to complete the four-phase handshaking cycle.
It should be noted that during the wake-up stage the “ Receivers” will need to be turned on for recovering the common-mode voltage of pairs with data and parity value $P=D$. The first token-cell of “RX Toke-Ring” will only take data and parity with $P=\overline{D}$. For an event data with even bit width, a safe approach is to fully recover both common-mode and differential values of previous bit by repeating the LSB of the previous event data with data $D=D_{0}<0>$ and party $P=D_{0}<0>$.
The mutually exclusive token-cells of “RX Token-Ring” will take data from receivers bit-by-bit. Each bit cycle is distinguished by either $P=\overline{D}$ or $P=D$. The output of each token-cell is latched, once the current token-cell is disabled by its successor. As soon as the last token-cell gets its bit, it will request “Output Buffer” to take the whole data packet from all token-cells and reset the “RX Token-Ring”. In this design the “RX Token-Ring” is required to have the highest throughput. The tunable delay $T_{d}$ is added in “TX Token-Ring” to enforce the timing assumptions that “RX Token-Ring” has a higher throughput than “TX Token-Ring”, to get sequence bit within one TX bit cycle.
Circuits Implementation {#sec:circ-impl}
=======================
Transmitter Token-Ring
----------------------
![Transmitter Token-Ring for encoding data into data/phase scheme in proposed bit-serial link. []{data-label="fig:TX_token_ring"}](TX_token_ring){width="44.00000%"}
![Circuit implementation of the TX token-cell based on a bit-buffer. Each token cell comprises “Handshaking”, “Validity Check”, “Bit Buffer”, “Data Buffer” and “Odd/Even Parity Buffer” blocks. []{data-label="fig:TX_token_cell"}](TX_token_cell){width="48.00000%"}
The block diagram of “TX Token-Ring” is shown in Fig. \[fig:TX\_token\_ring\]. A dual-rail asynchronous protocol and four-phase handshaking are used for processing input data. The “TX Token-Ring” comprises an input “Validity Check” block, “Token-Ring” with odd and even token-cells, “ Drivers” and a “Control Queue” block. The “Validity Check” block first checks and indicates a valid input event data by $TX.r$. The “ Drivers” can then be turned on by $TX.r$ for a valid input event. Meanwhile, the first token-cell starts to take the first bit value $TX.f<n-1>/TX.t<n-1>$ and push relative data and parity outputs to shared wires. For odd bits, data and parity outputs are $D=B$ and $P=\overline{D}$, respectively, while for even bits, they are $D=B$ and $P=D$, respectively. So the first odd token-cell will push $D=TX.t<n-1>$ to shared data wire and $P=TX.f<n-1>$ to shared parity wire. After a tunable delay $t_{wk}$ when the first token-cell successfully pushes data and parity value of MSB of input event to shared data and parity wires, the first token-cell will send the $enable$ signal to enable its successor for the next bit value. As a response, its successor will send back the $disable$ signal as soon as it successfully takes a bit value. After a set “bit cycle” time $t_{d}$ when the last token-cell pushes its output to shared data and parity wires, $Enc.a$ will be asserted to high to reset the whole “TX Token-Ring” and acknowledge “Input Buffer” to erase old data, and will be reset to low to acknowledge that old data has returned to zero ($TX.f<n-1:0>=0, TX.t<n-1:0>=0$). At this point the “TX Token-Ring” is free to take new data.
Figure \[fig:TX\_token\_cell\] shows the circuit implementation of the proposed token-cell, based on an asynchronous buffer following a dual-rail protocol and four-phase handshaking. The token-cell comprises a “Handshaking” block, a “Validity Check” block, “Bit Buffer”, “Data Buffer” and “Odd/Even Parity Buffer” blocks. The “Validity Check” block checks the validity of input bit value and indicates the state by signal $in.v$. The “Handshaking” block generates the acknowledge signal $in.a$ to acknowledge a valid bit input and control signal $en$ to enable Bit Buffer block for buffering current input bit value ($en=1$) or reset the “Bit Buffer” block for the next cycle ($en=0$). The “Data Buffer” and “Odd/Even Parity Buffer” blocks will convert buffered bit value $out.t$ and $out.f$ to data and parity value according to protocol and push them to shared $Data$ and $Parity$ wires. Once the current token-ring generates a valid output which is indicated by $out.v=1$, it will enable its successor and disable its predecessor for mutual exclusion.
LVDS Drivers
------------
![Circuit implementation of the “TX Driver”. []{data-label="fig:TX_driver"}](TX_driver.png){width="40.00000%"}
Current mode Drivers, shown in Fig. \[fig:TX\_driver\], are used to convert data and parity value on shared wires to pairs. The “ Driver” is implemented such that it can convert input value $D_{in}$ into signals for a valid input ($WKUP=1$) and fully turned off for a standby mode ($WKUP=0$). When no data is transmitted ($WKUP=0$), the “CM-FB” block is switched off. The signals $DN$ and $D$ are then both set to logic “1” to tune off their gating PMOS transistors and tune on their gating NMOS to pull both $LVDS\_f$ and $LVDS\_t$ down to *Gnd*, with a common-mode pair voltage $V_{CM}=0$. This will switch off its linked “ Receiver” block following the proposed instant on/off scheme. Once there is a valid input ($WKUP=1$), the “CM-FB” block will supply property common-mode pair voltage $V_{CM}=V_{ref}$ to switch on the “ Receiver” on the receiver side, and the “Driver” block will start to convert $D_{in}$ into . [The $V_{B1}$ and $V_{B2}$ signals are biases to generate proper tail currents for the “CM-FB” and “Driver” blocks.]{} Two resistors with value $R=50\Omega$ (with another two resistors placed at the input terminals of the “ Receiver”) are used to setup differential amplitude of pair.
Receiver Token-Ring
-------------------
![Receiver Token-Ring for decoding data/phase to event data in proposed bit-serial link. []{data-label="fig:RX_token_ring"}](RX_token_ring.png){width="44.00000%"}
![Circuit implementation of RX token-cell based on 1-bit buffer. Each token-cell comprises Handshaking block, Validity Check block and Bit Buffer block. []{data-label="fig:RX_Tcell"}](RX_Tcell.png){width="44.00000%"}
The architecture of the “RX Token-Ring” for processing and decoding pairs $LVDS\_D$ and $LVDS\_P$ is shown in Fig. \[fig:RX\_token\_ring\]. Following the dual-rail asynchronous protocol and four-phase handshaking, the “RX Token-Ring” comprises “ Receivers”, “Token-Ring” with odd and even token-cells and an “Output Buffer” block. The “ Receivers” first digitize the pairs $LVDS\_D/P$ to digital sequential bits $D.f/t$ and $P.f/t$, respectively. The token-cells then take bits one-by-one till the end of this event transmission. Once all token-cells take and buffer bits value, the following the “Output buffer” will buffer received event data $RX.f<n-1:0>$ and $RX.t<n-1:0>$ to output bus $AER.out$ and reset the “RX Token-Ring” for new data.
The circuit implementation of the “RX token-cell” following the dual-rail protocol and four-phase handshaking is shown in Fig. \[fig:RX\_Tcell\]. Each RX token-cell comprises a “Handshaking”, “Validity Check” and “Bit Buffer” block. The odd token-cell will only take and process bit value with $P=\overline{D}$ and even token-cell will only take and process bit value with $P=D$. When new bit value comes to the token-ring with proper data and parity value relationship, for example, $D.f=P.t$ and $D.t=P.f$ for $P=\overline{D}$, the current activated odd token-cell will take this bit value and buffer it with its “Bit Buffer” block. After generating a valid output bit value ($out.v=1$), internal signal $en$ will be set to logic “0” to latch the output bit value and block it to take new bit value. Meanwhile, this token-cell will enable its following token-cell for a new token.
LVDS Receivers
--------------
![LVDS Receiver for digitizing differential to digital signals. []{data-label="fig:RX_amp"}](RX_amp.png){width="44.00000%"}
In order to meet the requirement of instant on/off by means of the common-mode voltage, we implemented the amplifier-based receivers with NMOS inputs. The circuit implementation of the proposed “LVDS Receiver” is shown in Fig. \[fig:RX\_amp\]. It comprises an “Amp” block, a “Latch” block and a “Buffer” block. The “Amp” block is responsible for digitizing the signals. In standby mode, the “Amp” stage will be fully tuned off with $LVDS\_f/t=0$. Once there is data from transmitter side that needs to be transmitted (i.e., when $V_{CM}=V_{ref}$), the “Amp” stage will be tuned on instantly. A latch stage with dynamic biases is implemented to latch the last bit value of previous event data once the “Amp” stage is switched to sleep mode so that the receiver will not wake up with a random output bit value. After a successful event transmission, $V_{CM}$ of the pair will be switched from $Vref$ to *Gnd*, with $VP$ and $VN$ shifting to *Vdd* and *Gnd* respectively. This will strengthen the drive ability of the latch stage to store the current bit value when the “Amp” stage is turned off. As new event data arrives, the signals $VP$ and $VN$ will be shifted to near *Vdd/2* to tune the latch stage weaker, so that it can be modified by the new data. An active-low reset signal $RstB$ is used to reset the circuit outputs to a proper initial condition ($P=D$) when powering up the chip.
Experimental results {#sec:experiment}
====================
![Die photo of test chip with proposed event-driven bit-serial link in AMS 0.18um 1P6M process, in which TX block occupies an area of 0.08 $mm^{2}$ and RX block occupies an area of 0.06$mm^{2}$.[]{data-label="fig:die_photo"}](async_link_die){width="42.00000%"}
![The setup for testing links between two chips for bidirectional communication. []{data-label="fig:setup"}](setup){width="40.00000%"}
The proposed fully asynchronous event-driven bit-serial link was implemented using a standard 0.18$\mu$m 1P6M CMOS process, occupying a silicon area of 0.14$mm^{2}$. Figure \[fig:die\_photo\] shows the die photo of the fabricated test chip. The whole “Transmitter” block including the “TX\_Buffer” occupies an area of 0.08 $mm^{2}$, and the “Receiver” block including the “RX\_Buffer” occupies an area of 0.06$mm^{2}$. Additionally, a small spiking neural array with tunable output event rate is implemented to provide events for testing. A 32-bit router is implemented for routing events from Receiver to Transmitter to realize a transmission loop between 2 chips to explore peak transmission throughput.
![Transient signals of pairs captured on the receiver’s inputs: the traces $D.f$ and $D.t$ represent the differential signals for $LVDS\_D$; the traces $P.f$ and $P.t$ represent the differential signals for $LVDS\_P$; The $D\_Diff$ and $P\_Diff$ traces are differential voltages of two pairs; The $D\_CM$ and $P\_CM$ traces represent the common voltages of the two pairs; The last plot shows the $RX\_Ack$ signal, which is the acknowledge signal from the target chip to the source chip, representing a successful event transmission.[]{data-label="fig:experiment"}](experiment.png){width="46.00000%"}
![Transient signals of pairs at receiver inputs: (a) differential mode of signals, (b) acknowledge signal from the receiver, (c) details of single event transmission signals. []{data-label="fig:experiment2"}](experiment2){width="46.00000%"}
Figure \[fig:setup\] shows a setup with two chips placed side-by-side for the experiments. With this setup, we transmitted sequences of 32-bit AER events bi-directionally between two chips, through four pairs: The signals $LVDS1\_D$ and $LVDS 1\_P$ were used to transmit events from Chip1 to Chip2, and $LVDS2\_D$ and $LVDS 2\_P$ were used to transmit events from Chip2 to Chip1.
Transient Signals of pairs were observed and captured using a Tektronix DPO7000 Oscilloscope, from the input terminals of the “ Receivers”. As shown in Fig. \[fig:experiment\], the $LVDS\_D$ plot shows data from the pair with differential signals $D.f$ and $D.t$. The $LVDS\_P$ plot shows the parity pair with differential signals $P.f$ and $P.t$. The $D\_Diff$ and $P\_Diff$ traces in the $V_{Diff}$ plot are the differential voltages of the data and parity pairs, respectively. The $D\_CM$ and $P\_CM$ traces in the $V_{CM}$ plot are the common voltage of data and parity pairs, respectively. The $out.a$ plot shows the acknowledge signal from the target receiver chip for acknowledging a successfully event transmission. Sequence bits are presented bit-by-bit following the protocol, via the data and parity differential signals $D\_Diff$ and $P\_Diff$. The common-voltages of the two pairs $D\_CM$ and $P\_CM$ are reset to *Gnd* at the end of a successful event transmission and are quickly recovered with new coming events. During the recovery of common-mode voltages of the pairs, the LSB of previous event with $P=D$ is repeated for sufficient long time to guarantee that the receiver is fully and successfully switched on.
**[@Teifel_Manohar03]** **[@Zamarreno-Ramos_etal08]** **[@Zamarreno-Ramos_etal13a]** **This work**
---------------------------- ------------------------- ------------------------------- -------------------------------- ---------------
**Technology** 0.18$\mu$m 90nm 0.35$\mu$m 0.18$\mu$m
**Power Supply** 1.8V 1V 3.3V 1.8V
**Area** 0.016$mm^{2}$ 0.09$mm^{2}$ 0.352$mm^{2}$ 0.14$mm^{2}$
**Clocked CDR** No Yes Yes No
**Bit Rate** 3Gps 1Gps 0.64Gps 1.5Gps
**Event Rate** - 29.4MEvent/s 13.7MEvent/s 35.7MEvent/s
$\mathbf{P_{max}}$ 77mA 40.1mA 15.9mA 22.9mA
$\mathbf{P_{min}}$ - 40.1mA 0.4mA 0.122$\mu$A
$\mathbf{P_{max}/P_{min}}$ - 1 39 187.7k
\[table:performance\]
Figure \[fig:experiment2\] shows the transient signals of the pairs at the input terminals of receiver chip, captured by an probe TektronixP6880. The bit cycle is set to be around 0.67ns by tuning the delay cell $t_{d}$ in the “TX Token-Ring” to achieve a bit-rate of 1.5Gps. The observed switch on/off speed of the receiver is approximately 0.45ns and 0.5ns, respectively, leading to a smaller latency for transmission. As measured in Fig. \[fig:experiment2\](b), the latency needed for a successful transmission between chips (from switching on the Receiver to getting acknowledge signal $out.a$ from receiver chip) for a 1.5Gps bit-rate is 31ns. The period of a successive events transmission is 28ns. Since the transmitter has locally pre-stored the “out.a” signal in the “Control Queue” block (see Section \[sec:token\_ring\]), it will keep on sending event data without waiting for acknowledge signals from the receiver chip until the “Control Queue” is fully empty, to further decrease latency. This is evidenced in Fig. \[fig:experiment2\](a) and (b), as the second event transmission happens before the arrival of first acknowledge signal $out.a$. In Fig. \[fig:experiment2\](c) we can observe that, for transmitting a 32-bit event data with a bit-rate of 1.5Gps, the link will only be switched on for 25.6ns, and will be switched off instantly on both transmitter and receiver sides, leading to a pure event-rate related power consumption.
![Power consumption of asynchronous serial-bit link. []{data-label="fig:power"}](power){width="46.00000%"}
In Fig. \[fig:power\] we plot the measured power consumption for different event transmission rates. The peak event rate that can be achieved in our experimental setup is 35.7MEvents/second (32-bit) with current consumption of 19.3mA and 3.57mA for transmitter and receiver part, respectively. The power consumption of both transmitter and receiver part scales linearly with the event transmission rate. At a 10k event rate, the power consumption of the transmitter and receiver blocks are 5.2$\mu$A and 1.05$\mu$A, respectively. The power consumption can further go down to sub-$\mu$A for a lower event rates (<1kEvents/second), with a floor of 80nA for transmitter and 42nA for receiver which is mainly dominated by leakage current of circuits.
Table \[table:performance\] shows a performance comparison between different designs. However, area and power consumption of circuits employed in the designs of [@Zamarreno-Ramos_etal08; @Zamarreno-Ramos_etal13a] are not reported. So it may be that significant additional silicon area and power consumption are required for those designs.
Conclusions {#sec:conclusions}
===========
While neuromorphic electronic systems have the potential of solving the memory bottleneck problem [@Indiveri_Liu15], by construction they also face an important I/O bottleneck problem: large scale neuromorphic system implementations are typically composed of multiple cores and/or multiple chips tiled together, with grid-like communication networks. To transmit address-events across these cores and chips and to sustain the required bandwidth, current implementations use multiple parallel buses (e.g., for North-South, East-West, and possibly diagonal links). In this paper we argued that full parallel or even word-serial protocols are not scalable, as they require large number of pins/pads and large power consumption to quickly charge and discharge all these lines. To solve this problem, we proposed an ultra low-power fully asynchronous event-driven instant on/off bit-serial link, which is suitable for transmission in neuromorphic multi-chip systems. The proposed link uses encoding and a token-ring architecture to eliminate the need for clock-based blocks with expensive on chip DLL/PLL circuits, leading to a very compact and low-power circuit implementation. A novel scheme is proposed to implement a low-latency event-driven transmission with sub-ns instant on/off feature. Experimental results demonstrate how the proposed bit-serial link can achieve an event rate of 35.7MEvents/second with a bit-rate of 1.5Gps. The power consumption of the proposed link is pure rate-dependent, with a sub-$\mu$A power consumption for low event rates (e.g.,$\approx$1kEvents/second).
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported by the EU ERC grant “NeuroP” (257219) and by the EU ICT grant “NeuRAM$^3$” (687299).
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|
---
address: |
DESY, Hamburg, Germany\
E-mail: joerg.gayler@desy.de
author:
- Jörg Gayler
title: Jet Production in DIS at HERA
---
Introduction
============
Inclusive deep inelastic lepton nucleon scattering, where only the scattered lepton is detected, played an important role in establishing QCD and continues to provide a well defined testing ground of perturbative QCD (pQCD). The aim of measurements of final state jets is to relate them to final state quarks and gluons and thereby to gain additional insight in the dynamics of lepton nucleon scattering.
The data presented in this talk[^1] were recorded in the years 1995 to 1997 at HERA with the H1 and ZEUS detectors where positrons of 27.5 GeV collided with protons of 820 GeV.
Kinematics {#subsec:kin}
----------
The basic Feynman diagrams describing jet production in deep inelastic scattering are shown in Fig. \[fig:kin1\].
Standard kinematic quantities [^2] are $ Q^2 = -q^2 = -(l-l')^2$, the virtuality of the boson exchange and the Bjorken variable $x_{Bj} = Q^2/2 pq$. The momentum fraction entering the hard process of jet production with a jet-jet mass $M_{jj}$ (see Fig. \[fig:kin1\]) is given by $\xi = x_{Bj} (1 + M^2_{jj}/Q^2)$ of which the fraction $x_p = x_{Bj}/\xi$ interacts with the exchanged boson.
In most cases the data are analysed in the Breit frame defined by the condition $ 2 x_{Bj} \vec p + \vec q = 0$. Quark parton model like events (Fig. \[fig:kin1\], left) exhibit no $p_t$ in this frame apart from effects of fragmentation and decays. Jet finding is performed mostly using the inclusive $k_t$ algorithm [@incl].
Multi-Jet Production in pQCD {#subsec:pQCD}
----------------------------
Calculations at the parton level are available up to order $\alpha_s^2$, i.e. to next to leading order (NLO) (Fig. \[fig:kin1\] shows diagrams up to leading order (LO)). They can be compared with data after corrections for hadronisation are applied. DISENT [@dis] and DISASTER++ [@disas] have been shown [@Dup] to agree in the kinematic range of interest here. MEPJET [@mep] is the only program implementing also charged current reactions and JetVip [@jetv] allows resolved photon processes to be included.
A common ambiguity in these fixed order calculations is the choice of the renormalization scale $\mu_R^2$. Typical quantities characterizing the process are $Q^2$ and $E_t^2$ and the agreement with the data for these hard scales and the sensitivity to scale variations is studied.
Forward jets, i.e. jets close to the proton remnant, are of special interest, because they are expected [@muel] to be sensitive probes of the evolution of parton densities. In particular in $~\alpha_s \log(1/x)$ resummation (BFKL approach) one expects jets with larger $p_t$ (“$k_t$”) close to the proton remnant than in the standard $\alpha_s \log(Q^2)$ resummation (DGLAP approach), due to the strong $k_t$ ordering in the latter case.
$\phi$ Asymmetries
==================
A measurement of the $\phi$ distribution of charged particle tracks has been presented by the ZEUS [@zphi] collaboration, for different transverse momentum cuts. Here $\phi$ is the azimuthal angle of the hadron production plane with respect to the positron scattering plane in the hadronic centre of mass system. Finite terms $B < 0$ and $C > 0$ were measured in the angular distribution $d\sigma/d\phi = A + B \cos(\phi) + C \cos(2\phi)$ as expected in QCD-based calculations.
Jets in CC Interactions
=======================
Jet distributions in charged current (CC) interactions at high $Q^2$ are consistent with pQCD expectations (see Fig. \[fig:CC\]). The differences to neutral current (NC) jets are mainly due to the different boson propagators [@h1cc].
Jets at Low and High $Q^2$
==========================
The $E_T$ distribution in the Breit frame of single-inclusive jets is shown in Fig. \[fig:f3\] in different regions of $\eta_{lab}$. The discrepancies visible in the forward region, where the NLO corrections are huge, originate predominantly from small $Q^2$.
The $x_{Bj}$ dependence in the forward and central region (Fig. \[fig:xbj\]) cannot be described with the scale $\mu^2_R=E_T^2$. A consistent description is possible with $\mu^2_R=Q^2$, but the susceptibility to scale variations is vastly increased (shaded band in Fig. \[fig:xbj\]) [@h1low].
Such forward cross sections can be described by the NLO program JetVip and by DGLAP-based QCD Monte Carlo models if the hadronic structure of the interacting virtual photon is resolved (RAPGAP [@rapgap], dir+res in Fig. \[fig:zxbj\]), whereas inclusion of direct photon interactions only (RAPGAP, dir and LEPTO [@lepto]) is insufficient [@z896]. In the case of resolved photons, the strong $k_t$ ordering is effectively lost, leading to larger jet $E_T$ close to the proton remnant.
However, there are ambiguities in JetVip in the treatment of parton masses and no general solution has been found which is consistent with the H1 data in a large range of rapidities $\eta_{lab}$ [@h1low].
For detailed discussions of di-jet production at low $Q^2$ see the contributions [@h1dilow; @zdilow].
At high $Q^2$ there are precise high statistics data available from H1 and ZEUS which agree with NLO calculations on the 10% level in detailed comparisons (see Fig. \[fig:zdi\]) [@zdihi]. For inclusive jets ZEUS reports [@zinhi] at $Q^2 < 250$ GeV$^2$ some disagreement on the 15% level for $E_T^2$ and $Q^2$ scales (see Fig. \[fig:zincl\]), but otherwise the agreement of data and NLO calculations (DISENT) is very good [@h1fit].
Conclusion
==========
The description of the available jet data is considerably improved in going from LO to NLO ($\sim \alpha_s^2$) pQCD. However some definite discrepancies remain to be resolved. They are more pronounced choosing $E_T^2$ as renormalization scale than for $Q^2$. In the latter case the effects of scale variations are large. Forward jets are better described if the hadronic structure of the virtual photon is taken into account. At high $Q^2$ (${\raisebox{-0.5mm}{$\stackrel{>}{\scriptstyle{\sim}}$}}150$ GeV$^2$) the data are well described by NLO pQCD, the NLO corrections are moderate and hadronization corrections ${\raisebox{-0.5mm}{$\stackrel{<}{\scriptstyle{\sim}}$}}10$%. These data are well suited for quantitative QCD analyses [@tassi].
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to T. Schörner and M. Wobisch for discussions and to E. Elsen and B. Foster for comments on the manuscript.
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[^1]: ICHEP2000, Osaka, 2000
[^2]: Polar angles $\theta$ are measured with respect to the incident proton direction, the pseudo rapidity is given by $\eta = -\ln(\tan\theta/2)$.
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---
abstract: 'We study a quiet-Sun blowout jet which is observed on 2014 May 16 by the instruments on board *Solar Dynamics Observatory* (SDO). We find the twin CME as jet-like and bubble-like CME simultaneously as observed by LASCO C2 on board *Solar and Heliospheric Observatory* (SoHO), *Solar Terrestrial Relation Observatory* (STEREO\_A and STEREO\_B/COR2). They are respectively associated with the eruption of the northern and southern sections of the filament. A circular filament is rooted at the internetwork region at the base of the blowout jet. The collective magnetic cancellation is observed by SDO/HMI line of sight (LOS) magnetograms at the northern end of the filament, which makes this filament unstable and further makes it to erupt in two different stages. In the first stage, northern section of the filament is ejected, and causes an evolution of the northern part of the blowout jet. This part of the blowout jet is further extended as a collimated plasma beam to form a jet-like CME. We also observe the plasma blobs at the northern edge of the blowout jet resulting from Kelvin-Helmholtz (K-H) instability in its twisted magneto-plasma spire. In the second stage, southern section of the filament erupts in form of deformed/twisted magnetic flux rope which forms the southern part of the blowout jet. This eruption is most likely caused by the eruption of the northern section of filament, which removes the confined magnetic field of the southern section of the filament. Alternative scenarios may be a magnetic implosion between these magnetic structures confined in a much larger magnetic domain. This eruption of southern section of the filament further results in the bubble-like CME in the outer corona.'
author:
- 'R. Solanki'
- 'A. K. Srivastava'
- 'Y. K. Rao'
- 'B. N. Dwivedi'
title: 'Twin CME Launched by a Blowout Jet Originated from the Eruption of a Quiet-Sun Mini-filament '
---
Introduction {#sec:intro}
============
Solar coronal jets are the magnetically driven confined plasma eruptions which may be rooted in the lower solar atmosphere and further evolved in the solar corona. These dynamics occur in all solar environments such as quiet Sun, active region and coronal holes (e.g., Shimojo et al.1996, Nistico et al. 2009, Panesar et al. 2016, Raouafi et al. 2016). Solar coronal jets can be divided in two classes based on the temperature range of the ejected plasma as i) hot jets and ii) cool jets. In the hot jets, plasma is ejected into the solar corona along the open magnetic field lines as seen in UV, EUV, and X-ray wavelengths (e.g., Shibata et al. 1992, Shibata et al. 2007, Shen et al. 2012, Sterling et al. 2015, Liu et al. 2015, Shen et al. 2017). The temperature range of the ejected plasma is $\approx 4\times10^{6}$ K for solar X-ray jets and $\approx 10^{5}$ K for EUV jets (e.g., Shimojo et al. 1996, Shimojo et al. 1998). Jets observed in H$\alpha$ wavelength are termed as cool jets which are known as solar surges having the typical temperature $\approx 10^{4}$ K (e.g., Shibata et al. 1992, Yokoyama et al. 1995, Jiang et al. 2007, Uddin et al. 2012, Kayshap et al. 2013). Howerver cool coronal jets are also observed whose plasma is maintained at chromosphere, TR temperatues (e.g., Srivastava et al. 2011, Kayshap et al. 2013). Solar coronal jets have their width, which ranges from $5 \times 10^3$ $km$ to $1 \times 10^5$ $km$, the height ranges from $\approx 10^{4}$ km to $4\times10^{5}$ km, and the velocity ranges from $10$ $km sec^{-1}$ to $1000$ $km sec^{-1}$ (e.g., Shimojo et al. 1996, Nistico et al. 2009). The width of the jets can be determined by the length of the emerging bipoles as illustrated by Shen et al. 2011.
According to the magnetic topology of the coronal jets, they can be categorized in these following types- i) Eiffel Tower (ET) jets ii) $\lambda$ jets (e.g., Nistico et al. 2009) etc. In ET jets, photospheric bipolar magnetic field reconnects with the ambient open unipolar magnetic field of opposite polarity at the top of its loop. In the $\lambda$-shaped jets, the bipolar magnetic field reconnects with opposite polarity open magnetic field at their footpoints. Solar jets are classified as the anemone jets and two-sided loop jets based on their morphological structures and triggering mechanisms (e.g., Shibata et al. 1994, Yokoyama et al. 1995, Tian et al. 2017). There is another classification of solar coronal jets according to the standard model of the jet (Shibata et al. 1992). The jets which follow the standard model of the jets are called the standard jets and the others which do not follow such characteristics are termed as blowout jets. The concept of the blowout jets is firstly introduced by Moore et al. 2010 using Hinode/XRT observation of an X-ray jet. Blowout jets possess broad spire and bright magnetic base arch, while standard jets have narrow spire and dim magnetic base arch. In the blowout jet, the core field of the magnetic base arch carries the cool plasma ($\approx 10^{4}$-$10^{5}$ K) filaments. In the blowout jet, there is enough twist and shear in the base arch field, so when base arch field and ambient field reconnect at the current sheet, this sheared and twisted arch field is also erupted. On the contrary, in the standard jet, the base arch is inactive and does not participate in the eruption (e.g., Moore et al. 2010, Moore et al. 2013, Sterling et al. 2015). It should be noted that the blowout jet involves two times reconnection, while the standard jet have only one time reconnection. Sometimes, the eruption of base-arch field takes the large form of the eruption and drags CMEs (chen et al. 2011).
Hong et al. 2011 presented the observation of blowout jet in quiet Sun region where mini-filament is ejected during eruption and associated with a mini coronal mass ejections (CMEs). Shen et al. 2012 observed an active region blowout jet which is associated with two CMEs as one is bubble- like CME and another is jet-like CME, where bubble like CME relates cool component of the jet and jet-like CME relates hot componet of the jet. Pucci et al. 2013 have done the comparative study of the blowout jet and standard jet and found that blowout jets have ten times higher magnetic energy as compared to the standard jets. Adams et al. 2014 have analyzed the blow-out jet event which originates from the on-disk coronal hole and shows the different characteristics.
Miao et al. 2018 have analyzed a coronal blowout jet eruption which is associated with a EUV wave at its leading top and complex CME structures (jet-like and bubble-like CME) where filament eruption is observed during the blowout jet. Shen et al. 2018 have found the close relationship between the coronal jets and the EUV waves at different scales (spatial and temporal) in their observational event, where these EUV waves were propelled by the coronal jets. The observational results of Shen et al. 2018 have shown the presence of the EUV waves along with the coronal jets, where these EUV waves are generated by the lateral expansion of loop system due to the coronal jet eruption. Shen et al. 2018 have discussed the observations of arc-shaped EUV waves, a quasi-periodic fast propagating (QFP) wave, and a kink wave simultaneously with an active region coronal jet eruption. Zhu et al. 2017 have observed an active region blowout jet and investigated the 3D magnetic structure of the blowout jet and found that the kink instability is a possible triggering mechanism for this blowout jet. There are many research articles which deal with these interesting eruptive events and describe the different possible mechanisms of these eruptions (e.g., Liu et al. 2008, Murawski et al. 2011, Chen et al. 2015, Liu et al. 2015, Alzate et al. 2016). Wang et al. 1998 showed that EUV jets may be directly extended into the form of white-light jet-like CMEs. Coronal jets can cause multiple CMEs through interaction with remote structures (Jiang et al. 2008) or through the self-evolution of coronal blowout jets (Shen et al. 2012, Miao et al. 2018). So far, studies on this issue are still very scarce.
In this paper, we have studied a blowout jet eruption observed on 2014 May, 16 which is evolved due to the eruptions of the various segments of a quiescent filament. Here we observed the twin CME generation with this blowout jet. The jet-like CME is associated with the northern part of the blowout jet eruption and the bubble-like CME is driven from the eruption of the southern part of the blowout jet. Both the parts of the blowout jet are associated with successive eruptions of the various segments of a filament. The K-H unstable blobs are also observed in the northern part of the blowout jet on its spire. Observational data and its analyses are described in Section 2. In Section 3, we illustrate the observational results and driving mechanisms of the observed blowout jet, and the kinematics of the twin CME. In the last Section, discussion and conclusions are presented.
Analysis of Observational Data
==============================
Observations from *Solar Dynamics Observatory* (SDO)/*Atmospheric Imaging Assembly* (AIA)
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We use high temporal and spatial *Solar Dynamics Observatory* (SDO; Pesnell et al. 2012) data for the multi-wavelength study of the blowout jet. *Atmospheric Imaging Assembly* (AIA; Lemen et al. 2012) observes full disk Sun in transition region and coronal emissions upto 0.5 R$_\odot$ above the solar limb. SDO/AIA provides the full disk images of the Sun in three UV wavelength bands 1600 Å , 1700 Å , 4500 Å and in seven EUV wavelength bands 304 Å , 171 Å , 193 Å , 211 Å , 335 Å , 131 Å , 94 Å covering the temperature range from 0.6 MK to 16 MK with $1.5^{\prime\prime}$ spatial resolution, and $0.6^{\prime\prime}$ pixel width. SDO/AIA captures the full-disk images of the Sun with 12 s. cadence in EUV filters and 24 s. in UV filters. We have taken the SDO/AIA data on 2014 May, 16 during the time period of 03:30:00 UT - 05:10:00 UT for the selected region of $500^{\prime\prime}$ to $900^{\prime\prime}$ in the X-direction and $-300^{\prime\prime}$ to $0^{\prime\prime}$ in the Y-direction. We have downloaded SDO/AIA data from Joint Science Operation Center (JSOC) [^1]. Standard subroutines of SSWIDL (Freeland et al. 1998) are used for aligning and scaling AIA images as observed in different filters.
Observations from *Solar Dynamics Observatory* (SDO)/*Helioseismic Magnetic Imager* (HMI)
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We use *Helioseismic Magnetic Imager* (HMI; Scherrer et al. 2012) data to examine the morphology and topology of magnetic field at the footpoints of the observed blowout jet. SDO/HMI provides the full-disk line-of-sight (LOS) magnetic flux in the Fe<span style="font-variant:small-caps;">i</span> 6173 Å spectral line. It has 45-second temporal resolution, $0.5^{\prime\prime}$ pixel width and $1^{\prime\prime}$ spatial resolution. We have analyzed HMI magnetograms for the time period of 02:59:24 UT - 05:30:54 UT. SDO/HMI data has been rotated and aligned with the SDO/AIA data by using the standard subroutines of SSWIDL.
Observations from *Global Oscillation Network Group* (GONG)
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We use GONG H$\alpha$ (Harvey et al. 2011) data for study the dynamics of the filament eruption. We have downloaded H$\alpha$ data from GONG data archive [^2]. This gives the full disk H$\alpha$ data with 1 minute cadence and $1^{\prime\prime}$ spatial resolution in 6563 Å wavelength.
Observations from *Solar and Heliospheric Observatory* (SoHO)/*Large Angle and Spectrometric Coronagraph* (LASCO)
-----------------------------------------------------------------------------------------------------------------
We use LASCO CMEs data obtained from CME catalogue, which is available in CDA website [^3]. *Large Angle and Spectrometric Coronagraph* (LASCO; Brueckner et al. 1995) on board the *Solar and Heliospheric Observatory* (SoHO; Domingo et al. 1995) daily identify the coronal mass ejections (CMEs) in the images of the solar corona since 1996. LASCO has three telescopes named C1, C2, C3. LASCO observes the white light images of the solar corona from 1.1 R$_\odot$ to 30 R$_\odot$. LASCO C2 coronagraph images the solar corona from 1.5 R$_\odot$ to 6 R$_\odot$, while C3 coronagraph images the solar corona from 3.5 R$_\odot$ to 30 $R_\odot$. We use LASCO C2 and C3 data for detailed scientific investigation of narrow CME which is associated with some part of the blowout jet that further erupted in the outer corona after its origin into the quiet-Sun.
Observations from *Solar Terrestrial Relation Observatory* (STEREO)/*Sun Earth Connection Coronal and Heliospheric Investigation* (SECCHI)
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We use *Sun Earth Connection Coronal and Heliospheric Investigation* (SECCHI; Howard et al. 2008) on board STEREO-A and STEREO-B spacecraft data to analyse the kinematics of the blowout jet, jet-like and bubble-like CMEs. We use the images of the *Extreme Ultraviolet Imager* (EUVI; Wuelser et al. 2004) of SECCHI for determining the kinematics of the blowout jet, and the COR2 data of SECCHI for kinematics of jet-like and bubble-like CMEs. The field of view (FOV) of EUVI and COR2 is 1-1.7 $R_\odot$ and 2.5-15 $R_\odot$ respectively. We have downloaded the SECCHI data from the UKSSDC-STEREO archive [^4].
Observational Results
=====================
Source Location of the Blowout Jet Evolved due to Quiet-Sun Filament Eruption
-----------------------------------------------------------------------------
The blowout jet, which we have studied in the present paper was observed in the quiet Sun region on 2014, May 16. This quiet Sun region is near the western side of the NOAA AR12058 (S11W40), and lies in the fourth quadrant of the solar disk co-ordinates. The location of the blowout jet is Xcen = $660^{\prime\prime}$ and Ycen = $-150^{\prime\prime}$, and the initiation time is about 04:08:43 UT. This blowout jet is evolved due to the multiple stages of filament eruption which is rooted in the internetwork region of the quiet-Sun. We will describe physical picture of such unique plasma dynamics in the coming sub-sections.
The multi-wavelength behaviour of the blowout jet is seen in the composite image of different SDO/AIA filters (*cf.* upper panel of Figure 1). We have plotted the composite image of blowout jet and surrounding regions in AIA 1600 Å , AIA 304 Å and in HMI line of sight (LOS) magnetogram at 04:11 UT. This image collectively shows the behaviour of the blowout jet simultaneously in the emissions from UV continnumm to the transition region, and also the magnetic field polarites around its footpoint. The chosen field of view (FOV) for this composite image is $400^{\prime\prime} \times 300^{\prime\prime}$ as $500^{\prime\prime}$ to $900^{\prime\prime}$ in the X-direction and $-300^{\prime\prime}$ to $0^{\prime\prime}$ in the Y-direction. In the composite image of the blowout jet, orange colour represents AIA 304 Å , green colour reprents AIA 1600 Å , and blue colour represents HMI magnetogram.
The photospheric magnetic fields at the footpoint of the blowout jet and at its surroundings are shown in the HMI LOS (line of sight) magnetogram at 04:12:38 UT (*cf.* upper panel of Figure 2). The size of this magnetogram is $200^{\prime\prime} \times 200^{\prime\prime}$ with the co-ordinates of $550^{\prime\prime}$ to $750^{\prime\prime}$ in X-direction and $-200^{\prime\prime}$ to $0^{\prime\prime}$ in the Y-direction. We see that there are various quiet Sun magnetic networks in the neighbourhood of the blowout jet, and it is occured from an inter-network quiet Sun element where a circular filament is rooted (*cf.* bottom panels of Figure 3). The footpoint of the blowout jet is at the negative polarity (minority polarity) region in the vicinity of the positive polarity (majority polarity) region (*cf.* upper panel of Figure 2). For investigating the possible causes of the eruption of the filament and associated blowout jet and to understand the triggering mechanism for their eruptions, we examine the behaviour of the underlying magnetic field. In the initiation phase of the filament (*cf.* Figure 3), we have noticed the initial activity at the northern end of the filament which initiates the eruption of the filament. Therefore, we examine the time evolution of the magnetic flux at around the northern end of the filament. The time variation of the negative magnetic flux and the positive magnetic flux at the northern end of the filament is shown in the bottom panel of Figure 2. The negative and positive magnetic fluxes are extracted from the box shown by black solid line, which is overplotted on the HMI LOS (line of sight) magnetogram. The size of box is $20^{\prime\prime} \times 15^{\prime\prime}$ with the coordinates of $640^{\prime\prime}$ to $660^{\prime\prime}$ in X-direction and $-135^{\prime\prime}$ to $-120^{\prime\prime}$ in the Y-direction. The magnetic field intensities are extracted for the observational period of 02:59:24 UT - 05:30:54 UT. Negative magnetic flux has the order of $10^{21}$ Mx while positive magnetic flux has the order of $10^{22}$ Mx. The blue dashed line overplotted on this figure indicates the starting time (03:59 UT) of the slow rising phase of the filament. We have noticed that the positive flux shows the declined trend while the negative flux shows an increasing trend. The changing behaviour of the negative and positive magnetic fluxes suggests that the negative flux is emerging and at the same time the flux cancellation between positive and negative flux takes place. This is the confirmation of the magnetic cancellation at the northern end of the filament, where filament eruption and activation of the northern part of the blowout jet are observed. Therefore, we can infer that the magnetic flux cancellation at the northern end of the filament makes it eruptive in multiple parts, which further evolve the blowout jet eruption.
Time-intensity Profile at the Base of the Blowout Jet in Different SDO/AIA Filters
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The lightcurve is plotted in different SDO/AIA filters for analyzing the behaviour of the EUV brightening which is observed at the base of the blowout jet (*cf.* bottom panel of Figure 1). The intensity is extracted in different SDO/AIA filters from the white solid line box which is overplotted on the composite image of SDO/AIA filters over the observational period of 1 hour 40 minute from 03:30:00 UT to 05:10:00 UT. The white-line box size is $60^{\prime\prime} \times 60^{\prime\prime}$ and has the coordinates $630^{\prime\prime}$ to $690^{\prime\prime}$ in the X-direction and $-180^{\prime\prime}$ to $-120^{\prime\prime}$ in the Y-direction. Light curve is plotted in AIA 1600 Å (shown by black colour), AIA 304 Å (shown by red colour), AIA 171 Å (blue colour), AIA 335 Å (yellow colour), AIA 94 Å (violet colour) between the normalized intensity (maximum intensity/mean intensity) and the observational period of the evolution of the blowout jet. Light curve in different SDO/AIA filters show different behaviour i.e., there is no identical intensity peak for all filters. Intensity gets its peak value in AIA 1600 Å , AIA 304 Å earlier in comparison to other filters. For these two filters intensity peaks show nearly same behaviour as first peak observed at about 04:08:18 UT and second peak is observed at about 04:11:36 UT. This demonstrates that cool plasma is firstly evolved during the formation of the blowout jet’s spire. This is an opposite scenario as in typical coronal jets the hot plasma evolves first and the cool plasma thereafter (e.g., Jiang et al. 2007, Nishizuka et al. 2008, Solanki et al. 2018). This current observations reveal that some distinct mechanism is at work in the formation of this quiet-Sun blowout jet other than the typical magnetic reconnection in the corona. The most likely scenario is a collective small-scale flux emergences and subsequent cancellation with the neighbourhood at the boundary of magnetic network (*cf.* Figure 2). This launches the bulk plasma flows into the pre-existing blowout jet spire’s magnetic field in the upward direction. The cool plasma consists of the temperature range from $\approx 10^{4}$ - $10^{5}$ K. After these two filters intensity get its peak value in AIA 94 Å and in AIA 335 Å at about 04:20:00 UT and 04:23:18 UT respectively. AIA 171 Å shows slightly different kind of behaviour in intensity plot as there are many peaks in light curve of AIA 171 Å , first two small peaks match well with AIA 1600 Å and AIA 304 Å at 04:08:18 UT and at 04:11:36 UT with less intensity. Intensity gets its peak value at about 04:33:18 UT in AIA 171 Å . In the light curve, the shift is observed in the intensity peaks of different SDO/AIA filters, which emphasizes the time-lagging behaviour of the evolution of multi-temperature plasma throughout the entire period of the evolution of blowout jet at different time epoch . This time-lagging behaviour indicates the presence of flare evolution at the base of blowout jet. A weak flare is observed at about 04:08 UT near to the northern side of the blowout jet and the filament, which accelerates the plasma to the eruption. We relate this weak flare with the network flare (Krucker et al 1997, Krucker et al. 2000).
Evolution of the Blowout Jet due to the Eruption of Segments of a Filament as seen in Different SDO/AIA Filters and GONG H$\alpha$
----------------------------------------------------------------------------------------------------------------------------------
The initiation phase of the blowout jet in different SDO/AIA filters as seen in AIA 304 Å , AIA 171 Å and in GONG H$\alpha$ 6563 Å indicates the presence of the cool plasma and the filament at the base of the blowout jet (*cf.* bottom panels of Figure 3). This filament is embedded at the internetwork region, which can be seen in first image of bottom panel in H$\alpha$ at 04:03:54 UT. In the GONG H$\alpha$ observations this filament looks like circular shaped structure. Firstly a slow rise is observed in the filament. The cool plasma and ustable filament move up and the jet bright points are created at the filament root at 04:08:54 UT, which is a network flare (see Figure 3). The filament is erupted at about 04:10:54 UT. In the initiation phase, filament shows slow rise, ejection and evolution of the blowout jet. The evolution of the blowout jet can be seen in the animation Movie1.
The filament is ejected in two stages as the circular shaped structure of the filament divides into two parts. In the first stage, the eruption of northern part of the filament (first part) takes place and drives the blowout jet (*cf.* Figure 4). The hot plasma escapes out and moves linearly along the open magnetic field lines and form a broad and complex jet-like spire (*cf.* northern part of eruptions in Figure 4). In the blowout jet, eruption of the northern side of the blowout jet is most activated in it’s primary phase, and significant plasma dynamics is seen along it (*cf.* Figure 4). The significance of the formation of plasma blobs are clearly evident in this part, which we will describe in the forthcoming sub-section 3.4.
Formation of the Plasma Blobs in the Northern Spire of the Blowout Jet
----------------------------------------------------------------------
We have found the plasma blobs at the edge of the northern spire of the blowout jet. The formation and evolution of these plasma blobs can be seen in the time-sequence images of AIA 304Å in Figure 5, 6. The selected FOV for these images is $80^{\prime\prime} \times 50^{\prime\prime}$ from $670^{\prime\prime}$ to $750^{\prime\prime}$ in X-direction and from $-150^{\prime\prime}$ to $-100^{\prime\prime}$ in the Y-direction in Figure 5. These plasma blobs are identified as B1, B2, B3, B4 in the middle panel of Figure 5. The total time-duration of these plasma blobs are about 03 min 48 sec. The northern side of the blowout jet where these plasma blobs are formed is highlighted with the blue-lined box in Figure 6. The bottom panel of Figure 6 shows the zoomed picture of the plasma blobs. The formation of these plasma blobs results from the Kelvin-Helmholtz (K-H) instability. These hot and high-dense plasma blobs are moving along the northern spire of the blowout jet. The K-H instability arises due to the shear flows between the high speed blowout jet and the steady local plasma and results into the formation of the magnetic islands. These islands take the form of plasma blobs at the time of the evolution.
We have analyzed the velocity field at the northern side of the blowout jet by using Fourier ocal Correlation Tracking (FLCT; Fisher et al. 2008) method (*cf.* right panel of Figure 7). The velocity field is analyzed at the northern side of the blowout jet corresponding to the box which is overplotted on the AIA 1600 Å image having the coordinates of $500^{\prime\prime}$ to $900^{\prime\prime}$ in the X-direction and $-300^{\prime\prime}$ to $0^{\prime\prime}$ in the Y-direction (*cf.* left panel of Figure 7). We have selected two HMI LOS magnetograms first magnetogram at 03:37:36 UT and second magnetogram at 04:14:08 UT (at the timing of the onset of the blowout jet) to estimate the flow field velocity field. In the right panel of Figure 7, we see the base image of HMI magnetogram at 03:37:36 UT and the velocity field which is shown by orange arrows overplotted on the base image as taken from HMI magnetogram at 04:14:08 UT.
It is clear that the clock-wise plasma flows is evident centered at that particular quiet-Sun region from where the blowout jet is originated. The right panel of Figure 7 shows the partial field of view showing the footpoint of the northern side of the blowout jet (*X* = $640^{\prime\prime}$ - $710^{\prime\prime}$,*Y* = $-125^{\prime\prime}$ - $-100^{\prime\prime}$). Yellow contour which is overlaid on HMI magnetogram shows the AIA 304 Å intensities. This is clear that the clock-wise shearing flow-field is acting at the footpoint of the northern side of the blowout jet, which further launch the right-handed twist in the entire overlying plasma column/spire associated with the blowout jet. When this clock-wise shearing flow-field moves further and interact with the local stationary plasma field, it causes the K-H instability which results in the formation of four plasma blobs as B1, B2, B3, B4. The plasma blobs are less visible in AIA 304 Å channel due to the less spatial and time resolutions data. The northern spire of the jet also does untwisting/rotational motion and releases its twist. The jet’s spire shows the estimated twist about 1-1.5 turns (or $2\pi$-$3\pi$). Our twist value is found in good agreement with the results of the Pariat et al. 2009, which shows for the driving of a solar coronal jet the threshold value of the twist is 1.4 turns ($2.8\pi$). Our finding of the K-H unstable plasma blobs in the rotating/untwisting spire of the blowout jet supports the numerical results of the Ni et al. 2017, Zhelyazkov et al. 2018 and Zhelyazkov et al. 2018. This K-H unstable northern plasma spire further moves into higher coronal region and drives a jet-like CME, which we will discuss in the forthcoming Section 3.7.
Kinematics of the Northern Part of the Blowout Jet
--------------------------------------------------
We have done the height-time analysis of the northern part of the blowout jet. To calculate the height of the blowout jet, we have used the tie-pointing method of the Inhester et al. 2006. In this method the triangulation technique between the different view points of the STEREO\_A and SDO/AIA is used. In this triangulation technique, scc\_measure.pro is used which is available in solarsoft library for estimation of real height of blowout jet. We have calculated the height of the blowout jet by tracking the tip of the blowout jet in two different view points of the STEREO\_A and SDO/AIA (*cf.* upper panel of Figure 8). In the bottom panel of Figure 8, the height of the jet has been tracked using its tip starting at 04:06:09 UT when the jet is clearly visible in both the instruments (SDO/AIA 304 Å and STEREO\_A EUVI 304 Å ) while the initiation of jet takes place at 03:59 UT. We have calculated the height (in $R_\odot$) of the blowout jet at ten different times (in UT) as 04:06:19, 04:10:31, 04:14:07, 04:15:31, 04:16:19, 04:16:31, 04:20:31, 04:25:31, 04:26:15, 04:30:30. With this height-time array, we have calculated the velocity of the blowout jet which is $325$ $km sec^{-1}$ (*cf.* bottom panel of Figure 8). The calculated acceleration for this blowout jet is $-0.30$ $km sec^{-2}$ which is estimated by the second order fitting in the H-T plot of blowout jet.
Eruption of the Southern Part of the Filament and Associated Blowout Jet
------------------------------------------------------------------------
After the activation and eruption of the northern section of the filament and associated segment of the blowout jet, the southern section of the filament is also erupted in the form of a twisted/deformed flux rope (*cf.* H$\alpha$, AIA 304 Å images in bottom and middle panel of Figure 9 which makes the second stage of the eruption (*cf.* AIA 171Å images of Figure 9). The eruption of the southern part of the filament can be seen in the animation Movie1. This deformed/twisted magnetic flux rope moves further and releases its helicity and forms a rotating plasma spire of the southern part of the blowout jet. The eruption of the northern section of the filament destabilizes and removes the local magnetic field configurations and induces the eruption of the southern section of the filament. Alternatively we may adopt the magnetic implosion physical mechanism for the initiation of the eruption of the southern section of the filament. It may occur between neighbouring magnetic structures confined by a large magnetic structure (e.g., Hudson 2000, Liu et al. 2009, Shen et al. 2012). At the magnetic implosion site, the eruption takes place when the upward magnetic pressure decreases resulting into the contraction of overlying field and free magnetic energy release.
Jet-like and Bubble-like Twin CME
---------------------------------
In this analyzed event, we have found the generation of twin CME associated with the blowout jet eruption. Northern part of the filament erupts firstly and causes the evolution of the coronal blowout jet, which may also subject to the K-H instability. Thereafter, the southern part of the filament also erupts in form of magnetic flux rope, and form the full blowout jet eruption.
The northern and southern segments of the filament may be confined by the same magnetic field system. The magnetic flux cancellation is occured at the northern end of the filament which makes this filament unstable and erupts it in different stages. In first stage northern segment of filament ejects and initiates the eruption of the northern part of the blowout jet. The northern part of the blowout jet further drags the first CME which is a jet-like CME. This jet-like CME is the extension of the collimated plasma beam which is generated by the external magnetic reconnection (Shen et al. 2012).
The eruption of the northern section of the filament removes the confined magnetic field of the southern section of the filament and induces the eruption of the southern section of the filament. Alternatively we may adopt the magnetic implosion mechanism for the eruption of the southern section of the filament. The eruption of the southern section of the filament form the full blowout jet eruption and causes the second CME which is a bubble-like CME.
The line of sight (LOS) evolution of these twin CME are shown in running difference images of SoHO/LASCO C2 coronagraph (*cf.* Figure 10). The dynamics of these two CMEs can be seen in the animation Movie2. We have used the multi-scale gaussian normalization method of Morgan et al. 2014 in making these running difference images. We have marked the jet-like CME and bubble-like CME in Figure 10 in the image of 05:12:05 UT. The bubble-like CME has typical three part structures as bright core which consists of cool plasma material of the filament, dark cavity and the bright front of the CME. The bright core is at the northern side of the dark cavity, generally the bright core resides at the central of the dark cavity and the CME. The first appearance of the CMEs in the LASCO C2’s FOV is observed at about 04:38:53 UT. The spatial and the temporal relationship between the twin CME and the northern as well as southern part of the jet eruption relates to the eruptions of the northern and southern sections of the filament which determine jet-like and the bubble-like CME.
Kinematics of the Jet-like and Bubble-like CME
----------------------------------------------
We have done the height-time analysis for the study of the kinematics of jet-like and bubble-like CMEs. We have applied the same method for height-time measurements of these twin CME. We have measured the projected height for these CMEs with respect to the centre of the solar disk by using the tie-pointing method of the Inhester et al. 2006. We have tracked the tip of the CMEs in STEREO\_A COR2 and STEREO\_B COR2 simultaneously with the help of the triangulation technique. The separation angle between STEREO\_A and STEREO\_B on 2014 May 16 at 04:00 UT is about . Since longitude angle is high in measurements, we have estimated the projected height of the twin CME. In case of jet-like CME, we have estimated the projected height at five different times as 05:09:15 UT, 05:24 UT, 05:39 UT, 05:54 UT and 06:09:15 UT. At each time we have tracked the tip of the CME (measurement of projected height) at ten times to estimate the underlying uncertainty in the measurements. With these data sets of projected height and time we have plotted the H-T plot for jet-like CME (*cf.* Figure 11). The calculated velocity for jet-like CME is about $619$ $km sec^{-1}$. We have done the second order fitting on the H-T plot to calculate the acceleration of the CME and get the $0.35$ $km sec^{-2}$ acceleration value for the jet-like CME.
In case of bubble-like CME we have estimated the projected height at 04:54 UT, 05:09:15 UT, 05:24 UT, 05:39 UT, 05:54 UT, 06:09:15 UT, 06:24 UT, 07:09:15 UT. With the projected height and time data sets we have plotted the H-T plot for bubble-like CME and calculated the velocity of the bubble-like CME (*cf.* Figure 12). The calculated velocity is about $620$ $km sec^{-1}$. The calculated acceleration is about $-0.031$ $km sec^{-2}$. We have done the second order fitting on the H-T plot of bubble-like CME to get the value of this negative acceleration.
Discussion and Conclusions
==========================
There are many observational results which deal with the blow-out jet eruption from the active region of the Sun and their different characteristics and triggering mechanisms. Li et al. 2015 observed an active-region blow-out jet which is associated with a CME and a M-class solar flare , where the filament eruption triggers this blow-out jet. Li et al. 2017 observed a blow-out surge in coronal loops where he found that the magnetic reconnection between the erupting filament and the coronal loop is responsible cause for the eruption of blowout surge. Shen et al. 2017 observed an active region blow-out jet with SDO, this jet is associated with the filament eruption and consisted of hot and cool plasma structure, where cool plasma component preceeds further than the hot plasma component. Hong et al. 2017 studied an active region blow-out jet associated with C-class flare and a type-<span style="font-variant:small-caps;">iii</span> radio burst, where it is observed that the filament eruption triggers these eruptive events. Li et al. 2018 has recently discovered the Kelvin-Helmholtz Instability (KHI) in an active region penumbral structural blow-out jet in the high-resolution Interface Region Imaging Spectrograph (IRIS) observations.
In the present work, we describe the physical properties, triggering mechanism and kinematics of a quiet-Sun blowout jet which is observed by SDO/AIA in different wavelengths on 2014 May, 16. This blowout jet is initiated by the multi-section eruptions of the circular filament which is at the base of the blowout jet. Based on our observational results, we have also found the generation of the twin CME as jet-like and bubble-like CMEs which are associated with blowout jet eruption.
Here are some concluding points of this observed event in the studied observed baseline.
i\) In the time-intensity profile of the blowout jet it is observed that the cool plasma has the temperature range $\approx 10^{4}$-$10^{5}$ K, which is evolved earlier in the formation of the magnetized plasma spire of the blowout jet. This is due to the multiple filament ejection which further drags blowout jet. This is the unique scenario which shed the light that the formation mechanism of this blowout jet is entirely different from the typical coronal reconnection driven blowout jets. There is the presence of the multi-temperature plasma and the time-lagging behaviour is observed during the evolution period of the blowout jet, which is analogous with the flare eruption. We relate this flare with the network flare (Krucker et al. 1997, Krucker et al. 2000). This flare energy accelerates the blowout jet plasma. The energisation of this network flare is due to the activation of filament segments and its reconnection with the existing overlying fields.
ii\) The time-evolution of the magnetic flux at the northern end of the filament is analyzed which shows the magnetic cancellation signature. The magnetic cancellation destabilizes the filament and further makes it to erupt in different stages.
iii\) The complete evolution of the blowout jet and the filament is observed in AIA 304 Å (at TR temperature), AIA 171 Å (at inner coronal temperature) and in H$\alpha$ (at chromospheric temperature). This eruption goes through the different stages as the circular filament ejects in two stages. Firstly the northern section of the filament lifts up, ejects and drives the northern part of blowout jet. In second stage the southern section of the filament also erupts and forms the rotating plasma spire of the blowout jet, *i.e.* southward part of the blowout jet.
iv\) The plasma blobs are formed at the edge of the northern plasma spire of the blowout jet which are moving along the jet’s spire. These plasma blobs are most likely subjected to the K-H instability, which arises due to the interaction between the sheared motion of the northern part of the blowout jet and the local stationary plasma in the surrounding.
v\) The velocity field is analyzed at the footpoint of the northern part of the blowout jet using Fourier Local Correlation Technique (FLCT). The velocity field shows the clock-wise plasma flows is centered at the blowout jet triggering site, which enables the magnetic twists of similar sign in the whole northern spire of the blowout jet. This enables the sheared plasma motion in the jet’s spire and most likely the evolution of K-H unstable plasma blobs.
vi\) We have done the height-time analysis of the northern side of the blowout jet. The calculated velocity and acceleartion are found to be $325$ $km sec^{-1}$ and $-0.30$ $km sec^{-2}$ respectively.
vii\) The twin CME are observed associated with the blowout jet. The eruption of the northern part of the blowout jet drives the jet-like CME. The outward moving hot plasma on the disk is observed as northern part of the blowout jet and in the outer coronal region it is observed as the jet-like CME. The jet-like CME is the extension of the collimated plasma beam which is generated by the external magnetic reconnection (Shen et al. 2012). The eruption of the southern section of the filament enables the rotating spire of the blowout jet, which further drives the bubble-like CME. These twin CME are observed simultaneously.
viii\) The calculated velocity and acceleration for jet-like and bubble-like CMEs are found to be $619$ $km sec^{-1}$ and $0.35$ $km sec^{-2}$, $620$ $km sec^{-1}$ and $-0.031$ $km sec^{-2}$. respectively
In the observations of the Shen et al. 2012 the double CMEs are less distinguishable but in our case, we can easily distinguish the jet-like and bubble-like CME as these CMEs occur side by side. To the best of our knowledge, our observed event is the third event of the twin CME with blowout jet eruption after the observations of the Shen et al. 2012 and Miao et al. 2018.
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AKS acknowledges the joint research grant under the frame-work of UKIERI (UK-India Educative and Research Initiatives). R.S. thanks the Department of Physics, Indian Institute of Technology (BHU) for providing her Senior Research Fellowship (SRF) and computational facilities. We acknowledge the SDO/AIA, SDO/HMI, SoHO/LASCO, STEREO/SECCHI, GONG H$\alpha$ observations for this work. Authors acknowledge Alphonso Sterling, Navdeep Panesar, T. V. Zaqarashvili for their fruitful discussion at initial stage and suggestions. We thank the anonymous referee for his/her valuable comments and suggestions. We thank Sudheer K. Mishra for his help in using tie-pointing method for the kinematics of the blowout jet and twin CME.
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[^1]: <http://jsoc.stanford.edu.>
[^2]: <https://gong.nso.edu/.>
[^3]: <http://cdaw.gsfc.nasa.gov/CME_list.>
[^4]: <https://www.ukssdc.ac.uk/solar/stereo/data.html.>
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abstract: 'The strength functions of quadrupole modes in the unstable oxygen isotopes $^{22}$O and $^{24}$O are calculated using an extended version of the time-dependent Hartree-Fock theory known as the time-dependent density-matrix theory (TDDM). It is found that TDDM gives the lowest quadrupole states which are energetically shifted upward and become significantly collective due to the coupling to two-body configurations. It is pointed out that these features of the lowest quadrupole states are similar to those obtained in the quasi-particle random phase approximation.'
author:
- 'M. Tohyama'
- 'A. S. Umar'
title: 'Quadrupole resonances in unstable oxygen isotopes in time-dependent density-matrix formalism'
---
=10000 =10000
The study of low-lying quadrupole states in unstable oxygen isotopes has recently gained both experimental [@Jewell; @Khan1; @Thirolf; @Belleguic] and theoretical interest [@Alex; @Utsu; @Khan2; @Khan3; @Matsu]. One of the reasons for this is that the energy and transition strength of the first $2^+$ state are considered to be closely related to the change in the shell structure in this region of neutron rich nuclei [@Oza; @Otsu1]. Various theoretical approaches have been used to study the low-lying quadrupole states in unstable oxygen isotopes: The shell model [@Alex; @Utsu], the random-phase approximation (RPA) [@Yoko; @Lanza; @Hama], and the quasi-particle RPA (QRPA) [@Khan2; @Khan3; @Matsu]. It is shown by the QRPA calculations [@Khan2; @Khan3; @Matsu] that paring correlations are essential to reproduce the large $B(E2)$ values of the first $2^+$ states. Earlier TDDM calculations [@Toh1] did not include the spin-orbit splitting of the single-particle states, thereby the first $2^+$ states, which mainly consist of inner shell transitions of neutrons, could not be treated properly. The TDDM approach has recently been improved to include the spin-orbit force and now is able to handle low-lying quadrupole states in a more quantitative way [@Toh2]. In this letter we apply TDDM to study low-lying quadrupole states, especially, the first $2^+$ states and show that TDDM gives results similar to QRPA. The present formulation of TDDM assumes the Hartree-Fock (HF) ground state as a starting ground state as will be shown below. The quantitative application of TDDM, therefore, is restricted to the quadrupole states in $^{22}$O and $^{24}$O, where the first-order approximation of the sub-shell closure may be justified [@Yoko].
TDDM is an extended version of the time-dependent Hartree-Fock theory (TDHF) and is formulated to determine the time evolution of one-body and two-body density matrices $\rho$ and $\rho_2$ in a self-consistent manner [@Gon]. The equations of motion for $\rho$ and $\rho_2$ can be derived by truncating the well-known Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices [@Wan]. To solve the equations of motion for $\rho$ and $\rho_2$, $\rho$ and $C_2$ (the correlated part of $\rho_2$) are expanded using a finite number of single-particle states $\psi_{\alpha}$ which satisfy a TDHF-like equation, $$\begin{aligned}
\rho(11',t)=\sum_{\alpha\alpha'}n_{\alpha\alpha'}(t)\psi_{\alpha}(1,t)
\psi_{\alpha'}^{*}(1',t), \end{aligned}$$ $$\begin{aligned}
C_{2}(121'2',t)&=&\rho_{2} - {\cal A}(\rho\rho) \nonumber \\
&=&\sum_{\alpha\beta\alpha'\beta'}C_{\alpha\beta\alpha'\beta'}(t)
\nonumber \\
&\times&\psi_{\alpha}(1,t)\psi_{\beta}(2,t)
\psi_{\alpha'}^{*}(1',t)\psi_{\beta'}^{*}(2',t), \end{aligned}$$ where ${\cal A}$ is the antisymmetrization operator and the numbers denote space, spin, and isospin coordinates. Thus, the equations of motion of TDDM consist of the following three coupled equations [@Gon]: $$\begin{aligned}
i\hbar\frac{\partial}{\partial t}\psi_{\alpha}(1,t)=h(1,t)
\psi_{\alpha}(1,t),\end{aligned}$$ $$\begin{aligned}
i\hbar \dot{n}_{\alpha\alpha'}=\sum_{\beta\gamma\delta}
[\langle\alpha\beta|v|\gamma\delta\rangle C_{\gamma\delta\alpha'\beta}
-C_{\alpha\beta\gamma\delta}\langle\gamma\delta|v|\alpha'\beta\rangle],\end{aligned}$$ $$\begin{aligned}
i\hbar\dot{C}_{\alpha\beta\alpha'\beta'}=B_{\alpha\beta\alpha'\beta'}
+P_{\alpha\beta\alpha'\beta'}+H_{\alpha\beta\alpha'\beta'}, \end{aligned}$$ where $h$ is the mean-field Hamiltonian and $v$ the residual interaction. The term $B_{\alpha\beta\alpha'\beta'}$ on the right-hand side of Eq.(5) represents the Born terms (the first-order terms of $v$). The terms $P_{\alpha\beta\alpha'\beta'}$ and $H_{\alpha\beta\alpha'\beta'}$ in Eq.(5) contain $C_{\alpha\beta\alpha'\beta'}$ and represent higher-order particle-particle (and hole-hole) and particle-hole type correlations, respectively. Thus full two-body correlations including those induced by the Pauli exclusion principle are taken into account in the equation of motion for $C_{\alpha\beta\alpha'\beta'}$. The explicit expressions for $B_{\alpha\beta\alpha'\beta'}$, $P_{\alpha\beta\alpha'\beta'}$ and $H_{\alpha\beta\alpha'\beta'}$ are given in Ref.[@Gon]. Diverse configurations including excitations of the $^{16}$O core can be taken into account in Eqs.(3)-(5). In order to obtain a clear understanding of contributions of various configurations we solve the coupled equations in the following three schemes: 1) Two-body amplitudes $C_{\alpha\beta\alpha'\beta'}$ consisting of the neutron 2$s_{1/2}$, 1$d_{3/2}$ and 1$d_{5/2}$ states only are considered. We call this scheme TDDM1. 2) The neutron 2$p_{3/2}$ and 1$f_{7/2}$ are added to TDDM1. We call this version TDDM2. 3) All single-particle states up to the 2$p_{3/2}$ and 1$f_{7/2}$ are taken into account for both protons and neutrons. We call this scheme TDDM3. Two-body configurations corresponding to excitations of the $^{16}$O core are included in TDDM3. The number of independent $C_{\alpha\beta\alpha'\beta'}$’s drastically increases from 513 (TDDM1) to 50,727 (TDDM3).
The $E2$ strength function is calculated according to the following three steps :
1\) A static HF calculation is performed to obtain the initial ground state. The 1$d_{5/2}$ and 2$s_{1/2}$ orbits are assumed to be the last fully occupied neutron orbit of $^{22}$O and $^{24}$O, respectively. The Skyrme III with spin-orbit force is used as the effective interaction. The Skyrme III has been used as one of standard parameterizations of the Skyrme force in nuclear structure calculations even for very neutron rich nuclei [@Khan2; @Otsu]. The single-particle wavefunctions are confined to a cylinder with length 20 fm and radius 10 fm. (Axial symmetry is imposed to calculate the single-particle wavefunctions [@Uma].) The mesh size used is 0.5 fm.
2\) To obtain a correlated ground state, we evolve the HF ground state using the TDDM equations and the following time-dependent residual interaction $$\begin{aligned}
v(t)=(1-e^{-t/\tau})v(\bm{r}-\bm{r}').\end{aligned}$$ The time constant $\tau$ should be sufficiently large to obtain a nearly stationary solution of the TDDM equations [@Toh4]. We choose $\tau$ to be 150 fm/$c$. In a consistent calculation the residual interaction should be the same as that used to generate the mean field. However, a Skyrme-type force contains momentum dependent terms , which make the computation time of two-body matrix elements quite large. Therefore, we need to use a simple force of the $\delta$ function form $v\propto\delta^3(\bm{r} -\bm{r}')$. In order to make a comparison with the results of HFB and QRPA calculations, we use the following pairing-type residual interaction of the density-dependent $\delta$ function form [@Chas] $$\begin{aligned}
v(\bm{r}-\bm{r}')=v_{0}(1-\rho (\bm{r})/\rho_0)
\delta^3(\bm{r}-\bm{r}'),\end{aligned}$$ where $\rho (\bm{r})$ is the nuclear density. $\rho_0$ and $v_{0}$ are set to be 0.16fm$^{-3}$ and $-1200$ MeV fm$^3$, respectively. Similar values of $\rho_0$ and $v_{0}$ have been used in the Hartree-Fock-Bogoliubov (HFB) calculations [@Terasaki; @Duguet; @Yamagami] in truncated single-particle space. The time step size used to solve the TDDM equations is 0.75 fm/$c$.
3\) The quadrupole mode is excited by boosting the single-particle wavefunctions at $t=5\tau$ with the quadrupole velocity field: $$\begin{aligned}
\psi_{\alpha}(5\tau)\longrightarrow e^{ikQ}\psi_{\alpha}.\end{aligned}$$ We consider the isoscalar quadrupole mode and the proton quadrupole mode to calculate a $B(E2)$ value. For the former, $Q$ is equal to $r^2Y_{20}$ for both protons and neutrons, and for the latter only the proton single-particle states are boosted. When the boosting parameter $k$ is sufficiently small, the strength function defined by $$\begin{aligned}
S(E)=\sum_{n}|\langle\Phi_n|\hat{Q}|\Phi_0\rangle|^{2}\delta (E-E_{n})\end{aligned}$$ is obtained from the Fourier transformation of the time-dependent quadrupole moment $Q(t)$ as $$\begin{aligned}
S(E)=\frac{1}{\pi k\hbar}\int_{0}^{\infty}Q(t)\sin\frac{Et}{\hbar}
dt. \end{aligned}$$ In Eq.(9) $|\Phi_0\rangle$ is the total ground-state wavefunction and $|\Phi_n\rangle$ is the wavefunction for an excited state with excitation energy $E_n$, and $\hat{Q}$ the quadrupole operator. The TDDM calculations are stopped at $t=1500$ fm/$c$ and the upper limit of the time integration in Eq.(10) becomes 750 fm/$c$. To reduce fluctuations in $S(E)$, originating from the finite-time integration, the quadrupole moment is multiplied by a damping factor $e^{-\Gamma t/2\hbar}$ with $\Gamma=1$MeV before the time integration. Since the integration time is limited, the strength function in a very low energy region ($E<2\pi\hbar/750\approx2$MeV), which depends on long time behavior of $Q(t)$, is not well determined and somewhat affected by numerical inaccuracies. The energy-weighted sum rule (EWSR) for the isoscalar quadrupole mode is expressed as $$\begin{aligned}
\int S(E)EdE
&=& \frac{1}{2}\langle\Phi_0|[\hat{Q},[H,\hat{Q}]]|\Phi_0\rangle
\nonumber \\
&=& \frac{5\hbar^2}{4\pi m}A\langle r^2 \rangle,\end{aligned}$$ where $A$ is the mass number, $m$ is the nucleon mass and $\langle r^2 \rangle$ means the mean square radius of the ground state. For the proton mode the mass number is replaced by the atomic number and the mean square radius is taken for protons, and there appears an enhancement factor due to the momentum dependence of the Skyrme interaction which is expressed as $$\begin{aligned}
\frac{5}{8\pi}(t_1+t_2)\int r^2\rho_p(\bm{r})\rho_n(\bm{r})d\bm{r},\end{aligned}$$ where $t_1$ and $t_2$ are the parameters of the momentum dependent parts of the Skyrme force, and $\rho_p$ ($\rho_n$ ) is the proton (neutron) density distribution. In TDDM3 where proton single-particle states are also considered in two-body configurations, a term containing $C_{\alpha\beta\alpha'\beta'}$, which exhibits the effects of ground-state correlations appears in addition to Eq.(12) [@Toh2]. However, its contribution is less than 1% of the total EWSR value and is neglected. The contribution of Eq.(12) is about 15% of the total EWSR value. The strength function obtained in TDHF (without Eqs.(4) and (5)) is equivalent to that in RPA without any truncation of unoccupied single-particles states because the TDHF equations for the boosted single-particle wavefunctions $\psi_{\alpha}$ is solved in coordinate space. The boundary condition for the continuum states, however, is not properly taken into account in our calculation because all the single-particle wave functions are confined to the cylindrical geometry.
We first discuss some ground state properties of $^{22}$O and $^{24}$O evaluated at $t=5\tau$ in TDDM1 where only the neutron 1$d_{5/2}$, 2$s_{1/2}$ and 1$d_{3/2}$ orbits are use to evaluate $C_{\alpha\beta\alpha'\beta'}$. In $^{22}$O, the energies of the neutron 1$d_{5/2}$, 2$s_{1/2}$, and 1$d_{3/2}$ states are $-$6.8MeV, $-$2.3MeV, and 1.3MeV, respectively. Their occupation probabilities are 0.92, 0.14, and 0.05, respectively. In $^{24}$O, their energies and occupation numbers are $-$7.3MeV, $-$2.3MeV, 0.8MeV, 0.97, 0.89, and 0.10, respectively. The correlation energy $E_{\rm cor}$ defined by $$\begin{aligned}
E_{\rm cor}=\frac{1}{2}\sum\langle\alpha\beta|v|\gamma\delta\rangle C_{\gamma\delta\alpha\beta}\end{aligned}$$ is $-4.4$ MeV in $^{22}$O and $-3.5$ MeV in $^{24}$O. The correlation energy defined above includes effects of various two-body correlations. However, pairing correlations seem to dominate. This is because $E_{\rm cor}$ does not change much even if the sum over single-particle states in Eq.(13) is restricted to a pair of conjugate states under time reversal such as $$\begin{aligned}
\frac{1}{2}\sum\langle\alpha\beta|v|\gamma\delta\rangle C_{\gamma\delta\alpha\beta}
\longrightarrow
\frac{1}{2}\sum\langle\alpha\bar{\alpha}|v|\beta\bar{\beta}\rangle C_{\beta\bar{\beta}\alpha\bar{\alpha}},\end{aligned}$$ where $\bar{\alpha}$ is a state conjugate to $\alpha$ under time reversal. The correlation energies thus obtained for $^{22}$O and $^{24}$O are $-4.8$ MeV and $-3.9$ MeV, respectively. Since the pairing correlations seem dominant, it might be meaningful to estimate a quantity $\bar{\Delta}$ corresponding to an average pairing gap in HFB [@Duguet; @Yamagami]. We define $\bar{\Delta}$ as $$\begin{aligned}
\bar{\Delta}=\frac{\sum\langle\alpha\bar{\alpha}|v|\beta\bar{\beta}\rangle
C_{\beta\bar{\beta}\alpha\bar{\alpha}}}{\sum u_{\alpha}v_{\alpha}},\end{aligned}$$ where $v_{\alpha}$ and $u_{\alpha}$ are calculated as $\sqrt{n_{\alpha\alpha}}$ and $\sqrt{1-n_{\alpha\alpha}}$, respectively. The values of $\bar{\Delta}$ thus obtained are $-3.1$ MeV and $-2.7$ MeV for $^{22}$O and $^{24}$O, respectively. These values may be comparable to the empirical pairing gap of $\Delta\approx 12/\sqrt{A}\approx 2.5$ MeV. When the single-particle space is extended to the $2p_{3/2}$ and $1f_{7/2}$ in TDDM2, $\bar{\Delta}$ becomes slightly larger and its difference between $^{22}$O and $^{24}$O becomes smaller: The values of $\bar{\Delta}$ in TDDM2 are $-3.6$ MeV and $-3.5$ MeV for $^{22}$O and $^{24}$O, respectively, and become close to the values obtained from HFB calculations using a Woods-Saxon potential [@Matsu]. These values might be rather large as compared to experimental gaps obtained by combining the binding energies of neighbouring nuclei. However, only a semi-quantitative agreement with experimental values should be expected because $\bar{\Delta}$ in Eq.(15) has no direct relation to the definition of the experimental gap.
{height="6cm"}
The strength functions for the isoscalar quadrupole modes in $^{22}$O and $^{24}$O are shown in Figs.1 and 2, respectively. The dotted lines depict the results in TDDM1 and the solid lines those in TDDM2. The results in TDHF are also shown with the dot-dashed lines. The width of each peak is due both to the smoothing with $\Gamma=1$ MeV and to the finite-time integration of Eq.(10). In the very low energy region ($E<2$ MeV), $S(E)$ fluctuates slightly and has small negative components. This originates in uncertainties associated with the calculation of Eq.(10) as mentioned above. The fraction of the EWSR values depleted below 60MeV is about 98% in all calculations shown in Figs.1 and 2. Based on the single-particle energies of the neutron $2s$ and $1d$ states, we consider that the lowest state seen at $E=3.8$ MeV in TDHF for $^{22}$O originates in the transition of a neutron from the 1$d_{5/2}$ orbit to the 2$s_{1/2}$ orbit, while that at $E=2.3$ MeV in $^{24}$O comes from transition of a neutron from the 2$s_{1/2}$ orbit to the 1$d_{3/2}$ orbit. The second lowest states seen at $E=7.8$ MeV in the TDHF calculations for both nuclei are also considered to be from the inner shell transition of a neutron, that is, the transition from the 1$d_{5/2}$ to the 1$d_{3/2}$. When the two-body correlations among the neutron 1$d_{5/2}$, 2$s_{1/2}$ and 1$d_{3/2}$ states are included in TDDM1, the collectivity and excitation energies of the lowest states are increased and the strength of the giant quadrupole resonance (GQR) located at 20 MeV is slightly decreased correspondingly. The increase of the excitation energy of the lowest state is from 3.8 MeV to 4.7 MeV in $^{22}$O and from 2.3 MeV to 3.5 MeV in $^{24}$O. The increase of the collectivity of the lowest states is due to the mixing of quadrupole states consisting of two-body configurations and the upward shift in excitation energy is related to the lowering of the ground state due to two-body correlations. The paring correlations seem to dominate the two-body correlations because the two-body amplitudes of $C_{\alpha\bar{\alpha}\beta\bar{\beta}}$ type are most important as discussed above. These behaviors of the first $2^+$ states under the influence of the pairing correlations are similar to those obtained in QRPA [@Khan2; @Matsu]. In TDDM the pairing correlations are treated as two-body correlations, whereas they are considered in a mean field approximation in QRPA. The expansion of the single-particle space up to the neutron $2p_{3/2}$ and $1f_{7/2}$ states (TDDM2) further enhances the collectivity and the excitation energy: The excitation energy of the lowest state is increased to 5.0 MeV in $^{22}$O and to 4.4 MeV in $^{24}$O. The quadrupole strengths below 6 MeV in $^{22}$O are 214 fm$^4$, 271 fm$^4$ and 300 fm$^4$ in TDHF, TDDM1 and TDDM2, respectively. The quadrupole strengths below 6 MeV in $^{24}$O are 163 fm$^4$, 206 fm$^4$ and 249 fm$^4$ in TDHF, TDDM1 and TDDM2, respectively. The increase in the collectivity with increasing number of single-particle states seems to be slightly larger in $^{24}$O than in $^{22}$O.
{height="6cm"}
{height="6cm"}
The proton quadrupole modes in $^{22}$O and $^{24}$O are shown in Figs. 3 and 4, respectively. The fraction of the EWSR values depleted below 60MeV is about 93% in all calculations shown in Figs. 3 and 4. The strength distributions seen above 25 MeV correspond to isovector quadrupole resonances. The $B(E2)$ values obtained from the integration of $S(E)$ below 6 MeV in $^{22}$O are 11.0 $e^2$fm$^4$, 14.2 $e^2$fm$^4$ and 16.4 $e^2$fm$^4$ in TDHF, TDDM1 and TDDM2, respectively. The $B(E2)$ value of 16.4 $e^2$fm$^4$ in TDDM2 may be comparable to the observed value of $21\pm 8$ $e^2$fm$^4$ [@Thirolf] for the first $2^+$ state, although the excitation energy of 5.0 MeV is larger than the experimental value of 3.2 MeV [@Thirolf]. The QRPA calculations [@Khan2; @Matsu] also give the lowest quadrupole states in oxygen isotopes whose excitation energies are slightly larger than experimental values. The excitation energies of the first $2^+$ states sensitively depend on the single-particle energies of the $2s$ and $1d$ states. To reproduce the observed excitation energies, appropriate choice and adjustment of the Skyrme force parameters including the strength of spin-orbit force may be necessary [@Khan3]. It has also been pointed out [@Colo] that the energy of the $2s_{1/2}$ state in oxygen isotopes is significantly shifted downward due to the coupling to phonon states. The $B(E2)$ values obtained from the integration of $S(E)$ below 6 MeV in $^{24}$O are 3.5 $e^2$fm$^4$, 5.4 $e^2$fm$^4$ and 7.2 $e^2$fm$^4$ in TDHF, TDDM1 and TDDM2, respectively. The small $B(E2)$ value in $^{24}$O is variant from the QRPA calculation done by Matsuo [@Matsu] but consistent with other QRPA calculations [@Khan2; @Khan3] and the shell model calculations [@Alex; @Utsu].
{height="6cm"}
The strength function of the isoscalar quadrupole modes in $^{24}$O calculated in TDDM3 (solid line) where the excitations of the $^{16}$O core are included is shown in Fig.5. To obtain a stable ground state in TDDM3, we reduced the strength of the residual interaction and used $v_0=-800$ fm$^3$MeV in stead of $v_0=-1200$ fm$^3$MeV. The result in TDDM2 calculated with this value of $v_0$ is also shown in Fig.5 with the dotted line. The highest peak of GQR is reduced and GQR becomes broader in TDDM3. Thus the major effect of the excitations of the $^{16}$O core on the quadrupole mode is to modify the distribution of the GQR strength. However, the damping of GQR in $^{24}$O is modest as compared with that in $^{16}$O. To make this point clear, we show in Fig.6 the strength function for the isoscalar quadrupole mode in $^{16}$O. The solid line denotes the result in TDDM3 with $v_0=-800$ fm$^3$MeV and the dotted line that in TDHF. The EWSR value in TDDM3 is 7 % larger than that in TDHF. The splitting of the GQR strength obtained in TDDM3 is consistent with experimental observation [@Fritsch]. The damping of GQR in $^{16}$O is caused by its coupling to two-body configurations mainly consisting of two holes in the $1p$ states and two particles in the $2s$ or $1d$ states. In $^{24}$O, the coupling of GQR to two-body configurations is hindered by the spatial extension of neutron single-particle wavefunctions due to the neutron excess.
{height="6cm"}
In summary, the strength functions of the quadrupole resonances in $^{22}$O and $^{24}$O were studied using TDDM. In this approach, the correlated ground state was first obtained starting from the HF ground state with the subsequent excitation of the quadrupole mode. It is shown that the lowest quadrupole states are shifted upward and become significantly collective due to the coupling to two-body configurations. The obtained $B(E2)$ value for $^{22}$O was found comparable to the observed one. It was pointed out that the lowest quadrupole states obtained in TDDM have properties similar to those in QRPA. It was also found that the damping of GQR in $^{24}$O is modest as compared with that in $^{16}$O.
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abstract: 'We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley’s structure theorem for algebraic groups.'
address:
- |
Université de Grenoble I\
Département de Mathématiques\
Institut Fourier, UMR 5582 du CNRS\
38402 Saint-Martin d’Hères Cedex, France
- |
Facultad de Ciencias\
Universidad de la República\
Iguá 4225\
11400 Montevideo, Uruguay
author:
- Michel Brion and Alvaro Rittatore
title: The structure of normal algebraic monoids
---
Introduction {#sec:introduction}
============
A classical theorem of Chevalley asserts that any connected algebraic group is an extension of an abelian variety by a connected affine algebraic group. In this note, we obtain an analogous result for normal algebraic monoids. This reduces their structure to that of more familiar objects: abelian varieties, and affine (equivalently, linear) algebraic monoids. The latter have been extensively investigated, see the expositions [@Pu88; @Re05].
To state Chevalley’s theorem and our analogue in a precise way, we introduce some notation. We consider algebraic varieties and algebraic groups over an algebraically closed field ${{\Bbbk}}$ of arbitrary characteristic. By a variety, we mean a separated integral scheme of finite type $X$ over ${{\Bbbk}}$; by a point of $X$, we mean a closed point. An algebraic group is a smooth group scheme of finite type over ${{\Bbbk}}$.
Let $G$ be a connected algebraic group, then there exists a unique connected normal affine algebraic subgroup $G_{{\operatorname{aff}}}$ such that the quotient $G/G_{{\operatorname{aff}}}$ is an abelian variety. In other words, we have an exact sequence of connected algebraic groups $$\label{eqn:extension}
\CD
1 @>>> G_{{\operatorname{aff}}} @>>> G @>{\alpha_G}>> {{\mathcal A}}(G) @>>> 0
\endCD$$ where $G_{{\operatorname{aff}}}$ is affine and ${{\mathcal A}}(G)$ is projective (since the group ${{\mathcal A}}(G)$ is commutative, its law will be denoted additively). It follows that the morphism $\alpha_G$ is affine; hence the variety $G$ is quasi-projective (see [@Co02] for these developments and for a modern proof of Chevalley’s theorem).
Next, let $M$ be an irreducible *algebraic monoid*, i.e., an algebraic variety over ${{\Bbbk}}$ equipped with a morphism $M \times M \to M$ (the *product*, denoted simply by $(x,y) \mapsto xy$) which is associative and admits an identity element $1$. Denote by $G = G(M)$ the group of invertible elements of $M$. The *unit group* $G$ is known to be a connected algebraic group, open in $M$ (see [@Ri98 Thm. 1]).
Let $G_{{\operatorname{aff}}} \subseteq G$ be the associated affine group, and $M_{{\operatorname{aff}}}$ the closure of $G_{{\operatorname{aff}}}$ in $M$. Clearly, $M_{{\operatorname{aff}}}$ is an irreducible algebraic monoid with unit group $G_{{\operatorname{aff}}}$. By [@Ri06 Thm. 2], it follows that $M_{{\operatorname{aff}}}$ is affine. Also, note that $$\label{eqn:GM}
M = G M_{{\operatorname{aff}}} = M_{{\operatorname{aff}}} G$$ as follows from the completeness of $G/G_{{\operatorname{aff}}} = {{\mathcal A}}(G)$ (see Lemma \[lem:map\] for details).
We may now state our main result, which answers a question raised by D. A. Timashev (see the comments after Thm. 17.3 in [@Ti06]):
\[thm:main\] Let $M$ be an irreducible algebraic monoid with unit group $G$. If the variety $M$ is normal, then $\alpha_G : G \to {{\mathcal A}}(G)$ extends to a morphism of algebraic monoids $\alpha_M : M \to {{\mathcal A}}(G)$. Moreover, the morphism $\alpha_M$ is affine, and its scheme-theoretic fibers are normal varieties; the fiber at $1$ equals $M_{{\operatorname{aff}}}$.
In loose words, any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid.
For nonsingular monoids, Theorem \[thm:main\] follows immediately from Weil’s extension theorem: any rational map from a nonsingular variety to an abelian variety is a morphism. However, this general result no longer holds for singular varieties. Also, the normality assumption in Theorem \[thm:main\] cannot be omitted, as shown by Example \[ex:nonnormal\].
Some developments and applications of the above theorem are presented in Section 2. The next section gathers a number of auxiliary results to be used in the proof of that theorem, given in Section 4. The final Section 5 contains further applications of our structure theorem to the classification of normal algebraic monoids, and to their faithful representations as endomorphisms of homogeneous vector bundles on abelian varieties.
[**Acknowledgements.**]{} The second author would like to thank the Institut Fourier for its hospitality; his research was also partially supported by the Fondo Clemente Estable, Uruguay (FCE-10018).
Some applications {#sec:applications}
=================
With the notation and assumptions of Theorem \[thm:main\], observe that $\alpha_M$ is equivariant with respect to the action of the group $G \times G$ on $M$ via $$(g_1,g_2) \cdot m = g_1 m g_2^{-1},$$ and its action on ${{\mathcal A}}(G)$ via $$(g_1,g_2) \cdot a = \alpha(g_1) - \alpha(g_2) + a.$$ Since the latter action is transitive, $\alpha_M$ is a $G \times G$-homogeneous fibration. In particular, all fibers are isomorphic, and $\alpha_M$ is faithfully flat.
Also, each irreducible component of the closed subset $M_{{\operatorname{aff}}} \setminus G_{{\operatorname{aff}}} \subset M_{{\operatorname{aff}}}$ is of codimension $1$, since $G_{{\operatorname{aff}}}$ is affine. Together with the above observation, it follows that the same holds for the set $M \setminus G$ of non-units in $M$:
\[cor:noninvertible\] Each irreducible component of $M \setminus G$ has codimension $1$ in $M$.
Next, we obtain an intrinsic characterization of the morphism $\alpha_M$. To state it, recall from [@Se58] that any variety $X$ admits an *Albanese morphism*, i.e., a universal morphism to an abelian variety.
\[cor:albanese\] $\alpha_M$ is the Albanese morphism of the variety $M$.
Let $f : M \to A$ be a morphism (of varieties) to an abelian variety. Composing $f$ with a translation of $A$, we may assume that $f(1) = 0$. Then the restriction $f\vert_G : G \to A$ is a morphism of algebraic groups by [@Co02 Lem. 2.2]. So $f(G_{{\operatorname{aff}}})$ equals $0$ by [@Co02 Lem. 2.3]. It follows that $f\vert_G = \varphi \circ \alpha_G$, where $\varphi : {{\mathcal A}}(G) \to A$ is a morphism of algebraic groups. Hence $f$ equals $\varphi \circ \alpha_M$, since both morphisms have the same restriction to the open subset $G$.
Also, since the morphism $\alpha_M : M \to {{\mathcal A}}(G)$ is affine and the variety ${{\mathcal A}}(G)$ is projective (see e.g. [@Mi86 Thm. 7.1]), we obtain the following:
\[cor:quasi-projective\] $M$ is quasi-projective.
Another application of Theorem \[thm:main\] concerns the set $$E(M) = \{ e\in M ~\vert~ e^2 = e\}$$ of *idempotents*. Indeed, since $\alpha_M$ is a morphism of monoids and the unique idempotent of ${{\mathcal A}}(G)$ is the origin, we obtain:
\[cor:idempotents\] $E(M) = E(M_{{\operatorname{aff}}})$.
Next, recall that a monoid $N$ is said to be *regular* if given any $x \in N$, there exists $y \in N$ such that $x = x y x$.
\[cor:regular\] $M$ is regular if and only if $M_{{\operatorname{aff}}}$ is regular.
If $M_{{\operatorname{aff}}}$ is regular, then so is $M$ by (\[eqn:GM\]). Conversely, assume that $M$ is regular. Let $x \in M_{{\operatorname{aff}}}$ and write $x = x y x$, where $y \in M$. Then we obtain: $\alpha_M(y) = 0$, so that $y \in M_{{\operatorname{aff}}}$.
By [@Pu82 Thm. 13], every regular irreducible affine algebraic monoid $N$ is *unit regular*, i.e., given any $x \in N$, there exists $y \in G(N)$ such that $x = x y x$; equivalently, $N = G(N) E(N)$. Together with Corollary \[cor:regular\], this implies:
\[cor:unit\] If $M$ is regular, then it is unit regular.
Finally, we show that Theorem \[thm:main\] does not extend to arbitrary irreducible algebraic monoids:
\[ex:nonnormal\] Let $A$ be an abelian variety. Then $$M := A \times {{\mathbb A}}^1$$ is a commutative nonsingular algebraic monoid via the product $$(x,y) \; (x',y') = (x + x', yy'),$$ with unit group $$G:= A \times {{\mathbb G}}_m$$ and kernel $A \times \{ 0 \}$. The morphism $\alpha_G$ is the first projection $A \times {{\mathbb G}}_m \to A$; likewise, the Albanese morphism of the variety $M$ is just the first projection $$p: A \times {{\mathbb A}}^1 \to A.$$
Next, let $F \subset A$ be a non-trivial finite subgroup. Let $M'$ be the topological space obtained from $M$ by replacing the closed subset $A \times \{ 0 \}$ with the quotient $A/F \times \{ 0 \}$; in other words, each point $(x + f,0)$ (where $x \in A$ and $f \in F$) is identified with the point $(x, 0)$. Denote by $$q : M \to M'$$ the natural map. We claim that $M'$ has a structure of a irreducible, non-normal, commutative algebraic monoid with unit group $G$, such that $q$ is a morphism of monoids; furthermore, $\alpha_G: G \to A$ does not extend to a morphism $M' \to A$.
Indeed, one readily checks that $M'$ carries a unique product such that $q$ is a morphism of monoids. Moreover, the restriction $q\vert_G$ is an isomorphism onto $G(M')$, and we have a commutative square $$\CD
M \ @>{q}>> M' \\
@V{ p }VV @V{\alpha}VV \\
A @>{ }>> A/F \\
\endCD$$ where $\alpha$ is equivariant with respect to the action of $A$ on $M'$ via the product of $M'$, and the natural action of $A$ on $A/F$. Let $$N := \alpha^{-1}(0),$$ then the set $N$ is the image under $q$ of the subset $F \times {{\mathbb A}}^1 \subset A \times {{\mathbb A}}^1$. So $N$ is a union of copies of the affine line, indexed by the finite set $F$, and glued along the origin. Hence $N$ is a reduced affine scheme, and its product (induced by the product of $M'$) is a morphism: $N$ is a connected, reducible affine algebraic monoid. Furthermore, the natural map $$A \times^F N \to M'$$ is clearly an isomorphism of monoids, and the left-hand side is also an algebraic monoid. This yields the desired structure of algebraic monoid on $M'$. The map $q$ is induced from the natural map $F \times {{\mathbb A}}^1 \to N$, which is a morphism; hence so is $q$. Finally, the projection $p: A \times {{\mathbb G}}_m \to A$ cannot extend to a morphism $M' \to A$: such a morphism would be $A$-equivariant, and hence restrict to an $A$-equivariant morphism $A/F \times \{ 0 \} \cong A/F \to A$, which is impossible. This completes the proof of the claim.
Alternatively, this claim follows from a general result concerning the existence of pinched schemes (see [@Fe03 Thm. 5.4]). Indeed, $M'$ is obtained by pinching the quasi-projective variety $M$ along its closed subset $A \times \{ 0 \} \cong A$ via the finite morphism $A \to A/F$, in the terminology of [@Fe03].
One easily checks that $\alpha$ is the Albanese morphism of the variety $M'$, and $q$ is its normalization. Moreover, $$M'_{{\operatorname{aff}}} = q( \{ 0 \} \times {{\mathbb A}}^1) \cong {{\mathbb A}}^1$$ is strictly contained in $N$, and is nonsingular whereas $M'$ is non-normal.
Note finally that $M'$ is weakly normal, i.e., any finite bijective birational map from a variety to $M'$ is an isomorphism. Thus, Theorem \[thm:main\] does not extend to weakly normal monoids.
Auxiliary results {#sec:auxiliary}
=================
We consider a connected algebraic group $G$ and denote by $Z^0$ its connected center regarded as a closed reduced subscheme of $G$, and hence as a connected algebraic subgroup.
\[lem:product\] [(i)]{} The scheme-theoretic intersection $Z^0 \cap G_{{\operatorname{aff}}}$ contains $Z^0_{{\operatorname{aff}}}$ as a normal subgroup, and the quotient $(Z^0 \cap G_{{\operatorname{aff}}})/Z^0_{{\operatorname{aff}}}$ is a finite group scheme.
[(ii)]{} The product map $Z^0 \times G_{{\operatorname{aff}}} \to G$ factors through an isomorphism $$(Z^0 \times G_{{\operatorname{aff}}})/(Z^0 \cap G_{{\operatorname{aff}}}) \cong G,$$ where $Z^0 \cap G_{{\operatorname{aff}}}$ is embedded in $Z^0 \times G_{{\operatorname{aff}}}$ as a normal subgroup scheme via the identity map on the first factor and the inverse map on the second factor.
[(iii)]{} The natural map $Z^0/(Z^0 \cap G_{{\operatorname{aff}}}) \to G/G_{{\operatorname{aff}}} = {{\mathcal A}}(G)$ is an isomorphism of algebraic groups.
This easy result is proved in [@Br06 Sec. 1.1] under the assumption that ${{\Bbbk}}$ has characteristic zero; the general case follows by similar arguments. We will also need the following result, see e.g. [@Br06 Sec. 1.2]:
\[lem:action\] Let $G$ act faithfully on an algebraic variety $X$. Then the isotropy subgroup scheme of any point of $X$ is affine.
Next we consider an irreducible algebraic monoid $M$ with unit group $G$. If $M$ admits a zero element, then this point is fixed by $G$ acting by left multiplication, and this action is faithful. Thus, $G$ is affine by Lemma \[lem:action\]. Together with [@Ri06 Thm. 2], this yields:
\[cor: fixed\] Any irreducible algebraic monoid having a zero element is affine.
Returning to an arbitrary irreducible algebraic monoid $M$, recall that an *ideal* of $M$ is a subset $I$ such that $M I M \subseteq I$.
\[lem:kernel\] [(i)]{} $M$ contains a unique closed $G\times G$-orbit, which is also the unique minimal ideal: the kernel ${\operatorname{Ker}}(M)$.
[(ii)]{} If $M$ is affine, then ${\operatorname{Ker}}(M)$ contains an idempotent.
\(i) is part of [@Ri98 Thm. 1]. For (ii), see e.g. [@Re05 p. 35].
\[lem:map\] [(i)]{} Let $Z^0 \cap G_{{\operatorname{aff}}}$ act on $Z^0 \times M_{{\operatorname{aff}}}$ by multiplication on the first factor, and the inverse map composed with left multiplication on the second factor. Then the quotient $$Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}} :=
(Z^0 \times M_{{\operatorname{aff}}})/(Z^0 \cap G_{{\operatorname{aff}}})$$ has a unique structure of an irreducible algebraic monoid such that the quotient map is a morphism of algebraic monoids. Moreover, $$G\bigl(Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}}\bigr) =
Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} G_{{\operatorname{aff}}} \cong G.$$ Regarded as a $G$-variety via left multiplication, $Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}}$ is naturally isomorphic to the quotient $$G \times^{G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}} := (G \times M_{{\operatorname{aff}}})/G_{{\operatorname{aff}}},$$ where the action of $G_{{\operatorname{aff}}}$ on $G \times M_{{\operatorname{aff}}}$ is defined as above.
[(ii)]{} The product map $Z^0 \times M_{{\operatorname{aff}}} \to M$ factors uniquely through a morphism of algebraic monoids $$\label{eqn:map}
\pi: Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}} \to M.$$ Moreover, $\pi$ is birational and proper.
[(iii)]{} $M = Z^0 M_{{\operatorname{aff}}}$ and ${\operatorname{Ker}}(M) = Z^0 {\operatorname{Ker}}(M_{{\operatorname{aff}}})$.
\(i) and the first assertion of (ii) are straightforward. Also, the restriction of $\pi$ to the unit group is an isomorphism, and hence $\pi$ is birational. To show the properness, observe that $\pi$ factors as a closed immersion $$Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}} \to
Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M$$ (induced by the inclusion map $M_{{\operatorname{aff}}} \to M$), followed by an isomorphism $$Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M \to (Z^0/Z^0 \cap G_{{\operatorname{aff}}}) \times M
\cong {{\mathcal A}}(G) \times M$$ given by $(z,m)\mapsto \bigl(z(Z^0 \cap G_{{\operatorname{aff}}}), zm\bigr)$, followed in turn by the projection $${{\mathcal A}}(G) \times M \to M$$ which is proper, since ${{\mathcal A}}(G)$ is projective.
\(iii) By (ii), $\pi$ is surjective, i.e., the first equality holds. For the second equality, note that $Z^0 {\operatorname{Ker}}(M_{{\operatorname{aff}}})$ is closed in $M$ since $\pi$ is proper, and is a unique orbit of $G \times G$ by Lemma \[lem:product\].
Proof of the main result {#sec:proof}
========================
We begin by showing the following result of independent interest:
\[thm:map\] The morphism $$\pi: Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}} = G \times^{G_{{\operatorname{aff}}}} M
\to M$$ is an isomorphism for any normal irreducible monoid $M$.
We proceed through a succession of reduction steps.
1\) It suffices to show that $\pi$ is finite for any irreducible algebraic monoid $M$ (possibly non-normal). Indeed, the desired statement follows from this, in view of Lemma \[lem:map\](ii) and Zariski’s Main Theorem.
2\) Since $\pi$ is proper, it suffices to show that its fibers are finite. But the points of $M$ where the fiber of $\pi$ is finite form an open subset (by semicontinuity), which is stable under the action of $G \times G$. Thus, it suffices to check the finiteness of the fiber at some point of the unique closed $G \times G$-orbit, ${\operatorname{Ker}}(M)$. By Lemmas \[lem:kernel\] (ii) and \[lem:map\] (iii), ${\operatorname{Ker}}(M)$ contains an idempotent $e \in {\operatorname{Ker}}(M_{{\operatorname{aff}}})$. So we are reduced to showing that the set $\pi^{-1}(e)$ is finite.
3\) Consider the $Z^0$-orbit $Z^0 e$ and its inverse image under $\pi$, $$\pi^{-1}(Z^0 e) \cong
Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}}).$$ It suffices to check that the map $$p : Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}}) \to Z^0 e$$ (the restriction of $\pi$) has finite fibers. Since $p$ is surjective, it suffices in turn to show that $Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}})$ and $Z^0 e$ are algebraic groups of the same dimension, and $p$ is a morphism of algebraic groups.
4\) Since $e$ is idempotent and $Z^0$ is a central subgroup of $G$, the orbit $Z^0 e$ (regarded as a locally closed, reduced subscheme of $M$) is a connected algebraic group under the product of $M$, with identity element $e$. Moreover, the intersection $Z^0 e \cap M_{{\operatorname{aff}}}$ (also regarded as a locally closed, reduced subscheme of $M$) is a closed submonoid of $Z^0 e$, with the same identity element $e$. By [@Re05 Exer. 3.5.1.2], it follows that $Z^0 e \cap M_{{\operatorname{aff}}}$ is a subgroup of $Z^0 e$. Hence $Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}})$ is an algebraic group as well, and clearly $p$ is a morphism of algebraic groups.
5\) It remains to show that $$\label{eqn:dim}
\dim Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}})
= \dim (Z^0 e).$$
We first analyze the left-hand side. Since $Z^0 e \cap M_{{\operatorname{aff}}}$ is a quasi-affine algebraic group, it is affine. But the maximal connected affine subgroup of $Z^0 e$ is $Z^0_{{\operatorname{aff}}} e$, since $Z^0 e \cong Z^0/ {\operatorname{Stab}}_{Z^0}(e)$ as groups. It follows that $Z^0_{{\operatorname{aff}}} e \subseteq Z^0 e \cap M_{{\operatorname{aff}}}$ is the connected component of the identity. Hence $$\dim Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}}) =
\dim (Z^0) - \dim (Z^0 \cap G_{{\operatorname{aff}}})
+ \dim(Z^0_{{\operatorname{aff}}} e).$$ But $\dim(Z^0 \cap G_{{\operatorname{aff}}}) = \dim(Z^0_{{\operatorname{aff}}})$ by Lemma \[lem:product\], and $$\dim(Z^0_{{\operatorname{aff}}} e) = \dim (Z^0_{{\operatorname{aff}}}) - \dim {\operatorname{Stab}}_{Z^0_{{\operatorname{aff}}}}(e)$$ so that $$\dim Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}}(Z^0 e \cap M_{{\operatorname{aff}}}) =
\dim (Z^0) - \dim {\operatorname{Stab}}_{Z^0_{{\operatorname{aff}}}}(e).$$
On the other hand, $$\dim (Z^0 e) = \dim (Z^0) - \dim {\operatorname{Stab}}_{Z^0}(e),$$ and ${\operatorname{Stab}}_{Z^0}(e)$ is affine by Lemma \[lem:action\]. Hence $$\dim {\operatorname{Stab}}_{Z^0}(e) = \dim {\operatorname{Stab}}_{Z^0_{{\operatorname{aff}}}}(e).$$ This completes the proof of Equation (\[eqn:dim\]) and, in turn, of the finiteness of $\pi$.
We may now prove Theorem \[thm:main\]. Observe that the projection $$M \cong Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}} \to Z^0 /(Z^0 \cap G_{{\operatorname{aff}}})
\cong {{\mathcal A}}(G)$$ yields the desired extension $\alpha_M$ of $\alpha_G$. Clearly, the scheme-theoretic fiber of $\alpha_M$ at $1$ equals $M_{{\operatorname{aff}}}$.
To show that the morphism $\alpha_M$ is affine, oberve that $M_{{\operatorname{aff}}}$ is an affine variety equipped with an action of the affine group scheme $Z^0 \cap G_{{\operatorname{aff}}}$. Thus, $M_{{\operatorname{aff}}}$ admits a closed equivariant immersion into a $(Z^0 \cap G_{{\operatorname{aff}}})$-module $V$. Then $Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}}$ (regarded as a variety over ${{\mathcal A}}(G)$) admits a closed $Z^0$-equivariant immersion into $Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} V$, the total space of a vector bundle over ${{\mathcal A}}(G)$.
Finally, to show that the variety $M_{{\operatorname{aff}}}$ is normal, consider its normalization $\widetilde{M_{{\operatorname{aff}}}}$, an affine variety where $Z^0 \cap G_{{\operatorname{aff}}}$ acts such that the normalization map $f : \widetilde{M_{{\operatorname{aff}}}} \to M_{{\operatorname{aff}}}$ is equivariant. This defines a morphism $$\widetilde{\pi} : Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} \widetilde{M_{{\operatorname{aff}}}} \to M$$ which is still birational and finite. So $\widetilde{\pi}$ is an isomorphism by Zariski’s Main Theorem; it follows that $f$ is an isomorphism as well.
Classification and faithful representation {#sec:classification}
==========================================
We begin by reformulating our main results as a classification theorem for normal algebraic monoids:
\[thm:equivalence\] The category of normal algebraic monoids is equivalent to the category having as objects the pairs $(G,N)$, where $G$ is a connected algebraic group and $N$ is a normal affine algebraic monoid with unit group $G_{{\operatorname{aff}}}$.
The morphisms from such a pair $(G,N)$ to a pair $(G',N')$ are the pairs $(\varphi,\psi)$, where $\varphi : G \to G'$ is a morphism of algebraic groups and $\psi: N \to N'$ is a morphism of algebraic monoids such that $\varphi \vert_{G_{{\operatorname{aff}}}} = \psi \vert_{G_{{\operatorname{aff}}}}$.
By Theorem \[thm:map\], any normal irreducible algebraic $M$ with unit group $G$ is determined by the pair $(G,M_{{\operatorname{aff}}})$ up to isomorphism. Conversely, any pair $(G,N)$ as in the above statement yields a normal algebraic monoid $$M := G \times^{G_{{\operatorname{aff}}}} N$$ together with isomorphisms $G \to G(M)$ and $N \to M_{{\operatorname{aff}}}$, as follows from Lemmas \[lem:product\] and \[lem:map\].
Next, consider a morphism of normal algebraic monoids $$f: M \to M'.$$ Clearly, $f$ restricts to a morphism of algebraic groups $$\varphi : G(M) \to G(M').$$ Moreover, the universal property of the Albanese maps $\alpha_M$, $\alpha_{M'}$ yields a commutative diagram $$\CD
M @>{f}>> M' \\
@V{\alpha_M}VV @V{\alpha_{M'}}VV \\
{{\mathcal A}}\bigl(G(M)\bigr) @>{\alpha_f}>> {{\mathcal A}}\bigl(G(M')\bigr), \\
\endCD$$ where $\alpha_f$ is a morphism of varieties such that $\alpha_f(0) = 0$, and hence a morphism of abelian varieties (see e.g. [@Mi86 Cor. 3.6]). In turn, this yields a morphism of algebraic monoids $$\psi : M_{{\operatorname{aff}}} = \alpha_M^{-1}(0) \to \alpha_{M'}^{-1}(0) = M'_{{\operatorname{aff}}}$$ such that $\varphi\vert_{G(M)_{{\operatorname{aff}}}} = \psi\vert_{G(M)_{{\operatorname{aff}}}}$. Conversely, any such pair $(\varphi,\psi)$ defines a morphism $f$, as follows from Theorem \[thm:map\] again.
\(i) The irreducible affine algebraic monoids having a prescribed unit group $G$ are exactly the affine equivariant embeddings of the homogeneous space $(G \times G)/{\operatorname{diag}}(G)$, by [@Ri98 Prop. 1]. In the case where $G$ is reductive, such embeddings admit a combinatorial classification, see [@Ri98] and [@Ti06].
\(ii) The normality assumption in Theorem \[thm:equivalence\] cannot be omitted: with the notation of Example \[ex:nonnormal\], the monoids $M$ and $M'$ yield the same pair $(A \times {{\mathbb G}}_m,{{\mathbb A}}^1)$; but they are not isomorphic as varieties, since the image of their Albanese map is $A$, resp. $A/F$.
Next, we obtain faithful representations of normal algebraic monoids as endomorphisms of vector bundles over abelian varieties. For this, we need additional notation and some preliminary observations.
Let $A$ be an abelian variety, and $$p : E \to A$$ a vector bundle. Observe that $p$ is the Albanese morphism of the variety $E$ (as follows e.g. from [@Mi86 Cor. 3.9]). Thus, any morphism of varieties $f : E \to E$ fits into a commutative square $$\CD
E @>{f}>> E \\
@V{p}VV @V{p}VV \\
A @>{\alpha(f)}>> A, \\
\endCD$$ where $\alpha(f)$ is a morphism of varieties as well. By [@Mi86 Cor. 3.9] again, $\alpha(f)$ is the composition of a translation of $A$ with an endomorphism of the abelian variety $A$.
We say that $f$ is an *endomorphism* (resp. an *automorphism*) of $E$, if $\alpha(f)$ is the translation $$t_a : A \to A, \quad x \mapsto a +x$$ for some $a = a(f)\in A$, and the induced maps on fibers $$f_x: E_x \to E_{a+x} \quad (x \in A)$$ are all linear (resp. linear isomorphisms).
Clearly, the endomorphisms of $E$ form a monoid under composition, denoted by ${\operatorname{End}}(E)$; its unit group ${\operatorname{Aut}}(E)$ consists of the automorphisms. The map $$\label{eqn:alb}
\alpha : {\operatorname{End}}(E) \to A, \quad f \mapsto a(f)$$ is a morphism of monoids, and its fiber at a point $a \in A$ is isomorphic to the set of morphisms of vector bundles from $E$ to $t_a^* E$ (a finite-dimensional $k$-vector space).
In particular, the fiber at $0$ is the monoid ${\operatorname{End}}_A(E)$ of endomorphisms of $E$ regarded as a vector bundle over $A$. Moreover, ${\operatorname{End}}_A(E)$ is a finite-dimensional $k$-algebra; in particular, an irreducible affine algebraic monoid. Its unit group ${\operatorname{Aut}}_A(E)$ is the kernel of the restriction of $\alpha$ to ${\operatorname{Aut}}(E)$.
The vector bundle $E$ is called *homogeneous* if the restriction map ${\operatorname{Aut}}(E) \to A$ is surjective; equivalently, $E \cong t_a^* E$ for any $a \in A$.
For example, a line bundle is homogeneous if and only if it is algebraically equivalent to $0$ (see [@Mi86 Sect. 9]). More generally, the homogeneous vector bundles are exactly the direct sums of vector bundles of the form $L \otimes F$, where $L$ is an algebraically trivial line bundle, and $F$ admits a filtration by sub-vector bundles such that the associated graded bundle is trivial (see [@Mu78 Thm. 4.17]).
We are now in a position to state:
\[thm:rep\] [(i)]{} Let $p: E \to A$ be a homogeneous vector bundle over an abelian variety. Then ${\operatorname{End}}(E)$ has a structure of a nonsingular irreducible algebraic monoid such that its action on $E$ is algebraic. Moreover, the Albanese morphism of ${\operatorname{End}}(E)$ is the map of (\[eqn:alb\]), so that ${\operatorname{End}}(E)_{{\operatorname{aff}}} = {\operatorname{End}}_A(E)$.
[(ii)]{} Any normal irreducible algebraic monoid $M$ is isomorphic to a closed submonoid of ${\operatorname{End}}(E)$, where $E$ is a homogeneous vector bundle over the Albanese variety of $M$.
\(i) We claim that ${\operatorname{Aut}}(E)$ has a structure of a connected algebraic group such that its action on $E$ is algebraic and the map ${\operatorname{Aut}}(E) \to A$ is a (surjective) morphism of algebraic groups.
Indeed, any $f \in {\operatorname{Aut}}(E)$ extends uniquely to an automorphism of the projective completion ${{\mathbb P}}(E \oplus O_A)$ (regarded as an algebraic variety), where $O_A$ denotes the trivial bundle of rank $1$ over $A$. Moreover, every automorphism of ${{\mathbb P}}(E \oplus O_A)$ induces an automorphism of $A$, the Albanese variety of ${{\mathbb P}}(E \oplus O_A)$. It follows that ${\operatorname{Aut}}(E)$ may be identified to the group $G$ of automorphisms of ${{\mathbb P}}(E \oplus O_A)$ that induce translations of $A$, and commute with the action of the multiplicative group ${{\mathbb G}}_m$ by multiplication on fibers of $E$. Clearly, $G$ is contained in the connected automorphism group ${\operatorname{Aut}}^0 {{\mathbb P}}(E \oplus O_A)$ (a connected algebraic group) as a closed subgroup; hence, $G$ is an algebraic group. Moreover, the exact sequence $$1 \to {\operatorname{Aut}}_A(E) \to {\operatorname{Aut}}(E) \to A \to 0$$ implies that $G$ is connected; this completes the proof of our claim.
This exact sequence also implies that ${\operatorname{Aut}}(E)_{{\operatorname{aff}}} = {\operatorname{Aut}}_A(E)$. Hence the natural map $$\pi: {\operatorname{Aut}}(E) \times^{{\operatorname{Aut}}_A(E)} {\operatorname{End}}_A(E) \to {\operatorname{End}}(E)$$ is bijective, since the map $\alpha: {\operatorname{End}}(E) \to A$ is a ${\operatorname{Aut}}(E)$-homogeneous fibration. This yields a structure of algebraic monoid on ${\operatorname{End}}(E)$, which clearly satisfies our assertions.
\(ii) By [@Pu88 Thm. 3.15], the associated monoid $M_{{\operatorname{aff}}}$ is isomorphic to a closed submonoid of ${\operatorname{End}}(V)$, where $V$ is a vector space of finite dimension over ${{\Bbbk}}$. In particular, $V$ is a rational $G_{{\operatorname{aff}}}$-module. We may thus form the associated vector bundle $$p : E := G \times^{G_{{\operatorname{aff}}}} V = Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} V
\to {{\mathcal A}}(G).$$ Since the action of ${{\mathcal A}}(G)$ on itself by translations lifts to the action of $G$ on $E$, then $E$ is homogeneous. Moreover, one easily checks that the product action of $Z^0 \times M_{{\operatorname{aff}}}$ on $Z^0 \times V$ yields a faithful action of $M = Z^0 \times^{Z^0 \cap G_{{\operatorname{aff}}}} M_{{\operatorname{aff}}}$ on $E$.
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---
abstract: 'We theoretically analyze quasi-one-dimensional Bose-Einstein condensates under the influence of a harmonic trap and a narrow potential defect that moves through the atomic cloud. Performing simulations on the mean field level, we explore a robust mechanism in which a single dark soliton is nucleated and immediately pinned by the moving defect, making it possible to drag it to a desired position and release it there. We argue on a perturbative level that a defect potential which is attractive to the atoms is suitable for holding and moving dark solitons. The soliton generation protocol is investigated over a wide range of model parameters and its success is systematically quantified by a suitable fidelity measure, demonstrating its robustness against parameter variations, but also the need for tight focusing of the defect potential. Holding the soliton at a stationary defect for long times may give rise to dynamical instabilities, whose origin we explore within a Bogoliubov-de Gennes linearization analysis. We show that iterating the generation process with multiple defects offers a perspective for initializing multiple soliton dynamics with freely chosen initial conditions.'
author:
- 'I. Hans'
- 'J. Stockhofe'
- 'P. Schmelcher'
bibliography:
- 'soliton.bib'
title: |
Generating, dragging and releasing dark solitons\
in elongated Bose-Einstein condensates
---
Introduction
============
Solitonic wave excitations that maintain their shape during propagation are found in a large variety of physical systems, ranging from hydrodynamics to modern telecommunication systems and even biological molecules [@solitons]. The enormous technological advance of recent years has made it possible to prepare and observe solitons in ultracold atom experiments, primarily (but not only, see e.g. [@Yefsah2013]) in condensed bosonic ensembles near zero temperature [@Pethick_book; @book_kevrekidis]. On the mean-field level, a Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation (GPE) [@Pethick_book], a nonlinear Schrödinger equation with cubic nonlinearity induced by the interatomic interaction. In one spatial dimension (1D), this equation is well-known to feature dark (bright) solitons for defocusing (focusing) nonlinearities, respectively [@book_kevrekidis; @DJF]. Experimentally, a highly elongated quasi-1D regime can be reached by tightly confining the atoms in the radial direction, effectively freezing out the transverse dynamics. Early experiments succeeded in preparing bright [@L.Khaykovich05172002; @Strecker2002] and dark [@PhysRevLett.83.5198; @Science.287.5450] matter-wave solitons. Dark solitons (which are in the focus of this work) are characterized by a localized density minimum across which the phase changes by $\pi$. These can be created by manipulating the condensate phase [@PhysRevLett.83.5198; @Science.287.5450; @Becker2008; @PhysRevLett.101.120406] or density [@ZacharyDutton07272001] using external potentials, see also the discussion in [@DJF]. They can also form in the wake of a repulsive barrier dragged through the condensate [@Pavloff2002; @Radouani2004; @Engels2007] or in collisions of initially separated atomic clouds [@Weller2008]. Still, controllably creating a dark soliton at a desired position in a Bose-Einstein condensate is a challenging task in experiments.\
Once a soliton has formed, its dynamics can again be influenced with external potentials. A dark soliton in a harmonically trapped BEC performs particle-like harmonic oscillations around the trap center [@PhysRevLett.84.2298; @Becker2008; @PhysRevLett.101.120406; @Weller2008]. The emission of sound waves due to acceleration of a soliton has been studied in [@Parker2003a; @Parker2003; @Pelinovsky2005a; @Parker2010]. Thinking of the external force as a handle for controlling the soliton dynamics, a direct manipulation of the soliton motion with narrow potential defects has been suggested in [@Frantzeskakis2002], where the interaction of a dark soliton with a pointlike impurity was analyzed within perturbation theory (see also [@Kivshar1994; @Konotop1994; @Konotop1997] and the corresponding studies for bright [@Herring2005] and dark-bright solitons [@Achilleos2011]). Moreover, the possibility of dragging along dark solitons in a moving optical lattice potential has been demonstrated in [@Theocharis2005a; @Theocharis2005]. In a similar spirit, dragging of bright solitons in a discrete lattice model has been discussed recently [@Brazhnyi2011], while pinning and transporting quantum vortices with focused external potentials has been shown in [@Tung2006; @Davis2009].\
Here, we describe a method for controllably creating, dragging and releasing a dark soliton in a repulsively interacting Bose-Einstein condensate employing a tightly focused red-detuned laser beam (acting as an attractive potential for the atoms via the dipole force). Specifically, we study a trapped quasi-1D BEC under the influence of a moving Gaussian potential defect of attractive sign. When entering the atomic cloud from outside with this defect, a dark soliton can be created and at the same time pinned by the defect, such that it can subsequently be placed and released at an arbitrary position. Motivated by an analysis of the instantaneous energy levels in a single particle model, a similar scheme for exciting nodes in the condensate wave function by traversing it with an attractive defect has been proposed in [@Karkuszewski2001; @Damski2001]. A related protocol for the extraction of bright solitons from an attractive BEC has been suggested in [@Carpentier2008] (see also [@Carpentier2006]), but there the defect was not assumed to act as an external potential, but instead to cause a local variation of the effective atomic interaction through the mechanism of optical Feshbach resonance [@Theis2004]. In our work, the defect acts as a single particle potential for the atoms. Dynamically manipulating (e.g. splitting) the entire BEC cloud with such optical “tweezers” is nowadays well established [@Gustavson2001; @Boyer2006]. If, instead, the light is focused to the comparably short length scale of a dark soliton, the same type of technology can be employed to manipulate the dynamics of a localized solitonic excitation.\
Our presentation is structured as follows. In Sec. \[perturbth\] we introduce the theoretical mean-field framework our study is based on and give a discussion of the results from a perturbative treatment of the potential defect, revealing in particular that a potential that is attractive to the individual atoms is effectively also attractive to a dark soliton. In Sec.\[sec:generation\] we show results from numerical simulations, demonstrating the robust creation, dragging and release of a dark soliton. We quantify the fidelity of the creation process and vary the defect parameters to explore the robustness of this protocol. If the dark soliton is pinned to the defect for long times, a dynamical instability may occur which we address in Sec.\[sec:instab\], making the connection to a Bogoliubov-de Gennes linearization around the corresponding stationary solution of the GPE. We briefly conclude and point to further perspectives in Sec.\[conclu\] . Details of the perturbation theory are defered to appendix \[app\].
Setup and results from perturbation theory {#perturbth}
==========================================
We investigate a quasi-1D Bose-Einstein condensate of a single atomic species with repulsive interaction at zero temperature, described by the Gross-Pitaevskii equation [@Pethick_book]. We assume strong harmonic confinement in two spatial directions, taking the trapping potential to be of the form $V_{\text{3D}}(\mathbf r, t)=\frac{m \omega_\perp^2}{2}(y^2+z^2)+V(x,t)$, where $m$ denotes the atomic mass, $\omega_\perp$ the frequency of the transverse oscillator potential and $V(x,t)$ models a potential in the longitudinal direction. Assuming that the transverse dynamics is fully frozen out and thus transversally the condensate wave function remains in the oscillator ground state, one can integrate out the $y$- and $z$-directions and is left with the effectively 1D Gross-Pitaevskii equation [@book_kevrekidis] $$\begin{aligned}
-\dfrac{\hbar^2}{2m} \partial_{x}^2 \psi + V(x,t) \psi + g_{\text{1D}} |\psi|^2 \psi = i \hbar \partial_{t}\psi ,
\label{eq:GPEdimensions}\end{aligned}$$ where $\psi=\psi(x,t)$ denotes the longitudinal part of the wave function and the nonlinearity coefficient $g_{\text{1D}}=2\alpha\hbar \omega_{\perp}$ with the $s$-wave scattering length $\alpha>0$ [@Pethick_book]. Measuring length, time, energy and density $|\psi (x,t)|^2$ in units of $a_{\perp}=\sqrt{\hbar/m\omega_{\perp}}$, $\omega_{\perp}^{-1}$, $\hbar \omega_{\perp}$ and $(2\alpha)^{-1}$, respectively, Eq. (\[eq:GPEdimensions\]) is cast into the dimensionless form $$\begin{aligned}
-\dfrac{1}{2} \partial_{x}^2 \psi + V(x,t) \psi + |\psi|^2 \psi = i \partial_t \psi \label{dimlesstimedepGPE1}\end{aligned}$$ which we will work with in the following.
The corresponding stationary equation is obtained by factorizing $\psi(x,t)=\phi(x) \exp(-i\mu t)$ with $\mu$ the chemical potential. Our focus here will be on a longitudinal potential that consists of a static harmonic part (whose frequency $\omega_\parallel$ is small compared to that of the transverse trap) plus a Gaussian of fixed height and width, but moving in time, i.e. in dimensionless units $$\begin{aligned}
V(x,t)=\dfrac{1}{2} \Omega^2 x^2 + V_0 \exp \left[-\frac{1}{2} \frac{(x-x_G(t))^2}{\sigma ^2}\right] \label{potential}.\end{aligned}$$ Here, $x_G(t)$ specifies the trajectory of the Gaussian impurity, while $\sigma$ and $V_0$ set its width and amplitude, respectively. For convenience, the aspect ratio will be fixed to $\Omega^2=\omega_\parallel^2/\omega_\perp^2=0.04$ in the following. Our results can be transfered to other aspect ratios by a straightforward rescaling of Eqs. (\[dimlesstimedepGPE1\],\[potential\]). We simulate the time evolution of the condensate by propagating Eq. (\[dimlesstimedepGPE1\]) with a fourth-order Runge-Kutta integrator. The initial state is chosen to be the ground state at a given $\mu$, obtained from the stationary 1D-GPE by an adapted Newton method [@kelley].\
The main objective of this work is the controlled formation and dragging of a dark soliton. By analogy with previous work on vortices in 2D [@Davis2009], and the somewhat intuitive picture that a density dip such as the dark soliton may offer the possibility of pinning it with a narrow repulsive barrier in its center, one may conjecture that $V_0 > 0$ will be the favorable parameter regime for our purposes. It turns out, however, that this is not the case and an attractive Gaussian impurity with $V_0 < 0$ is much more adequate to drag along the soliton. This can be seen on the level of dark soliton perturbation theory [@Kivshar1994], resulting in an approximate particle-like equation of motion for the soliton center in the presence of a weak external potential. This has been worked out for a dark soliton perturbed by a $\delta$-shaped impurity potential in [@Frantzeskakis2002], with the result that the impurity is attractive (repulsive) to the soliton if it is attractive (repulsive) to the atoms in the condensate. We have extended this analysis to our impurities of Gaussian shape (see appendix \[app\] for the details) and find that the overall result persists: Dark solitons are effectively attracted by a Gaussian of $V_0 < 0$. Specifically, the soliton center $x_0$ follows the equation of motion $$\begin{aligned}
\dfrac{\mathrm{d}^2x_0}{\mathrm{d}t^2}=-\dfrac{\mathrm{d}W}{\mathrm{d}x_0},\end{aligned}$$ where the effective potential $W(x_0)$ is predicted from perturbation theory, see appendix \[app\] for the details. Fig. \[effpot\] shows the resulting effective potentials for varying widths of the Gaussian impurity (located at $x=0$), keeping the amplitude fixed. Clearly, a potential minimum for the soliton dynamics is found for all cases with $V_0 <0$, while it turns into an unstable maximum for $V_0 > 0$ (black circles in Fig. \[effpot\]). In the vicinity of the fixed point $x=0$, the effective potential is strongly shaped by the Gaussian, while further away from the center it asymptotes to $\Omega^2 x^2/4$, half the bare potential of the trap, yielding the characteristic soliton oscillation at $\Omega/\sqrt{2}$ as expected in the absence of the Gaussian [@PhysRevLett.84.2298; @DJF].
![[(Color online) Effective potentials $W(x_0)$ for the soliton position coordinate $x_0$ as obtained from soliton perturbation theory. In each case, the underlying atomic potential consists of a harmonic trap and a Gaussian of variable width $\sigma$ and fixed attractive amplitude $V_0=-1$ centered at $x=0$ (black circles for repulsive potential $V_0=1$), $\mu=1$ throughout.]{}[]{data-label="effpot"}](figure1.pdf){width="45.00000%"}
This perturbative treatment suggests that for dragging along a dark soliton with a Gaussian impurity, one should choose the impurity as attractive for the atoms. Consequently, we will focus on $V_0 < 0$ in the following. It should be noted that the parameters used in the rest of this work are mostly out of the range of validity of the soliton perturbation theory.
Generating, pinning and dragging a dark soliton {#sec:generation}
===============================================
In this section, we demonstrate the possibility to generate, pin and drag along a dark soliton with an attractive Gaussian impurity entering the BEC cloud from outside. An example of this is shown in Fig. \[gendrag\](a), displaying the spatio-temporal evolution of the atomic density under the influence of the impurity potential. The dimensionless chemical potential is chosen as $\mu=1$, corresponding for instance to a condensate of around 3300 sodium atoms under a transverse confinement of $\omega_\perp=2\pi\times 200$ Hz. For these parameters, the resulting healing length in the center of the cloud is close to one micron. The white line indicates the trajectory of the Gaussian that moves linearly into the BEC cloud towards $x=1$. After staying stationary at this point for a time interval of $\Delta t=10$, it is switched off. The generated soliton can be identified already at an early stage. When the impurity enters the cloud, the characteristic density minimum as well as the phase shift close to $\pi$ (see Fig. \[gendrag\](b)) are created almost immediately. The soliton follows the motion of the impurity and is dragged along towards $x=1$ where it is held for $\Delta t=10$. When the Gaussian potential is switched off, the soliton is released and starts to oscillate in the harmonic trap. On a perfect Thomas-Fermi background, the frequency of this solitonic oscillation is expected to be $\Omega/\sqrt{2}$ [@PhysRevLett.84.2298; @DJF]. In the present simulation, the Gaussian impurity also slightly excites the collective dipole mode, causing a center-of-mass oscillation of the entire cloud at the trap frequency $\Omega$. A two-sine fit to the soliton trajectory reveals a superposition between the particle-like soliton oscillation at $\Omega/\sqrt{2}$ and the collective dipole oscillation at $\Omega$, validating the dark soliton character of the created excitation.
![[(Color online) Generating, dragging and releasing a dark soliton. The white line indicates the trajectory of the Gaussian impurity (parameters $V_0=-12, \sigma=0.1$). At $t=129$ the impurity is switched off. (a) Density $|\psi(x,t)|^2$. (b) Snapshots of the phase angle profile at different times.]{}[]{data-label="gendrag"}](figure2a.pdf "fig:"){width="45.00000%"} ![[(Color online) Generating, dragging and releasing a dark soliton. The white line indicates the trajectory of the Gaussian impurity (parameters $V_0=-12, \sigma=0.1$). At $t=129$ the impurity is switched off. (a) Density $|\psi(x,t)|^2$. (b) Snapshots of the phase angle profile at different times.]{}[]{data-label="gendrag"}](figure2b.pdf "fig:"){width="45.00000%"}
We have confirmed in our simulations that the generation, immediate pinning and dragging of a dark soliton as shown in Fig. \[gendrag\] is successful in a wide range of parameters. Specifically, while the velocity of the impurity is relevant (see also below), the exact trajectory is not, and the process works equally well with curved trajectories of the impurity and shorter or longer hold times prior to release (see, however, the discussion in Sec. \[sec:instab\]).\
To develop a fidelity measure for the dark soliton generation process, we need to identify a scenario in which a precise definition of the ideal desired outcome can be given. While on a homogeneous background there is a notion of a perfect grey (moving) soliton, an extension of this accounting for the inhomogeneous density induced by the trap is available on an approximate level only. In contrast, the profile of a fully stationary (black) soliton in the trap can be obtained unambiguously by solving the stationary GPE. Moreover, it can be expected that at parameters, at which an undisturbed initialization of a black soliton is possible, an off-center release from the impurity and the subsequent acceleration due to the trap (similar as seen in Fig. \[gendrag\]) will produce a clean grey soliton. Thus, we proceed to quantitatively evaluate the success of the soliton generation process by focusing on the preparation of a black soliton in the trap center. To do so, we choose Gaussians of different parameters that enter the cloud at a given velocity and move towards $x=0$, are held there for a while and then turned off. For each of these runs, we compare the resulting final state to that of a stationary black soliton, our target state. This target state $\phi_\text{BS}(x)$ (with the same squared norm $N$ as the wave function in the simulation) is computed separately by numerically solving the time-independent GPE. Then, for any time $t$ after the potential has been turned off, we can calculate the overlap $$\begin{aligned}
S(t)=\dfrac{1}{N^2} \bigg\vert \int \text{d}x \,\phi_\text{BS}^*(x) \psi(x,t)\bigg\vert^2.\end{aligned}$$ This quantity is then averaged over a time-interval $[t_r,t_f]$, ranging from the release time $t_r$ (when the impurity is switched off) to a final time $t_f$: $$\begin{aligned}
\overline{S}=\dfrac{1}{t_f-t_r}\int_{t_r}^{t_f}\mathrm{d}t\,S(t)\label{overlapeq}.\end{aligned}$$ By the Cauchy-Schwarz inequality, $0\leq \overline S \leq 1$. If the dynamics caused by the impurity results in a perfect stationary black soliton, one has $|\psi(x,t)|=|\phi_\text{BS}(x)|$ for all $t>t_r$ (taking advantage of the fact that the target state is stationary), and thus $\overline S = 1$ would correspond to a perfect fidelity of the creation process. Smaller deviations from this indicate dynamics of the soliton and/or the bulk of the cloud after the impurity has been turned off, while $\overline S \ll 1$ suggests that the generation of a single dark soliton has completely failed. Fig. \[overlap\] shows results for the fidelity $\overline S$ as a function of width $\sigma$ and height $V_0 <0$ of the Gaussian impurity. The $1/e^2$ width of the Gaussian is given by $w=4 \sigma$ and ranges from $0.24$ to $1.2$ here. This is to be compared to the healing length in the center of the cloud given by $\xi \approx 0.7$. The impurity moves towards the trap center on a linear trajectory, as in Fig. \[gendrag\], at a velocity that is fixed to $v=0.0925$ in this set of simulations (measured in units of $a_\perp/\omega_\perp$). It is then suddenly stopped and held at $x=0$ for $\Delta t=10$, before being switched off. The quantity $\overline S$ is then obtained by averaging $S(t)$ from the subsequent dynamics over an interval $t_f - t_r = 481$.
![[ Black soliton generation fidelity $\overline{S}$ as a function of the potential strength $V_0$ and width $\sigma$. The Gaussian enters the cloud at a velocity $v=0.0925$, the initial chemical potential $\mu =1$. ]{}[]{data-label="overlap"}](figure3.pdf){width="45.00000%"}
\
Most notably, there is an extended parameter region of substantial fidelity $\overline S \gtrsim 0.9$. A small or intermediate potential width $\sigma$ together with a large or intermediate potential strength $|V_0|$ is applicable for the controlled generation, dragging and holding of a single dark soliton. An exemplary plot at the same parameter values as in Fig. \[gendrag\] (but now with the final position of the Gaussian at $x=0$) is shown in Fig. \[genex\](a) for the parameter set $(V_0=-12,\sigma=0.1)$. For this comparably narrow impurity, the final state is close to a stationary black soliton, but both the dipole mode and the soliton oscillation mode are slightly excited. In contrast, turning to the parameters $(V_0=-7,\sigma=0.28)$, the fidelity is roughly the same, $\overline S \approx 0.94$, but the deviations from the black soliton state are of a different kind, see Fig. \[genex\](b). Here, the soliton itself is closer to stationary than in Fig. \[genex\](a), but the background is excited at higher frequency modes and more disturbed by density waves. The comparison of Figs. \[genex\](a) and (b) illustrates that the fidelity $\overline S$ is sensitive to different types of remaining excitations around the target state (both particle-like oscillations of the soliton and collective oscillations in the bulk), and that one has some freedom in reducing either the particle-type soliton dynamics or the background excitations by tuning the parameters of the Gaussian. In both regions of parameter space, the dark soliton characteristics of the induced density minimum are clearly observed; specifically, we have checked the oscillation at $\Omega/\sqrt{2}$ for off-centered release. Finally, Fig. \[genex\](c) shows a simulation at parameters $(V_0=-8,\sigma=0.19)$ that yields a particularly large fidelity of $ \overline S \approx 0.98$. Here, the evolution is similar to that shown in Fig. \[genex\](b), but the background excitations are further suppressed.
![[(Color online) Density evolution $|\psi(x,t)|^2$ for some of the simulations underlying Fig. \[overlap\]. The parameters of the Gaussians are $(V_0, \sigma)= (-12,0.1)$ (a), $(-7,0.28)$ (b) and $(-8,0.19)$ (c), respectively]{}.[]{data-label="genex"}](figure4a.pdf "fig:"){width="45.00000%"} ![[(Color online) Density evolution $|\psi(x,t)|^2$ for some of the simulations underlying Fig. \[overlap\]. The parameters of the Gaussians are $(V_0, \sigma)= (-12,0.1)$ (a), $(-7,0.28)$ (b) and $(-8,0.19)$ (c), respectively]{}.[]{data-label="genex"}](figure4b.pdf "fig:"){width="45.00000%"} ![[(Color online) Density evolution $|\psi(x,t)|^2$ for some of the simulations underlying Fig. \[overlap\]. The parameters of the Gaussians are $(V_0, \sigma)= (-12,0.1)$ (a), $(-7,0.28)$ (b) and $(-8,0.19)$ (c), respectively]{}.[]{data-label="genex"}](figure4c.pdf "fig:"){width="45.00000%"}
So far, we have not addressed the role of the velocity at which the Gaussian defect is moved through the condensate. For the two parameter sets $(V_0, \sigma)=(-12,0.1), (-7,0.28)$, we have performed simulations for a range of defect velocities $v$. The resulting fidelity $\overline S$ for varying $v$ is shown in Fig. \[velocity\].
![(Color online) Black soliton creation fidelity as a function of the impurity velocity for two different sets of parameters of the Gaussian potential.[]{data-label="velocity"}](figure5.pdf){width="45.00000%"}
For both parameter sets, we find a robust plateau of large $\overline S$ at small $v$. In particular for the deep and narrow Gaussian with $(V_0, \sigma)=(-12,0.1)$ the fidelity is close to $1$ for $v \lesssim 0.08$. Increasing the velocity, for this parameter set a sharp drop of $\overline S$ is observed at $v \approx 0.16$. Beyond this velocity, in addition to a rather strong excitation of the background, further dark solitons are generated of which at most one is pinned by the Gaussian, which drastically reduces the overlap with the single black soliton state. This parameter regime could be of interest in its own right when moving the focus towards multiple soliton physics, but the high degree of control over the single soliton creation process is lost there. In contrast, a comparable critical drop in the fidelity is not observed for the Gaussian with $(V_0, \sigma)=(-7,0.28)$, where $\overline S$ remains relatively large for an extended range of $v$. For these parameters, going to even larger velocities than shown in Fig. \[velocity\] we observe a trend of decreasing $\overline S$, but caused by enhanced excitation of the background and the soliton oscillation mode, instead of multiple soliton formation.\
In this work, we do not aim to explore the full velocity dependence and the dynamical details of the nucleation process in which the dark soliton is created when the defect enters the cloud. Generally, the formation of excitations in a superfluid simultaneously exposed to a moving defect and a trapping potential (which leads to an inhomogeneous background density) is an interesting and timely subject to study in its own right, see for instance the recent experiment on vortex shedding in quasi-2D condensates [@Kwon2015]. Even on a homogeneous background, predicting the critical velocity above which a defect causes the creation of nonlinear excitations is an intricate problem (see for instance the extensive discussion of vortex nucleation in [@Barenghi2001]) that is subject to ongoing research [@Pinsker2014; @Kunimi2015]. Some related results on soliton formation in 1D (mostly focusing on repulsive defects) are available, e.g. [@Hakim1997; @Pavloff2002; @Radouani2004; @Kamchatnov2012], but in our setup additional complications due to the inhomogeneous background density traversed by the defect, the attractive sign and the comparably soft boundary of the Gaussian potential [@Kwon2015] and possibly also the 1D reduction [@Fedichev2001] will require separate investigations that are beyond the scope of the present study.
Instabilities for long hold times {#sec:instab}
=================================
In the above simulations, the soliton was created and dragged by the moving impurity, then placed at a desired position in the trap and held there for a comparably short time ($\Delta t=10$) before being released. Substantially extending this hold time reveals an additional effect. For certain parameter values we observe instability phenomena in the dynamics of the pinned soliton, see Fig. \[instab\]. We find that the soliton performs micro-oscillations around the Gaussian potential during the hold interval (similar oscillations are observed in all our simulations). Here, however, at $t \approx 140$ the amplitude of the micro-oscillations starts to increase strongly, before it decays again after $t \approx 210$. A similar increase and decrease is observed again at around $t\gtrsim 300$. We conclude from this that long hold times may give rise to undesired effects when aiming for a stationary soliton. Inadvertently releasing the soliton during a period of enhanced micro-oscillation will yield a comparably large momentum of its particle-like motion and correspondingly a large amplitude of its subsequent oscillations in the trap. Thus, the hold interval may crucially affect the final state of the soliton preparation process. Comparable dynamical instabilities of a dark soliton under the influence of a narrow external potential have been related to sound emission caused by the repeated asymmetric deformation of the oscillating soliton due to the impurity [@Parker2003a], see also [@Parker2003; @Yulin2003; @Proukakis2004; @Parker2004; @Theocharis2005a; @Allen2011].
![[(Color online) Controlled creation of a dark soliton, followed by a long hold time. The parameters of the Gaussian are $V_0=-12$, $\sigma=0.08$, $v=0.0925$ (before reaching $x=0$). (a) Density evolution $|\psi(x,t)|^2$. The piece-wise linear trajectory of the impurity is marked by a white line. The white box highlights the first region of dynamical instability. (b) Position of the density minimum (as a measure of the soliton center) as a function of time in the time interval marked by the box in (a).]{}[]{data-label="instab"}](figure6a.pdf "fig:"){width="45.00000%"} ![[(Color online) Controlled creation of a dark soliton, followed by a long hold time. The parameters of the Gaussian are $V_0=-12$, $\sigma=0.08$, $v=0.0925$ (before reaching $x=0$). (a) Density evolution $|\psi(x,t)|^2$. The piece-wise linear trajectory of the impurity is marked by a white line. The white box highlights the first region of dynamical instability. (b) Position of the density minimum (as a measure of the soliton center) as a function of time in the time interval marked by the box in (a).]{}[]{data-label="instab"}](figure6b.pdf "fig:"){width="45.00000%"}
To obtain further insights into the dynamical instabilities due to the Gaussian potential, we employ a linearization analysis. Let us assume that the state that originates from the Gaussian entering the cloud and moving towards $x=0$ is close to the corresponding stationary black soliton state (now in the presence of the Gaussian). Then, information about its stability is encoded in the Bogoliubov-de Gennes (BdG) excitation spectrum, obtained by adding a small deviation $\delta \psi(x,t)=e^{-i \mu t} \left[ u(x) e^{-i \omega t}+ v^* (x)e^{i \omega^* t}\right]$ to the stationary dark soliton state, linearizing the GPE in $\delta \psi$ and solving the ensuing eigenvalue problem for $\omega$ [@book_kevrekidis]. Here, $\mu$ denotes the chemical potential of the stationary solution. Frequencies having nonzero imaginary part indicate an instability of the intial state as they induce exponential growth of a generic small perturbation. Such complex modes may emerge from collisions of normal and anomalous modes (modes with positive or negative energy/Krein signature, respectively) [@MacKay; @book_kevrekidis]. Here, the BdG spectrum of the dark soliton state is expected to exhibit a single anomalous mode that is related to its particle-like motion [@DJF]. If this becomes resonant with one of the background modes as a parameter is tuned (for instance the width of the Gaussian), this may lead to instability of the state. Indeed, we observe these effects in the BdG spectrum of the black soliton state with a Gaussian potential placed in its center. We fix the norm of the wave function and the amplitude $V_0 = -12$ of the Gaussian to the same values as in Fig. \[instab\] and scan $\sigma$. The resulting spectrum as a function of $\sigma$ is shown in Fig. \[BdG\](a) (by the Hamiltonian symmetry, if $\omega$ is in the BdG spectrum, then so are $-\omega$, $\omega^*$ and $-\omega^*$, so we can restrict to positive real and imaginary parts in the figure). Here, even for moderately small $\sigma$ the large amplitude of the Gaussian strongly shifts the anomalous solitonic oscillation mode away from its value $\Omega/\sqrt{2}$ expected in the harmonic trap only. Increasing $\sigma$ leads to a further increase of the anomalous mode frequency, causing subsequent collisions with background modes that result in complex quartets, signalling oscillatory instability. The width $\sigma=0.08$, as used in the simulation of Fig. \[instab\] indeed lies at the edge of such a region of instability.
![[(Color online) BdG linearization spectra of the dark soliton state in the presence of a Gaussian potential in its center. All BdG frequencies are given in units of $\omega_\perp$. Blue crosses, red circles and green asterisks denote the normal, anomalous and complex modes, respectively. The chemical potential is fixed at $\mu=1.15$, yielding the same norm as the ground state at $\mu=1$. (a) $V_0=-12$, revealing a complex mode at $\sigma=0.08$, corresponding to the parameter set of Fig. \[instab\]. (b) $V_0 = -0.25$, including also the linearization frequency predicted from the soliton perturbation theory (black dots). ]{}[]{data-label="BdG"}](figure7a.pdf "fig:"){width="23.00000%"} ![[(Color online) BdG linearization spectra of the dark soliton state in the presence of a Gaussian potential in its center. All BdG frequencies are given in units of $\omega_\perp$. Blue crosses, red circles and green asterisks denote the normal, anomalous and complex modes, respectively. The chemical potential is fixed at $\mu=1.15$, yielding the same norm as the ground state at $\mu=1$. (a) $V_0=-12$, revealing a complex mode at $\sigma=0.08$, corresponding to the parameter set of Fig. \[instab\]. (b) $V_0 = -0.25$, including also the linearization frequency predicted from the soliton perturbation theory (black dots). ]{}[]{data-label="BdG"}](figure7b.pdf "fig:"){width="23.00000%"}
Let us at this point return to the perturbative regime of small $|V_0|$ that was discussed in Sec. \[perturbth\]. Numerically calculating the BdG spectrum as a function of $\sigma$ also reveals the emergence of complex instability bubbles in this regime, see Fig. \[BdG\](b) for an example. The solitonic perturbation theory does not account for background excitation modes and is not capable of predicting these instabilities. Interestingly, however, the linearization frequencies predicted from the perturbation theory (by linearizing the effective potential $W(x_0)$ around $x_0=0$) quite accurately capture the real part of the unstable BdG modes even in the instability regions. In regions of stability, the frequency from the perturbative approach is close to the anomalous mode in the BdG spectrum, as expected. As is to be expected, the agreement slightly deteriorates for large $\sigma$, where the overall perturbation due to the Gaussian effectively becomes stronger.
Perspectives and conclusions {#conclu}
============================
We theoretically investigated the possibility to controllably generate, drag, hold and release a dark soliton in a quasi-1D Bose-Einstein condensate using a Gaussian-shaped impurity as could be implemented with a focused laser beam. The time-dependent Gross-Pitaevskii equation containing the trap potential as well as the Gaussian impurity was propagated in time to obtain the spatio-temporal evolution of the condensate wave function when disturbed by the moving defect. On a perturbative level, we found that if the Gaussian is attractive (repulsive) to the atoms, then it is effectively attractive (repulsive) to the dark soliton as well, thus suggesting the use of an attractive impurity (red-detuned focused laser) for holding and dragging dark solitons, in contrast to the pinning of vortices at repulsive barriers.\
We demonstrated that by entering the atomic cloud with an attractive Gaussian one can create a single dark soliton that immediately sticks to the defect and can be controllably placed and released at a desired position in the condensate, showing the expected characteristics of a dark soliton after release. Detailed investigations revealed an extended range of model parameters (such as the width and amplitude of the Gaussian) for which this mechanism is successful, thereby underlining its robust nature. As a drawback, the width of the Gaussian must be relatively small, comparable in size to the soliton healing length, requiring a much tighter focus than in previous experiments [@Engels2007]. For instance, in the case of ${}^{23}\mathrm{Na}$ atoms and a transverse confinement of $\omega_\perp=2\pi\times 200$ Hz, a dimensionless value of $\sigma=0.2$ in our simulations (for which we observe particularly successful soliton generation and control) translates into a $1/e^2$ beam width of $w=4\sigma a_\perp \approx 1.2 \mu$m, close to the central healing length, while the wave function norm in our simulations translates into a relatively small number of around 3300 atoms. This is a challenging requirement, but not out of reach, given that optical systems with sub-micron resolution are already employed in present-day cold atom experiments [@Bakr2009; @Zimmermann2011].\
Moreover, our studies suggest that long stationary hold times of the defect are not favorable for the controlled generation of black solitons, due to the possibility of dynamical instabilities that may lead to a spontaneous increase of the micro-oscillation amplitude of the soliton around the Gaussian. This was related to corresponding complex modes arising in the Bogoliubov-de Gennes spectra, and linked to the linearization results from the perturbative approach.\
The protocol for simultaneous generation and holding of the dark soliton as described here is appealing, since capturing an existing soliton in a BEC cloud would be a much more difficult task. Even if a suitable pinning potential is available, catching the soliton requires information about its time-dependent position, which is hard to obtain given the destructive measurement schemes. Furthermore, we point out that if more than one laser beam is available, the soliton creation scheme described herein can immediately be cascaded to generate two or more dark solitons at predefined positions as shown in Fig. \[twosols\], cf. also [@Damski2001]. The two dark solitons are created and trapped by their respective defect potential, and when released they exhibit the expected particle-like collision dynamics, cf. [@Weller2008; @theocharis2010].\
![[(Color online) Generating two solitons with two impurity potentials moving independently on the trajectories indicated by white lines. Colors encode the density $|\psi(x,t)|^2$. For both Gaussian potentials, $V_0=-14$, $\sigma=0.09$.]{}[]{data-label="twosols"}](figure8.pdf){width="45.00000%"}
In this work, we have not addressed the details of the soliton nucleation process occuring in the low density wings of the cloud and its dependence on the defect velocity. This aspect of the problem promises to be an interesting topic for future studies. Also, it would be desirable to perform simulations of the soliton creation and dragging protocol in the framework of the full three-dimensional Gross-Pitaevskii equation, checking for possible transverse excitation effects that are not captured within the dimensional reduction.
J. S. gratefully acknowledges support from the Studienstiftung des deutschen Volkes. P. S. acknowledges financial support by the Deutsche Forschungsgemeinschaft via the contract Schm 885/26-1.
Soliton perturbation theory {#app}
===========================
In this appendix, we outline the derivation of the effective potential for the soliton in the harmonic trap, perturbed by a Gaussian impurity. We follow the presentation in [@Frantzeskakis2002], but generalize the Dirac $\delta$-potential used in that work to a Gaussian as in Eq. (\[potential\]). By the same arguments as in [@Frantzeskakis2002], the Thomas-Fermi-like background profile $u_b(x)$ of the condensate ground state is approximated by $u_b(x)=u_0+f_{\text{trap}}(x)+f_g(x)$, where $u_0$ is the maximum background amplitude ($u_0 = \sqrt{\mu}$ in the Thomas-Fermi limit considered here), $f_\text{trap}(x)$ accounts for the modified shape due to the harmonic trap and $f_g(x)$ incorporates the perturbation by the Gaussian impurity. Explicitly, $f_{\text{trap}}(x)=-\dfrac{1}{2u_0}V_{\text{trap}}(x)$ with $V_{\text{trap}}=\Omega^2 x^2/2$, as in [@Frantzeskakis2002], and $$\begin{aligned}
f_g(x)&= \notag \dfrac{V_0\sigma}{2}\sqrt{\dfrac{\pi}{2}} e^{2u_0^2\sigma^2} \\ \notag
&\times \left \{\left(-1+\text{erf}\left[\dfrac{\sigma}{\sqrt{2}}\left(2u_0+ \frac{x}{\sigma^2}\right)\right]\right)e^{2u_0x} \right. \\
& \,\, \left. + \left(-1-\text{erf}\left[\dfrac{\sigma}{\sqrt{2}}\left(-2u_0+ \frac{x}{\sigma^2}\right)\right]\right)e^{-2u_0x} \right \}.\end{aligned}$$ Here, we can recover the result of [@Frantzeskakis2002] by taking the Dirac limit $V_0=b/(\sqrt{2\pi}\sigma)$ and $\sigma \rightarrow 0$ for a fixed $b$, resulting in $f_g(x)= -(b/2)\exp(-2u_0|x|)$.\
With the generalized $u_b(x)$, we follow the further steps in [@Frantzeskakis2002]. The dynamics of the dark soliton on top of the Thomas-Fermi-like background is investigated with the Ansatz $$\begin{aligned}
\psi(x,t)=u_b(x)e^{-iu_0^2t}v(x,t) \label{ansatz}\end{aligned}$$ where $v(x,t)$ represents a dark soliton on this background. Inserting Eq. (\[ansatz\]) into the time-dependent GPE leads to a perturbed defocusing nonlinear Schrödinger equation for the soliton function $v$, with a perturbation term that depends on $f_\text{trap}$ and $f_g$. Making an Ansatz for $v$ in the form of a dark soliton solution of the defocusing nonlinear Schrödinger equation, but with its position and phase angle slowly varying in time, one can employ the adiabatic perturbation theory for dark solitons of [@Kivshar1994] to obtain the desired equation of motion of the soliton center $x_0 (t)$, which in our case reads as $$\begin{aligned}
\label{eq:effpot}
\dfrac{\mathrm{d^2}x_0}{\mathrm{d}t^2}&=\notag -\dfrac{1}{2}\dfrac{\mathrm{d}V_{\text{trap}}}{\mathrm{d}x}\bigg\vert_{x=x_0} \\
&+ u_0^3 e^{2u_0^2\sigma^2}\sqrt{\dfrac{\pi}{2}} \dfrac{V_0\sigma}{2}\int_{-\infty}^{\infty}\mathrm{d}x \left[ F_1(x) + F_2(x) \right] \notag \\
&=:- \dfrac{\mathrm{d}W}{\mathrm{d}x_0},\end{aligned}$$ where the integrands $$\begin{aligned}
F_1(x) &=\left(-1-\text{erf}\left[\dfrac{\sigma}{\sqrt{2}} \left(-2u_0+\frac{x}{\sigma^2}\right)\right] \right)e^{-2u_0x} \notag \\
& \times \left\{\tanh [u_0(x-x_0)]-1\right\} \text{sech}^4 \left[u_0(x-x_0)\right], \\
F_2(x) &=\left(-1+\text{erf}\left[\dfrac{\sigma}{\sqrt{2}} \left(2u_0+\frac{x}{\sigma^2}\right)\right] \right)e^{2u_0x} \notag \\
& \times \left\{\tanh [u_0(x-x_0)]+1\right\} \text{sech}^4 \left[u_0(x-x_0)\right],\end{aligned}$$ From Eq. \[eq:effpot\], we can numerically compute the effective potential $ W(x_0)$ (as shown in Fig. \[effpot\]) and the linearization frequency around its fixed point at $x_0 = 0$.
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author:
- 'Sabine Hossenfelder [^1]\'
title: At the Frontier of Knowledge
---
At any time, there are areas of science where we are standing at the frontier of knowledge, and can wonder whether we have reached a fundamental limit to human understanding. What is ultimately possible in physics? I will argue here that it is ultimately impossible to answer this question. For this, I will first distinguish three different reasons why the possibility of progress is doubted and offer examples for these cases. Based on this, one can then identify three reasons for why progress might indeed be impossible, and finally conclude that it is impossible to decide which case we are facing.
Doubt {#doubt .unnumbered}
=====
There are three different reasons why scientists question whether progress in a particular direction is possible at all.
- [**There exists a proof or no-go theorem for theoretical impossibility.**]{}
Modern theoretical physics is formulated in the language of mathematics, and consequently subject to mathematical proof. Such proof can be in the form of excluding particular scenarios due to inconsistency.
A basic example is that in Special Relativity it is not possible for massive particles to travel faster than the speed of light. Other examples are the Weinberg-Witten theorem that shows the incompatibility of massless gravitons with any Lorentz covariant renormalizable quantum field theory and with that constrains approaches to emergent gravity [@Weinberg:1980kq], or the no-go theorems on gravitational theories with more than one interacting metric tensor [@Boulanger:2000rq].
Physicists have a love-hate relationship with no-go theorems. They love them for the power to sort out possible options and reduce the space of theories that have to be considered. No-go theorems also clarify why a particular direction is not promising. Physicists hate no-go theorems for the same reasons.
- [**There exists an argument for practical impossibility.**]{}
Even though progress may not be impossible on theoretical grounds, it may be impossible for all practical purposes. An example may be testing quantum gravity. To present day we have no experimental evidence for quantum gravity. And as if that wasn’t depressing enough already, it has been shown [@grav] that even with a detector the size of Jupiter we would not be able to measure a single graviton if we waited the lifetime of the universe, and any improvement in the detector would let it collapse to a black hole. It is of little comfort that we could test particle scattering in the regime where quantum gravitational effects are expected to become important with a collider the size of the Milky Way.
Another example for questioning practical possibility is the emergence of structures on increasingly macroscopic levels. While most particle physicists believe in reductionism and would insist the atomic structures, molecule properties, and chemical reactions can in principle all be derived from the Standard Model of Particle Physics, we are far off achieving such a derivation. Even more glaring gaps arise on higher levels. Can one derive all of biology from fundamental physics? What about psychology? Sociology, anybody? A hardcore believer in reductionism will think it possible.
It has been shown in a specific setting that more really [*is*]{} different [@different] and a derivation of emergent from fundamental properties impossible even theoretically. This specific setting is an infinitely extended, and thus unphysical, system but nevertheless sharpens the question for practical possibility even for finite systems. This example is still under debate, but it might turn out to also represent a case in which for all practical purposes a derivation is impossible to achieve.
- [**Despite long efforts, no progress has been made.**]{}
This situation is one that seems to bother physicists today more than ever due to the lack of breakthroughs in fundamental physics that has lasted several decades now. This is even more frustrating since meanwhile the world around us seems to change in a faster pace every day.
As an example for doubt of this category may serve the understanding of quantum mechanics, in particular its measurement process and interpretation. “Shut up and calculate” is a still prevalent pragmatic approach frequently complemented by the attitude that there is nothing more to understand than what our present theories describe, and all questioning of the foundations of quantum mechanics is eventually nothing but a waste of time, or a pastime for philosophers, or maybe both amounts to the same.
Another example is instead of attempting to explain the parameters of the Standard Model to conjecture there is no explanation other than that we just happen to live in a part of the “multiverse” – a structure containing universes with all possible choices of parameters – in which the parameters are suitable for the existence of life. After all, if life wasn’t possible with the parameters we observe, then we wouldn’t be here to observe them. While this is an expression of doubt of category 3, it is not to say invoking such reasoning, known as the “anthropic principle,” is necessarily scientifically empty. We have observational evidence that our universe allows for the existence of life, and given a useful quantification of “existence of life,” the requirement of its possibility constrains the parameters in the standard model. The controversy remains though whether or not to give up searching for a more satisfactory explanation [@Smolin:2004yv] simply on the basis that none has been found for many decades.
Impossibilities {#impossibilities .unnumbered}
===============
The previous section categorized causes to suspect fundamental limits to our knowledge; the following categorizes actual reasons for impossibilities corresponding to the above mentioned three reasons of doubt.
- [**Impossible because the laws of Nature do not allow it.**]{}
That is D1 is indeed true. This implies D2 is also true.
- [**Possible in theory, but impossible in practice.**]{}
That is though D1 is not true, D2 is true.
- [**Possible both in theory and in practice, but not yet possible**]{}
Though progress is not excluded neither in theory nor practice, it might not be possible at a given time because theoretical knowledge is still missing, or necessary data has not yet been obtained. Scientific insight builds upon previous knowledge. Progress is thus gradual and can stagnate if an essential building block is missing.
Since impossibility of the type I3 can be overcome, we will not consider it to be a fundamental impossibility.
Though physicists do not usually include this in discussions about fundamental limits, any question for what is possible should take into account constraints set by the human brain. It is in our nature to overestimate the human capability to understand the world and to exert control about it. However, the capacity and ability of our brains is finite. It is limited in the processes it can perform, and it is limited in speed. There will thus be problems the human brain in its present form will not be able to solve at all, or not in a limited amount of time. And since solving a problem might be necessary to sustain an environment in which solving of problems can be pursued, a problem that cannot be solved in a limited amount of time might turn into one that cannot be solved at all.
This limit could be overcome with improvements of the human brain, either by evolution or design. It is far from clear though whether such an improvement can be limitless. Though not in the realm of physics, the possibility of enhancements of human cognition is another question at the frontier of knowledge to which we presently have no answer. The above cases I2 and I3 both should be understood as including this potential limit to human ingenuity: An experiment that we cannot think of cannot be realized, and a problem whose solution takes more time than has passed will not yet be solved.
Where are we? {#where-are-we .unnumbered}
=============
Let us now investigate whether from any of the three doubts one can conclude a fundamental impossibility of type I1 or I2.
First, we note that proofs are only about the mathematical properties of certain objects in their assumptions. A physical theory that describes the real world necessarily also needs a connection between these mathematical objects and the corresponding objects of the real world. While evidence might be abundant that a particular mathematical description of reality is excellent, it can never be verified and shown to be true. Consequently, it is impossible to know whether a particular mathematical representation is indeed a true description of reality, and it cannot be concluded a mathematical proof based on it must necessarily apply to the physical world. We can thus never know whether D1 is caused by an actual fundamental impossibility I1.
Another way to put this is that no proof is ever better than its assumptions. A loophole in the Coleman-Mandula theorem [@Coleman:1967ad] feeds today a huge paper-producing industry called Supersymmetry, and bi-metric theories may be non-interacting [@Hossenfelder:2009ne].
Turning to doubt D2, no argument for practical impossibility can be obtained without a theoretical basis quantifying this practicability. Since the theoretical basis can never be verified, neither can the practical impossibility. Thus, I2 cannot be followed from D2, and neither of the both fundamental impossibilities can ever be identified with certainty.
Coming back to our earlier D2 examples, despite all ridicule about “Chaoplexity” [@plex] scientists still investigate emergence in complex systems with the hope to achieve a coherent understanding, and during the last decade an increasing amount of tests of quantum gravity has been proposed. These proposals have in common a modification in the assumptions that lead to the conclusion quantum gravity might be practically untestable. Two scenarios that have obtained particularly much attention are higher dimensional gravity, in which case quantum gravity might become testable at the Large Hadron Collider [@Landsberg:2008ax], and deviations from Lorentz invariance resulting in modified dispersion relations [@AmelinoCamelia:2008qg], potentially observable in gamma ray bursts [@AmelinoCamelia:2009pg]. Depending on your attitude you might call these studies interesting or a folly, but what they are for certain is possibilities.
Finally, let us discuss doubt D3. If we assume knowledge discovery is pursued as a desirable activity then doubt D3 is equivalent to impossibility I3, since in this case the only reason for lacking progress can be that it has not been possible. With hindsight one often wonders why a particular conclusion was not drawn earlier, even though the pieces were all there already. But since we included limitations of the human brain, slow insights represent imperfections in scientists’ thought processes that are part of I3. So long as increasing the understanding of Nature continues to be part of our societies’ pursuits, it can then never be concluded from D3 that an impossibility is fundamental.
What we can thus state with certainty at any time is merely “To our best current knowledge...” To our best current knowledge it is not possible to travel faster than the speed of light. To our best current knowledge we cannot see beyond the black hole horizon. To our best current knowledge the measurement process in quantum mechanics is non-deterministic.
It remains to be said however that progress on fundamental questions becomes impossible indeed if we do not pursue it. And one reason for not pursuing it would be the mistaken conviction that it is impossible.
Scientific progress is driven by curiosity, and the desire to contribute a piece to mankind’s increasing body of knowledge. It lives from creativity, from stubbornness, and from hope. What I have shown here is that there is always reason to hope.
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[^1]: hossi@nordita.org
|
---
abstract: 'We carry out 2-D high resolution numerical simulations of type I planet migration with different disk viscosities. We find that the planet migration is strongly dependent on disk viscosities. Two kinds of density wave damping mechanisms are discussed. Accordingly, the angular momentum transport can be either viscosity dominated or shock dominated, depending on the disk viscosities. The long term migration behavior is different as well. Influences of the Rossby vortex instability on planet migration are also discussed. In addition, we investigate very weak shock generation in inviscid disks by small mass planets and compare the results with prior analytic results.'
author:
- 'C. Yu, H. Li, S. Li, S. H. Lubow, D.N.C. Lin'
title: 'TYPE I PLANET MIGRATION IN NEARLY LAMINAR DISKS – LONG TERM BEHAVIOR'
---
INTRODUCTION
============
The discovery of close orbiting extrasolar planets led to extensive studies of disk planet interactions and the forms of migration that can explain their location. Early theoretical work established the so-called type I and type II migration regimes for low mass embedded planets and high mass gap forming planets (Goldreich & Tremaine 1980; Lin & Papaloizou 1986; Ward 1997), respectively. Although it is suggested that migration is necessary to account for the observed distribution of planets (Ida & Lin 2008), the problem is that analytic theories and numerical simulations have shown that migration timescales of type I are quite short (Tanaka et al. 2002) so that the planet tends to migrate to its central star before it has time to become massive enough to open a gap in the disk. This problem thus becomes a competition between two timescales: type I migration and core accretion for planet mass growth (Pollack et al. 1996; Hubickyj et al. 2005). Several mechanisms have been suggested to address this challenging problem, which include thermal effects of the disk (Jang-Condell & Sasselov 2004), radial opacity jump (Menou & Goodman 2004), magnetic turbulent fluctuations (Nelson & Papaloizou 2004) and effects of co-orbital material (Masset et al. 2006). Non-isothermal slowing down of type I migration is studied by Paardekooper & Mellema (2006), Baruteau & Masset (2008), and Kley et al. (2009).
Recently, Li et al. (2009; hereafter Paper I) found that the low mass planet migration can have a strong dependence on the disk viscosity. They found that the type I migration is halted in disks of sufficiently low viscosity. This is caused by a density feedback effect which results in a mass redistribution around the planet. The simulations confirm the existence of a critical mass ($M_{cr} \sim 10 M_{\oplus}$) beyond which migration is halted in nearly laminar disks. The critical masses are in good agreement with the analytic model of Rafikov (2002).
This paper is a follow-up study to Paper I. By performing a series of high resolution, 2-D hydrodynamic simulations, we present a more detailed analysis on the density feedback effect, and describe the long term ($> 10^4$ orbits) behavior of migration. The paper is organized as follows. In §2 we give a brief description of our simulations. In §3 we discuss the density wave damping mechanism for different disk viscosities and the consequent long term migration behavior, including the density feedback and the Rossby vortex instability (RVI). Possible 3-D effects are discussed in §4. Summary and discussions are given in §5. A study on the shock excitation in inviscid disks is given in the Appendix.
Simulations
===========
The 2-D hydrodynamic simulation set-up and the numerical methods we used here are the same as that in Paper I (more details on the code are given in Li & Li 2009). We choose an initial surface density profile normalized to the minimum mass solar nebulae model (Hayashi 1981) as $\Sigma(r) = 152 f (r/5 AU)^{-3/2} \rm{gm \ cm}^{-2}$, where $f=1$ in this paper. (The migration dependence on $f$ in the low viscosity limit has been explored in Paper I so we will not vary $f$ in this study.) The disk is assumed to be isothermal throughout the simulated region, having a constant sound speed $c_s$. The dimensionless disk thickness $c_s/v_{\phi}(r=r_p)$ is set as $0.035$, where $v_{\phi}$ is the Keplerian velocity at the initial planet location $r_p$. (Simulations with higher $c_s$ were given in Paper I.) The dimensionless kinematic viscosity $\nu$ (normalized by $\Omega^2 r$ at the planet’s initial orbital radius) is taken as a spatial constant and ranges from $0$ to $10^{-6}$. This corresponds to an effective viscous $\alpha = 1.5\nu/h^2 = 1.2\times 10^3 \nu$. For most runs, we have chosen the planet mass to be $10 M_{\oplus}$. The planet’s Hill radius is $r_H =0.0215 r_p$, which is $\sim 0.6 h$. A pseudo-3D softening is used (Li et al. 2005). Fully 2-D disk-self gravity is included (Li, Buoni, & Li 2009). The disk is simulated with $0.4 \leq r \leq 2$. Runs are made typically using a radial and azimuthal grid of $(n_r\times n_\phi) = 800\times
3200$, though we have used higher resolution to ensure convergence on some runs. Simulations typically last more than ten thousand orbits so that we can study the long term behavior of migration.
Results
=======
Figure \[09v\_runs\] shows the orbital radius evolution of a $10
M_{\oplus}$ protoplanet in a 2-D laminar disk with different disk viscosities. When the disk viscosity is relatively large ($\nu = 10^{-6}$ or $\alpha = 1.2\times 10^{-3}$), the migration rate agrees well with the theoretical results given by Tanaka et al. (2002) for type I migration. When the disk viscosity is low, the migration behavior differs markedly from the usual type I migration (see also Fig. 1 in Paper I). Such slowing down behavior was explained in terms of the density feedback effect (Ward 1997; Rafikov 2002) in Paper I. Here, we have extended the evolution to be about ten times longer ($> 10^4$ orbits) than those in Paper I. But before we discuss the long term behavior in detail, we present some additional analysis of the density feedback effect first.
Density Wave Damping Mechanism for Different Viscosities
--------------------------------------------------------
The critical physics issue in deciding the density feedback effect is the density wave damping mechanism. Where and how the density waves generated by the protoplanet damp will contribute critically to the torque on the planet. Furthermore, such damping process will modify the density distribution around the planet, which directly affects the torque as well. This effect was partially analyzed in Paper I. In principle, the density wave can damp both due to disk viscosity (a viscous process) and by shocks (a nonlinear process). The relative importance of these processes will naturally depend on the disk viscosity.
To quantify the damping process, we have evaluated the Reynolds stress and viscous stress. An effective, azimuthally averaged $\alpha$ based on the Reynolds stress can be defined as: $$\alpha_{Rey}= \ \left\langle\frac{\Sigma v_r
\delta v_{\phi}}{P}\right\rangle~~,$$ where $\langle ... \rangle$ indicates the azimuthal average, $\delta v_{\phi} =
v_{\phi} - \langle v_{\phi}\rangle$, $\Sigma$ and $P$ are disk surface density and pressure, respectively. This method was previously discussed in Balbus & Hawley (1998) and Li et al. (2001). Similarly, the azimuthally averaged $\alpha$ based on the viscous stress can be defined as: $$\alpha_{vis}= \ \left\langle\frac{\nu \Sigma r
\frac{d\Omega}{dr}} {P}\right\rangle~~,$$ and this quantity scales as $r^{-3/2}$.
In Figure \[reynoldsviscous\], we present both $\alpha_{Rey}$ and $\alpha_{vis}$ as a function of disk radius. The results are based on the runs at 400, 1000, 760 and 700 orbits for $\nu = 10^{-6}, 10^{-7}, 10^{-8}$ and $0$, respectively. These times are chosen so that the planet is at roughly the same orbital radius in all the runs. For $\nu=10^{-6}$, the viscous transport is much bigger than the Reynolds transport (by more than a factor of 2 around the planet). For $\nu=10^{-7}$, $ \alpha_{vis} \approx 1.2\times 10^{-4}$, which is smaller than $\alpha_{Rey}$ around the planet. This means that the dominant wave damping mechanism around the planet changes from being viscous damping to being shock-dominated damping when the disk viscosity changes from $10^{-6}$ to $10^{-7}$. For even smaller disk viscosity, shocks dominate the wave damping. The peaks of $\alpha_{Rey}$ are approximately $h (=1.6 r_H)$ away from the planet, consistent with the excitation of shocks. Note that as disk viscosity changes, the shock strength and structure will be changed somewhat. This could account for the changes in $\alpha_{Rey}$ for $\nu \leq 10^{-7}$.
The wave damping by shocks causes the density profiles at the shocks to be significantly modified. To confirm this effect further, we analyze the torque density profiles by examining $dT/dM(r)$ where $T$ is the torque on the planet by disk material and $M$ is the mass within each radial ring. We choose three runs with $\nu = 10^{-6}, 10^{-7},
10^{-8}$ and pick the planet radial location at $r_p = 0.945$ to compare (this corresponds to $t=400, 1000,$ and $760$ orbits for these runs respectively). From Figure \[09v\_runs\], their migration trend at this location is quite different (i.e., the total torque on the planet is very different). The $dT/dM(r)$ profiles are given in the top panel of Figure \[dTdMcmp09v1v2v3\]. For different viscosities, the difference in $dT/dM$ is not large, within a factor of 2. But the torque amplitude on the planet in the $\nu=10^{-6}$ case is about a factor of 100 larger than that of the two cases for $\nu = 10^{-7}$ and $10^{-8}$. This shows that the difference should be caused by the density variations. The bottom panel of Figure \[dTdMcmp09v1v2v3\] shows the radial disk density profile around the planet. The density imbalance between the inner Lindblad and outer Lindblad regions are much stronger for the lower viscosity cases than the case for $\nu = 10^{-6}$.
In the usual picture of wave damping, as the viscosity decreases, the density waves are expected to propagate farther away (Takeuchi et al. 1996). As a result, the peak positions of $dT/dM$ are expected to be farther away from the planet. We did not find such behavior in the simulation results because the shock dissipation dominates the wave damping when the viscosity is sufficiently small. This also indicates that, for these choices of planet mass and disk sound speed, shocks are always produced.
We want to emphasize that, even though the effective “viscosity” caused by shocks is not high (see Figure \[reynoldsviscous\]), it is the density imbalance due to the angular momentum transport by the shocks that causes a big change in the total torque on the planet. For planet’s mass above the critical values described in Rafikov (2002) and Paper I, the above analysis indicates that there exists a critical disk viscosity, below which the density wave damping will be dominated by the shock dissipation and the density feedback effect can slow down (or halt) the migration. For planet mass less than the critical values, however, even when the disk viscosity is low enough so that the shock dissipation is dominant, the density imbalance caused by the shock dissipation is too weak or taking too long to be able to change the migration behavior.
Long Term Evolution
-------------------
The long term evolution ($\sim 10^4$ orbits) for different disk viscosity (Figure \[09v\_runs\]) is complicated. For three low viscosity cases, $\nu = 10^{-7}, 10^{-8}$, and $0$, the migration is significantly slowed down or even reversed, but the detailed behavior is different. (Note that for $\nu=0$, there is still some low level of numerical viscosity, which we estimate to be roughly equivalent to $\nu < 10^{-9}$.) We now discuss each case in detail.
### The $\nu = 10^{-7}$ case
For $\nu = 10^{-7}$, Figure \[09v\_runs\] shows that the planet has a steady migration rate. Figure \[vistimescale\] shows the comparison between the simulation and the viscous drift rate calculated using ${\dot r} = 3\nu/2 r_p$. It looks like that the density feedback effects take the planet migration into a “viscous” limit, where the migration is consistent with being on the viscous time scale after about 2500 orbits. The corresponding surface density profile evolution is shown in Figure \[denevol1e-7\]. It can be verified that the density distribution remains smooth and evolves on the viscous timescale as well. This implies that the shock damping of the density waves causes the planet and surrounding disk material to migrate with approximately the same timescale. This situation is similar to the previous type II migration study where a gap has formed in the disk. Upon more detailed analysis, however, the accretion rate throughout the disk is not quite a constant. This suggests that the steady migration observed so far could change if we follow it to even longer timescales.
Figure \[denevol1e-7\] shows that a wide density “depression” (not quite a gap) is forming. Given the wide gap, one might have expected the excitation of the secondary instability such as the Rossby vortex instability (RVI), but this instability is suppressed in this case by the disk viscosity.
### The $\nu = 10^{-8}$ case
For $\nu = 10^{-8}$, the planet’s migration is essentially halted and gradually going in the reverse direction at late stages. Figure \[den1e-8\] shows the density distribution at $t=800$ orbits. The density “depression” is steeper than what was seen in the $\nu =
10^{-7}$ case. For such a low viscosity, the RVI is also excited at a low level. Figure \[den1e-8\] shows that the azimuthal density variation is more pronounced in the low azimuthal wave number $m$. This is because, during the nonlinear stage of RVI, vortices will merge (Li et al. 2005), and one is often left with only large scale variations in disk surface density. (More detailed discussions on RVI will be given below.)
Because of the RVI, the behavior of $dT/dM$ becomes more complicated. This is shown in Figure \[dTdmcmp09v3\] where we have plotted the evolution of $dT/dM$ over a period between $\sim 3000 - 3400$ orbits. This coincides with a period when the RVI is mildly excited (see Figure \[09v\_runs\]). The peaks around $\pm h$ in $dT/dM$ are still consistent with the shock damping. The drastic changes around $\pm (2-3) h$ are due to the azimuthal asymmetries in surface density caused by RVI, which give rise to the sign change in $dT/dM$. We have confirmed that the Lindblad resonance positions for these low $m$ modes are coincident with the positions where $dT/dM$ change dramatically around $r-r_p = \pm 2.5 h$, as shown in Figure \[dTdmcmp09v3\].
When averaged over a few hundred orbits, however, the changes in $dT/dM$ cancel out as evident in Figure \[dTdMcmp09v1v3\]. The averaged profile, when compared with that from the $\nu =10^{-6}$ case, shows that RVI causes the torque contribution to extend to a larger radial extent (the tails between $\pm (2.5-4) h$), though the peak amplitudes of $dT/dM$ at $\pm
h$ are smaller by about a factor of 4. This implies that the excitation of RVI has a minor impact on the overall migration in this case.
It is not clear why the migration is slowly going outward, nor whether this trend will continue at much longer times than what was simulated. This is a regime where both the density feedback effects by shock dissipation and the influence by mild RVI are playing some roles in planet migration. Though it seems reasonable to expect that the planet migration is significantly slowed down when compared to the usual type I rate, it is difficult to get a definite answer.
### The $\nu = 0$ case
For $\nu = 0$, the planet migrates in a more complex way, now strongly influenced by the RVI. Large amplitude oscillations in the semi-major axis evolution appear and sometimes exhibit rapid radial drops. Figure \[vort\] presents several snapshots of the disk surface density, showing the evolution of RVI. The vortices exert strong torques to the planet as they move past the planet. It seems that the planet’s migration is still inward overall, though it undergoes many oscillations, reversals, and fast drops (see the black curve in Figure 1). Several factors could have contributed to this type of evolution. First, the low disk viscosity makes the shocks stronger (cf. Figure \[reynoldsviscous\]), causing a stronger disk response and faster disk density evolution. The excitation of RVI is associated with the inflexion points (which are regions of density depressions) in the radial profile of potential vorticity (Lovelace et al. 1999; Li et al. 2000). Second, these vortices tend to have slightly different azimuthal speeds so they will merge (Li et al. 2001; 2005), forming large scale density structures azimuthally. Third, they are anti-cyclones with high densities, so they produce their own spiral shocks around them. Their influence on the surrounding flow and the existence of spiral shocks lead to an effective angular momentum transport (mostly outward) so these vortices will gradually migrate inwards (see results also in Li et al. 2001). This is seen in the current simulations as well. Fourth, when the vortices migrate away inwardly (on a relatively fast timescale), the planet migration is subsequently affected because the disk density profile is significantly changed by these vortices. Fifth, because the shocks by the planet is strong, new generations of vortices are produced after the previous generation has migrated away. All these highly non-linear effects, unfortunately, make it very difficult to predict the behavior of planet migration.
One curious observation is the fast radial drop during the planet migration, as indicated, say, between $t = 6400$ and $8000$ orbits (cf. Figure \[09v\_runs\]). In Figure \[rho-turn398\], we show the density distribution at the time of a rapid drop (7960 orbits). It is interesting to see that the drop is coincident with this close encounter between the planet and the density blob. The mass of the dense blob is estimated to be the same order of the planet, about $3\times 10^{-5}$ or $10 M_{\oplus}$. The $dT/dM$ profiles at the time around the rapid drop are shown in Figure \[dTdMcmp397-399\]. We can identify that at the time of 7960 orbits, a huge negative torque occurs and could contribute to the rapid drop of the planet.
Possible 3-D Effects {#sec:3-D}
====================
Our results indicate that the disk gas density distribution near the planet sensitively controls migration. Migration stoppage in low viscosity disks is a consequence of a systematic mild redistribution of gas mass near the planet, favoring outward over inward torques. It does not require complete removal of gas near the planet, as in the type 2 regime (Li et al 2009). The redistribution is in turn controlled by shocks. The location and structure of these shocks have an important influence on the feedback torque on the planet. The nature of the shocks that occur in 3-D can be quite different from the 2-D case analyzed in this paper. In a 2-D isothermal disk with pressure, only one type of wave is excited, a rotationally modified acoustic wave. In a 3-D disk that is not vertically isothermal and/or has a nonzero vertical buoyancy frequency, this wave is modified and other types of waves may be excited (Lubow & Pringle 1993; Korycansky & Pringle 1995). Their damping properties differ from the 2-D case. If the disk is not vertically isothermal, as suggested by steady-state models of dead zones (e.g., Terquem 2008), then the main wave that is excited, the $f$ mode, becomes more confined near the disk surface as it propagates, through ’wave channelling’ (Lubow & Ogilvie 1998; Bate et al 2002). The wave becomes more nonlinear as it propagates and undergoes more rapid shock damping than in the 2-D case. Since the material that gets shocked lies above the disk midplane, it is not clear how effective the breaking surface waves will be in affecting migration in comparison to the isothermal case. But, the rate of change of disk angular momentum produced by waves is determined by the angular momentum flux they carry. For given disk surface density near a resonance, the $f$ mode carries about the same amount of angular momentum flux as the 2-D acoustic mode. So if the $f$ mode damps closer to the planet than the 2-D acoustic mode, then its effects on the migration torque could be more important. It is possible that the upper layers are successively shocked from the outside-in towards the midplane and displaced radially. The process may become less effective, as the remaining gas becomes less optically thick and more isothermal. These suggestions are speculative. Further analysis is required to determine the importance of the $f$ mode effects on migration.
Modes other than the $f$ mode that are excited in a 3-D disk can damp rapidly. For a vertically isothermal disk undergoing adiabatic wave perturbations, the fraction of the wave energy that goes into these alternative modes is given by $1-\sqrt{\gamma(2-\gamma)}$ \[see Eq. (B4) in Bate et al 2002\]. For $\gamma = 1.4$, this fraction is only about $8\%$. It is possible that the damping of these waves may produce a feedback torque that is more significant in strength than $8\%$ of the total feedback torque. The reason is that the wave damping will likely occur closer to the planet than the 2-D mode investigated in this paper. For example, vertically propagating gravity waves are produced that damp in the disk atmosphere. For a $\gamma = 5/3$ gas, the ratio is $25\%$ and the damping effects of these modes are more important. The damping of these waves occurs well above the disk midplane and it is not clear how much the feedback torque on the planet is modified. As discussed above, the disk may be affected from the outside-in, towards the midplane. A proper 3D analysis of the disk evolution in the low viscosity case is required.
Summary and Discussion {#sec:diss}
======================
We have carried out 2-D global hydrodynamic simulations to study the migration of a $10 M_{\oplus}$ protoplanet in a protoplanetary disk. The disk surface density is taken to have the same value in the minimum mass solar nebula model, but we have taken the normalized disk sound speed to be relatively low, $c_s = 0.035$. In Paper I, we have shown the existence and the concrete values for critical planet masses (depending on the disk mass and sound speed) above which the density feedback effects will slow down the type I migration significantly. Here, we have mainly focused on the long term behavior of planet migration in such low viscosity disks. We find the following results:
1\) When the disk viscosity is high (e.g., $\nu \geq 10^{-6}$, or $\alpha
\geq 10^{-3}$), the density wave damping is dominated by the disk viscosity. The migration can be described as the typical type I migration.
2\) When the viscosity is relatively low (e.g., $\nu$ is between $\sim
10^{-8}$ and $10^{-6}$, or $\alpha
\sim 10^{-5}$ and $10^{-3}$), the density wave damping is dominated by shocks. This then modifies the disk surface density profile quite significantly, which produces a density feedback effect that alters the planet migration, slowing it down into a viscous time scale or halting the migration altogether. The new migration timescale, $t
\geq 1/\nu \sim 10^6$ orbits, is considerably longer than the usual type I migration time. This range of the disk $\alpha$ is interestingly consistent with the expected values in the “dead zone” of protoplanetary disks where protoplanet cores are believed to arise. If the cores of protoplanets can manage to grow above the critical masses (as given in Paper I) without migrating away, then these cores can spend a long time in the dead zone (essentially the disk lifetime).
3\) When the disk viscosity is even lower (e.g., $\alpha < 10^{-5}$), the density feedback effect is still present but the RVI starts to dominate the nonlinear evolution of the disk. The planet migration is severely affected by the RVI. Large amplitude oscillations appear in the planet semi-major axis evolution and the rapid drop of the planet occurs sometimes as RVI-induced density blob experiences close encounters with the planet. The overall migration seems still inward and becomes unpredictable. It is not quite clear whether realistic disks will ever have such low viscosities.
We have only studied the long term migration behavior of a $10
M_{\oplus}$ protoplanet. This mass is above the critical mass limit discussed in Paper I. For lower planet masses, however, even for low viscosity disks, the shocks produced by the planet will tend to be weak, so the density feedback effects discussed in this paper and Paper I will not be strong enough to slow down the migration significantly. In this limit, the Type I migration still poses a serious threat to the survivability of these small mass protoplanets (say, $< 3 M_{\oplus}$), if no other mechanisms can stop the migration.
The critical masses for stopping planet migration are sensitive to the disk interior temperature. Dead zones may have higher temperatures than assumed here. For a steady state disk, the surface density varies inversely with $\nu$. The higher surface density, due to the lower $\nu$ in a dead zone, gives rise to a higher optical depth and therefore higher temperature at the disk midplane (e.g., Terquem 2008). If the disk temperatures reach a value corresponding to $H/r \ga 0.1$, then the critical masses can become substantially higher than determined here and the effects of the feedback effect on planet migration become much less important. In addition, realistic 3-D simulations are certainly desirable to address how layered vertical structures (with both magnetically active and less-active regions) will affect the wave damping and planet migration.
The research at LANL is supported by a Laboratory Directed Research and Development program. C.Y. thanks the support from National Natural Science Foundation of China (NSFC, 10703012) and Western Light Young Scholar Program. S.L. acknowledges support from NASA Origins grant NNX07AI72G.
Shock Damping for Low Mass Proto-Planets in Inviscid Disks
==========================================================
In this paper, we have studied the density wave damping by shocks when the disk viscosity is low and how this damping affects the disk density evolution and planet migration over $10^4$ orbits. The existence of the shocks, however, deserves further analysis. Prior studies (e.g., Goodman & Rafikov 2001) have suggested that shocks will always be produced in an inviscid disk, even for very small planet masses. Other nonlinear studies (e.g., Korycansky & Papaloizou 1996) have examined similar issues, pointing out the importance of parameter ${\it M} = r_H/h = (\mu/3)^{1/3}/h$, where $\mu$ is ratio of planet mass to the central star. (Their definition omitted the factor of 3.) For $c_s=0.035 -
0.05$, the planet masses considered in Korycansky & Papaloizou (1996) are all larger than $10 M_{\oplus}$, which is above the critical mass limit shown in Li et al. (2009). This means that shocks should always be produced in an inviscid disk for this planet mass. Here, we present some numerical results using 2-D global hydrodynamic simulations, extending the planet mass to values smaller than one Earth mass. To isolate the effects of shock production, we place the planet on a fixed circular orbit and turn off the disk self-gravity.
Such simulations are numerically challenging because the simulations have to capture and resolve very weak shocks. We have used an initial disk surface density profile that goes as $r^{-3/2}$ and the disk is isothermal and has a constant $c_s$ so that the initial disk potential vorticity (PV) profile is nearly flat (the weak radial dependence of PV from the pressure gradient is fully captured). We then monitor the changes in PV. Figure \[shockresolution\] shows the azimuthally averaged radial PV profile around a protoplanet of $0.5 M_{\oplus}$. The normalized disk scale height is $h=0.035$. This gives ${\it M} = r_H/h = 0.23$. We used a constant softening distance of $\epsilon = 1.0 h$.
In an ideal flow, PV should be conserved. There are (at least) two non-ideal regions in our simulations. One is the co-orbital region where the planet mass is introduced. This process is necessarily non-ideal, no matter how slowly the planet mass is introduced. This will produce changes in PV. This is because, when the planet is introduced, the flow lines around the planet change from being non-interacting to horseshoe orbits. This non-ideal process could produce changes in PV. Judging from Figure \[shockresolution\], this process affects the PV in a region spreading over $\sim \pm 0.4 h$. The other region is associated with the density wave propagation where the waves could steepen into weak shocks (e.g., Goodman & Rafikov 2001). We believe this is represented by the PV changes at $r \geq 3.2 h$ and $r \leq -
2.7 h$ using the highest resolution run. It is difficult to determine the exact values of the starting locations of the shocks. Based on Figure \[shockresolution\], we identify $r = 3-3.2 h$ and $r=-(2.6-2.7) h$ as the shock starting positions at the right and left side of the planet, respectively. Note that on both sides of the planet, there exists an “ideal” flow region where the PV remains largely unchanged ($\sim 0.4 - 2.0 h$). Even with these high resolutions, the simulations have not completely converged, though the shock locations are roughly consistent among the two highest resolution runs. (Note that for higher planet masses, such as $10 M_{\oplus}$, the shocks are much stronger and the convergence can be achieved with these resolutions.)
It is interesting to see that even for ${\it M} = r_H/h$ as low as $0.23$, shocks are clearly produced. We have also found that the shock location and strength depend on the softening distance we use. This is not surprising because larger softening distance will weaken the strength of wave excitation. Since we do not really know what is the most appropriate softening to use in a 2-D simulation, we have tried three values, $\epsilon = 0.6, 0.8$ and $1.0 h$. Figure \[shocksoftening\] shows the PV profile produced by a $0.5
M_{\oplus}$ planet in a disk with $h=0.035$. The time is 146 orbits. The shock gets stronger when the softening is smaller, as indicated by the magnitude in PV. In addition, the starting location of the shock moves further away from the planet when the softening distance increases, as indicated by the PV profiles to the right of the planet.
We have also tried to make quantitative comparison between our simulations and the results by Goodman & Rafikov (2001). Figure \[shocklocation\] shows our best estimates from simulations along with their predictions \[eq. (30)\] in Goodman & Rafikov (2001). (Note that their study was done in a shearing sheet configuration.) For these simulations we have used a softening of $1.0 h$. The agreement is quite amazing, at least for this choice of softening. In their analysis, they matched a linear wave excitation process with the nonlinear propagation. It is hard for us to ensure that the wave excitation in simulations is strictly in the linear regime (though increasing softening distance seems to be going in that direction). We have tried to extend our simulations to $0.1 M_{\oplus}$ planet but we are less confident about the numerical effects, so we omit that result.
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|
---
abstract: 'The method of brackets is a method of integration based upon a small number of heuristic rules. Some of these have been made rigorous. An example of an integral involving the Bessel function is used to motivate a new evaluation rule.'
address:
- |
Instituto de Fisica y Astronomia\
Universidad de Valparaiso\
Chile
- 'Department of Mathematics, Tulane University, New Orleans, LA 70118'
- 'Department of Mathematics, Tulane University, New Orleans, LA 70118'
author:
- Ivan Gonzalez
- Lin Jiu
- 'Victor H. Moll'
title: |
Pochhammer symbol with negative indices.\
A new rule for the method of brackets.
---
\[section\] \[Definition\][**Theorem**]{} \[Definition\][**Example**]{} \[Definition\][**Lemma**]{} \[Definition\][**Corollary**]{} \[Definition\][**Proposition**]{}
Introduction {#sec-intro}
============
The evaluation of definite integrals is connected, in a surprising manner, to many topics in Mathematics. The last author has described in [@moll-2002a] and [@moll-2010c] some of these connections. Many of these evaluations appear in the classical table [@gradshteyn-2015a] and proofs of these entries have appeared in a series of papers starting with [@moll-2007a] and the latest one is [@amdeberhan-2015c]. An interesting new method of integration, developed in [@gonzalez-2007a] in the context of integrals coming from Feynman diagrams, is illustrated here in the evaluation of entry $6.671.7$ in [@gradshteyn-2015a]: $$I:=\int_{0}^{\infty} J_{0}(ax) \sin(bx) \, dx = \begin{cases}
0, & \quad \text{ if } 0< b < a, \\
1/\sqrt{b^{2}-a^{2}}, & \quad \text{ if } 0 < a < b.
\end{cases}
\label{entry1}$$
This so-called *method of brackets* is based upon a small number of heuristic rules. These are described in the next section. Sections \[gr-entry1\] and \[sec-alternative\] present the evaluation of . The conflict between these two evaluations is resolved in Section \[sec-extension\] with the proposal of a new rule dealing with the extension of the Pochhammer symbol to negative integer indices.
The method of brackets {#sec-method}
======================
A method to evaluate integrals over the half line $[0, \, \infty)$, based on a small number of rules has been developed in [@gonzalez-2007a; @gonzalez-2008a; @gonzalez-2009a]. This method of brackets is described next. The heuristic rules are currently being made rigorous in [@amdeberhan-2012b] and [@jiu-2015e]. The reader will find in [@gonzalez-2014a; @gonzalez-2010a; @gonzalez-2010b] a large collection of evaluations of definite integrals that illustrate the power and flexibility of this method.
For $a \in \mathbb{C}$, the symbol $$\langle a \rangle \mapsto {\int_{0}^{\infty}}x^{a-1} \, dx$$ is the [*bracket*]{} associated to the (divergent) integral on the right. The symbol $$\phi_{n} := \frac{(-1)^{n}}{\Gamma(n+1)}$$ is called the [*indicator*]{} associated to the index $n$. The notation $\phi_{i_{1}i_{2}\cdots i_{r}}$, or simply $\phi_{12 \cdots r}$, denotes the product $\phi_{i_{1}} \phi_{i_{2}} \cdots \phi_{i_{r}}$.
[**[*Rules for the production of bracket series*]{}**]{}
${\mathbf{Rule \, \, P_{1}}}$. If the function $f$ is given by the formal power series $$f(x)=\sum_{n=0}^{\infty}a_{n}x^{\alpha n+\beta-1},$$ then the improper integral of $f$ over the positive real line is formally written as the *bracket series* $$\int_{0}^{\infty}f(x)dx=\sum_{n}a_{n}\left\langle \alpha n+\beta\right\rangle .$$
${\mathbf{Rule \, \, P_{2}}}$. For $\alpha \in \mathbb{C}$, the multinomial power $(a_{1} + a_{2} + \cdots + a_{r})^{\alpha}$ is assigned the $r$-dimension bracket series $$\sum_{n_{1}} \sum_{n_{2}} \cdots \sum_{n_{r}}
\phi_{n_{1}\, n_{2} \, \cdots n_{r}}
a_{1}^{n_{1}} \cdots a_{r}^{n_{r}}
\frac{\langle -\alpha + n_{1} + \cdots + n_{r} \rangle}{\Gamma(-\alpha)}.$$
[**[*Rules for the evaluation of a bracket series*]{}**]{}
${\mathbf{Rule \, \, E_{1}}}$. The one-dimensional bracket series is assigned the value $$\sum_{n} \phi_{n} f(n) \langle an + b \rangle =
\frac{1}{|a|} f(n^{*}) \Gamma(-n^{*}),$$ where $n^{*}$ is obtained from the vanishing of the bracket; that is, $n^{*}$ solves $an+b = 0$.
The next rule provides a value for multi-dimensional bracket series where the number of sums is equal to the number of brackets.
${\mathbf{Rule \, \, E_{2}}}$. Assume the matrix $A = (a_{ij})$ is non-singular, then the assignment is $$\sum_{n_{1}} \cdots \sum_{n_{r}} \phi_{n_{1} \cdots n_{r}}
f(n_{1},\cdots,n_{r})
\langle a_{11}n_{1} + \cdots + a_{1r}n_{r} + c_{1} \rangle \cdots
\langle a_{r1}n_{1} + \cdots + a_{rr}n_{r} + c_{r} \rangle
\nonumber$$ $$= \frac{1}{| \text{det}(A) |} f(n_{1}^{*}, \cdots n_{r}^{*})
\Gamma(-n_{1}^{*}) \cdots \Gamma(-n_{r}^{*})
\nonumber$$ where $\{ n_{i}^{*} \}$ is the (unique) solution of the linear system obtained from the vanishing of the brackets. There is no assignment if $A$ is singular.
${\mathbf{Rule \, \, E_{3}}}$. Each representation of an integral by a bracket series has associated an [*index of the representation*]{} via $$\text{index } = \text{number of sums } - \text{ number of brackets}.$$ It is important to observe that the index is attached to a specific representation of the integral and not just to integral itself. The experience obtained by the authors using this method suggests that, among all representations of an integral as a bracket series, the one with [*minimal index*]{} should be chosen.
The value of a multi-dimensional bracket series of positive index is obtained by computing all the contributions of maximal rank by Rule $E_{2}$. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded.
A systematic procedure in the simplification of the series obtained by this procedure has been used throughout the literature: express factorials in terms of the gamma function and the transform quotients of gamma terms into Pochhammer symbol, defined by $$(a)_{k} = \frac{\Gamma(a+k)}{\Gamma(a)}.$$ Any presence of a Pochhammer with a negative index $k$ is transformed by the rule $$(a)_{-k} = \frac{(-1)^{k}}{(1-a)_{k}}.
\label{rule-11}$$
The example discussed in the next two section provides motivation for an additional evaluation rule for the method of brackets.
A first evaluation of entry $\mathbf{6.671.7}$ in Gradshteyn and Ryzhik {#gr-entry1}
=======================================================================
The evaluation of uses the series $$J_{0}(ax) = \sum_{m=0}^{\infty} \phi_{m} \frac{a^{2m}}{\Gamma(m+1) 2^{2m}} x^{2m}$$ and $$\sin(bx) = \sum_{n=0}^{\infty} \phi_{n} \frac{\Gamma(n+1)}{\Gamma( 2 n + 2)} b^{2 n +1} x^{2 n+1}.$$ Therefore the integral in is given by $$I = \sum_{m,n} \phi_{m,n} \frac{a^{2m} b^{2n+1} \Gamma(n+1)}{2^{2m} \Gamma(m+1) \Gamma(2 n + 2)}
\langle 2 m + 2n + 2 \rangle.$$ The duplication formula for the gamma function transforms this expression to $$I = \frac{\sqrt{\pi}}{2} \sum_{m,n} \phi_{m,n} \frac{a^{2m} b^{2n+1} }{2^{2m+2 n} \Gamma(m+1) \Gamma( n+ 3/2)}
\langle 2 m + 2n + 2 \rangle.$$
Eliminating the parameter $n$ using Rule $E_{1}$ gives $n^{*} = - m -1$ and produces $$\begin{aligned}
I & = & \frac{\sqrt{\pi}}{b} \sum_{m=0}^{\infty} \phi_{m} \frac{1}{\Gamma \left(-m + \tfrac{1}{2} \right)} \left( \frac{a}{b} \right)^{2m} \\
& = & \frac{1}{b} \sum_{m = 0}^{\infty} \frac{(-1)^{m}}{m!} \frac{ \left( \frac{a^{2}}{b^{2}} \right)^{m}}{\left( \tfrac{1}{2} \right)_{-m} }\\
& = & \frac{1}{b} \sum_{m=0}^{\infty} \left( \frac{1}{2} \right)_{m} \frac{1}{m!} \left( \frac{a^{2}}{b^{2}} \right)^{m} \\
& = & \frac{1}{\sqrt{b^{2} - a^{2}}}.\end{aligned}$$ The condition $|b|>|a|$ is imposed to guarantee the convergence of the series on the third line of the previous argument.
The series obtained by eliminating the parameter $m$ by $m^{*} = -n -1$ vanishes because of the factor $\Gamma(m+1)$ in the denominator. The formula has been established.
An alternative evaluation {#sec-alternative}
=========================
A second evaluation of begins with $$\int_{0}^{\infty} J_{0}(ax) \sin(bx) \, dx = \sum_{m,n} \phi_{m,n} \frac{a^{2m} b^{2n+1}}{2^{2m} m! (2n+1)!}
\langle 2m+2n+2 \rangle.$$ The evaluation of the bracket series is described next.
*Case 1*. Choose $n$ as the free parameter. Then $m^{*} = -n-1$ and the contribution to the integral is $$I_{1} = \frac{1}{2} \sum_{n=0}^{\infty} \phi_{n}
\left( \frac{a}{2} \right)^{-2n-2} b^{2n+1} \frac{1}{\Gamma(-n)} \frac{n!}{(2n+1)!} \Gamma(n+1).$$ Each term in the sum vanishes because the gamma function has a pole at the negative integers.
*Case 2*. Choose $m$ as a free parameter. Then $2m+2n+2=0$ gives $n^{*} = -m-1$. The contribution to the integral is $$I_{2} = \frac{1}{2} \sum_{m} \phi_{m} \left( \frac{a}{2} \right)^{2m} b^{-2m-1} \frac{1}{m!} \cdot
\frac{n!}{(2n+1)!}\Big{|}_{n=-m-1} \Gamma(m+1).
\label{int-1}$$ Now write $$\frac{n!}{(2n+1)!} = \frac{1}{(n+1)_{n+1}}$$ and becomes $$I_{2} = \frac{1}{2b} \sum_{m=0}^{\infty} \frac{(-1)^{m}}{m!} \left( \frac{a}{2b} \right)^{2m} \frac{1}{(-m)_{-m}}.
\label{int-2}$$ Transforming the term $(-m)_{-m}$ by using $$(x)_{-n} = \frac{(-1)^{n}}{(1-x)_{n}} \label{ivan-rule1}$$ gives $$(-m)_{-m} = \frac{(-1)^{m}}{(1+m)_{m}}.$$ Replacing in produces $$\begin{aligned}
I_{2} & = & \frac{1}{2b} \sum_{k=0}^{\infty} \left( \frac{1}{2} \right)_{k} \frac{1}{k!} \left( \frac{a^{2}}{b^{2}} \right)^{k} \\
& = & \frac{1}{2} \frac{1}{\sqrt{b^{2}-a^{2}}}.
\end{aligned}$$
The method of brackets produces *half of the expected answer*. Naturally, it is possible that the entry in [@gradshteyn-2015a] is erroneous (this happens once in a while). Some numerical computations and the evaluation in the previous section, should convince the reader that this is not the case. The source of the error is the use of for the evaluation of the term $(-m)_{-m}$. A discussion is presented in Section \[sec-extension\].
Extensions of the Pochhammer symbol {#sec-extension}
===================================
Rule $E_{1}$ of the method of brackets requires the evaluation of $f(n^{*})$. In many instances, this involves the evaluation of the Pochhammer symbol $(x)_{m}$ for $m \not \in \mathbb{N}$. In particular, the question of the value $$(-m)_{-m} \quad \text{ for } \quad m \in \mathbb{N}$$ is at the core of the missing factor of $2$ in Section \[sec-alternative\].
The first extension of $(x)_{m}$ to negative values of $n$ comes from the identity $$(x)_{-m} = \frac{(-1)^{m}}{(1-x)_{m}}.
\label{poch-neg}$$ This is obtained from $$(x)_{-m} = \frac{\Gamma(x-m)}{\Gamma(x)} = \frac{\Gamma(x-m)}{(x-1)(x-2) \cdots (x-m) \Gamma(x-m)}$$ and then changing the signs of each of the factors. This is valid as long as $x$ is not a negative integer. The limiting value of the right-hand side in as $x \to -km$, with $k \in \mathbb{N}$, is $$(-km)_{-m} = \frac{(-1)^{m} \, (km)!}{((k+1)m)!}.$$
On the other hand, the limiting value of the left-hand side is $$\begin{aligned}
\lim\limits_{\varepsilon \to 0} \left( -k(m+ \varepsilon) \right)_{-(m+ \varepsilon)} & = &
\lim\limits_{\varepsilon \to 0} \frac{\Gamma(-(k+1)m -(k+1) \varepsilon)}{\Gamma(-km - k \varepsilon )} \\
& = & \lim\limits_{\varepsilon \to 0}
\frac{\Gamma( -(k+1) \varepsilon) (-(k+1) \varepsilon)_{-(k+1)m} }
{\Gamma(- k \varepsilon) ( - k \varepsilon)_{-km}} \\
& = & \lim\limits_{\varepsilon \to 0}
\frac{\Gamma( -(k+1) \varepsilon)}{\Gamma(-k \varepsilon)}
\frac{(-1)^{(k+1)m}}{(1+ (k+1) \varepsilon)_{(k+1)m}} \cdot \frac{(1+ k \varepsilon)_{km}}{(-1)^{km} } \\
& = & \frac{(-1)^{m} (km)!}{((k+1)m)!} \cdot \frac{k}{k+1}.\end{aligned}$$
Therefore the function $(x)_{-m}$ is discontinuous at $x = -km$, with $$\frac{\text{Direct } (-km)_{-m}}{\text{Limiting } (-km)_{-m}} = \frac{k+1}{k}.$$ For $k=1$, this ratio becomes $2$. This explains the missing $\tfrac{1}{2}$ in the calculation in Section \[sec-alternative\]. Therefore it is the discontinuity of at negative integer values of the variables, what is responsible for the error in the evaluation of the integral .
This example suggest that the rules of the method of brackets should be supplemented with an additional one:
${\mathbf{Rule \, \, E_{4}}}$. Let $k \in \mathbb{N}$ be fixed. In the evaluation of series, the rule $$(-km)_{-m} = \frac{k}{k+1} \, \frac{(-1)^{m} \, (km)!}{((k+1)m)!}$$ must be used to eliminate Pochhammer symbols with negative index and negative integer base.
A variety of other examples confirm that this heuristic rule leads to correct evaluations.
**Acknowledgments**. The last author acknowledges the partial support of NSF-DMS 1112656. The second author is a graduate student, partially supported by the same grant.
[10]{}
T. Amdeberhan, A. Dixit, X. Guan, L. Jiu, A. Kuznetsov, V. Moll, and C. Vignat. The integrals in [G]{}radshteyn and [R]{}yzhik. [P]{}art 30: [T]{}rigonometric integrals. , To appear, 2015.
T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Harrison, V. Moll, and A. Straub. Ramanujan [M]{}aster [T]{}heorem. , 29:103–120, 2012.
I. Gonzalez, K. Kohl, and V. Moll. Evaluation of entries in [G]{}radshteyn and [R]{}yzhik employing the method of brackets. , 25:65–84, 2014.
I. Gonzalez and V. Moll. Definite integrals by the method of brackets. [P]{}art 1. , 45:50–73, 2010.
I. Gonzalez, V. Moll, and A. Straub. The method of brackets. [P]{}art 2: Examples and applications. In T. Amdeberhan, L. Medina, and Victor H. Moll, editors, [*Gems in Experimental Mathematics*]{}, volume 517 of [*Contemporary Mathematics*]{}, pages 157–172. American Mathematical Society, 2010.
I. Gonzalez and I. Schmidt. Optimized negative dimensional integration method ([NDIM]{}) and multiloop [F]{}eynman diagram calculation. , 769:124–173, 2007.
I. Gonzalez and I. Schmidt. Modular application of an integration by fractional expansion ([IBFE]{}) method to multiloop [F]{}eynman diagrams. , 78:086003, 2008.
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V. Moll. The evaluation of integrals: a personal story. , 49:311–317, 2002.
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|
---
abstract: 'We investigate the scattering cross section of aligned dipolar molecules in low-temperature gases. Over a wide range of collision energies relevant to contemporary experiments, the cross section declines in inverse proportion to the collision speed, and is given nearly exactly by a simple semiclassical formula. At yet lower energies, the cross section becomes independent of energy, and is reproduced within the Born approximation to within corrections due to the s-wave scattering length. While these behaviors are universal for all polar molecules, nevertheless interesting deviations from universality are expected to occur in the intermediate energy range.'
address: |
${}^1$ JILA, NIST and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA\
${}^2$ Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055 USA\
${}^3$ ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Hawthorn, Vic 3122, Australia
author:
- 'J. L. Bohn${}^1$, M. Cavagnero${}^2$, and C. Ticknor${}^3$'
title: 'Quasi-Universal Dipolar Scattering in Cold and Ultracold Gases'
---
Introduction
============
The energy dependence of scattering cross sections for atoms at ultralow collision energies is very simple. Either the cross section is nearly independent of energy, for distinguishable particles or identical bosons; or else the cross section vanishes altogether, for identical fermions. This behavior emerges in the limit where the deBroglie wavelength exceeds any natural length scale of the interatomic interaction, and the scattering is characterized by a single quantity, the s-wave scattering length $a$ (alternatively, the p-wave scattering volume $V_p$ for fermions). Although $a$ dominates the threshold scattering, nevertheless its value is not immediately obvious from the interaction potential, and must be determined painstakingly from experiments.
By contrast, for low-energy collisions between polarized dipolar molecules, the near-threshold scattering [*is*]{} often approximately determined directly by parameters of the interaction potential. The interaction between two molecules of reduced mass $M$ and dipole moment $\mu$ is characterized by a dipole length, given by $D =M \mu^2 / \hbar^2$. [^1] Under a wide variety of circumstances, to be discussed in this paper, $D$ is the dominant length scale and sets the threshold cross section, i.e., $\sigma \sim D^2$. Moreover, this circumstance holds for identical fermions as well as bosons. In this sense, the scattering of two dipoles is nearly universal at threshold, apart from possible modifications arising from s-wave scattering.
For realistic collisions in present-day experiments, however, collision energies are not always in this threshold region. When the threshold region is left behind, there is significant numerical evidence to suggest that a universal behavior still emerges, and that the cross section scales as $\sigma \sim D/K$, where $K$ is the wave number of the relative motion [@CTuniversal]. The switch between the two types of behavior corresponds roughly to the natural energy scale of the dipolar interaction, $E_D = \mu^2/D^3
=\hbar^6/M^3 \mu^4$. Below this energy nonzero partial waves contribute only perturbatively, and only at large intermolecular separation; whereas at higher energies, many partial waves contribute and the scattering is semiclassical.
These two behaviors may thus be said to define the operational limit between “cold” and “ultracold” regimes of scattering for dipoles. We propose that the onset of cold collisions occurs when the temperature of the gas passes well below the molecule’s rotational constant $B_e$ (or else its $\Lambda$-doublet splitting) so that the orientational degrees of freedom freeze out. These temperatures are typically in the mK-K range. This cold collision regime, in which semiclassical scattering occurs, persists until the temperature gets as low as $E_D$ (typically nK-$\mu$K temperatures for dipolar molecules). Temperatures below $E_D$ define the ultracold regime, where true threshold scattering is apparent. Actual values of these temperatures depend strongly on the species considered and on the applied electric field.
Our goal in this article is to make these ideas precise. We will illustrate the universal behavior of dipole-dipole scattering at low temperatures in the two regimes, and, more importantly, we will see the circumstances under which this universality fails. Figure 1 shows the basic elements of universality for dipole-dipole scattering. This figure plots the total scattering cross section $\sigma$, averaged over all incident directions, versus collision energy. Both quantities are presented in terms of the “natural” units given above. The black curve is a complete numerical close-coupling calculation. At low energies, $E<E_D$, $\sigma$ approaches a constant value that is well-approximated using the Born approximation (blue). At higher energies $E>E_D$, $\sigma$ falls off as $1/\sqrt{E}$, and is given by the semiclassical eikonal approximation (red). Both approximations will be derived below.
{width="90.00000%"}
In this article, we present alternative close-coupling calculations that reveal deviations from this universal behavior. For example, at threshold, the cross section can deviate substantially from the Born result. This occurs when threshold resonances in s-wave scattering dramatically increase the cross section above the $D^2$ value [@Ticknor05_PRA; @Bohn05_ICLP; @mike; @mike2]. More interestingly, in the intermediate energy regime $E \approx E_D$ where neither approximation holds, our close-coupling calculations show that the behavior of cross section versus energy depends on details of the physics on length scales small compared to $D$. For this reason, it is conceivable that elastic scattering experiments that probe deviations from universality will be an important tool in unraveling information on the short-range physics of close encounters. Finally, we also discuss the angle-dependence of cold collisions, in which the incident direction of the collision partners is varied with respect to the polarization axis; this is a measure of the importance of the anisotropy of the dipole-dipole interaction. These results should serve as simple and accurate guidelines to low-energy collision cross sections needed to understand scattering or thermal equilibration in contemporary experiments.
Formulation of the problem
==========================
For a pair of dipoles with reduced mass $M$ and polarized in the ${\hat z}$ direction by an external field, the two-body Schrödinger equation reads $$\begin{aligned}
\label{raw_Schrodinger}
\left[ - {\hbar ^2 \over 2 M} \bigtriangledown ^2 + \langle \mu_1
\rangle \langle \mu_2 \rangle {1 - 3\cos \theta \over R^3} + V_{SR}
\right] \psi = E \psi.\end{aligned}$$ Here ${\vec R} = (R, \theta, \phi)$ is the relative displacement of the two dipoles, and $\langle \mu_1 \rangle$ and $\langle \mu_2
\rangle$ are their induced dipole moments. These dipoles are field dependent, and their values are set by the field and the internal structure of the dipole. In the present context, we will take them as fixed quantities, and observe their influence on the scattering. By ignoring internal molecular structure, we are in effect modeling molecules trapped in their absolute ground state, as in an optical dipole trap. In particular, we assume that inelastic scattering is disallowed, a topic taken up in the next article of this issue [@Newell].
The potential term $V_{SR}$ represents all the short-range physics, i.e., the potential energy surface of chemical significance when the dipoles are close together. This detail will be specific to each pair of collision partners considered. To simplify the discussion, we will replace the complex details of $V_{SR}$ by imposing a boundary condition at a fixed interparticle separation $R_0$. We will assert that the wave function $\psi$ vanishes uniformly on this boundary, although alternative boundary conditions are certainly possible and indeed desirable [@Ticknor05_PRA; @Bohn05_ICLP]. Having made this approximation, we ignore $V_{SR}$ in the following.
The resulting Shrödinger equation then admits a natural length scale $D = M \langle \mu_1 \rangle \langle \mu_2 \rangle /
\hbar^2$, and a natural energy scale $E_D = \hbar^6 / M^3 \langle \mu_1
\rangle^2 \langle \mu_2 \rangle^2$. By recasting (\[raw\_Schrodinger\]) in the scaled units $r=R/D$, $\epsilon = E/E_D$, and by ignoring $V_{SR}$, we arrive at the universal Schrödinger equation (assuming that the molecules have the same orientation relative to the field axis) $$\begin{aligned}
\label{scaled_Schrodinger}
\left[ - {1 \over 2} \bigtriangledown^2 - {2 C_{20} \over r^3}
\right] \psi = \epsilon \psi,\end{aligned}$$ where $C_{20}(\theta,\phi) = (3 \cos^2 \theta - 1)/2$ is the standard reduced spherical harmonic [@brink]. Although the [*equation*]{} (\[scaled\_Schrodinger\]) has a universal form, nevertheless its [*solutions*]{} may depend on details of the short-range physics, represented here by the cutoff radius $r_0 =
R_0/D$. Our main objective is to explore circumstances under which solutions are universal, i.e., independent of $r_0$, under the conditions of modern cold molecule experiments.
As a preliminary argument in this direction, let us consider the relative sizes of the two characteristic lengths $D$ and $R_0$. For most atomic and molecular species, the size scale $R_0$ below which short-range physics can matter is on the order of the van der Waals length, which is typically $\sim 100$ $ a_0$, which is also the distance at which the internal fields generated by the dipoles are comparable to the applied field (at least for typical laboratory field strengths). By contrast, for molecules with typical 1 Debye dipole moments, $D$ will be orders of magnitude larger. For example, in the OH radical, with $\mu =1.68$ Debye, $D_{\rm OH} = 6770$ $ a_0$; for the representative alkali dimer KRb, with $\mu = 0.566$ Debye, $D_{\rm KRb} = 5740$ $a_0$; and for the highly polar LiCs molecule with $\mu =5.5$ Debye, we find $D_{\rm LiCs} = 6 \times 10^5$ $a_0$. Correspondingly, the characteristic energies are low: $E_{D,{\rm OH}} = 445$ nK, while $E_{D,{\rm KRb}} = 83$ nK, and $E_{D,{\rm LiCs}} = 7$ pK. In making these estimates, we assume the molecules are completely polarized; $D$ gets shorter, and $E_D$ higher, if they are only partially polarized. [^2]
The finite size of $R_0$ can destroy universality in three ways. First, at low energies, we will see that scattering cross sections are of order $D^2$. This would be spoiled by the geometric cross section $\propto R_0^2$, if $R_0$ were comparable to, or larger than, the dipole length $D$ itself. However, we have just argued that this is usually not the case for polar molecules. Second, there is the possibility that the s-wave scattering length $a$ is larger than $D$, and this would also alter the universal result at low energy. Indeed, this very idea has been invoked as a means of tuning the interaction between dipolar molecules [@You; @Ronen; @Derevianko; @Wang; @Blume] or else as a tool for probing details of these interactions [@Ticknor05_PRA]. Often, the effect of the scattering length is non-negligible, as we will see below.
The third instance where $R_0$ may matter is in the extreme high energy limit, where the universal cross section falls as $D/K$, where $K$ is the wave number. In this case the cross sections will again tend to the geometrical $\sim R_0^2$ for our artificially imposed hard wall, and will dominate at energies where $R_0^2 > D/K$, which translates to about 4.5 Kelvin in OH, well above Stark decelerator energies, and at which point other degrees of freedom of the molecule are relevant. Thus non-universal behavior may not be a concern at “high” energies either, from the standpoint of current experimental investigations.
In this article we compute scattering cross sections for dipoles over a wide range of collision energies. We do this in three ways: 1) a fully numerical close-coupling expansion of the wave function in partial waves; 2) a Born approximation that exhibits the correct universal behavior ($\sigma \sim D^2$) in the ultracold limit ($E<E_D$); and 3) a semiclassical eikonal approximation that exhibits the correct universal behavior ($\sigma \sim D/K$) in the cold collision regime ($E>E_D$). We briefly describe these methods in the following subsections.
Close-coupling formalism
------------------------
We expand the total wave function into partial waves in the conventional way: $$\begin{aligned}
\label{partial_wave_expansion}
\psi(r,\theta, \phi) = {1 \over r} \sum_{lm} F_{lm}(r) Y_{lm}
(\theta ,\phi),\end{aligned}$$ where the radial functions $F_{lm}$ satisfy a set of coupled-channel Schrödinger equations $$\begin{aligned}
\label{coupled_Schrodinger}
-{1 \over 2} {d^2 F_{lm} \over dr^2} + {l(l+1) \over 2 r^2}F_{lm} -
{2 \over r^3} \sum_{l^{\prime}} C_{l l^{\prime}}^{(m)} F_{l^\prime
m} = \epsilon F_{lm},\end{aligned}$$ and the coupling matrix element is given by [@brink] $$\begin{aligned}
\label{coupling_matrix}
C_{l l^{\prime}}^{(m)} &=& \langle lm | C_{20} | l^{\prime} m
\rangle \\
&=& (-1)^m \sqrt{ (2l+1)(2l^{\prime}+1) } \left(
\begin{array}{ccc}
l & 2 & l^{\prime} \\
-m & 0 & m \end{array} \right) \left( \begin{array}{ccc} l & 2 &
l^{\prime} \\ 0 & 0 & 0 \end{array}\right) \nonumber.\end{aligned}$$ Owing to the cylindrical symmetry of the Hamiltonian, the angular momentum projection $m$ is a good quantum number, and we can solve a separate set of coupled equations for each value of $m$. However, the boundary conditions of the wave function may not respect this symmetry, i.e., the incident wave could arrive from any direction, not just along the symmetry ($z$) axis. Thus we consider a complete sum over $m$ in the wave function (\[partial\_wave\_expansion\]). Similarly, symmetries of the 3-$j$ symbols in (\[coupling\_matrix\]) guarantee that each partial wave $l$ is coupled only to the partial waves $l^{\prime}=l$, $l \pm 2$ by the dipole interaction. Thus we can consider even partial waves separately from odd partial waves, and will do so in the following. For identical particles in the same internal state, these situations correspond to bosons and fermions, respectively.
The equations (\[coupled\_Schrodinger\]) admit as many linearly independent solutions as there are channels $(lm)$. Individually, they are defined by the boundary conditions (for each $m$) $$\begin{aligned}
\label{boundary_conditions}
F_{lm}^{l^{\prime} m}(r=r_0) &=& 0 \nonumber \\
F_{lm}^{l^{\prime} m} (r \rightarrow \infty) &=& \delta_{l
l^{\prime}}e^{-i(kr - l^{\prime} \pi /2)} - S_{l
l^{\prime}}^{(m)}e^{i(kr - l^{\prime} \pi /2)},\end{aligned}$$ where $k = \sqrt{ 2 \epsilon}=DK$ is the wave number in dipole units. These scattering boundary conditions serve to define the scattering matrix $S_{l l^{\prime}}^{(m)}$. From this matrix one can construct the scattering amplitude describing scattering in direction ${\hat
k}_f = (\theta, \phi)$ from an incident direction ${\hat k}_i$ [@Mott; @Bohn00_PRA]: $$\begin{aligned}
\label{scattering_amplitude}
f({\hat k}_i,{\hat k}_f) = -{2 \pi \over k} \sum_{l l^{\prime} m}
i^l Y_{lm}^*({\hat k}_i) T_{l l^{\prime}}^{(m)}
i^{-l^{\prime}}Y_{l^{\prime} m}({\hat k}_f),\end{aligned}$$ in terms of the $T$ matrix, $T = i(S-I)$. By integrating over the final directions, we arrive at the total cross section for dipoles incident along ${\hat k}_i$: $$\begin{aligned}
\label{sigma_tot}
\frac{\sigma_{{\rm tot}}({\hat k}_i)}{D^2} &=& \int d \phi d(\cos \theta) |f|^2
\nonumber \\
&=& {4 \pi \over k} \Im f({\hat k}_i,{\hat k}_i).\end{aligned}$$ This last line is the familiar optical theorem result. This is the type of cross section that can be measured in cold beam experiments, where, say, one species is trapped and the other is incident on the trap from the terminus of a Stark decelerator. If the molecules are magnetically trapped, then an electric field can be applied an at arbitrary angle relative to the collision axis [@Sawyer07_PRL].
Finally, the total cross section, integrated over an assumed isotropic distribution of initial directions, is $$\begin{aligned}
\frac{\sigma}{D^2} &=& \int d {\hat k}_i \frac{\sigma_{{\rm tot}}({\hat k}_i)}{D^2}
\nonumber
\\ &=& {\pi \over k^2} \sum_{l l^{\prime} m} |T_{l l^{\prime}}^{(m)}|^2.\end{aligned}$$ This cross section is more relevant to [*in situ*]{} collisions in a trap, which serve to re-thermalize the gas and provide evaporative cooling.
Numerical solutions to the coupled-channel equations (\[coupled\_Schrodinger\]) are determined using a variable stepsize version of Johnson’s algorithm [@johnson]. To ensure convergence of total cross sections, we include partial waves up to $l \sim 100$ at the highest collision energies of $\epsilon = 10^4$. Vice versa, at the lowest collision energies we can get away with partial waves up to $l \sim 30$, but must apply the boundary conditions (\[boundary\_conditions\]) as far out as $r = 20,000$.
Born approximation
------------------
At the lowest collision energies, the Wigner threshold laws are well known to be different for dipolar interactions than for, say, van der Waals interactions. The elastic scattering phase shift $\delta_l$ in partial wave $l>0$, due to a potential with $1/r^s$ long-range behavior, is [@Sadeghpour00_JPB] $$\begin{aligned}
\label{general_Wigner}
\tan \delta_l \sim A k^{2l+1} + B k^{s-2},\end{aligned}$$ for some constants $A$ and $B$ that depend on short-range details. The first term in (\[general\_Wigner\]) arises from the action of the short range potential, while the second is due to purely long-range scattering [*outside*]{} the centrifugal barrier. For the van der Waals potential ($s=6$), both contributions go to zero faster than $\sim k$ at zero energy, and do not contribute to the threshold cross section. However, for the dipole-dipole interaction ($s=3$), the second term is $\sim k$ for all partial waves. The contribution to the cross section, $\propto \sin^2 \delta_l/k^2$, is then independent of energy in all partial waves, and this cross section arises from long-range scattering.
This circumstance leads to the applicability of the Born approximation in threshold scattering of dipoles [@Deb; @Derevianko; @Wang; @Li01_PRA; @Avdeenkov05_PRA; @Blume]. At ever lower collision energy, scattering occurs at ever larger values of $r$ outside the barrier. But at long range the dipole-dipole interaction $\propto 1/r^3$ is weak, and the perturbative Born approximation is applicable. These remarks do not apply, however, to s-wave scattering, where there is no barrier.
The Born approximation for the scattering amplitude reads $$\begin{aligned}
f({\hat k}_i,{\hat k}_f) = -{1\over 2 \pi} \int d^3 r e^{-i{\vec
k}_f \cdot {\vec r}} V_d({\vec r}) e^{i{\vec k}_i \cdot {\vec r}}.\end{aligned}$$ Replacing each plane wave by its standard partial wave expansion and re-arranging yields $$\begin{aligned}
f({\hat k}_i,{\hat k}_f) = -{1 \over 2 \pi} \int d^3r \left( - {2
\over r^3} \right) C_{20}({\hat r}) && 4 \pi
\sum_{l^{\prime}m^{\prime}} i^{-l^{\prime}} Y_{l^{
\prime}m^{\prime}}({\hat k}_f)
Y_{l^{\prime}m^{\prime}}^*({\hat r}) j_{l^{\prime}}(kr) \nonumber \\
& \times & 4 \pi \sum_{lm} i^l Y_{lm}^*({\hat k}_i) Y_{lm}({\hat r})
j_l(kr).\end{aligned}$$ Consolidating the integrals into radial and angular varieties, we arrive at $$\begin{aligned}
\label{Born_f}
f({\hat k}_i,{\hat k}_f) = 2 \pi \sum_{l l^{\prime} m} i^l
Y_{lm}^*({\hat k}_i) C_{l l^{\prime}}^{(m)} \Gamma_{l l^{\prime}}
i^{-l^{\prime}} Y_{l^{ \prime}m^{\prime}}({\hat k}_f),\end{aligned}$$ where $C_{l l^{\prime}}^{(m)}$ is the angular integral (\[coupling\_matrix\]) defined above, and $\Gamma_{l l^{\prime}}$ is the radial integral $$\begin{aligned}
\Gamma_{l l^{\prime}} &=& 8 \int_0^{\infty} r^2 dr {j_{l}(kr)
j_{l^{\prime}}(kr) \over r^3} \nonumber \\
&=& {\pi \Gamma((l+l^{\prime})/2) \over \Gamma((-l+l^{\prime}+3)/2)
\Gamma((l+l^{\prime}+4)/2)
\Gamma((l-l^{\prime}+3)/2) }. \nonumber \\
&=& \left\{ \begin{array}{l} {32 \over l(l+1)}, \;\;\;\;\;\;\;\;\; l^{\prime}=l \\
{32 \over 3(l+1)(l+2)}, \;\;\; l^{\prime} = l + 2 \end{array}
\right.\end{aligned}$$ Comparing the Born result (\[Born\_f\]) with the expression (\[scattering\_amplitude\]) (the factor of 8 is intended to simplify this) identifies the $T$-matrix in the Born approximation as $$\begin{aligned}
T_{l l^{\prime}}^{(m),{\rm Born}} = - k C_{l
l^{\prime}}^{(m)}\Gamma_{l l^{\prime}}.\end{aligned}$$
The Born approximation must be applied with a caveat. For purely s-wave scattering, where $l = l^{\prime} = 0$, the matrix element $C_{00}^{(0)}$ vanishes, and so therefore does the $T$-matrix element $T_{00}^{(0),{\rm Born}}$. The Born approximation is therefore mute on the question of s-wave scattering. As argued above, s-wave scattering, described by a scattering length $a$, is part of the non-universal behavior of scattering anyway. To produce realistic scattering results, it is possible to supplement $T^{{\rm
Born}}$ with an empirical s-wave contribution, which is determined from the full close-coupling calculations [@You; @Derevianko; @Blume; @Wang].
Summarizing all these results, the threshold cross section in the Born approximation, averaged over incident directions, is given by the incoherent sum $$\begin{aligned}
\frac{\sigma_{{\rm Born}}}{D^2} = {\pi \over k^2} \sum_{ll^{\prime}m}
|T_{ll^{\prime}}^{(m),{\rm Born}}|^2\end{aligned}$$ Evaluating the sums, the even and odd partial wave cross sections at threshold can be given as $$\begin{aligned}
\label{born_cross_sections}
\sigma_{{\rm Born}}^e = && 1.117 D^2 + 4 \pi a^2 \nonumber \\
\sigma_{{\rm Born}}^o=&& 3.351 D^2 ,\end{aligned}$$ where $4 \pi a^2$ allows for the existence of a scattering length $a$, which is not determined in the Born approximation. For identical particles, the cross sections (\[born\_cross\_sections\]) must be multiplied by 2, as usual; for distinguishable particles, both even and odd partial waves are possible, and these cross sections are to be added.
The low-energy limits (\[born\_cross\_sections\]) were independently verified in near-threshold calculations using a coupled-channel adiabatic representation [@mike; @mike2], which also illustrated how threshold angular distributions are affected by the competetion between long-range (Born) coupling and s-wave scattering resonances, as discussed in Section 4.
Eikonal approximation
---------------------
At sufficiently high energies, a semi-classical analysis yields the simple $D/K$ scaling of the cross section. Note, in reference to Fig. (1), that the deBroglie wavelength becomes smaller than the natural dipole length scale, $2\pi/K<D$, when $E>2\pi^2E_D$. This marks the onset of semi-classical scattering. This semi-classical onset can lie at $\mu K$ temperatures, or even colder, owing to the large dipole length scale and therefore the small value of $E_D$. Many partial waves contribute to the scattering amplitude in the semi-classical regime, and differential cross-sections are increasingly concentrated in the forward direction.
The eikonal method was long ago developed to find approximate scattering solutions of wave equations such as (\[scaled\_Schrodinger\]) valid in the semi-classical or ray-optics limit, in which the potential is assumed to vary little on the scale of the wavelength. A derivation of the eikonal wavefunction will not be given here, as it can be found in familiar texts [@Bransden-Joachain] and in a comprehensive review article by Glauber [@Glauber]. Suffice it to say that a phase-amplitude [*ansatz*]{}, coupled with the assumption of a slowly-varying amplitude, leads directly to an approximate wavefunction $$\begin{aligned}
\psi(\vec r) = e^{i\vec k_i\cdot\vec r}\exp\left[-{i\over k}\int^z V(b,\phi,z^{\prime}) dz^{\prime}\right]\end{aligned}$$ Following Glauber, this wavefunction is expressed in cylindrical coordinates with a new quantization (or $z$) axis aligned with the average collision momentum, $\vec k_{\rm avg} = (\vec k_i+\vec
k_f)/2$. The cylindrical radius about this axis, $b$, can be associated with a classical impact parameter: $\phi$ is the azimuthal angle about the quantization axis. Due to the shift of quantization axis away from the direction of the applied field, $\vec {\cal E}$, we now note the field direction explicitly in the potential $V(\vec r) = [1-3(\hat r\cdot\hat{\cal E})^2]/r^3$.
The two factors in the eikonal wavefunction are familiar in the context of the one-dimensional WKB method, where they coincide with an expansion of the WKB phase $i\int dz~\sqrt{2(\epsilon-V)}\approx
ikz ~-~ (i/k)\int~dz~V$ to first order in $V/\epsilon$. Accordingly, we anticipate that the eikonal method will be most accurate when the incident energy is large compared to the magnitude of the dipole-dipole interaction, a more stringent criterion than the semi-classical constraint noted above.
The analysis of scattering amplitudes associated with the approximate eikonal wavefunction is simplified by a judicious choice of coordinates. Note, in particular, that the momentum transfer $\vec q = \vec k_i -\vec k_f$ is orthogonal to the quantization axis defined by $\vec k_{\rm avg}$. For simplicity, we define an $x$-axis along $\vec q$, in which case the $y$-axis is orthogonal to the collision plane and lies along $\vec k_{\rm avg}
\times \vec q$. In this reference frame, the impact parameter is written in vector form as $\vec b = b\cos(\phi)\hat x +
b\sin(\phi)\hat y$, and the relative displacement of the dipoles is $\vec r = \vec b + z\hat k_{\rm avg}$.
Insertion of the eikonal wavefunction, valid where the potential is non-negligible, into the integral equation for scattering leads to the eikonal scattering amplitude $$\begin{aligned}
f^{Ei}(\vec k_f,\vec k_i) = \frac{k}{2\pi i} \int~b ~ db~ d\phi
~e^{iqb\cos(\phi)} \left[e^{i\chi(\vec b)}-1\right]\end{aligned}$$ where $k=\vert k_i\vert = \vert k_f\vert=\sqrt{2\epsilon}$, $q=\vert\vec q\vert = 2k\sin(\theta_s/2)$, and where the eikonal phase is $$\begin{aligned}
\chi(\vec b) = -\frac{1}{k}
\int_{-\infty}^{\infty} dz^{\prime}~V(b,\phi,z^{\prime})\end{aligned}$$
With the explicit form of the dipole-dipole potential $$\begin{aligned}
V(b,\phi,z) = \frac{1}{(b^2+z^2)^{3/2}}
\left[
1-3\frac{(\vec b\cdot\hat {\cal E}+z\hat k_{\rm avg}\cdot\hat{\cal E})^2}
{b^2+z^2}\right]\end{aligned}$$ the phase is readily evaluated; setting $\sigma = z/b$ one finds $$\begin{aligned}
\chi &=& -\frac{1}{k b^2}\left[
\int_{-\infty}^{\infty}\frac{d\sigma}{(1+\sigma^2)^{3/2}}\right.\nonumber \\
&-&3(\hat b\cdot\hat{\cal E})^2
\int_{-\infty}^{\infty}\frac{d\sigma}{(1+\sigma^2)^{5/2}}\nonumber \\
&-&3(\hat k_{\rm avg}\cdot\hat{\cal E})^2
\left.\int_{-\infty}^{\infty}\frac{\sigma^2 d\sigma}{(1+\sigma^2)^{5/2}}\right]\end{aligned}$$ The integrals are straightforward, giving $$\begin{aligned}
\chi = -\frac{2}{k b^2}
\left[1-(\hat k_{\rm avg}\cdot\hat{\cal E})^2-2(\hat b\cdot\hat{\cal E})^2\right]\end{aligned}$$ Referring the electric field to our coordinate axes $(\hat x=\hat q, \hat y=\hat k_{\rm avg}\times\hat q,\hat k_{\rm avg})$ $$\begin{aligned}
\hat{\cal E} = \sin\alpha\cos\beta\hat x + \sin\alpha\sin\beta\hat y + \cos\alpha\hat k_{\rm avg}\end{aligned}$$ the phase is simply $$\begin{aligned}
\chi(b,\phi) = \frac{2}{k b^2}\sin^2\alpha \cos(2\phi-2\beta)\end{aligned}$$
The eikonal amplitude now has the form $$\begin{aligned}
f^{Ei} = \frac{k}{2\pi i}
\int~b~db~d\phi~e^{iqb\cos\phi}
\left[
\exp{\left\{i\frac{2}{k b^2}\sin^2\alpha\cos(2\phi-2\beta)\right\}}-1\right]\end{aligned}$$ Explicit evaluation of the resulting integrals has proven quite difficult. However, to extract total cross-sections, Glauber’s general proof of unitarity of the eikonal approximation [@Glauber] permits use of the optical theorem, (\[sigma\_tot\]).
For forward scattering, $q=0$ and the first phase vanishes identically. Expressing the result in terms of the orbital angular momentum $l=kb$ gives $$\begin{aligned}
f^{Ei}(\hat k_i,\hat k_i) = \frac{1}{2\pi i k}
\int~\ell~d\ell~d\phi~
\left[
\exp{\left\{i\frac{2k}{\ell^2}\sin^2\alpha\cos(2\phi-2\beta)\right\}}-1\right]\end{aligned}$$ but note that this appears undetermined since $\hat q$ and, consequently, $\hat x$ and $\hat y$ are not defined when $q=0$! In this limit, $\alpha = \arccos(\hat k\cdot\hat {\cal E})$ is well-defined, but $\beta = \arctan(\hat y\cdot\hat{\cal E}/\hat x\cdot\hat{\cal E})$ is not. Fortunately, it is easy to show that the azimuthal integral is independent of $\beta$, with the result $$\begin{aligned}
f^{Ei}(\hat k_i,\hat k_i) = \frac{1}{ik}\int~\ell~d\ell~\left[
J_0\left(\frac{2k}{\ell^2}\sin^2\alpha\right)-1\right]\end{aligned}$$
From the optical theorem, the total cross section is then $$\begin{aligned}
\frac{\sigma^{Ei}}{D^2} = \frac{4\pi}{k^2}\int_0^{\infty}~\ell~d\ell
\left[1-J_0\left(\frac{2k}{\ell^2}\sin^2\alpha\right)\right]\end{aligned}$$ This result provides some insight into partial wave analysis in the semi-classical regime, which approximately separates into two regions: For $l<\sqrt{k}\sin(\alpha)$, the integrand is nearly linear in $l$, while for larger $l$ it declines steeply as $k^2 \sin^4(\alpha)/l^3$. Using these approximations, the integral evaluates to $k\sin^2(\alpha)$, the exact result given below.
More carefully, we set $$\begin{aligned}
s = \frac{2k}{\ell^2}\sin^2\alpha
~~~,~~~\ell~d\ell = -k\sin^2\alpha \frac{ds}{s^2}\end{aligned}$$ and so express the total eikonal cross section in terms of a dimensionless integral $$\begin{aligned}
\frac{\sigma^{Ei}}{D^2} = \frac{4\pi\sin^2\alpha}{k}
\int_0^{\infty}~\frac{ds}{s^2}\left[1-J_0(s)\right]\end{aligned}$$ The integral evaluates to unity, with the result $$\begin{aligned}
\label{full_eikonal}
\frac{\sigma^{Ei}_{\rm tot}({\hat k}_i)}{D^2} =
\frac{4\pi}{k}\left[1-\left(\hat k_i\cdot\hat {\cal
E}\right)^2\right]\end{aligned}$$ Remarkably, the cross section is identically zero when the direction of incidence is aligned with the field axis. Since, as will be discussed below, this semi-classical cross section accurately describes dipolar collisions in the temperature range currently accessible experimentally, there are a variety of observables which might test this angle-dependence of the total elastic cross section. While equilibrium properties of the gas would not be sensitive to the angle-dependence, non-equilibrium properties, such as transmission of fast dipoles through a trapped dipolar gas, would be expected to show strong dependence on the alignment of the beam with the field axis.
Averaged over incident directions within a confined gas, one then expects $$\begin{aligned}
\label{eikonal_sigma}
\frac{\sigma_{\rm Ei}}{D^2} = \frac{8\pi}{3k} = \frac{8\pi}{3KD}\end{aligned}$$ which is the final result. As shown in Fig. 1, Glauber’s method yields not only the correct scaling, but quantitatively reproduces the universal results discovered in close-coupling calculations [@CTuniversal].
The basic structure of this result was surmised by Gallagher [@Gallagher] in a study of Rydberg-Rydberg collisions: from the uncertainty principle, an interaction of energy $1/b^3$ and lasting for a duration $b/k$, should satisfy $1/b^2k\sim 1$ in scaled units, so that $\sigma\sim b^2 = 1/k$. Using an isotropic $-1/r^3$ interaction, DeMille [@DeMille] also applied the eikonal approximation and determined a cross section $\sigma/D^2 =
2\pi^2/k$. Kajita’s Fourier technique [@kajita] yields the high-velocity result $\sigma/D^2 = 40\pi\sqrt{2}/3k$.
The rise and fall of universal scattering
=========================================
Figure 1 has already presented the message of universality, in that the Born and eikonal limits are achieved in the appropriate energy ranges. However, the calculations in this figure were carefully selected to have zero scattering length, by choosing an appropriate cutoff radius $r_0$. By changing $r_0$, we are able to generate any scattering length $a$. Experimentally, the value of $a$ can be altered by changing the electric or magnetic field strength. Changing $r_0$ could also change the scattering phase shift of any other partial wave. However, this is only likely for those partial waves whose centrifugal barrier lie below the collision energy, because these partial waves are the only ones to probe physics at the scale of $r_0$.
The importance of the centrifugal barriers in determining the range of applicability of the low-energy (Born) scaling was emphasized in [@mike; @mike2], through adiabatic calculations which converge much more rapidly than close-coupling calculations at the lowest energies. The adiabatic curves have pronounced barriers (in all but the s-wave channel) separating repulsive centrifugal behavior at large-$r$ from attractive dipolar behavior at small-$r$. The heights of the lowest barriers (approximately coincident with $E_D$) determine the range of energy over which the threshold behavior of the cross section is approximately constant. They also suggest the sensitivity to $r_0$ at intermediate energies, where incident flux can surmount the barrier.
Figure 2 shows the effect of changing $r_0$ on the “universal” cross section from Fig. 1. In Fig. 2a) is shown the result for even partial waves. The solid line is the $a=0$ result, and it amicably reaches the universal Born and eikonal limits. The other curves employ values of $r_0$ that produce scattering lengths of $a = 0.1
D$ (red) and $a = -0.1 D$ (blue). This change has made a significant difference in the low-energy limit, where now the Born approximation is merely a lower limit to the cross section. Notice also that for $a<0$ the cross section initially decreases with increasing energy, just as it does for alkali atoms [@Bohn99_PRA]
However, at higher energies $E>E_D$, this change in $r_0$ has no effect on $\sigma$. One way to look at this is that the phase shifts have changed for many partial waves, but because there are so many of them added together to get the cross section, these changes average out. Another point of view is that the semiclassical scattering occurs at high impact parameter, and is thus indifferent to what happens at $r = r_0$. In the intermediate energy range, the change is still quite significant, since phase shifts are changing for only those few partial waves that skip over their centrifugal barriers.
{width="90.00000%"}
Figure 2b) shows the same circumstance, but for odd partial waves. The same three values of $r_0$ are employed here, and so the three curves are labeled by the scattering lengths from part a). In this case the cross sections always approach the Born limit, since there is no aberrant s-wave scattering to derail them. In the high-energy limit, too, $\sigma$ is again insensitive to $r_0$. It is only in the intermediate energy range that a small deviation is seen. This suggests that for odd partial waves the behavior of the cross section is indeed nearly universal. This would be true, for example, in collisions of fermionic molecules (e.g., $^{40}$K$^{87}$Rb [@gpm]) in identical hyperfine states.
To further emphasize the consequence of an s-wave scattering length, Figure 3 reports the cross section $\sigma$ for the even partial waves as a function of $r_0$, in the threshold limit, using $E/E_D =
10^{-3}$. The range of $r_0$ shown here corresponds to a complete cycle of the scattering length from zero, through infinity, and back to zero again. Consequently, the numerically evaluated cross section shows a resonance, at which point the cross section is determined by $a^2$, not $D^2$. Even away from the resonance peak, the s-wave contribution can significantly increase the cross section. We are thus led to conclude that universality at low-energies, in cases where s-wave scattering is allowed, is similar to the universality for atoms. Namely, the form of the cross section (independent of energy) is universal, but its value relies on a (field-dependent) scattering length that must be determined empirically. The Born approximation does provide a useful lower limit, however. For odd partial waves, while resonances exist, they are shape resonances, hence narrow at low energies and less likely to destroy universal behavior.
The s-wave contribution to the Hamiltonian nominally vanishes, since $C_{00}^{(0)}=0$. However, it is not unreasonable that s-wave scattering has a strong influence near threshold. To see this, we evaluate an effective s-wave interaction at long range, via its coupling to the $l=2$ partial wave, in second-order perturbation theory (compare Ref. [@Avdeenkov02_PRA]): $$\begin{aligned}
V_0(r) \approx -{ |2C_{02}^{(0)}/r^3|^2 \over l(l+1)/2r^2} = - {C_4
\over r^4},\end{aligned}$$ where, in dipole units and using $l=2$, the coefficient is $C_4 =
4/3\sqrt{5}$. This in turn leads to a characteristic length scale for the s-wave interaction, analogous to the characteristic van der Waals length, $$\begin{aligned}
r_4 = \left( {2 \mu C_4 \over \hbar^2} \right)^{1/2} = \left( 2C_4
\right)^{1/2}.\end{aligned}$$ In dipole units, this is $r_4 = 1.09 D$, comparable to the dipole length itself. Based on this consideration, it is perhaps not too surprising that s-wave scattering plays a significant role.
{width="90.00000%"}
Dependence on incident angle
============================
Thus far we have focused on the cross section as integrated over all incident angles. One of the interesting aspects of the dipole-dipole interaction, however, is its anisotropy. The cross section $\sigma_{{\rm tot}}(\theta_i)$ may therefore depend on the angle $\theta_i=\arccos(\hat k_i\cdot\hat {\cal E})$ of the incident collision axis, with respect to the polarization direction. This cross section is easily calculated numerically from (\[sigma\_tot\]), and also from the useful eikonal estimate (\[full\_eikonal\]).
To show the utility of the eikonal expression, we present in Figure 4 $\sigma_{{\rm tot}}$ versus $\theta_i$, for a collision energy $E/E_D=10^{4}$ where the eikonal approximation should be fairly accurate. The total close-coupled cross section (black solid line) is the sum of contributions from even and odd partial waves. Note that the contributions from these two sets of partial waves are nearly equal here in the semiclassical limit where many partial waves contribute, and effects of dipole-indistinguishability are small. The angular distribution of each shows oscillations, but in the sum, representing distinguishable particles, the angular dependence is smooth. Moreover, for angles where the collision axis is orthogonal to the polarization axis, $\theta_i \approx \pi/2$, the eikonal approximation (dotted line) is quite good.
A major deviation occurs, however, for dipoles aligned parallel to the collision axis. Here the eikonal result calls for vanishing cross section, whereas the close coupling calculation yields a non-zero cross section. For $\theta_i=0$, the incident wave $e^{i\vec k_i\cdot\vec r}$ is invariant under rotations about the field axis, and so contains only $m=0$ partial waves. It is not too surprising that a semi-classical analysis will break down for low-$m$ states. Furthermore, for $m=0$ states, the wavefunction is large where the potential is strongest (near $r=0$), so the eikonal assumption $V/\epsilon<<1$ is no longer valid. Most interesting about this deviation from eikonal behavior, is the importance of back-scattering from the strong potential in this geometry, as indicated by the pronounced difference between even and odd partial waves. Observations near $\theta_i=0$ will accordingly be most sensitive to exchange scattering, and, presumably to short range physics.
![Dependence of scattering cross sections $\sigma_{{\rm tot}}$ on the incident angle $\theta_i$ between the collision axis and the polarization axis of the dipoles. This calculation is performed at a high collision energy, $E/E_D = 10^{4}$. The distribution is well-approximated using the eikonal result (\[full\_eikonal\]), except when the collision and polarization axes nearly coincide.](Fig4.eps){width="90.00000%"}
Cross section variations with the angle of incidence have also been studied at low energies [@mike; @mike2]. When the $s$-wave scattering length is negligible, a universal anisotropic distribution is obtained, entirely due to long-range scattering. However, when the $s$-wave scattering length dominates, near the peak in Figure 3, a completely isotropic distribution is found, as in the case of ultracold atomic collisions. Interestingly, this implies that effects of anisotropy are to be seen at the lowest temperatures only when the scattering length is small, and cross sections are accurately represented by the Born approximation.
What this means for you
=======================
Scaled units are fine for proving a theoretical point, as we have hoped to do here. However, since dipole length scales vary widely between different molecules and at different electric field strengths, it is also useful to consider specific examples that measure cross sections in cm$^2$.
Before presenting such an example, we first recapitulate our main results, cast in terms of the explicit dimensionful factors. The Born result, valid in the ultracold limit, is $$\begin{aligned}
\sigma_{\rm Born}^e &=& 1.117 {M^2 \langle \mu_1 \rangle^2 \langle
\mu_2 \rangle^2 \over \hbar^4} + 4 \pi a^2 \nonumber \\
\sigma_{\rm Born}^o &=& 3.351 {M^2 \langle \mu_1 \rangle^2 \langle
\mu_2 \rangle^2 \over \hbar^4}.\end{aligned}$$ The semiclassical result, valid for cold collisions, $E>E_D =
\hbar^6 / M^3 \langle \mu_1 \rangle^2 \langle \mu_2 \rangle^2$, is $$\begin{aligned}
\sigma_{\rm Ei} = {8 \pi \over 3} { \langle \mu_1 \rangle \langle
\mu_2 \rangle \over \hbar} \sqrt{ {M \over 2 E}}.\end{aligned}$$ (Recall that our eikonal derivation does not distinguish between even and odd partial wave contributions. To a good approximation, both the even and odd contributions would be half this value.) These formulas show explicitly that the cross sections at low and high energy differ not only in their dependence on energy, but also in their dependence on the parameters – reduced mass and dipole moments – of the molecules.
{width="90.00000%"}
To give a concrete example, consider the ground-state $^{40}$K$^{87}$Rb molecules that were recently produced at temperatures of several hundred nK [@gpm]. This molecule has a dipole moment of 0.566 Debye, and, being a fermion, would collide only in odd partial waves if it is trapped in a single quantum state. We therefore plot in Fig. 5 the odd partial wave cross section computed above, but cast in realistic units for this molecule. The largest cross section corresponds to the full dipole moment, $\langle \mu \rangle = 0.566$ Debye. Each successively lower curve divides the dipole moment in half from the previous one. For this reason, each low-energy cross section drops by a factor of 16 from the one above, while at high energy each cross section drops by a factor of 4. Because the dipole moment is something that can be changed by the application of a greater or lesser electric field, cross sections spanning this stunning range of magnitudes should be observable in experiments.
Also interesting is the energy scale encompassed by this figure. In the experiment, the gas is trapped at a temperature of 350 nK. At low electric field values, hence low dipole moments, these molecules are in the ultracold regime, and scatter according to the Born prescription. At higher fields, however, $E_D$ approaches the temperature of the gas, and experiments might start to observe the non-universal behavior of the scattering.
In summary, we have characterized the total scattering cross section for dipolar molecules, both in the cold limit $E_D < E <B_e$, and in the ultracold limit $E<E_D$. The behavior of this scattering is universal for cold collisions, and nearly so for ultracold collisions. In the temperature regime intermediate between these two, universality breaks down, and empirical cross sections will likely reveal information about the intermolecular potential energy surface.
The authors acknowledge the financial support from the NSF (J. L. B.), the ARC (C. T.), and the University of Kentucky (M. C.)
References {#references .unnumbered}
==========
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[^1]: $D$ is determined, roughly, by equating a typical centrifugal energy, $\hbar^2/MD^2$, to a typical dipolar energy, $\mu^2/D^3$. This is the same reasoning that leads to the definition of the Bohr radius, by equating centrifugal and Coulomb energies for hydrogen.
[^2]: For species whose [*magnetic*]{} dipole moment mediates the interaction, these scales can be quite different. For atomic chromium, for instance, one finds $D_{Cr}=27$ $a_0$, far smaller than its natural scattering length. A substantial body of literature now treats the complete details of the Cr-Cr interaction potential, for example [@Werner05_PRL; @Pavlovic05_PRA; @Stuhler07_JMO].
|
---
abstract: 'The correspondence of the Gauss-Bonnet (GB) and its modification (MGB) models of dark energy with the standard and generalized Chaplygin gas-scalar field models (SCG and GCG) have been studied in a flat universe. The exact solution of potentials and scalar fields, which describe the accelerated expansion of the universe, are reconstructed. According to the evolutionary behavior of the GB and MGB models, the same form of dynamics of scalar field and potential for different SCG and GCG models are derived. By calculating the squared sound speed of the MGB, GB model as well as the SCG, GCG, and investigating the GB-Chaplygin gas from the viewpoint of linear perturbation theory, we find that the best results which is consistent with the observation, may be appeared by considering the MGB-GCG. Also we find out some bounds for parameters.'
author:
- 'A. Khodam-Mohammadi[^1], E. Karimkhani[^2] and A. Alaei'
title: 'Gauss-Bonnet dark energy Chaplygin Gas Model '
---
Introduction
============
Astrophysical data which is out coming from distant Ia supernova [@A.J.Riess; @S.Perlmutter; @M.; @Hicken], Large Scale Structure (LSS) [@Tegmark; @K.; @Abazajian] and Cosmic Microwave Background (CMB)[@spergel; @komatsu], indicate that our universe undergoes with an accelerating expansion. This kind of expansion may be arisen by a mysterious energy component with negative pressure, so called, dark energy (DE).
However in the last decades, other models based on modified gravity ($F(R), F(G), F(R,\phi,X), F(T)$,...) have been proposed that have given another description of acceleration expansion of the universe. In these models, many authors have showed that all models of DE can be resolved by modifying the curvature term $R$ (Ricci scalar) of Einstein-Hilbert action with another curvature scalars such as any scalar function of $R$, Gauss-Bonnet term ($G$), torsion ($T$), scalar-tensor (X,$\phi$) and etc. (details are in Ref. [@modG1; @modG2; @modG3; @modG4; @modG5; @modG6; @modG7; @modG8; @modG9; @modG10; @modG11; @modG12; @modG13; @modG14] and references there in). Even, some authors found that, the early inflation, the intermediate decelerating expansion and late time acceleration expansion, could be described together in one model [@Chavanis].\
Lately, among many models of DE, dynamical models, which are considering a time dependent component of energy density and equation of state, have attracted a great deal of attention. Also, among many dynamical models, ones that represented by a power series of Hubble parameter and its derivative (*i.e.* $\dot{H},
H\dot{H},H^{2},...$) have been interested [@sola1409; @sola1412]. Also authors in [@sola31; @Perico; @Basilakos13; @bleem; @Lima] have shown that terms of the form $H^{3}$, $\dot{H}H^{2}$ and $H^{4}$ can be important for studying of the early universe. Hence, it would not be some thing strange to consider a DE density proportional to the Gauss-Bonnet (GB) term which is invariant in 4-dimensional. Besides, in geometrical meaning, the GB invariant has a valid dimension of energy density [@GRANDA]. Also authors in [@kofinas; @brown] showed that a unification between early time inflation and late time acceleration in a viable cosmology can be described by a coupling between GB term and a time varying scalar field [@nojiri222].\
The other successful model of DE is Chaplygin Gas model. The standard Chaplygin Gas model (SCG), first proposed by [@chaplygin; @kamens; @bento], regards as a perfect fluid which plays a dual role in the history of the universe: it behaves as dark matter in the first epoch of evolution of the universe and as a dark energy at the late time. Unfortunately this model has some inconsistency with observational data like SNIa, BAO, CMB [@gorini; @zhu; @bento2]. So Generalized Chaplygin Gas (GCG) [@bilic] and Modified Chaplygin Gas (MCG) models [@denbath; @Brown1; @Cai1] have been introduced in order to establish a viable cosmological model. It would be beneficial to study any relationship between SCG model and its modification while DE density behaves like GB invariant term as mentioned above. In this paper we would show that it leads interesting cosmological implications.\
As we would show in this paper, the EoS parameter of GB DE model on its own does not give rise to phantom phase of the universe. Besides in [@GRANDA], author shows that presence of matter drastically converts Friedmann equation into a nonlinear differential equation which alters the behavior of the EoS parameter which can lead to $w_{\circ }\sim -1.17$ and allows for quintom behavior. However, in this paper, we incorporate GB dark energy density with a SCG component without adding any matter content. In addition, corporation GB or MGB with different CG models (*i.e.* SCG and GCG) would be help full in order to obtain exact solution for scalar field and potential and would relieve us in order to determine some bounds for free parameters of models. So considering the cosmological solution for different compositions of GB and CG models could show the importance of each one. Also, we would succeed in the frame work where $\kappa ^{2}=8\pi G=M_{p}^{-2}=1$ and in the natural unit where $(\hbar=c =1)$.
The outline of this paper is as follows: In next section, we introduce the GB dark energy and calculate the deceleration and EoS parameters. Then, in subsections 2.1 , 2.2 and 2.3 we investigate corporation GB with SCG and GCG, in turn and then scalar field and scalar potential are obtained by exact solution. In section III, the same procedure has done for MGB energy density. In section IV, we would investigate Adiabatic Sound Speed, $v^{2}$, which is one of the critical physical quantity in the theory of linear perturbation. In section V, we discuss on behavior of scalar field, scalar potential and deceleration parameter *versus* $x$ for GB and MGB models and we gain some bounds for free parameters of models. Finally, we summarize our results in Sec. VI.
Gauss-Bonnet Dark energy in a flat universe
===========================================
The energy density of GB-DE is given by $$\rho _{D}=\alpha \mathcal{G} \label{1-1}$$where $\alpha $ is a positive dimensionless parameter [@GRANDA]. Gauss-Bonnet invariant $\mathcal{G}$ is topological invariant in four dimensions and may lead to some interesting cosmological effects in higher dimensional brane-world (for a review, see [@nojirii]). It is defined as $$\mathcal{G}=R^{2}-4R_{\mu \nu }R^{\mu \nu }+R_{\mu \nu \eta \gamma
}R^{\mu \nu \eta \gamma } \label{1-2}$$where $R$, $R_{\mu \nu }$ and $R_{\mu \nu \eta \gamma }$ are scalar curvature, Ricci curvature tensor and Riemann curvature tensor, respectively. In a spatially flat FRW universe $$d^{2}s=-dt^{2}+a^{2}\left( t\right) \left[ dr^{2}+r^{2}d\theta ^{2}
+r^{2}\sin ^{2}\theta d\phi^{2} \right] \label{1-3}$$the Eq. (\[1-1\]) takes the form $$\rho _{D}=24\alpha H^{2}\left( H^{2}+{\dot{H}}\right). \label{1-4}$$By using the energy density $\rho _{D}$, without any matter component, the Friedmann equation in flat universe in reduced Planck mass unit ($8\pi G=\hbar=c=1$) is $$H^{2}=\frac{1}{3}\rho _{D}=8\alpha H^{2}\left( H^{2}+{\dot{H}}
\right). \label{1-5}$$Defining the e-folding $x$ with definition $x=lna=-ln(1+z)$, where $z$ is the redshift parameter and using $d/d(x)=\frac{1}{H}d/d(t)$, we get the following differential equation $$H^{2}+\frac{1}{2}\frac{dH^{2}}{dx}-\frac{1}{8\alpha }=0,
\label{1-6}$$which immediately gives the solution $$H( x) =\sqrt{\frac{1}{8\alpha }(1+\xi e^{-2x})}. \label{1-8}$$The parameter $\xi$ is a constant of integration which is obtained by $\xi =8\alpha H_{0}^{2}-1$. Also it gives $\alpha=(1+\xi)/(8
H_0^2)$. Using the continuity equation $$\overset{\cdot }{\rho _{D}}+3H\left( 1+w_{D}\right) \rho _{D}=0
\label{1-9}$$and Eqs. (\[1-4\]),(\[1-5\]), the equation of state (EoS) parameter yields $$w_{D}=-1-\frac{\dot{\rho_D}}{3H\rho_{D}}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}}=-1-\frac{2}{3}(\frac{1}{8\alpha
H^2}-1). \label{1-11}$$ It is more preferable to write above equation in term of e-folding, $x$. Hence, by using Eq. (\[1-8\]), the EoS parameter can be rewritten as $$w_{D}=-1+\frac{2}{3}\left( \frac{\xi e^{-2x}}{1+\xi e^{-2x}}\right).
\label{1-12}$$We see that the constant $\xi $ plays a crucial role in the behavior of the EoS parameter. For $ \xi =0$ (*i.e.* $8\alpha
H_{0}^{2}=1$), the EoS parameter for $\Lambda CDM$ model ($w_{\Lambda }=-1$) is retrieved. For $\xi >0$ the expanding universe accelerates in quintessence phase ($-1<w_{D}<-1/3$). Using Eqs. (\[1-5\]) and (\[1-8\]), the deceleration parameter is calculated as $$q=-1-\frac{{\dot {H}}}{H^{2}}=-\frac{1}{8\alpha
H^{2}}=-\frac{1}{1+\xi e^{-2x} }. \label{1-13}$$ Since $\alpha $ and $H_{0}^{2}$ are positive parameters, so $\xi $ always must be greater than $-1$. Therefore, the deceleration parameter is always negative except for $-1<\xi <0$. In this way, the universe which is characterize by GB dark energy model could not exhibit a transition from deceleration to acceleration phase for $\xi\geq0 $, against what we expect from observations.
Gauss Bonnet Standard Chaplygin Gas {#GB-SCG}
-----------------------------------
The SCG is a perfect fluid with an equation of state as $$p_{SCG}=-\frac{A}{\rho }, \label{2-1}$$where $p$, $\rho $ and $A$ are pressure, energy density and a positive constant respectively. By substituting Eq.(\[2-1\]) into the continuity equation (\[1-9\]), the energy density immediately solved $$\rho_{SCG} =\sqrt{A+Be^{-6x}}, \label{2-2}$$where $B$ is an integration constant [@khodam2641]. Using the standard scalar field DE model in which the energy density and pressure are defined as $$\begin{aligned}
\rho _{\phi } &=&\frac{1}{2}{\dot {\phi }}^{2}+V\left( \phi \right)
= \sqrt{A+Be^{-6x}}, \label{2-3} \\
p_{\phi } &=&\frac{1}{2}{\dot {\phi} }^{2}-V\left( \phi \right)
=\frac{-A}{\sqrt{A+Be^{-6x}}}, \label{2-4}\end{aligned}$$and equating $p_{SCG}=p_{\phi }$ and $\rho _{SCG }=\rho _{\phi }$, the scalar potential and kinetic energy term of SCG model are given as $$V\left( \phi \right) =\frac{2A+Be^{-6x}}{2\sqrt{A+Be^{-6x}}}
\label{2-5}$$$${\dot {\phi }}^{2}=\frac{Be^{-6x}}{\sqrt{A+Be^{-6x}}}. \label{2-6}$$Also the EOS parameter becomes $$w_{SCG}=\frac{p}{\rho }=-\frac{A}{A+Be^{-6x}} \label{2-8}$$ Equating the energy densities (*i.e.*, $\rho_{SCG}=\rho_{D}$) and EoS parameters (*i.e.*, $w_{SCG}=w_{D}$), after using the Friedmann equation (\[1-5\]), constants $A$ and $B$ immediately given by $$\begin{aligned}
A &=& \frac{3}{(8\alpha )^{2}}\left[ (2+\xi e^{-2x})^{2}-1\right],
\label{2-9} \\
B &=& e^{6x}\left[ \left( \frac{3}{8\alpha }(1+\xi
e^{-2x})\right) ^{2}-A\right], \label{2-7}\end{aligned}$$ and hence the scalar potential and kinetic energy term rewritten as $$\begin{aligned}
V\left(x\right) &=& \frac{1}{8\alpha}\left( 3+2\xi
e^{-2x}\right)=\frac{H_0^2}{1+\xi}\left( 3+2\xi e^{-2x}\right),
\label{2-10}\\
\dot {\phi} &=& \frac{1}{2}\sqrt{\frac{\xi e^{-2x}}{\alpha }}.
\label{2-11}\end{aligned}$$ By inserting $\phi ^{\prime }=\dot {\phi}/H$ , where prime means derivative with respect to $x=\ln a$, the differential equation (\[2-11\]) gives the normalized scalar field ($\phi =1$ at present, $x=0$) in terms of $x$ as $$\phi =1-\frac{\sqrt{2}}{2}\ln\left( \frac{1+2\xi e^{-2x}+2\sqrt{\xi
e^{-2x}(1+\xi e^{-2x})}}{1+2\xi +2\sqrt{\xi(1+\xi)}}\right).
\label{2-14}$$It is easy to see that from Eq. (\[1-8\]), at present, we must have $1+\xi \geq 0$ and from (\[2-14\]), it must be required that $\xi(1+\xi) \geq 0$. Therefore in this model, we must have $\xi \geq
0$. As it is shown in Fig. \[fig11\], the normalized scalar field grows up to a saturated value at late time in such a way that this value exceeds for larger values of $\xi$. Also Eq. (\[2-10\]) shows that the universe goes to a stable equilibrium at infinity where $V(\infty)=3H_0^2/(1+\xi)$ and from (\[2-14\]), the scalar field reaches to $\phi(\infty)=1+(\sqrt{2}/2)\ln\left(1+2\xi
+2\sqrt{\xi(1+\xi)}\right).$
Gauss Bonnet Generalized Chaplygin Gas
--------------------------------------
The equation of state of generalized Chaplygin gas (GCG) defined as [@bento4] $$p=-\frac{A}{\rho ^{\delta -1}}, \label{3-1}$$where $A$ is a constant and $1\leq \delta \leq 2$. For $\delta =2$, it reaches to SCG model. The energy density, similar to previous case, is given by $$\rho_{GCG} =\left( A+Be^{\left( -3\delta x\right)
}\right) ^{\frac{1}{\delta }}, \label{3-2}$$and the scalar field model gives energy density and pressure of GCG as $$\rho _{\phi }=\frac{1}{2}{\dot {\phi} }^{2}+V\left( \phi \right)
=\left( A+Be^{-3\delta x}\right) ^{\frac{1}{\delta }}, \label{3-3}$$ $$p_{\phi }=\frac{1}{2}{\dot {\phi} }^{2}-V\left( \phi \right)
=-A\left( A+Be^{-3\delta x}\right) ^{-\frac{\delta -1}{\delta }}.
\label{3-4}$$After forward calculation, three quantities: the scalar potential, kinetic term and EoS parameter are given by $$V\left( x\right) =\frac{2A+Be^{-3\delta x}}{2\left( A+Be^{-3\delta
x}\right) ^{\frac{\delta -1}{\delta }}}, \label{3-5}$$$${\dot {\phi} }^{2}=\frac{Be^{-3\delta x}}{\left( A+Be^{-3\delta
x}\right) ^{\frac{\delta -1}{\delta }}}, \label{3-7}$$$$w_{GCG}=\frac{p}{\rho }=-\frac{A}{A+Be^{-3\delta x}}. \label{1-10}$$Also same as previous, the constants $A$ and $B$ reconstructed as $$\begin{aligned}
A&=&\frac{3+\xi e^{-2x}}{\left( 8\alpha \right) ^{\delta }}\left[
3\left( 1+\xi e^{\left( -2x\right) }\right) \right] ^{\delta -1},
\label{3-10} \\
B&=&e^{3\delta x}\left[ \left( \frac{3}{8\alpha }(1+\xi
e^{-2x})\right) ^{\delta }-A\right] \label{3-8}\end{aligned}$$and the potential and dynamics of GB-GCG can be written as $$\begin{aligned}
V\left( x\right) &=&\frac{3+2\xi e^{-2x}}{8\alpha }, \label{3-11}
\\
{\dot {\phi} }&=&\frac{1}{2}\sqrt{\frac{\xi e^{-2x}}{\alpha }}.
\label{3-12}\end{aligned}$$As it is seen, the potential and dynamics of GB-GCG are not a function of parameter $\delta $ and are exactly similar to previous case (see Eqs. (\[2-10\]) and (\[2-11\])). Therefore the reconstructed scalar potential and scalar field obtained by previous Eqs. (\[2-10\]) and (\[2-14\]). It is worthwhile to mention that both models that we have been studied, encourage with an essential problem. Despite of observational predictions, the phase transition between deceleration to acceleration expansion did not happen in GB-DE model. Therefore we will study on the MGB model, which may alleviate this problem.
Modified Gauss Bonnet Dark Energy
=================================
The energy density MGB has been defined by $$\rho _{D}=3H^{2}(\gamma H^{2}+\lambda \overset{\cdot }{H}),
\label{4-1}$$where $\gamma $ and $\lambda $ are dimensionless constants [@GRANDA]. The Friedmann equation in dark dominated flat universe gives $$\gamma H^{2}+\frac{1}{2}\lambda \left( \frac{dH^{2}}{dx}\right)
-1=0, \label{4-2}$$and the Hubble parameter given by $$H(x)=\sqrt{\frac{1}{\gamma }(1+\eta e^{-\frac{2\gamma x}{\lambda
}})}, \label{4-3}$$ where $\eta$ is an integration constant which is obtained by $\eta
=\gamma H_{0}^{2}-1$. The EoS parameter becomes $$w_{D}=-1-\frac{2}{3}\frac{\overset{\cdot }{H}}{H^{2}}=-1+\frac{2\gamma }{%
3\lambda }\left( \frac{\eta e^{-\frac{2\gamma x}{\lambda }}}{%
1+\eta e^{-\frac{2\gamma x}{\lambda }}}\right) \label{4-4}$$and deceleration parameter is obtained as follows $$q=-1-\frac{\overset{\cdot }{H}}{H^{2}}=-1+\frac{\gamma }{\lambda
}\left(
\frac{\eta e^{-\frac{2\gamma x}{\lambda }}}{1+\eta e^{-\frac{%
2\gamma x}{\lambda }}}\right). \label{4-21}$$ For positive values of $\gamma$ and $\lambda $, from Eq. (\[4-3\]), it is easy to see that $\eta $ must be always greater than $-1$ and from Eq. (\[4-21\]), a transition from deceleration to acceleration is expected provided that $\eta \geq 0$. Detailed discussion were transferred to section \[discussion\].
Modified Gauss Bonnet And SCG
-----------------------------
Now we want to investigate on the correspondence between MGB-SCG models and reconstruct the potential and dynamics of scalar field. Same as before, equating energy densities (*i.e.*, Eqs. (\[2-2\] ) and (\[4-1\])) and EoS parameters (*i.e.*, (\[2-8\]) and (\[4-4\])), yield $$\begin{aligned}
A&=&\frac{9}{\gamma ^{2}}\left( 1+\eta e^{-\frac{2\gamma x}{\lambda }%
}\right) \left[ 1+(1-\frac{2\gamma }{3\lambda })\eta e^{
-\frac{2\gamma x}{\lambda }}\right], \label{4-7}\\
B&=&e^{6x}\left[ \left( \frac{3}{\gamma }(1+\eta e^{-\frac{2\gamma x}{%
\lambda }})\right) ^{2}-A\right]. \label{4-5}\end{aligned}$$By substituting $A$ and $B$ in Eqs. (\[2-5\]) and (\[2-6\]), we find $$\begin{aligned}
V\left( x\right) &=&\frac{3}{\gamma }\left[ 1+(1-\frac{\gamma }{3\lambda }%
)\eta e^{-\frac{2\gamma x}{\lambda }}\right], \label{4-8}\\
\dot\phi&=&\sqrt{\frac{2\eta }{\lambda }e^{-\frac{%
2\gamma x}{\lambda }}} , \label{4-9}\end{aligned}$$which immediately gives the normalized scalar field as $$\phi =1-\frac{\sqrt{2}}{2}\sqrt{\frac{\gamma}{\lambda}}\ln\left(
\frac{1+2\eta e^{-\frac{2\gamma x}{\lambda }}+2\sqrt{\eta
e^{-\frac{2\gamma x}{\lambda }}(1+\eta e^{-\frac{2\gamma x}{\lambda
}})}}{1+2\eta +2\sqrt{\eta(1+\eta)}}\right). \label{4-10}$$The behavior of scalar field in this model is the similar to GB-DE model as discussed in Sec. \[GB-SCG\].
Modified Gauss Bonnet And GCG
-----------------------------
As previous, the constants $A$ and $B$ are $$\begin{aligned}
A&=&(\frac{3}{\gamma })^{\delta }\left( 1+\eta e^{^{-\frac{2\gamma x}{\lambda }%
}}\right) ^{\delta -1}\left[ 1+\left( 1-\frac{2\gamma }{3\lambda
}\right) \eta e^{-\frac{2\gamma x}{\lambda }}\right], \label{5-3}\\
B&=&e^{3\delta x}\left[ \left( \frac{3}{\gamma }(1+\eta e^{-\frac{2\gamma x}{%
\lambda }}\right) ^{\delta }-A\right]. \label{5-1}\end{aligned}$$ and the potential and dynamics of MGB-GCG are given by $$\begin{aligned}
V(\phi )&=&\frac{3}{\gamma }\left[ 1+\left( 1-\frac{\gamma
}{3\lambda }\right) \eta e^{^{-\frac{2\gamma x}{\lambda }}}\right]
\label{5-4}\\
\dot\phi&=&\sqrt{\frac{2\eta }{\lambda }e^{^{-\frac{2\gamma x}{%
\lambda }}}} \label{5-5}\end{aligned}$$ which are exactly similar to (\[4-8\]) and (\[4-9\]) in previous model. Therefore the behavior of normalized scalar field and potential are the same as MGB-SCG model.
Adiabatic Sound Speed
=====================
Investigation of the squared of sound speed, $v^{2}$, would help us to determine the growth of perturbation in linear theory [@peebleratra]. The sign of $v_{s}^{2}$ plays a crucial role in determining the stability of the background evolution. Positive sign of $v^{2}$ shows the periodic propagating mode for a density perturbation and probably represents an stable universe against perturbations. The negative sign of it shows an exponentially growing/decaying mode in density perturbation, and can show sounds of instability for a given model. The squared of sound speed is defined as [@peebleratra] $$v^{2}=\frac{dP}{d\rho }=\frac{\overset{\cdot }{P}}{\overset{\cdot
}{\rho }} \label{11-1}$$ In a dark dominated flat universe, it can be written as $$v^{2}=-1-\frac{1}{3}\left( \frac{\overset{\cdot \cdot }{H}}{\overset{\cdot }{%
H}H}\right). \label{11-3}$$and it immediately gives a constant squared of sound speed for GB-DE as $v^{2}=-1/3$. Therefore it may reveal an instability against the density perturbation in GB-DE model. For MGB-DE, Eq.(\[11-3\]) gives $$v^{2}=-1+\frac{2\gamma }{3\lambda }. \label{11-4}$$ It shows that $v^{2}$ can be positive provided that $\gamma/\lambda>3/2$. Thus an stable DE dominated universe may be achieved in this model. In the next section we would improve this bound for $\gamma/\lambda$ in a proper way.
Discussion
==========
We are interesting to focus on MGB-DE model. At first, we start with Eq. (\[4-21\]) and plot the deceleration parameter with respect to $x$ in Fig. \[fig1\]. It shows that the deceleration parameter transits from deceleration ($q>0$) to acceleration ($q<0$) in some point at the past. The parameters $\eta $ and $\gamma/\lambda$ play a crucial rule for this point. As $\eta $ or $\gamma/\lambda$ adopt bigger values, the transition point approaches to present time. By choosing the best values of $q_{0}~(\sim -0.6)$ and inflection point as $(x\simeq -0.5)$ which has been parameterized recently [@pavon; @et; @al; @kh-ka; @daly], we obtain some bounds for $\eta $ and $\gamma/\lambda$ as follow $$0<\eta <2.5~~~~~~ 1.5\leq \frac{\gamma }{\lambda }\leq 3$$ Using Eq. (\[4-8\]) for MGB model, we plot $\overset{\sim }{V(\phi
)}=\gamma V(\phi )$ *versus* $x$ for different values of $\gamma/\lambda$ and $\eta =1.5$ in Fig. \[fig2\]. This figure shows that as time goes, $\overset{\sim }{V(\phi )}$ is decreasing to small values and the potential will reach to a constant at infinity. In addition, by increasing the ratio of $\gamma/\lambda$, the tracking potential adopts bigger values at future.
As it is seen, the potential describes a tracker solution. According to the quintessential tracker solution, our universe undergoes a phase from $w=0$ to $w=-1$ and the effective EoS is $w_{eff}=-0.75$ [@steinhardt]. The huge advantage of the tracker solution is that it allows the quintessence model to be insensitive to initial conditions [@YOO]. So we use this feature in order to improve obtained bounds of parameters. In this way, Eqs. (\[4-4\]) and (\[5-4\]), for matter dominated universe ($w=0$), leads to $
V(\phi )=3/(2\gamma -3\lambda )$. On the other hand the quantity $
V(\phi )$ for quintessence barrier ($w=-1/3$) reach to $V(\phi )=2/(
\gamma -\lambda )$, so that the value $\gamma/\lambda=1$ is illegal. It is also consistent with what we got from investigation of the deceleration parameter. Finally, the potential might give a tracking solution provided that $1.5<\gamma/\lambda\leq 3$.
conclusion
==========
In this paper, the reconstruction of GB-DE and some variety of Chaplygin gas have been studied. We obtained exact solutions for reconstructed scalar field and its potential in each models (GB-SCH, GB-MCG, MGB-SCG and MGB-MCG). According to cosmological predictions and historical evolutions, some models should be rejected (*i.e.*, models combined with GB-DE) and another models which have been combined with MGB-DE can be permitted to express the evolution of the universe. The equation of state and deceleration parameters for both GB and MGB models were calculated. In GB-DE model, the deceleration parameter was always negative except for $-1<\xi <0$. This fact was shown that a transition from deceleration to acceleration expansion could not have happened in the past that is contrary to the facts of cosmology. Also it was easily shown that the EoS parameter in GB-DE model would not ever reach to phantom phase (*i.e.* $w_D<-1$). We showed that for $\xi =0$ (*i.e.* $8\alpha H_{0}^{2}=1$), the EoS parameter for $\Lambda$CDM model was retrieved. Investigation on the squared of sound speed, revealed an instability of model against density perturbation in GB-DE model.
In MGB-DE model, we found that the transition from deceleration to acceleration is permitted just for a limited range of values of $\eta $ and $\gamma/\lambda $. Choosing the best values for deceleration parameter at present and deflection point, according to observations, some bounds of $0<\eta <2.5$ and $ 1.5\leq
\gamma/\lambda\leq 3$ were obtained. We showed that by redefining $\gamma V(\phi )=\overset{\sim }{ V(\phi )}$, the scalar potential decreased to smaller values and will reach to a saturated constant at late time. Our investigation on $ V(\phi )$ for two phases, matter dominate and quintessence, showed that $\gamma/\lambda$, could not take two values $1$ and $3/2$.
It will be interesting to find the constraints of these models against the data of cosmological observations and structure formation. We hope to discuss these issues in the future.
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[^1]: `Email`:
[^2]: `Email`:
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=1
Introduction
============
Graphical methods have proven useful for many branches of theoretical and mathematical physics. First of all, it is the method of Feynman diagrams which is the main working tool of quantum field theory [@Fey49; @HooVel73]. Rather developed graphical methods are used in the quantum theory of angular momentum [@BazKas72; @VarMosKhe88; @YutLevVan62], the general relativity [@Pen04; @PenRin84; @PenRin86], and physical applications of the group theory [@Cvi08]. The graphical methods used in the theory of quantum integrable models of statistical physics [@Bax82] were successfully applied to the problems of enumerative combinatorics [@Ava09; @AvaDuc09; @BehFisKon17; @Gra17; @HagMor16; @Kup96; @Kup02; @RazStr04; @RazStr06b; @RazStr06c].
In this paper, we systematise and develop the graphical approach to the investigation of integrable vertex statistical models and the corresponding quantum spin chains. Here the most common vertex model is a two-dimensional quadratic lattice formed by vertices connected by edges. The vertices have weights determined by the states of the adjacent edges. The consideration of such systems begins with the definition of suitable integrability objects that possess necessary properties. The initial objects here are $R$-operators and basic monodromy operators encoding the weights of the vertices. An $R$-operator acts in the tensor square of a vector space called the auxiliary space, and a monodromy operator acts in the tensor product of the auxiliary space and an additional one called the quantum space. To ensure integrability, the $R$-operator must satisfy the Yang–Baxter equation, and the monodromy operator the so-called $RMM$-equation which, in the case when the auxiliary space coincides with the quantum one, reduces to the Yang–Baxter equation [@Bax82]. The necessary equations are satisfied automatically if one obtains integrability objects using the quantum group approach formulated in the most clear form by Bazhanov, Lukyanov and Zamolodchikov [@BazLukZam96; @BazLukZam97; @BazLukZam99]. The method proved to be efficient for the construction of $R$-operators [@BooGoeKluNirRaz10; @BooGoeKluNirRaz11; @BraGouZha95; @BraGouZhaDel94; @KhoTol92; @LevSoiStu93; @MenTes15; @TolKho92; @ZhaGou94], monodromy operators and $L$-operators [@BazLukZam96; @BazLukZam97; @BazLukZam99; @BazTsu08; @BooGoeKluNirRaz10; @BooGoeKluNirRaz11; @BooGoeKluNirRaz13; @BooGoeKluNirRaz14b; @BooGoeKluNirRaz14a; @NirRaz16a; @Raz13], and for the proof of functional relations [@BazHibKho02; @BazLukZam99; @BazTsu08; @BooGoeKluNirRaz14b; @BooGoeKluNirRaz14a; @Koj08; @NirRaz16c; @NirRaz16a].
A quantum group is a special kind of a Hopf algebra arising as a deformation of the universal enveloping algebra of a Kac–Moody algebra. The concept of the quantum group was introduced by Drinfeld [@Dri87] and Jimbo [@Jim85]. Any quantum group possesses the universal $R$-matrix connecting its two comultiplications. The universal $R$-matrix is an element of the tensor square of two copies of the quantum group. In the framework of the quantum group approach, the integrability objects are obtained by choosing representations for the factors of that tensor product and applying them to the universal $R$-matrix. Here one identifies the representation space of the first factor with the auxiliary space, and the representation space of the second one with the quantum space. However, the roles of the factors can be interchanged. The universal $R$-matrix satisfies the universal Yang–Baxter equation. This leads to the fact that the received objects have certain required properties. Besides, such integrability objects satisfy some additional relations, such as unitarity and crossing relations, which follow from the general properties of the universal $R$-matrix and used representations.
The structure of the paper is as follows. In Section \[section2\] we give the definition of the class of quantum groups, called quantum loop algebras, used in the quantum group approach to the study of integrable vertex models of statistical physics. Then we discuss properties of integrability objects and introduce their graphical representations.
Section \[section3\] is devoted to the case of quantum loop algebras ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$. We describe some finite-dimensional representations and derive an expression for the $R$-operator associated with the first fundamental representation of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$. Explicit forms of the unitarity and crossing relations are discussed.
The graphical methods of Section \[section2\] are used in Section \[section4\] to derive the commutativity conditions for the transfer matrices of lattices with boundary. Such conditions are relations connecting the corresponding $R$-operator with left and right boundary operators. For the first time the commutativity conditions for lattices with boundary were given by Sklyanin in paper [@Skl88] based on a previous work by Cherednik [@Che84]. In paper [@Skl88] rather restrictive conditions on the form of the $R$-operators were imposed. In a number of subsequent works [@deVGon93; @FanShiHouYan97; @MezNep91] these limitations were weakened with the corresponding modification of the commutativity conditions. Finally, Vlaar [@Vla15] gave the commutativity condition in the form which requires no essential limitations on the $R$-operator. It is this form which is obtained by using the graphical method.
We use the standard notations for $q$-numbers $$\begin{gathered}
[\nu]_q = \frac{q^\nu - q^{-\nu}}{q - q^{-1}}, \qquad \nu \in {\mathbb C}, \qquad [n]_q! = \prod_{k = 1}^n [k]_q, \qquad n \in {\mathbb Z}_{\ge 0}.\end{gathered}$$ Depending on the context, the symbol $1$ means the unit of an algebra or the unit matrix.
Quantum loop algebras and integrability objects {#section2}
===============================================
Quantum loop algebras
---------------------
### Some information on loop algebras {#s:siola}
Let ${\mathfrak g}$ be a complex finite-dimensional simple Lie algebra of rank $l$ [@Hum80; @Ser01], ${\mathfrak h}$ a Cartan subalgebra of ${\mathfrak g}$, and $\Delta$ the root system of ${\mathfrak g}$ relative to ${\mathfrak h}$. We fix a system of simple roots $\alpha_i$, $i \in {[1 \, . \, . \, l]}$. It is known that the corresponding coroots $h_i$ form a basis of ${\mathfrak h}$, so that $$\begin{gathered}
{\mathfrak h}= \bigoplus_{i = 1}^l {\mathbb C}h_i.\end{gathered}$$ The Cartan matrix $A = (a_{i j})_{i, j \in {[1 \, . \, . \, l]}}$ of ${\mathfrak g}$ is defined by the equation $$\begin{gathered}
a_{i j} = \langle \alpha_j, h_i \rangle. \label{dcm}\end{gathered}$$ Note that any Cartan matrix is symmetrizable. It means that there exists a diagonal matrix $D = \operatorname{diag}(d_1, \ldots, d_l)$ such that the matrix $D A$ is symmetric and $d_i$, $i \in {[1 \, . \, . \, l]}$, are positive integers. Such a matrix is defined up to a nonzero scalar factor. We fix the integers $d_i$ assuming that they are relatively prime.
Denote by $(\cdot | \cdot)$ an invariant nondegenerate symmetric bilinear form on ${\mathfrak g}$. Any two such forms are proportional one to another. We will fix the normalization of $(\cdot | \cdot)$ below. The restriction of $(\cdot | \cdot)$ to ${\mathfrak h}$ is nondegenerate. Therefore, one can define an invertible mapping $\nu \colon {\mathfrak h}\to {\mathfrak h}^*$ by the equation $$\begin{gathered}
\langle \nu(x), y \rangle = (x | y),\end{gathered}$$ and the induced bilinear form $(\cdot | \cdot)$ on ${\mathfrak h}^*$ by the equation $$\begin{gathered}
(\lambda | \mu) = \big(\nu^{-1}(\lambda) | \nu^{-1}(\mu)\big).\end{gathered}$$ We use one and the same notation for the bilinear form on ${\mathfrak g}$, for its restriction to ${\mathfrak h}$ and for the induced bilinear form on ${\mathfrak h}^*$.
Using the mapping $\nu$, given any root $\alpha$ of ${\mathfrak g}$, one obtains the following expression for the corresponding coroot $$\begin{gathered}
{\alpha \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} = \frac{2}{(\alpha | \alpha)} \nu^{-1}(\alpha). \label{alphavee}\end{gathered}$$ Hence, we can write $$\begin{gathered}
a_{i j} = \frac{2}{(\alpha_i | \alpha_i)} (\alpha_j | \alpha_i) =\frac{2}{(\alpha_i | \alpha_i)} (\alpha_i | \alpha_j).\end{gathered}$$ It is clear that the numbers $(\alpha_i | \alpha_i)/2$ are proportional to the integers $d_i$. We normalize the bilinear form $(\cdot | \cdot)$ assuming that $$\begin{gathered}
\frac{1}{2} (\alpha_i | \alpha_i) = d_i. \label{di}\end{gathered}$$
Denote by $\theta$ the highest root of ${\mathfrak g}$ [@Hum80; @Ser01]. Remind that the extended Cartan matrix $A^{(1)} = (a_{ij})_{i, j \in {[0 \, . \, . \, l]}}$ is defined by relation (\[dcm\]) and by the equations $$\begin{gathered}
a_{0 0} = \langle \theta, {\theta \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} \rangle, \qquad a_{0 j} = - \langle \alpha_j, {\theta \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} \rangle, \qquad a_{i 0} = - \langle \theta, h_i \rangle, \label{decm}\end{gathered}$$ where $i, j \in {[1 \, . \, . \, l]}$. We have $$\begin{gathered}
\theta = \sum_{i = 1}^l a_i \alpha_i, \qquad {\theta \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} = \sum_{i = 1}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} h_i\end{gathered}$$ for some positive integers $a_i$ and ${a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}}$ with $i \in {[1 \, . \, . \, l]}$. These integers, together with $$\begin{gathered}
a_0 = 1, \qquad {a \ifthenelse{\equal{0}{\null}}{\,}{} \check{{}_{0}} \ifthenelse{\equal{0}{\null}}{\,}{}} = 1,\end{gathered}$$ are the Kac labels and the dual Kac labels of the Dynkin diagram associated with the extended Cartan matrix $A^{(1)}$. Recall that the sums $$\begin{gathered}
h = \sum_{i = 0}^l a_i, \qquad {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} = \sum_{i = 0}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}}\end{gathered}$$ are called the Coxeter number and the dual Coxeter number of ${\mathfrak g}$. Using (\[alphavee\]), one obtains $$\begin{gathered}
{\theta \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} = \frac{2}{(\theta | \theta)} \nu^{-1}(\theta) = \frac{2}{(\theta | \theta)} \sum_{i = 1}^l a_i \nu^{-1}(\alpha_i) = \sum_{i = 1}^l \frac{(\alpha_i | \alpha_i)}{(\theta | \theta)} a_i h_i.\end{gathered}$$ It follows that $$\begin{gathered}
{a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} = \frac{(\alpha_i | \alpha_i)}{(\theta | \theta)} a_i\end{gathered}$$ for any $i \in {[1 \, . \, . \, l]}$.
It is clear that $$\begin{gathered}
a_{0 0} = 2, \qquad a_{0 j} = - \sum_{i = 1}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} a_{i j}, \qquad j \in {[1 \, . \, . \, l]}, \qquad a_{i 0} = - \sum_{j = 1}^l a_{i j} a_j, \qquad i \in {[1 \, . \, . \, l]}. \label{aaa}\end{gathered}$$ We see that for the extended Cartan matrix $A^{(1)}$ one has $$\begin{gathered}
\sum_{j = 0}^l a_{i j} a_j = 0, \qquad i \in {[0 \, . \, . \, l]}, \qquad \sum_{i = 0}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} a_{i j} = 0, \qquad j \in {[0 \, . \, . \, l]}.\end{gathered}$$
Since the Cartan matrix of ${\mathfrak g}$ is symmetrizable, so is the extended Cartan matrix. Indeed, using relation (\[alphavee\]), one can rewrite equations (\[decm\]) as $$\begin{gathered}
a_{0 0} = 2, \qquad a_{0 j} = - 2 \frac{(\alpha_j | \theta)}{(\theta | \theta)}, \qquad
a_{i 0} = - 2 \frac{(\theta | \alpha_i)}{(\alpha_i | \alpha_i)}.\end{gathered}$$ We see that the symmetricity condition $$\begin{gathered}
d_i a_{i j} = d_j a_{j i}, \qquad i, j \in {[0 \, . \, . \, l]},\end{gathered}$$ for the extended Cartan matrix is equivalent to the equations $$\begin{gathered}
\frac{d_0}{(\theta | \theta)} = \frac{d_i}{(\alpha_i | \alpha_i)}, \qquad d_i a_{i j} = d_j a_{j i}, \qquad i, j \in {[1 \, . \, . \, l]}.\end{gathered}$$ We take as $d_i$, $i \in {[1 \, . \, . \, l]}$, the relatively prime positive integers symmetrizing the Cartan matrix $A$ of ${\mathfrak g}$, then, using (\[di\]), we see that $$\begin{gathered}
d_0 = \frac{1}{2} (\theta | \theta). \label{dz}\end{gathered}$$ Note that, for our normalization of the quadratic form, $(\theta | \theta) = 4$ for the types $B_l$, $C_l$ and $F_4$, $(\theta | \theta) = 6$ for the type $G_2$, and $(\theta | \theta) = 2$ for all other cases. Therefore, we have relatively prime positive integers $d_i$, $i \in {[0 \, . \, . \, l]}$, which define the diagonal matrix symmetrizing the extended Cartan matrix $A^{(1)}$.
Following Kac [@Kac90], we denote by ${{\mathcal L}(\mathfrak g)}$ the loop algebra of ${\mathfrak g}$, by ${\tilde{\mathcal L}(\mathfrak g)}$ its standard central extension by a one-dimensional centre ${\mathbb C}K$, and by ${\widehat{\mathcal L}(\mathfrak g)}$ the Lie algebra obtained from ${\tilde{\mathcal L}(\mathfrak g)}$ by adding a natural derivation $d$. By definition $$\begin{gathered}
{\widehat{\mathcal L}(\mathfrak g)}= {{\mathcal L}(\mathfrak g)}\oplus {\mathbb C}K \oplus {\mathbb C}d,\end{gathered}$$ and we use as a Cartan subalgebra of ${\widehat{\mathcal L}(\mathfrak g)}$ the space $$\begin{gathered}
{\widehat{\mathfrak h}}= {\mathfrak h}\oplus {\mathbb C}K \oplus {\mathbb C}d.\end{gathered}$$ Introducing an additional coroot $$\begin{gathered}
h_0 = K - \sum_{i = 1}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} h_i,\end{gathered}$$ we obtain $$\begin{gathered}
{\widehat{\mathfrak h}}= \bigoplus_{i = 0}^l {\mathbb C}h_i \oplus {\mathbb C}d.\end{gathered}$$ It is worth to note that $$\begin{gathered}
K = h_0 + \sum^l_{i = 1} {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} h_i = \sum_{i = 0}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} h_i.\end{gathered}$$
We identify the space ${\mathfrak h}^*$ with the subspace of $\widehat {\mathfrak h}^*$ defined as $$\begin{gathered}
{\mathfrak h}^* = \big\{\lambda \in \widehat {\mathfrak h}^* \,|\, \langle \lambda, K \rangle = 0,
\, \langle \lambda, d \rangle = 0 \big\}.\end{gathered}$$ It is also convenient to denote $$\begin{gathered}
{\widetilde{\mathfrak h}}= {\mathfrak h}\oplus {\mathbb C}K\end{gathered}$$ and identify the space ${\mathfrak h}^*$ with the subspace of $\widetilde {\mathfrak h}^*$ which consists of the elements $\widetilde \lambda \in \widetilde {\mathfrak h}^*$ satisfying the condition $$\begin{gathered}
\langle \widetilde \lambda, K \rangle = 0. \label{lambdac}\end{gathered}$$ Here and everywhere below we mark such elements of $\widetilde {\mathfrak h}^*$ by a tilde. Explicitly the identification is performed as follows. The element $\widetilde \lambda \in \widetilde {\mathfrak h}^*$ satisfying (\[lambdac\]) is identified with the element $\lambda \in {\mathfrak h}^*$ defined by the equations $$\begin{gathered}
\langle \lambda, h_i \rangle = \langle \widetilde \lambda, h_i \rangle, \qquad i \in {[1 \, . \, . \, l]}.\end{gathered}$$ In the opposite direction, given an element $\lambda \in {\mathfrak h}^*$, we identify it with the element $\widetilde \lambda \in \widetilde {\mathfrak h}^*$ determined by the relations[$$\begin{gathered}
\langle \widetilde \lambda, h_0 \rangle = - \sum_{i = 1}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} \langle \lambda, h_i \rangle, \qquad
\langle \widetilde \lambda, h_i \rangle = \langle \lambda, h_i \rangle, \qquad i \in {[1 \, . \, . \, l]}.\end{gathered}$$ It is clear that $\widetilde \lambda$ satisfies (\[lambdac\]).]{}
After all we denote by $\delta$ the element of ${\widehat{\mathfrak h}}^*$ defined by the equations $$\begin{gathered}
\langle \delta, h_i \rangle = 0, \qquad i \in {[0 \, . \, . \, l]}, \qquad \langle \delta, d \rangle = 1,\end{gathered}$$ and define the root $\alpha_0 \in {\widehat{\mathfrak h}}^*$ corresponding to the coroot $h_0$ as $$\begin{gathered}
\alpha_0 = \delta - \theta,\end{gathered}$$ so that for the entries of the extended Cartan matrix we have $$\begin{gathered}
a_{i j} = \langle \alpha_j, h_i \rangle, \qquad i, j \in {[0 \, . \, . \, l]},\end{gathered}$$ see equations (\[dcm\]) and (\[decm\]). We stress that in the above relation $\langle \cdot , \cdot \rangle$ means the pairing of the spaces ${\widehat{\mathfrak h}}^*$ and ${\widehat{\mathfrak h}}$, while in equations (\[dcm\]) and (\[decm\]) it means the pairing of the spaces ${\mathfrak h}^*$ and ${\mathfrak h}$.
Thus, the elements $\alpha_i$, $i \in {[0 \, . \, . \, l]}$, are the simple roots and $h_i$, $i \in {[0 \, . \, . \, l]}$, are the corresponding coroots forming a minimal realization of the generalized Cartan matrix $A^{(1)}$ [@Kac90]. Let $\Delta_+$ be the full system of positive roots of ${\mathfrak g}$, then the full system $\widehat \Delta_+$ of positive roots of the Lie algebra ${\widehat{\mathcal L}(\mathfrak g)}$ is $$\begin{gathered}
\widehat \Delta_+ = \{\gamma + n \delta \,|\, \gamma \in \Delta_+, \ n \in {\mathbb Z}_{\ge 0} \} \cup \{n \delta \,|\, n \in {\mathbb Z}_{>0} \} \cup \{(\delta - \gamma) + n \delta \,|\, \gamma \in \Delta_+, \ n \in {\mathbb Z}_{\ge 0}\}.\end{gathered}$$ The system of negative roots $\widehat \Delta_-$ is $\widehat \Delta_- = - \widehat \Delta_+$, and the full system of roots is $$\begin{gathered}
\widehat \Delta = \widehat \Delta_+ \sqcup \widehat \Delta_-
= \{ \gamma + n \delta \,|\, \gamma \in \Delta, \ n \in {\mathbb Z}\} \cup \{n \delta \,|\, n \in {\mathbb Z}\setminus \{0\} \}.\end{gathered}$$ Recall that the roots $\pm n \delta$ are imaginary, all other roots are real [@Kac90]. It is worth to note here that the set formed by the restriction of the simple roots $\alpha_i$ to ${\widetilde{\mathfrak h}}$ is linearly dependent. In fact, we have $$\begin{gathered}
\delta|_{{\widetilde{\mathfrak h}}} = \sum_{i = 0}^l a_i \alpha_i|_{{\widetilde{\mathfrak h}}} = 0. \label{dth}\end{gathered}$$ This is the main reason to pass from ${\tilde{\mathcal L}(\mathfrak g)}$ to ${\widehat{\mathcal L}(\mathfrak g)}$.
We fix a non-degenerate symmetric bilinear form on ${\widehat{\mathfrak h}}$ by the equations $$\begin{gathered}
(h_i | h_j) = a^{}_{i j} d^{-1}_j, \qquad
(h_i | d) = \delta^{}_{i 0} d^{-1}_0, \qquad (d | d) = 0,\end{gathered}$$ where $i, j \in {[0 \, . \, . \, l]}$. Then, for the corresponding symmetric bilinear form on ${\widehat{\mathfrak h}}^*$ one has $$\begin{gathered}
(\alpha_i | \alpha_j) = d_i a_{i j}.\end{gathered}$$ It follows from this relation that $$\begin{gathered}
(\delta | \gamma) = 0, \qquad (\delta | \delta) = 0\end{gathered}$$ for any $\gamma \in \Delta$.
### Definition of a quantum loop algebra
Let $\hbar$ be a nonzero complex number such that $q = \exp \hbar$ is not a root of unity. For each $i \in {[0 \, . \, . \, l]}$ we set $$\begin{gathered}
q_i = q^{d_i}.\end{gathered}$$ and assume that $$\begin{gathered}
q^\nu = \exp (\hbar \nu)\end{gathered}$$ for any $\nu \in {\mathbb C}$.
The quantum loop algebra ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is a unital associative ${\mathbb C}$-algebra generated by the elements $$\begin{gathered}
e_i, \quad f_i, \quad i = 0, 1,\ldots,l, \quad q^x, \quad x \in {\widetilde{\mathfrak h}},\end{gathered}$$ satisfying the relations $$\begin{gathered}
q^{\nu K} = 1, \qquad \nu \in {\mathbb C}, \qquad q^{x_1} q^{x_2} = q^{x_1 + x_2}, \label{djra} \\
q^x e_i q^{-x} = q^{\langle \alpha_i, x \rangle} e_i, \qquad q^x f_i q^{-x} = q^{- \langle \alpha_i, x \rangle} f_i, \label{djrb} \\
[e_i, f_j] = \delta_{i j} \frac{q_i^{h_i} - q_i^{- h_i}}{q^{\mathstrut}_i - q_i^{-1}}, \label{djrc} \\
\ \sum_{n = 0}^{1 - a_{i j}} (-1)^n \frac{e_i^{1 - a_{i j} - n}}{[1 - a_{i j} - n]_{q^i}!} e^{\mathstrut}_j \frac{e_i^n}{[n]_{q^i}!} = 0, \qquad \sum_{n = 0}^{1 - a_{i j}} (-1)^n \frac{f_i^{1 - a_{i j} - n}}{[1 - a_{i j} - n]_{q^i}!} f^{\mathstrut}_j \frac{f_i^n}{[n]_{q^i}!} = 0. \label{djrd}\end{gathered}$$ Here, relations (\[djrb\]) and (\[djrc\]) are valid for all $i, j \in {[0 \, . \, . \, l]}$. The last line of the relations is valid for all distinct $i, j \in {[0 \, . \, . \, l]}$.
The quantum loop algebra ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is a Hopf algebra. Here the multiplication mapping $\mu \colon {{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}\to {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is defined as $$\begin{gathered}
\mu(a \otimes b) = a b,\end{gathered}$$ and for the unit mapping $\iota \colon {\mathbb C}\to {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ we have $$\begin{gathered}
\iota(\nu) = \nu 1.\end{gathered}$$ The comultiplication $\Delta$, the antipode $S$, and the counit $\varepsilon$ are given by the relations $$\begin{gathered}
\Delta(q^x) = q^x \otimes q^x, \qquad \Delta(e^{}_i) = e^{}_i \otimes 1 + q_i^{h_i} \otimes e^{}_i, \qquad \Delta(f^{}_i) = f^{}_i \otimes q_i^{- h_i} + 1 \otimes f^{}_i, \label{hsa} \\
S(q^x) = q^{- x}, \qquad S(e^{}_i) = - q_i^{- h_i} e^{}_i, \qquad S(f^{}_i) = - f^{}_i q_i^{h_i}, \label{sg} \\
\varepsilon(q^h) = 1, \qquad \varepsilon(e^{}_i) = 0, \qquad \varepsilon(f^{}_i) = 0. \label{hsc}\end{gathered}$$ For the inverse of the antipode one has $$\begin{gathered}
S^{-1}(q^x) = q^{-x}, \qquad S^{-1}(e^{}_i) = -e^{}_i q_i^{-h_i}, \qquad S^{-1}(f^{}_i) = - q_i^{h_i} f^{}_i. \label{isg}\end{gathered}$$
### Poincaré–Birkhoff–Witt basis
The abelian group $$\begin{gathered}
\widehat Q = \bigoplus_{i = 0}^l {\mathbb Z}\alpha_i\end{gathered}$$ is called the root lattice of ${\widehat{\mathcal L}(\mathfrak g)}$. The algebra ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ can be considered as $\widehat Q$-graded if we assume that $$\begin{gathered}
e_i \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_{\alpha_i}, \qquad f_i \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_{- \alpha_i}, \qquad q^x \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_0\end{gathered}$$ for any $i \in {[0 \, . \, . \, l]}$ and $x \in {\widetilde{\mathfrak h}}$. An element $a$ of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is called a root vector corresponding to a root $\gamma$ of ${\widehat{\mathfrak h}}^*$ if $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_\gamma$. In particular, the generators $e_i$ and $f_i$ are root vectors corresponding to the roots $\alpha_i$ and $- \alpha_i$.
One can construct linearly independent root vectors corresponding to all roots from $\widehat \Delta$, see, for example, papers [@KhoTol92; @KhoTol93; @KhoTol94; @TolKho92], and papers [@Bec94a; @Dam98] for an alternative approach. If some ordering of roots is chosen, then appropriately ordered monomials constructed from these vectors form a Poincaré–Birkhoff–Witt basis of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$. In fact, in applications to the theory of quantum integrable systems one uses the so-called normal orderings. The definition and an example for the case of ${\mathfrak g}= {\mathfrak{sl}_{l + 1}}$ is given in Section \[s:pbwb\].
### Universal $\boldsymbol{R}$-matrix
Let $\Pi$ be the automorphism of the algebra ${{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ defined by the equation $$\begin{gathered}
\Pi (a \otimes b) = b \otimes a,\end{gathered}$$ see Appendix \[a:tpsg\]. One can show that the mapping $$\begin{gathered}
\Delta' = \Pi \circ \Delta\end{gathered}$$ is a comultiplication in ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ called the opposite comultiplication.
Let ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ be a quantum loop algebra. There exists an element ${\mathcal R}$ of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ connecting the two comultiplications in the sense that $$\begin{gathered}
\Delta'(a) = {\mathcal R}\Delta(a) {\mathcal R}^{-1} \label{dpx}\end{gathered}$$ for any $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$, and satisfying in ${{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ the equations $$\begin{gathered}
(\Delta \otimes {\mathrm{id}})({\mathcal R}) = {\mathcal R}^{(13)} {\mathcal R}^{(23)}, \qquad ({\mathrm{id}}\otimes \Delta)({\mathcal R}) = {\mathcal R}^{(13)} {\mathcal R}^{(12)}. \label{drrr}\end{gathered}$$ The meaning of the superscripts in the above relations is explained in Appendix \[a:tpsg\]. The element ${\mathcal R}$ is called the universal $R$-matrix. One can show that it satisfies the universal Yang–Baxter equation $$\begin{gathered}
{\mathcal R}^{(12)} {\mathcal R}^{(13)} {\mathcal R}^{(23)} = {\mathcal R}^{(23)} {\mathcal R}^{(13)} {\mathcal R}^{(12)} \label{uybe}\end{gathered}$$ in ${{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}$.
It should be noted that we define the quantum loop algebra as a ${\mathbb C}$-algebra. It can be also defined as a ${\mathbb C}[[\hbar]]$-algebra, where $\hbar$ is considered as an indeterminate. In this case one really has a universal $R$-matrix. In our case, the universal $R$-matrix exists only in some restricted sense, see, for example, paper [@Tan92], and the discussion in Section \[sss:urm\] for the case of ${\mathfrak g}= {\mathfrak{sl}_{l + 1}}$.
There are two main approaches to the construction of the universal $R$-matrices for quantum loop algebras. One of them was proposed by Khoroshkin and Tolstoy [@KhoTol92; @KhoTol93; @KhoTol94; @TolKho92], and another one is related to the names Beck and Damiani [@Bec94a; @Dam98].
### Modules and representations
Let $\varphi$ be a representation of a quantum loop algebra ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$, and $V$ the corresponding ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module. The generators $q^x$, $x \in {\widetilde{\mathfrak h}}$, form an abelian group in ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$. Let vector $v \in V$ be a common eigenvector for all operators $\varphi(q^x)$, then $$\begin{gathered}
q^x v = q^{\langle \mu, x \rangle} v\end{gathered}$$ for some unique element $\mu \in {\widetilde{\mathfrak h}}^*$. Using the first relation of (\[djra\]), we obtain $$\begin{gathered}
q^{\nu K} v = q^{\nu \langle \mu, K \rangle} v = v\end{gathered}$$ for any $\nu \in {\mathbb C}$. Therefore, the element $\mu$ satisfies the equation $$\begin{gathered}
\langle \mu, K \rangle = 0,\end{gathered}$$ and there is a unique element $\lambda \in {\mathfrak h}^*$ such that $\mu = \widetilde \lambda$. For the definition of $\widetilde \lambda$ see Section \[s:siola\].
A ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module $V$ is said to be a weight module if $$\begin{gathered}
V = \bigoplus_{\lambda \in {\mathfrak h}^*} V_\lambda, \label{vvl}\end{gathered}$$ where $$\begin{gathered}
V_\lambda = \big\{v \in V \,|\, q^x v
= q^{\langle \widetilde \lambda, x \rangle} v \mbox{ for any } x \in {\widetilde{\mathfrak h}}\big\}.\end{gathered}$$ This means that any vector of $V$ has the form $$\begin{gathered}
v = \sum_{\lambda \in {\mathfrak h}^*} v_\lambda,\end{gathered}$$ where $v_\lambda \in V_\lambda$ for any $\lambda \in {\mathfrak h}^*$, and $v_\lambda = 0$ for all but finitely many of $\lambda$. The space $V_\lambda$ is called the weight space of weight $\lambda$, and a nonzero element of $V_\lambda$ is called a weight vector of weight $\lambda$. We say that $\lambda \in {\mathfrak h}^*$ is a weight of $V$ if $V_\lambda \ne \{0\}$.
We say that a ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module $V$ is in the category ${\mathcal O}$ if
- $V$ is a weight module all of whose weight spaces are finite-dimensional;
- there exists a finite number of elements $\lambda_1, \ldots, \lambda_s \in {\mathfrak h}^*$ such that every weight of $V$ belongs to the set $$\begin{gathered}
\bigcup_{i = 1}^s \{\lambda \in {\mathfrak h}^* \,|\, \lambda \leq \lambda_i \},\end{gathered}$$ where $\leq$ is the usual partial order in ${\mathfrak h}^*$ [@Hum80].
In this paper we deal only with ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules in the category ${\mathcal O}$ and in its dual ${\mathcal O}^\star$, see Section \[s:cr\].
Let $V_1$, $V_2$ be two ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules, and $\varphi_1$, $\varphi_2$ the corresponding representations. The tensor product of the vector spaces $V_1$ and $V_2$ can be supplied with the structure of a ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module corresponding to the representation $$\begin{gathered}
\varphi_1 \otimes_\Delta \varphi_2 = (\varphi_1 \otimes \varphi_2) \circ \Delta.\end{gathered}$$ We denote the obtained ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module as $V_1 \otimes_\Delta V_2$.
Using the opposite comultiplication, one can construct another representation $$\begin{gathered}
\varphi_1 \otimes_{\Delta'} \varphi_2 = (\varphi_1 \otimes \varphi_2) \circ {\Delta'}\end{gathered}$$ and define the corresponding ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module $V_1 \otimes_{\Delta'} V_2$. However, one can show that there is a natural isomorphism $$\begin{gathered}
\varphi_1 \otimes_\Delta \varphi_2 \cong \varphi_2 \otimes_{\Delta'} \varphi_1.\end{gathered}$$
### Spectral parameter
In applications to the theory of quantum integrable systems, one usually considers families of representations of a quantum loop algebra parametrized by a complex parameter called a spectral parameter. We introduce a spectral parameter in the following way. Assume that a quantum loop algebra ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is ${\mathbb Z}$-graded, $$\begin{gathered}
{{\mathrm U}_q(\mathcal L(\mathfrak g))}= \bigoplus_{m \in {\mathbb Z}} {{\mathrm U}_q(\mathcal L(\mathfrak g))}_m, \qquad {{\mathrm U}_q(\mathcal L(\mathfrak g))}_m {{\mathrm U}_q(\mathcal L(\mathfrak g))}_n \subset {{\mathrm U}_q(\mathcal L(\mathfrak g))}_{m + n},\end{gathered}$$ so that any element of $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ can be uniquely represented as $$\begin{gathered}
a = \sum_{m \in {\mathbb Z}} a_m, \qquad a_m \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_m.\end{gathered}$$ Given $\zeta \in {\mathbb C}^\times$, we define the grading automorphism $\Gamma_\zeta$ by the equation $$\begin{gathered}
\Gamma_\zeta(a) = \sum_{m \in {\mathbb Z}} \zeta^m a_m.\end{gathered}$$ It is worth noting that $$\begin{gathered}
\Gamma_{\zeta_1 \zeta_2} = \Gamma_{\zeta_1} \circ \Gamma_{\zeta_2} \label{gzz}\end{gathered}$$ for any $\zeta_1, \zeta_2 \in {\mathbb C}^\times$. Now, for any representation $\varphi$ of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ we define the corresponding family $\varphi_\zeta$ of representations as $$\begin{gathered}
\varphi_\zeta = \varphi \circ \Gamma_\zeta.\end{gathered}$$ If $V$ is the ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module corresponding to the representation $\varphi$, we denote by $V_\zeta$ the ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module corresponding to the representation $\varphi_\zeta$.
A common way to endow ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ by a ${\mathbb Z}$-gradation is to assume that $$\begin{gathered}
q^x \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_0, \qquad e_i \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_{s_i},
\qquad f_i \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}_{-s_i},\end{gathered}$$ where $s_i$ are arbitrary integers. We denote $$\begin{gathered}
s = \sum_{i = 0}^l a_i s_i, \label{ds}\end{gathered}$$ where $a_i$ are the Kac labels of the Dynkin diagram associated with the extended Cartan matrix $A^{(1)}$ and assume that $s$ is non-zero. It is clear that for such a ${\mathbb Z}$-gradation one has $$\begin{gathered}
\Gamma_\zeta(q^x) = q^x, \qquad \Gamma_\zeta(e_i) = \zeta^{s_i} e_i, \qquad \Gamma_\zeta(f_i) = \zeta^{-s_i} f_i. \label{gzqx}\end{gathered}$$ Further, it follows from the explicit expression for the universal $R$-matrix [@Bec94a; @Dam98; @KhoTol92; @KhoTol93; @KhoTol94; @TolKho92] that $$\begin{gathered}
(\Gamma_\zeta \otimes \Gamma_\zeta)({\mathcal R}) = {\mathcal R}\label{gagar}\end{gathered}$$ for any $\zeta \in {\mathbb C}^\times$. Besides, equations (\[sg\]) and (\[isg\]) give $$\begin{gathered}
S \circ \Gamma_\zeta = \Gamma_\zeta \circ S, \qquad S^{-1} \circ \Gamma_\zeta = \Gamma_\zeta \circ S^{-1}. \label{sgamma}\end{gathered}$$
Integrability objects and their graphical representations
---------------------------------------------------------
In this section we use the Einstein summation convention: if the same index appears in a single term exactly twice, once as an upper index and once as a lower index, summation is implied. Some additional information on integrability objects can be found in the remarkable paper by Frenkel and Reshetikhin [@FreRes92] and in papers [@BooGoeKluNirRaz13; @BooGoeKluNirRaz14a].
### Introductory words
What we mean by integrability objects are certain linear mappings acting between representation spaces of quantum groups, which are, in general, tensor products of some basic representation spaces. Certainly, the simplest mapping is the unit operator on a basic representation space. We use for its matrix elements the depiction given in Fig. \[f:uo\].
![[]{data-label="f:uo"}](Figures/UnitOperator)
In fact, we associate with a basic representation space an oriented line, which can be single, double, etc. The direction of a line is represented as an arrow. The arrowhead corresponds to the input, and the tail to the output of the operator. The spectral parameter associated with the representation is placed in the vicinity of the line. The unit operator acting on a tensor product of representation spaces is depicted as a bunch of oriented lines corresponding to the factors of the tensor product.
### $\boldsymbol{R}$-operators
A more complicated object is an $R$-operator. It depends on two spectral parameters and is defined as follows. Let $V_1$, $V_2$ be two ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules, $\varphi_1$, $\varphi_2$ the corresponding representations of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$, and $\zeta_1$, $\zeta_2$ the spectral parameters associated with the representations. We define the $R$-operator $R_{V_1 | V_2}(\zeta_1 | \zeta_2)$[^1] by the equation $$\begin{gathered}
\rho_{V_1 | V_2}(\zeta_1 | \zeta_2) R_{V_1 | V_2}(\zeta_1 | \zeta_2) = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}) ({\mathcal R}), \label{rhor}\end{gathered}$$ where $\rho_{V_1 | V_2}(\zeta_1 | \zeta_2)$ is a scalar normalization factor. It follows from (\[gzz\]) and (\[gagar\]) that $$\begin{gathered}
(\varphi_{1 \zeta_1 \nu} \otimes \varphi_{2 \zeta_2 \nu}) ({\mathcal R}) = ((\varphi_1 \otimes \varphi_2) \circ (\Gamma_{\zeta_1} \otimes \Gamma_{\zeta_2}) \circ (\Gamma_\nu \otimes \Gamma_\nu))({\mathcal R}) = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}) ({\mathcal R})\end{gathered}$$ for any $\nu \in {\mathbb C}^\times$. We will assume that the normalization factor in equation (\[rhor\]) is chosen in such a way that $$\begin{gathered}
\rho_{V_1 | V_2}(\zeta_1 \nu| \zeta_2 \nu) = \rho_{V_1 | V_2}(\zeta_1 | \zeta_2) \label{znu}\end{gathered}$$ for any $\nu \in {\mathbb C}^\times$. In this case $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 \nu | \zeta_2 \nu) = R_{V_1 | V_2}(\zeta_1 | \zeta_2),\end{gathered}$$ and one has $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2) = R_{V_1 | V_2}\big(\zeta_1 (\zeta_2)^{-1}| 1\big) = R_{V_1 | V_2}\big(\zeta_1 (\zeta_2)^{-1}\big), \label{rrr}\end{gathered}$$ where $$\begin{gathered}
R_{V_1 | V_2}(\zeta) = R_{V_1 | V_2}(\zeta | 1).\end{gathered}$$ Below we sometimes use the notation $$\begin{gathered}
\zeta_{i j} = \zeta_i (\zeta_j)^{-1}.\end{gathered}$$ Using this notation, we can, for example, write (\[rrr\]) as $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2) = R_{V_1 | V_2}(\zeta_{1 2}| 1) = R_{V_1 | V_2}(\zeta_{1 2}).\end{gathered}$$
It is clear that the operator $R_{V_1 | V_2}(\zeta_1 | \zeta_2)$ acts on $V_1 \otimes V_2$. Fixing bases, say $(e_\alpha)$ and $(f_\beta)$, of $V_1$ and $V_2$ we can write $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2) (e_{\alpha_2} \otimes f_{\beta_2}) = (e_{\alpha_1} \otimes f_{\beta_1}) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_1 \beta_1}{}_{\alpha_2 \beta_2}.\end{gathered}$$ We use for the matrix elements of $R_{V_1 | V_2}(\zeta_1 | \zeta_2)$ the depiction which can be seen in Fig. \[f:ro\].
![[]{data-label="f:iro"}](Figures/ROperator)
![[]{data-label="f:iro"}](Figures/InverseROperator)
Here we associate with $V_1$ and $V_2$ a single and a double line respectively. It is worth to note that the indices in the graphical image go clockwise.
For the matrix elements of the inverse $R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$ of the $R$-operator $R_{V_1 | V_2}(\zeta_1 | \zeta_2)$ we use the depiction given in Fig. \[f:iro\]. Here we use a grayed circle for the operator and the counter-clockwise order for the indices. This allows one to have a natural graphical form of the equations $$\begin{gathered}
\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{\alpha_1 \beta_1}{}_{\alpha_2 \beta_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_2 \beta_2}{}_{\alpha_3 \beta_3} = \delta^{\alpha_1}{}_{\mathstrut \alpha_3} \delta^{\beta_1}{}_{\mathstrut \beta_3}, \\
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_1 \beta_1}{}_{\alpha_2 \beta_2} \big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{\alpha_2 \beta_2}{}_{\alpha_3 \beta_3} = \delta^{\alpha_1}{}_{\mathstrut \alpha_3} \delta^{\beta_1}{}_{\mathstrut \beta_3},\end{gathered}$$ see Figs. \[f:irr\] and \[f:rir\].
![[]{data-label="f:rir"}](Figures/InverseROperatorROperator)
![[]{data-label="f:rir"}](Figures/ROperatorInverseROperator)
One can see that to represent a product of operators we connect outcoming and incoming lines corresponding to the indices common for the operators. It is clear that the notation used for the indices and spectral parameters are arbitrary. Therefore, when it does not lead to a misunderstanding, we do not write them explicitly in pictures. In fact, in such a case we obtain a depiction not for a matrix element, but for an operator itself. For example, we associate Figs. \[f:irrwi\] and \[f:rirwi\] with the operator equations $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} R_{V_1 | V_2}(\zeta_1 | \zeta_2) = 1, \qquad
R_{V_1 | V_2}(\zeta_1 | \zeta_2) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} = 1.\end{gathered}$$
![[]{data-label="f:rirwi"}](Figures/InverseROperatorROperatorWithoutIndices)
![[]{data-label="f:rirwi"}](Figures/ROperatorInverseROperatorWithoutIndices)
It is worth to note that the modules $V_1$ and $V_2$ are arbitrary. Therefore the above equations remain valid if we interchange them. Respectively, the graphical equations represented by Figs. \[f:irrwi\] and \[f:rirwi\] also remain valid if we interchange the single and double lines. This remark is applicable to all similar situations. \[p:sdl\]
It is in order to formulate some general rules. To obtain a graphical representation of an operator, we first specify the types of lines corresponding to the basic vector spaces and associate with each basic vector space a spectral parameter. Then we choose some shape which will represent the operator. This shape with the appropriate number of outcoming and incoming lines depicts the matrix element, or the operator itself. To depict the matrix element of the product of two operators we connect the lines corresponding to the common indices over which the summation is carried out.
It turns out to be useful to introduce new $R$-operators, which, at first sight, drop out of the general scheme described above.[^2] We denote these operators by $\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)$ and their inverses by $\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$. As the usual $R$-operators, they act on the tensor product $V_1 \otimes V_2$. The depiction of the corresponding matrix elements can be seen in Figs. \[f:tro\] and \[f:itro\].
![[]{data-label="f:itro"}](Figures/TildeROperator)
![[]{data-label="f:itro"}](Figures/InverseTildeROperator)
We require the operator $\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$ to be the ‘skew inverse’ of the operator $R_{V_1 | V_2}(\zeta_1 | \zeta_2)$. By this we mean the validity of the graphical equation given in Fig. \[f:itroro\].
![[]{data-label="f:itroroi"}](Figures/InverseTildeROperatorROperator)
![[]{data-label="f:itroroi"}](Figures/InverseTildeROperatorROperatorInd)
Marking out this figure with indices, we come to Fig. \[f:itroroi\]. We see that in terms of matrix elements the equation given in Fig. \[f:itroro\] has the form $$\begin{gathered}
\big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{\alpha_2 \beta_1}{}_{\alpha_3 \beta_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_1 \beta_2}{}_{\alpha_2 \beta_3} = \delta_{\alpha_3}{}^{\alpha_1} \delta^{\beta_1}{}_{\beta_3}.\end{gathered}$$ One can rewrite this as $$\begin{gathered}
\big(\big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}\big)_{\alpha_3}{}^{\beta_1}{}^{\alpha_2}{}_{\beta_2} \big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1}\big)_{\alpha_2}{}^{\beta_2}{}^{\alpha_1}{}_{\beta_3} = \delta_{\alpha_3}{}^{\alpha_1} \delta^{\beta_1}{}_{\beta_3}.\end{gathered}$$ Here $t_1$ denotes the partial transpose with respect to the space $V_1$, see Appendix \[a:pt\]. Note that $R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1}$ and $\big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}$ are linear operators on $V_1^\star \otimes V_2^{}$.[^3] Thus, we have the following operator equation $$\begin{gathered}
\big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1} = 1 \label{rtr}\end{gathered}$$ on $V_1^\star \otimes V_2^{}$, and we come to the equation $$\begin{gathered}
\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2) = \big(\big((R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1})^{-1}\big)^{t_1}\big)^{-1}.\end{gathered}$$ Certainly, equation (\[rtr\]) can be also written as $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1} \big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1} = 1. \label{trr}\end{gathered}$$ The corresponding graphical image is given in Fig. \[f:roitro\].
![[]{data-label="f:roitro"}](Figures/ROperatorInverseTildeROperator)
Transposing equations (\[rtr\]) and (\[trr\]), we obtain $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2} \big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2} = 1, \qquad \big(\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2} = 1,\end{gathered}$$ where $t_2$ denotes the partial transpose with respect to the space $V_2$, see again Appendix \[a:pt\]. One can get convinced that this does not lead to new pictures. However, using any of these equations, we obtain $$\begin{gathered}
\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2) = \big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2}\big)^{-1}\big)^{t_2}\big)^{-1}. \label{trrtiti}\end{gathered}$$
For completeness we introduce the $R$-operators denoted by ${\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)$, with the inverses ${\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$, acting on $V_1 \otimes V_2$ and depicted by Figs. \[f:dtro\] and \[f:idtro\].
![[]{data-label="f:idtro"}](Figures/DTildeROperator)
![[]{data-label="f:idtro"}](Figures/InverseDTildeROperator)
Now we require the operator ${\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)$ to be the ‘skew inverse’ of the operator $R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$. By this we mean the validity of the graphical equation given in Fig. \[f:dtroiro\].
![[]{data-label="f:irodtro"}](Figures/DTildeROperatorInverseROperator)
![[]{data-label="f:irodtro"}](Figures/InverseROperatorDTildeROperator)
Similarly as above, we determine that it is equivalent to the following operator equation $$\begin{gathered}
{\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1} \big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1} = 1, \label{dtrir}\end{gathered}$$ and, therefore, $$\begin{gathered}
{\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2) = \big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}\big)^{-1}\big)^{t_1}. \label{dt}\end{gathered}$$ Rewriting equation (\[dtrir\]) as $$\begin{gathered}
\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1} {\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1} = 1, \label{irdtr}\end{gathered}$$ we come to the graphical equation given in Fig. \[f:irodtro\]. After all, transposing equations (\[dtrir\]) and (\[irdtr\]), we obtain $$\begin{gathered}
\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2} {\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2} = 1, \qquad {\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2} \big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2} = 1.\end{gathered}$$ Using any of these equations, we obtain $$\begin{gathered}
{\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2) = \big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2}\big)^{-1}\big)^{t_2}.\end{gathered}$$
### Unitarity relations {#s:ur}
Applying the mapping $\Pi$ to both sides of the equation $$\begin{gathered}
\Pi(\Delta(a)) = {\mathcal R}\Delta(a) {\mathcal R}^{-1}, \qquad a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))},\end{gathered}$$ and using again the same equation, we obtain $$\begin{gathered}
\Delta(a) = \Pi({\mathcal R}) \Pi(\Delta(a)) \Pi\big({\mathcal R}^{-1}\big) = \Pi({\mathcal R}) {\mathcal R}\Delta(a) {\mathcal R}^{-1} \Pi({\mathcal R})^{-1}.\end{gathered}$$ Therefore, $$\begin{gathered}
\Delta(a) \Pi({\mathcal R}) {\mathcal R}= \Pi({\mathcal R}) {\mathcal R}\Delta(a). \label{dxprr}\end{gathered}$$ Let $\varphi_1$ and $\varphi_2$ be representations of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ on the vector spaces $V_1$ and $V_2$ respectively. For any $v \in V_1$, $w \in V_2$ and $a, b \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ one has $$\begin{aligned}
(\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})(\Pi(a \otimes b))(v \otimes w)
&= (\varphi_{1 \zeta_1}(b) \otimes \varphi_{2 \zeta_2}(a)) (v \otimes w)\\
& = (\varphi_{1 \zeta_1}(b))(v) \otimes (\varphi_{2 \zeta_2}(a))(w) \\
& = P_{V_2 | V_1} ((\varphi_{2 \zeta_2}(a))(w) \otimes (\varphi_{1 \zeta_1}(b))(v)) \\
& = (P_{V_2 | V_1} ((\varphi_{2 \zeta_2} \otimes \varphi_{1 \zeta_1})(a \otimes b)) P_{V_1 | V_2})(v \otimes w).\end{aligned}$$ It follows that $$\begin{aligned}
(\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})(\Pi({\mathcal R})) & = P_{V_2 | V_1} ((\varphi_{2 \zeta_2} \otimes \varphi_{1 \zeta_1}) ({\mathcal R})) P_{V_1 | V_2} \\
& = \rho_{V_2 | V_1}(\zeta_2 | \zeta_1) (P_{V_2 | V_1} R_{V_2 | V_1}(\zeta_2 | \zeta_1) P_{V_1 | V_2}).\end{aligned}$$ Now, applying to both sides of equation (\[dxprr\]) the mapping $\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}$, we see that for any $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ one has $$\begin{gathered}
\begin{split}&
(\varphi_{1 \zeta_1} \otimes_\Delta \varphi_{2 \zeta_2})(a) (P_{V_2 | V_1} R_{V_2 | V_1}(\zeta_2 | \zeta_1)) (P_{V_1 | V_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2)) \\
& \qquad{} = (P_{V_2 | V_1} R_{V_2 | V_1}(\zeta_2 | \zeta_1)) (P_{V_1 | V_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2)) (\varphi_{1 \zeta_1} \otimes_\Delta \varphi_{2 \zeta_2})(a).
\end{split}\end{gathered}$$ Hence, if the representation $\varphi_{1 \zeta_1} \otimes_\Delta \varphi_{2 \zeta_2}$ is irreducible for a general value of the spectral parameters,[^4] then $$\begin{gathered}
\check R_{V_2 | V_1}(\zeta_2 | \zeta_1) \check R_{V_1 | V_2}(\zeta_1 | \zeta_2) = C_{V_1 | V_2}(\zeta_1 | \zeta_2) \, {\mathrm{id}}_{V_1 \otimes V_2}, \label{uri}\end{gathered}$$ where $ C_{V_1 | V_2}(\zeta_1 | \zeta_2)$ is a scalar factor, and we use the notation $$\begin{gathered}
\check R_{V_2 | V_1}(\zeta_2 | \zeta_1) = P_{V_2 | V_1} R_{V_2 | V_1}(\zeta_2 | \zeta_1), \qquad \check R_{V_1 | V_2}(\zeta_1 | \zeta_2) = P_{V_1 | V_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2).\end{gathered}$$ Equation (\[uri\]) is called the unitarity relation. Since the representations and spectral parameters in (\[uri\]) are arbitrary, we also have $$\begin{gathered}
\check R_{V_1 | V_2}(\zeta_1 | \zeta_2) \check R_{V_2 | V_1}(\zeta_2 | \zeta_1) = C_{V_2 | V_1}(\zeta_2 | \zeta_1) \, {\mathrm{id}}_{V_2 \otimes V_1}. \label{urii}\end{gathered}$$ From the other hand, multiplying (\[uri\]) from the left by $ \check R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{-1}$ and from the right by $\check R_{V_2 | V_1}(\zeta_2 | \zeta_1)$ we obtain $$\begin{gathered}
\check R_{V_1 | V_2}(\zeta_1 | \zeta_2) \check R_{V_2 | V_1}(\zeta_2 | \zeta_1) = C_{V_1 | V_2}(\zeta_1 | \zeta_2) \, {\mathrm{id}}_{V_2 \otimes V_1}.\end{gathered}$$ It follows from the last two equations that $$\begin{gathered}
C_{V_1 | V_2}(\zeta_1 | \zeta_2) = C_{V_2 | V_1}(\zeta_2 | \zeta_1). \label{cvvcvv}\end{gathered}$$
Again fixing bases $(e_\alpha)$ and $(f_\beta)$ of $V_1$ and $V_2$ we write $$\begin{gathered}
\check R_{V_2 | V_1}(\zeta_2 | \zeta_1) (f_{\beta_2} \otimes e_{\alpha_2}) = (e_{\alpha_1} \otimes f_{\beta_1}) \check R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{\alpha_1 \beta_1}{}_{\beta_2 \alpha_2}, \\
\check R_{V_1 | V_2}(\zeta_1 | \zeta_2) (e_{\alpha_2} \otimes f_{\beta_2}) = (f_{\beta_1} \otimes e_{\alpha_1}) \check R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\beta_1 \alpha_1}{}_{\alpha_2 \beta_2}.\end{gathered}$$ It is easy to see that $$\begin{gathered}
\check R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{\alpha_1 \beta_1}{}_{\beta_2 \alpha_2} = R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{\beta_1 \alpha_1}{}_{\beta_2 \alpha_2}, \\
\check R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\beta_1 \alpha_1}{}_{\alpha_2 \beta_2} = R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_1 \beta_1}{}_{\alpha_2 \beta_2}.\end{gathered}$$ Hence, in terms of matrix elements equations (\[uri\]) and (\[urii\]) look as $$\begin{gathered}
R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{\beta_1 \alpha_1}{}_{\beta_2 \alpha_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_2 \beta_2}{}_{\alpha_3 \beta_3} = C_{V_1 | V_2}(\zeta_1 | \zeta_2) \delta^{\alpha_1}{}_{\alpha_3} \delta^{\beta_1}{}_{\beta_3}, \\
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{\alpha_1 \beta_1}{}_{\alpha_2 \beta_2} R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{\beta_2 \alpha_2}{}_{\beta_3 \alpha_3} = C_{V_2 | V_1}(\zeta_2 | \zeta_1) \delta^{\alpha_1}{}_{\alpha_3} \delta^{\beta_1}{}_{\beta_3}.\end{gathered}$$ These two equations are depicted in Figs. \[f:ui\] and \[f:uii\].
![[]{data-label="f:uii"}](Figures/UnitarityI)
![[]{data-label="f:uii"}](Figures/UnitarityII)
Instead of (\[uri\]) and (\[urii\]) we can also write $$\begin{gathered}
\check R_{V_1 | V_2}(\zeta_1 | \zeta_2) = C_{V_1 | V_2}(\zeta_1 | \zeta_2) \check R_{V_2 | V_1}(\zeta_2 | \zeta_1)^{-1}, \\
\check R_{V_2 | V_1}(\zeta_2 | \zeta_1) = C_{V_2 | V_1}(\zeta_2 | \zeta_1) \check R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}.\end{gathered}$$ These equations can be recognized in Figs. \[f:uiii\] and \[f:uiv\].
![[]{data-label="f:uiv"}](Figures/UnitarityIII)
![[]{data-label="f:uiv"}](Figures/UnitarityIV)
For completeness, we also redraw Figs. \[f:uiii\] and \[f:uiv\] in the form of Figs. \[f:uv\] and \[f:uvi\].
![[]{data-label="f:uvi"}](Figures/UnitarityV)
![[]{data-label="f:uvi"}](Figures/UnitarityVI)
### Crossing relations {#s:cr}
Let $V$ be ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module in the category ${\mathcal O}$. Define two dual modules $V^*$ and ${}^*V$. As vector spaces both $V^*$ and ${}^*V$ coincide with the restricted dual space $$\begin{gathered}
V^\star = \bigoplus_{\lambda \in {\mathfrak h}^*} (V_\lambda)^*\end{gathered}$$ of $V$. This means that any element $\mu \in V^\star$ has the form $$\begin{gathered}
\mu = \sum_{\lambda \in {\mathfrak h}^*} \mu_\lambda,\end{gathered}$$ where $\mu_\lambda \in (V_\lambda)^*$ for any $\lambda \in {\mathfrak h}^*$, and $\mu_\lambda = 0$ for all but finitely many of $\lambda$. The action of an element $\mu \in V^\star$ on a vector $v \in V$ is given by the equation $$\begin{gathered}
\langle \mu, v \rangle = \sum_{\lambda \in {\mathfrak h}^*} \langle \mu_\lambda, v_\lambda \rangle,\end{gathered}$$ where the sum in the right hand side is finite. If $V$ is a finite-dimensional module, the restricted dual space coincides with the usual dual space. The module operation for the module $V^*$ is defined by the equation $$\begin{gathered}
\langle a \mu, v \rangle = \langle \mu, S(a) v \rangle, \qquad \mu \in V^\star, \qquad v \in V\end{gathered}$$ and for ${}^*V$ by the equation $$\begin{gathered}
\langle a \mu, v \rangle = \big\langle \mu, S^{-1}(a) v \big\rangle, \qquad \mu \in V^\star, \qquad v \in V.\end{gathered}$$ For any two ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules $V$ and $W$ there are natural isomorphisms $$\begin{gathered}
(V \otimes_\Delta W)^* \cong V^* \otimes_{\Delta'} W^* \cong W^* \otimes_\Delta V^*, \qquad {}^*(V \otimes_\Delta W) \cong {}^*V \otimes_{\Delta'} {}^*W \cong {}^*W \otimes_\Delta {}^*V.\end{gathered}$$
We now define the category ${\mathcal O}^\star$ containing the dual modules $V^*$ and ${}^*V$. We say that a ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module $V$ is in the category ${\mathcal O}^\star$ if
- $V$ is a weight module all of whose weight spaces are finite-dimensional;
- there exists a finite number of elements $\lambda_1, \ldots, \lambda_s \in {\mathfrak h}^*$ such that every weight of $V$ belongs to the set $$\begin{gathered}
\bigcup_{i = 1}^s \{\lambda \in {\mathfrak h}^* \,|\, \lambda_i \leq \lambda \},\end{gathered}$$ where, as in the definition of the category ${\mathcal O}$, $\leq$ is the usual partial order in ${\mathfrak h}^*$.
It is clear that for any module $V$ in the category ${\mathcal O}$ the modules $V^*$ and ${}^*V$ are objects of the category ${\mathcal O}^\star$.
Let $V$ be in the category ${\mathcal O}$, and $\varphi$ the corresponding representation of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$. For any $M \in \operatorname{End}(V)$ one defines the transpose of $M$ as an element $M^t \in \operatorname{End}(V^\star)$ defined by the equation $$\begin{gathered}
\langle M^t \mu, v \rangle = \langle \mu, M v \rangle, \qquad \mu \in V^\star, \qquad v \in V.\end{gathered}$$ Denote by $\varphi^*$ and ${}^* \! \varphi$ the representations of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ corresponding to the modules $V^*$ and ${}^*V$ respectively. Now one has $$\begin{gathered}
\varphi^*(a) = \varphi(S(a))^t, \qquad {}^* \! \varphi(a) = \varphi\big(S^{-1}(a)\big)^t, \qquad a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}. \label{fsx}\end{gathered}$$ Note that, using equation (\[sgamma\]), one obtains $$\begin{gathered}
(\varphi^*)_\zeta = (\varphi_\zeta)^*, \qquad ({}^* \! \varphi)_\zeta = {}^*(\varphi_\zeta) \label{fsz}\end{gathered}$$ for any $\zeta \in {\mathbb C}^\times$. Therefore, we write instead of $(\varphi^*)_\zeta$ and $(\varphi_\zeta)^*$ just $\varphi_\zeta^*$, and instead of $({}^* \! \varphi)_\zeta$ and ${}^*(\varphi_\zeta)$ just ${}^* \! \varphi_\zeta$.
Let $V$ be a ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module in the category ${\mathcal O}$. Define a mapping $\eta_V \colon V \to V^{\star \star}$ by the equality $$\begin{gathered}
\langle \eta_V(v), \mu \rangle = \langle \mu, v \rangle\end{gathered}$$ for all $v \in V$ and $\mu \in V^\star$. It can be shown that $\eta_V$ is an isomorphism of vector spaces. It is easy to see that for any $M \in \operatorname{End}(V)$ one has $$\begin{gathered}
\eta_V^{-1} \big(M^t\big)^t \eta_V = M. \label{evi}\end{gathered}$$ In what follows we identify the spaces $V$ and $V^{\star \star}$, and whether an element belongs to the space $V$ or to the space $V^{\star \star}$ will be determined by the context. Equation (\[evi\]) becomes the identification $$\begin{gathered}
\big(M^t\big)^t = M. \label{mtt}\end{gathered}$$
Consider now the modules ${}^*(V^*)$ and $({}^* V)^*$. These modules as vector spaces are identical to the vector space $V^{\star \star} = V$. Equations (\[fsx\]) and (\[mtt\]) give $$\begin{gathered}
{}^* (\varphi^*) = \varphi, \qquad ({}^* \! \varphi)^* = \varphi,\end{gathered}$$ and we have the identification of the corresponding modules $$\begin{gathered}
{}^* (V^*) = V, \qquad ({}^* V)^* = V.\end{gathered}$$ Similarly as above, we see that the notations ${}^* \! \varphi^*_\zeta$ and ${}^* V^*_\zeta$ have a unique sense.
According to the definition of an $R$-operator (\[rhor\]), we write $$\begin{gathered}
\rho_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2) R_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2) = (\varphi_{1 \zeta_1}^* \otimes \varphi^{}_{2 \zeta_2})({\mathcal R}).\end{gathered}$$ Using the decomposition $$\begin{gathered}
{\mathcal R}= \sum_i a_i \otimes b_i,\end{gathered}$$ we determine that $$\begin{aligned}
(\varphi_{1 \zeta_1}^* \otimes \varphi^{}_{2 \zeta_2})({\mathcal R}) & = \sum_i \varphi_{1 \zeta_1}^*(a_i) \otimes \varphi^{}_{2 \zeta_2}(b_i) = \sum_i \varphi_{1 \zeta_1}(S(a_i))^t \otimes \varphi^{}_{2 \zeta_2}(b_i) \\
& = \bigg( \sum_i \varphi_{1 \zeta_1}(S(a_i)) \otimes \varphi^{}_{2 \zeta_2}(b_i) \bigg)^{t_1} = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}) ((S \otimes {\mathrm{id}})({\mathcal R}))^{t_1}.\end{aligned}$$ Now, using the equation $$\begin{gathered}
(S \otimes {\mathrm{id}})({\mathcal R}) = {\mathcal R}^{-1},\end{gathered}$$ see, for example, [@ChaPre94 p. 124], we come to the equation $$\begin{gathered}
(\varphi_{1 \zeta_1}^* \otimes \varphi^{}_{2 \zeta_2})({\mathcal R}) = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}) \big({\mathcal R}^{-1}\big)^{t_1}.\end{gathered}$$ We have $$\begin{gathered}
1 = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})\big({\mathcal R}{\mathcal R}^{-1}\big) = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})({\mathcal R}) (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})\big({\mathcal R}^{-1}\big),\end{gathered}$$ therefore, $$\begin{gathered}
(\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}) \big({\mathcal R}^{-1}\big) = ((\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2}) ({\mathcal R}))^{-1} = \rho_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}.\end{gathered}$$ Hence, we obtain $$\begin{gathered}
R_{V_1^* | V^{}_2}(\zeta_1 | \zeta_2) = D (\zeta_1 | \zeta_2) \big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}, \label{cri}\end{gathered}$$ where $$\begin{gathered}
D (\zeta_1 | \zeta_2) = \rho_{V_1^* | V^{}_2}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}.\end{gathered}$$ We call relation (\[cri\]), and any similar to it, a crossing relation.
Any crossing relation has the form of an equation whose left and right hand sides contain an $R$-operators or the inverse of an $R$-operator. The right hand side contains also a scalar coefficient $D$ whose concrete form is determined by the following rules. If the left hand side contains an $R$-operator $R_{V_1 | V_2}(\zeta_1 | \zeta_2)$ or its inverse, the factor $D$ contains the factor $\rho_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$ or $\rho_{V_1 | V_2}(\zeta_1 | \zeta_2)$. Respectively, if the right hand side contains an $R$-operator $R_{W_1 | W_2}(\eta_1 | \eta_2)$ or its inverse, the factor $D$ contains the factor $\rho_{W_1 | W_2}(\eta_1 | \eta_2)$ or $\rho_{W_1 | W_2}(\eta_1 | \eta_2)^{-1}$.
In the same way as above, using the identity $$\begin{gathered}
\big({\mathrm{id}}\otimes S^{-1}\big)({\mathcal R}) = {\mathcal R}^{-1},\end{gathered}$$ one comes to the equation $$\begin{gathered}
R_{V_1 | {}^*V_2}(\zeta_1 | \zeta_2) = D(\zeta_1 | \zeta_2) \big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2}. \label{crii}\end{gathered}$$ The graphical representation of the crossing relations (\[cri\]) and (\[crii\]) are given in Figs. \[f:cri\] and \[f:crii\].
![[]{data-label="f:crii"}](Figures/CrossingI)
![[]{data-label="f:crii"}](Figures/CrossingII)
Here and below, for the representation $\varphi^*$ we use the dotted variant of the line used for the representation $\varphi$, and for the representation ${}^* \! \varphi$ we use the dashed variant of that line.
Further, we have $$\begin{gathered}
({}^* \! \varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})\big({\mathcal R}^{-1}\big) = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})\big(\big(S^{-1} \otimes {\mathrm{id}}\big)\big({\mathcal R}^{-1}\big)\big)^{t_1} = (\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2})({\mathcal R})^{t_1}.\end{gathered}$$ It follows from this equation that $$\begin{gathered}
R_{{}^*V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} = D(\zeta_1 | \zeta_2) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1}. \label{criii}\end{gathered}$$ Similarly, $$\begin{gathered}
R_{V_1 | V_2^*}(\zeta_1 | \zeta_2)^{-1} = D(\zeta_1 | \zeta_2) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2}. \label{criv}\end{gathered}$$ One can see that Figs. \[f:criii\] and \[f:criv\] are the depiction of the crossing relations (\[criii\]) and (\[criv\]).
![[]{data-label="f:criv"}](Figures/CrossingIII)
![[]{data-label="f:criv"}](Figures/CrossingIV)
Concluding this section, we give two crossing relations obtained as a result of combining the crossing relations given above. They are $$\begin{gathered}
R_{V_1^* | V_2^*}(\zeta_1 | \zeta_2) = D(\zeta_1 | \zeta_2) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^t, \label{crv}\end{gathered}$$ and $$\begin{gathered}
R_{{}^*V_1 | {}^*V_2}(\zeta_1 | \zeta_2) = D(\zeta_1 | \zeta_2) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^t. \label{crvi}\end{gathered}$$ One can see the graphical representation of these relations in Figs. \[f:crv\] and \[f:crvi\].
![[]{data-label="f:crvi"}](Figures/CrossingV)
![[]{data-label="f:crvi"}](Figures/CrossingVI)
For completeness we give in Figs. \[f:crvii\] and \[f:crviii\] the graphical images of the crossing relations obtained from (\[crv\]) and (\[crvi\]) by inversion.
![[]{data-label="f:crviii"}](Figures/CrossingVII)
![[]{data-label="f:crviii"}](Figures/CrossingVIII)
### Double duals {#s:dd}
Let us proceed to the module $V^{**}$. Certainly, as a vector space it is again the vector space $V^{\star \star}$. Now for any $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ we have $$\begin{gathered}
\langle a \eta_V(v), \mu \rangle = \langle \eta_V(v), S(a) \mu \rangle = \langle S(a) \mu, v \rangle = \big\langle \mu, S^2(a) v \big\rangle = \big\langle \eta_V\big(S^2(a)v\big), \mu \big\rangle.\end{gathered}$$ It means that $\eta_V$ intertwines the representations $\varphi^{**}$ and $\varphi \circ S^2$, and since $\eta_V$ is an isomorphism of vector spaces we have the isomorphism of representations $$\begin{gathered}
\varphi^{**} \cong \varphi \circ S^2.\end{gathered}$$ In the same way we prove the isomorphism $$\begin{gathered}
{}^{**} \! \varphi \cong \varphi \circ S^{- 2}.\end{gathered}$$ It follows from (\[fsz\]) that $$\begin{gathered}
(\varphi^{**})_\zeta = (\varphi_\zeta)^{**}, \qquad ({}^{**} \! \varphi)_\zeta = {}^{**} \! (\varphi_\zeta)\end{gathered}$$ for any $\zeta \in {\mathbb C}^\times$, hence, the notations $\varphi^{**}_\zeta$ and ${}^{**}\varphi_\zeta$ are unambiguous.
Using (\[sg\]), we obtain $$\begin{gathered}
S^2(q^x) = q^x, \qquad S^2(e_i) = q^{- 2 d_i} e_i, \qquad S^2(f_i) = q^{2 d_i} f_i.\end{gathered}$$ For the image of $S^2(e_i)$ in the representation $\varphi_\zeta$ we have $$\begin{gathered}
\varphi_\zeta\big(S^2(e_i)\big) = \varphi_\zeta\big(q^{- 2 d_i} e_i\big).\end{gathered}$$ Thus, in this representation the action of $S^2$ on $e_i$ is realized as a rescaling. Looking at (\[djrb\]), one can try to perform such rescaling by conjugation with an appropriate element $q^x$. One has $$\begin{gathered}
q^x e_i q^{-x} = q^{\langle \alpha_i, x \rangle} e_i = q^{\mu_i} e_i,\end{gathered}$$ where $$\begin{gathered}
\mu_i = \langle \alpha_i, x \rangle, \label{mi}\end{gathered}$$ and, using relation (\[dth\]), we obtain $$\begin{gathered}
\sum_{i = 0}^l a_i \mu_i = 0. \label{sxiai}\end{gathered}$$ It is clear that it impossible to find an element $x \in {\widetilde{\mathfrak h}}$ such that the similarity transformation determined by $q^x$ gives the desired result. However, one can simultaneously with such transformation modify the spectral parameter. Let $\widetilde \zeta$ be a new spectral parameter. We have to satisfy the equation $$\begin{gathered}
\zeta^{s_i} q^{- 2 d_i} = \widetilde \zeta^{s_i} q^{\mu_i}. \label{zsitszi}\end{gathered}$$ It follows from equation (\[sxiai\]) that $$\begin{gathered}
\zeta^s q^{- 2 \sum\limits_{i = 0}^l a_i d_i} = \widetilde \zeta^s,\end{gathered}$$ where $s$ is defined by equation (\[ds\]). Using equations (\[dz\]) and (\[di\]), we find $$\begin{gathered}
\sum_{i = 0}^l a_i d_i = \frac{1}{2} \left[ (\theta | \theta) + \sum_{i = 1}^l a_i (\alpha_i | \alpha_i) \right] = \frac{(\theta | \theta)}{2} \sum_{i = 0}^l {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} = \frac{(\theta | \theta)}{2} {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}}.\end{gathered}$$ Hence, we come to the following expression for the new spectral parameter $$\begin{gathered}
\widetilde \zeta = q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta. \label{tzeta}\end{gathered}$$ Finally, we find that equation (\[zsitszi\]) is satisfied if $$\begin{gathered}
\mu_i = - 2 d_i + (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} s_i / s. \label{hia}\end{gathered}$$ Note that in the case where $s_i = d_i$ we have $\mu_i = 0$, and, therefore, $x = 0$.
The element $x$ can be written as $$\begin{gathered}
x = \sum_{i = 0}^l \lambda_i h_i \label{xlh}\end{gathered}$$ for some numbers $\lambda_i \in {\mathbb C}$. Using (\[mi\]) and (\[dcm\]), we obtain the following system of equations for $\lambda_i$: $$\begin{gathered}
\sum_{j = 0}^l \lambda_j a_{j i} = \mu_i. \label{ljaji}\end{gathered}$$ The solution of this equation is not unique. We fix the ambiguity by the condition $$\begin{gathered}
\lambda_0 = 0,\end{gathered}$$ and, using (\[sxiai\]), rewrite the system (\[ljaji\]) as $$\begin{gathered}
\sum_{j = 1}^l \lambda_j a_{j 0} = - \sum_{i = 1}^l a_j \mu_j, \label{ljaj0} \\
\sum_{j = 1}^l \lambda_j a_{j i} = \mu_i, \qquad i \in {[1 \, . \, . \, l]}. \label{ljajim}\end{gathered}$$ The system (\[ljajim\]) has the unique solution $$\begin{gathered}
\lambda_i = \sum_{j = 1}^l \mu_j b_{j i}, \qquad i \in {[1 \, . \, . \, l]}, \label{lmb}\end{gathered}$$ where $b_{i j}$ are the matrix elements of the matrix $B$ inverse to the Cartan matrix $A = (a_{i j})_{i, j \in {[1 \, . \, . \, l]}}$ of the Lie algebra ${\mathfrak g}$. Substituting this solution into (\[ljaj0\]) and taking into account the last equation of (\[aaa\]), we see that equation (\[ljaj0\]) is satisfied identically.
Let us obtain another expression for the element $x$, cf. paper [@FreRes92]. To this end recall that the elements $\omega_i \in {\mathfrak h}$, $i \in {[1 \, . \, . \, l]}$, defined by the equation $$\begin{gathered}
\langle \omega^{}_i, h_j \rangle = \delta_{i j}\end{gathered}$$ are called the fundamental weights. Their sum $$\begin{gathered}
\rho = \sum_{i = 1}^l \omega_i\end{gathered}$$ satisfies the equation $$\begin{gathered}
\langle \rho, h_i \rangle = 1\end{gathered}$$ for any $i \in {[1 \, . \, . \, l]}$. We obtain $$\begin{gathered}
\big\langle \alpha_i, \nu^{-1}(\rho) \big\rangle = \big(\nu^{-1}(\alpha_i) | \nu^{-1}(\rho)\big) = \frac{(\alpha_i | \alpha_i)}{2}\big(h_i | \nu^{-1}(\rho)\big) = \frac{(\alpha_i | \alpha_i)}{2} \langle \rho, h_i \rangle = d_i\end{gathered}$$ and $$\begin{gathered}
\big\langle \alpha_0, \nu^{-1}(\rho) \big\rangle = - \sum_{i = 1}^l a_i \big\langle \alpha_i, \nu^{-1}(\rho) \big\rangle = - \sum_{i = 1}^l a_i d_i = - \frac{(\theta | \theta)}{2} ({h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} - 1).\end{gathered}$$ This gives $$\begin{gathered}
x = - 2 \nu^{-1}(\rho) + y, \label{hrho}\end{gathered}$$ where for the components $$\begin{gathered}
\nu_i = \langle \alpha_i, y \rangle\end{gathered}$$ we have the expressions $$\begin{gathered}
\nu_0 = (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} (s_0 - s) / s, \qquad \nu_i = (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} s_i / s.\end{gathered}$$ In the case $$\begin{gathered}
s_0 = 1, \qquad s_i = 0, \qquad i \in {[1 \, . \, . \, l]},\end{gathered}$$ we see that $y = 0$.
Thus, for any $i \in {[0 \, . \, . \, l]}$ we have $$\begin{gathered}
\varphi^{**}_\zeta(e_i) = \varphi_{\widetilde \zeta}\big(q^x e_i q^{-x}\big),\end{gathered}$$ where the new spectral parameter $\widetilde \zeta$ is given by (\[tzeta\]) and element $x$ is determined either by equation equations (\[xlh\]), (\[lmb\]) and (\[hia\]), or by equation (\[hrho\]). In a similar way we obtain $$\begin{gathered}
\varphi^{**}_\zeta(f_i) = \varphi_{\widetilde \zeta}\big(q^x f_i q^{-x}\big)\end{gathered}$$ for any $i \in {[0 \, . \, . \, l]}$. Summarizing, we see that $$\begin{gathered}
\varphi^{**}_\zeta(a) = \varphi\big(q^x\big) \varphi_{\widetilde \zeta}(a) \varphi\big(q^{- x}\big). \label{fsszx}\end{gathered}$$ for any $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$. This means that we have the isomorphism $$\begin{gathered}
V^{**}_\zeta \cong V_{\widetilde \zeta}.\end{gathered}$$
In a similar way we obtain the equation $$\begin{gathered}
{}^{**} \! \varphi_\zeta(a) = \varphi\big(q^{-x}\big) \varphi_{\widetilde \zeta}(a) \varphi\big(q^{x}\big),\end{gathered}$$ where $x$ is determined again either by equations (\[xlh\]), (\[lmb\]) and (\[hia\]), or by equation (\[hrho\]), while $\widetilde \zeta$ is now defined as $$\begin{gathered}
\widetilde \zeta = q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}}/s} \zeta.\end{gathered}$$
Using equation (\[cri\]), we obtain $$\begin{gathered}
R_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2) = \rho_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \big(R_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}. \label{rvssviii}\end{gathered}$$ Using (\[cri\]) again, we come to the equation $$\begin{gathered}
R_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2) = \rho_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}(\zeta_1 | \zeta_2) \big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}\big)^{-1}\big)^{t_1}. \label{rvssvi}\end{gathered}$$ Comparing it with (\[dt\]), we see that ${\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)$ is proportional to $R_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2)$. It follows from (\[fsszx\]) and (\[tzeta\]) that $$\begin{gathered}
R_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2) = \rho_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) \nonumber\\
\hphantom{R_{V_1^{**} | V_2^{}}(\zeta_1 | \zeta_2) =}{} \times ({\mathbb X}_{V_1}^{} \otimes {\mathrm{id}}_{V_2}) R_{V_1 | V_2}\big(q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) \big({\mathbb X}_{V_1}^{-1} \otimes {\mathrm{id}}_{V_2}\big). \label{rvssvii}\end{gathered}$$ Here and below for a ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module $V$ and the corresponding representation $\varphi$ we denote $$\begin{gathered}
{\mathbb X}_V = \varphi\big(q^x\big)\end{gathered}$$ for $x$ given either by equations (\[xlh\]), (\[lmb\]) and (\[hia\]), or by equation (\[hrho\]). Comparing equations (\[rvssvi\]) and (\[rvssvii\]), we come to the equation $$\begin{gathered}
\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}\big)^{-1}\big)^{t_1} = \rho_{V_1^{} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) \\
\hphantom{\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}\big)^{-1}\big)^{t_1} =}{} \times ({\mathbb X}_{V_1}^{} \otimes {\mathrm{id}}_{V_2}) R_{V_1 | V_2}\big(q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) \big({\mathbb X}_{V_1}^{-1} \otimes {\mathrm{id}}_{V_2}\big).\end{gathered}$$ Further, comparing equations (\[rvssvii\]) and (\[rvssviii\]), we come to the crossing relation $$\begin{gathered}
R_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2)^{-1} = D(\zeta_1 | \zeta_2) \big(\big({\mathbb X}_{V_1}^{-1}\big)^t \otimes {\mathrm{id}}_{V_2}\big) R_{V_1 | V_2}\big(q^{-(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big)^{t_1} \big({\mathbb X}_{V_1}^t \otimes {\mathrm{id}}_{V_2}\big), \label{crix}\end{gathered}$$ where $$\begin{gathered}
D(\zeta_1 | \zeta_2) = \rho_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2) \rho_{V_1^{} | V_2^{}}\big(q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) = \rho_{V_1^* | V_2^{}}(\zeta_1 | \zeta_2) \rho_{V_1^{} | V_2^{}}\big(\zeta_1 | q^{ (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big).\end{gathered}$$ Here equation (\[znu\]) is used.
Starting with the $R$-matrix $R_{V_1^{} | {}^{**} V_2}(\zeta_1 | \zeta_2)$, we come to the equation $$\begin{gathered}
\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2}\big)^{-1}\big)^{t_2}
= \rho_{V_1^{} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(\zeta_1 | q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big) \\
\hphantom{\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2}\big)^{-1}\big)^{t_2} =}{} \times \big({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2}^{-1}\big) R_{V_1 | V_2}\big(\zeta_1 | q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big)^{t_2} ({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2})\end{gathered}$$ and to the crossing relation $$\begin{gathered}
R_{V_1^{} | {}^* V_2^{}}(\zeta_1 | \zeta_2)^{-1} = D(\zeta_1 | \zeta_2) \big({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2}^t\big) R_{V_1 | V_2}\big(\zeta_1 | q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big)^{t_2} \big({\mathrm{id}}_{V_1} \otimes \big({\mathbb X}_{V_2}^{-1}\big)^t\big), \label{crx}\end{gathered}$$ where $$\begin{gathered}
D(\zeta_1 | \zeta_2) = \rho_{V_1 | {}^* V_2}(\zeta_1 | \zeta_2) \rho_{V_1^{} | V_2^{}}\big(\zeta_1 | q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big) = \rho_{V_1 | {}^* V_2}(\zeta_1 | \zeta_2) \rho_{V_1^{} | V_2^{}}\big(q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big).\end{gathered}$$
![[]{data-label="f:iho"}](Figures/HOperator)
![[]{data-label="f:iho"}](Figures/InverseHOperator)
Let us give a graphical representation of the crossing relations (\[crix\]) and (\[crx\]). For the matrix elements of the operator ${\mathbb X}_V$ and its inverse we use the depiction given in Figs. \[f:ho\] and \[f:iho\].
Note that the equation $$\begin{gathered}
\varphi^*\big(q^x\big) = {}^* \! \varphi\big(q^x\big) = \big(\varphi\big(q^{x}\big)^{-1}\big)^t\end{gathered}$$ results in four graphical equations given in Figs. \[f:hoihoi\]–\[f:hoihoiv\].
![[]{data-label="f:hoihoii"}](Figures/HOperatorInverseHOperatorI)
![[]{data-label="f:hoihoii"}](Figures/HOperatorInverseHOperatorII)
![[]{data-label="f:hoihoiv"}](Figures/HOperatorInverseHOperatorIII)
![[]{data-label="f:hoihoiv"}](Figures/HOperatorInverseHOperatorIV)
It can be demonstrated now that Figs. \[f:crix\] and \[f:crx\] represent the crossing relations (\[crix\]) and (\[crx\]).
![[]{data-label="f:crx"}](Figures/CrossingIX)
![[]{data-label="f:crx"}](Figures/CrossingX)
Finally, starting with the $R$-operators $R_{{}^{**} V_1 | V_2^{}}(\zeta_1 | \zeta_2)$ and $R_{V_1^{} | V_2^{**}}(\zeta_1 | \zeta_2)$, we obtain two more equations $$\begin{gathered}
\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1}\big)^{-1}\big)^{t_1}\big)^{-1} = \rho_{V_1^{} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big( q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) \\
\hphantom{\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1}\big)^{-1}\big)^{t_1}\big)^{-1} =} {} \times \big({\mathbb X}_{V_1}^{-1} \otimes {\mathrm{id}}_{V_2}\big) R_{V_1 | V_2}\big(q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big) ({\mathbb X}_{V_1}^{} \otimes {\mathrm{id}}_{V_2}), \\
\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2}\big)^{-1}\big)^{t_2}\big)^{-1} = \rho_{V_1^{} | V_2^{}}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(\zeta_1 | q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big) \\
\hphantom{\big(\big(\big(R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_2}\big)^{-1}\big)^{t_2}\big)^{-1} =}{} \times ({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2}^{}) R_{V_1 | V_2}\big(\zeta_1 | q^{-(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big) \big({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2}^{-1}\big),\end{gathered}$$ and two more crossing relations $$\begin{gathered}
R_{{}^*V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1} = D(\zeta_1 | \zeta_2) \big({\mathbb X}_{V_1}^{-1} \otimes {\mathrm{id}}_{V_2}\big) R_{V_1 | V_2}\big(q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big)^{-1} \big({\mathbb X}_{V_1}^{} \otimes {\mathrm{id}}_{V_2}\big) \label{crxi}\end{gathered}$$ and $$\begin{gathered}
R_{V_1^{} | V_2^*}(\zeta_1 | \zeta_2)^{t_2} = D(\zeta_1 | \zeta_2) ({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2}^{}) R_{V_1 | V_2}\big(\zeta_1 | q^{-(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big)^{-1} \big({\mathrm{id}}_{V_1} \otimes {\mathbb X}_{V_2}^{-1}\big), \label{crxii}\end{gathered}$$ where $$\begin{aligned}
D(\zeta_1 | \zeta_2) &= \rho_{{}^* V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big)^{-1} \\
& = \rho_{{}^* V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(\zeta_1 | q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big)^{-1}\end{aligned}$$ and $$\begin{aligned}
D(\zeta_1 | \zeta_2) & = \rho_{V_1^{} | V_2^*}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(\zeta_1 | q^{-(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_2\big)^{-1} \\
& = \rho_{V_1^{} | V_2^*}(\zeta_1 | \zeta_2)^{-1} \rho_{V_1^{} | V_2^{}}\big(q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} \zeta_1 | \zeta_2\big)^{-1}\end{aligned}$$ respectively. The crossing relations (\[crxi\]) and (\[crxii\]) are depicted in Figs. \[f:crxi\] and \[f:crxii\].
![[]{data-label="f:crxii"}](Figures/CrossingXI)
![[]{data-label="f:crxii"}](Figures/CrossingXII)
Now, similarly as for the case of ${\overset{\scriptstyle \approx}{\rule{0em}{.6em} \smash{R}}}_{V_1 | V_2}(\zeta_1 | \zeta_2)$, one can demonstrate that $\widetilde R_{V_1 | V_2}(\zeta_1 | \zeta_2)$ is proportional to $R_{{}^{**} V_1 | V_2^{}}(\zeta_1 | \zeta_2)$.
### Yang–Baxter equation
Now, let $V_1$, $V_2$, $V_3$ be ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules, $\varphi_1$, $\varphi_2$, $\varphi_3$ the corresponding representations of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$, and $\zeta_1$, $\zeta_2$, $\zeta_3$ the spectral parameters associated with the representations. We associate with $V_1$, $V_2$ and $V_3$ a single, double and triple lines, respectively. Applying to both sides of equation (\[uybe\]) the mapping $\varphi_{1 \zeta_1} \otimes \varphi_{2 \zeta_2} \otimes \varphi_{3 \zeta_3}$ and using the definition of an $R$-operator (\[rhor\]), we obtain the Yang–Baxter equation $$\begin{gathered}
R_{V_1 | V_2}^{(1 2)}(\zeta_1 | \zeta_2) R_{V_1 | V_3}^{(1 3)}(\zeta_1 | \zeta_3) R_{V_2 | V_3}^{(2 3)}(\zeta_2 | \zeta_3) = R_{V_2 | V_3}^{(2 3)}(\zeta_2 | \zeta_3) R_{V_1 | V_3}^{(1 3)}(\zeta_1 | \zeta_3) R_{V_1 | V_2}^{(1 2)}(\zeta_1 | \zeta_2).\end{gathered}$$ It is natural, slightly abusing notation, to denote $R_{V_i | V_j}(\zeta_i | \zeta_j)^{(i j)}$ simply by $R_{V_i | V_j}(\zeta_i | \zeta_j)$. Now the above equation takes the form $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2) R_{V_1 | V_3}(\zeta_1 | \zeta_3) R_{V_2 | V_3}(\zeta_2 | \zeta_3) = R_{V_2 | V_3}(\zeta_2 | \zeta_3) R_{V_1 | V_3}(\zeta_1 | \zeta_3) R_{V_1 | V_2}(\zeta_1 | \zeta_2). \label{yb}\end{gathered}$$
![[]{data-label="f:wyb"}](Figures/YangBaxterI)
![[]{data-label="f:wyb"}](Figures/WrongYangBaxter)
=-1 One can recognize the graphical image of this equation in Fig. \[f:ybi\]. As one can see, we have three external arrowheads and three external arrowtails there. It is worth to stress that the heads and the tails are grouped together, and the graphical equation given in Fig. \[f:wyb\], where there is no such grouping, is not a true Yang–Baxter equation. However, as it is shown below, in the case when the corresponding $R$-operators satisfy the unitarity relations this equation is also true.
Multiplying both sides of equation (\[yb\]) on the left and right by $R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$, we obtain $$\begin{gathered}
R_{V_1 | V_3}(\zeta_1 | \zeta_3) R_{V_2 | V_3}(\zeta_2 | \zeta_3) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} \nonumber\\
\qquad {} = R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1} R_{V_2 | V_3}(\zeta_2 | \zeta_3) R_{V_1 | V_3}(\zeta_1 | \zeta_3). \label{rrir}\end{gathered}$$ It is instructive to obtain this equation by the graphical method. It is clear that the multiplication of (\[yb\]) by $R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{-1}$ is equivalent to transition from the equation given in Fig. \[f:ybi\] to the equation given in Fig. \[f:tyb\]. Now, using the graphical equations given in Figs. \[f:irrwi\] and \[f:rirwi\], we come to the graphical image of equation (\[rrir\]) given in Fig. \[f:ybii\].
![[]{data-label="f:tyb"}](Figures/TransformingYangBaxter)
![[]{data-label="f:ybvii"}](Figures/YangBaxterII)
![[]{data-label="f:ybvii"}](Figures/YangBaxterVII)
In a similar way one can obtain a lot of graphical versions of the Yang–Baxter equation. We give here only one additional example, shown in Fig. \[f:ybvii\]. One can get convinced that the analytical form of that graphical equation is $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1} R_{V_2 | V_3}(\zeta_2 | \zeta_3) \big(\widetilde R_{V_1 | V_3}(\zeta_1 | \zeta_3)^{-1}\big)^{t_1} \nonumber\\
\qquad{}= \big(\widetilde R_{V_1 | V_3}(\zeta_1 | \zeta_3)^{-1}\big)^{t_1} R_{V_2 | V_3}(\zeta_2 | \zeta_3) R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{t_1}, \label{rrtr1}\end{gathered}$$ or, equivalently, $$\begin{gathered}
\big(\widetilde R_{V_1 | V_3}(\zeta_1 | \zeta_3)^{-1}\big)^{t_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2) R_{V_2 | V_3}(\zeta_2 | \zeta_3)^{t_2} \nonumber\\
\qquad{} = R_{V_2 | V_3}(\zeta_2 | \zeta_3)^{t_2} R_{V_1 | V_2}(\zeta_1 | \zeta_2) \big(\widetilde R_{V_1 | V_3}(\zeta_1 | \zeta_3)^{-1}\big)^{t_2}. \label{rrtr2}\end{gathered}$$
Let us demonstrate how equations (\[rrtr1\]) and (\[rrtr2\]) can be obtained analytically. For simplicity, we denote $R_{V_i | V_j}(\zeta_i | \zeta_j)$ just by $R_{i j}$. Transposing the Yang–Baxter equation (\[yb\]) with respect to $V_1$ and using equations (\[mtm\]) and (\[mmt\]), we come to the equation $$\begin{gathered}
(R_{1 2} R_{1 3})^{t_1} R_{2 3} = R_{2 3} (R_{1 3} R_{1 2})^{t_1}.\end{gathered}$$ Taking into account equation (\[mtmt\]), we see that it is equivalent to $$\begin{gathered}
(R_{1 3})^{t_1} (R_{1 2})^{t_1} R_{2 3} = R_{2 3} (R_{1 2})^{t_1} (R_{1 3})^{t_1}.\end{gathered}$$ Multiplying both sides of this equation on the left and right by $\big((R_{1 3})^{t_1}\big)^{-1}$ we obtain $$\begin{gathered}
(R_{1 2})^{t_1} R_{2 3} \big((R_{1 3})^{t_1}\big)^{-1} = \big((R_{1 3})^{t_1}\big)^{-1} R_{2 3} (R_{1 2})^{t_1}.\end{gathered}$$ It follows from (\[trrtiti\]) that this equation is equivalent to (\[rrtr1\]). Equation (\[rrtr2\]) can be obtained in a similar way.
More examples of graphical Yang–Baxter equations can be found in Section \[s:coto\]. One should keep in mind that initially we have only one Yang–Baxter equation with many analytical and graphical reincarnations.
![[]{data-label="f:ybxi"}](Figures/YangBaxterX)
![[]{data-label="f:ybxi"}](Figures/YangBaxterXI)
Let us show now that if the corresponding $R$-operators satisfy the unitarity relations we can obtain the Yang–Baxter equation given in Fig. \[f:wyb\]. We start with the Yang–Baxter equation depicted in Fig. \[f:ybx\]. Using the crossing relation in Fig. \[f:crii\], we come to the Yang–Baxter equation in Fig. \[f:ybxi\]. Now if the corresponding $R$-operators satisfy the unitarity relations described in Section \[s:ur\] we obtain the Yang–Baxter equation given in Fig. \[f:wyb\].
It is not difficult to demonstrate that if the corresponding unitarity relations are satisfied, all possible Yang–Baxter equations with all possible types of the vertices and directions of the arrows are correct. However, if the unitarity relations are not true, one has to check whether the used Yang–Baxter equations can be obtained without them.
### Monodromy operators
The representation spaces of the basic modules used to construct integrability objects are of two types: auxiliary and quantum spaces. Although the boundary between these two types is rather conventional, such a division proves useful. When both spaces used to define a basic integrability object are auxiliary, we call it an $R$-operator. When one of the spaces is auxiliary and another one is quantum, we say about a monodromy operator. In this paper we use for auxiliary spaces as before a single line, double lines, etc., and indices $\alpha$, $\beta$, $\gamma$, etc. For quantum spaces we use waved lines, and indices $i$, $j$, $k$, etc.
The definition of a basic monodromy operator is in fact the same as the definition of an $R$-operator, except the notation. Let $V$ and $W$ be two ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules, and $\varphi$ and $\psi$ the associated representations, corresponding to auxiliary and quantum spaces respectively. We define a monodromy operator $M_{V | W}(\zeta | \eta)$ as $$\begin{gathered}
\rho_{\varphi | \psi}(\zeta | \eta) M_{V | W}(\zeta | \eta) = (\varphi_\zeta \otimes \psi_\eta)({\mathcal R}). \label{mpp}\end{gathered}$$ The graphical representation of the matrix elements of $M_{V | W}(\zeta | \eta)$ and its inverse is practically the same as for $R$-matrices. However, for completeness, we present it in Figs. \[f:mo\] and \[f:imo\].
![[]{data-label="f:imo"}](Figures/MOperator)
![[]{data-label="f:imo"}](Figures/InverseMOperator)
From the point of view of spin chains, the monodromy operator $M_{V | W}(\zeta | \eta)$ corresponds to a chain of one site. For a general spin chain we use instead of the module $W_\eta$ the tensor product $$\begin{gathered}
W_{\eta_1} \otimes_\Delta W_{\eta_2} \otimes_\Delta \cdots \otimes_\Delta W_{\eta_N},\end{gathered}$$ and define the monodromy operator $M_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N)$ as $$\begin{gathered}
\rho_{V | W} (\zeta | \eta_1) \rho_{V | W} (\zeta | \eta_2) \cdots \rho_{V | W} (\zeta | \eta_N) M_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N) \\
\qquad{} = (\varphi_\zeta \otimes (\psi_{\eta_1} \otimes_\Delta \psi_{\eta_2} \otimes_\Delta \cdots \otimes_\Delta \psi_{\eta_N}))({\mathcal R}).\end{gathered}$$
Let us establish the connection of the general monodromy operator with the basic one. Consider the case of $N = 2$. By definition, we have $$\begin{gathered}
\rho_{V | W} (\zeta | \eta_1) \rho_{V | W} (\zeta | \eta_2) M_{V | W}(\zeta | \eta_1, \eta_2) = (\varphi_\zeta \otimes (\psi_{\eta_1} \otimes_\Delta \psi_{\eta_2}))({\mathcal R}) \nonumber\\
\qquad{}= (\varphi_\zeta \otimes \psi_{\eta_1} \otimes \psi_{\eta_2}) (({\mathrm{id}}\otimes \Delta)({\mathcal R})). \label{gmomomo}\end{gathered}$$ Using the second equation of (\[drrr\]) in (\[gmomomo\]), we obtain $$\begin{gathered}
M_{V | W}(\zeta | \eta_1, \eta_2) = R^{(1 3)}_{V | W}(\zeta | \eta_2) R^{(1 2)}_{V | W}(\zeta | \eta_1).\end{gathered}$$ It is not difficult to see that for a general $N$ one has $$\begin{gathered}
M_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N) = R^{(1, N + 1)}_{V | W}(\zeta | \eta_N) \cdots R^{(1 3)}_{V | W}(\zeta | \eta_2) R^{(1 2)}_{V | W}(\zeta | \eta_1), \label{gmo}\end{gathered}$$ or in terms of matrix elements $$\begin{gathered}
M_{V | W}(\zeta | \eta_1, \eta_2, \ldots,
\eta_N)^{\alpha i_1 i_2 \ldots i_N}{}_{\beta j_1 j_2 \ldots j_N} \\
\qquad{} = R_{V | W}(\zeta | \eta_N)^{\alpha i_N}{}_{\gamma_1 j_N} \cdots
R_{V | W}(\zeta | \eta_2)^{\gamma_{N - 2} i_2}{}_{\gamma_{N - 1} j_2} R_{V | W}(\zeta | \eta_1)^{\gamma_{N - 1} i_1}{}_{\beta j_1}.\end{gathered}$$ Now it is clear that these matrix elements can be depicted as in Fig. \[f:gmo\].
![[]{data-label="f:gmo"}](Figures/MonodromyOperator)
The inverse monodromy operator has the form $$\begin{gathered}
M_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N)^{-1} = R^{(1 2)}_{V | W}(\zeta | \eta_1)^{-1} R^{(1 3)}_{V | W}(\zeta | \eta_2)^{-1} \cdots R^{(1, N + 1)}_{V | W}(\zeta | \eta_N)^{-1},\end{gathered}$$ and the graphical representation of its matrix elements can be found in Fig. \[f:igmo\].
![[]{data-label="f:igmo"}](Figures/InverseMonodromyOperator)
A graphical exposition of the fact that the operators depicted in Figs. \[f:gmo\] and \[f:igmo\] are mutually inverse is given in Figs. \[f:mimoi\]–\[f:mimoiv\].
![[]{data-label="f:mimoii"}](Figures//MonodromyInverseMonodromyOperatorsI)
![[]{data-label="f:mimoii"}](Figures//MonodromyInverseMonodromyOperatorsII)
![[]{data-label="f:mimoiv"}](Figures//MonodromyInverseMonodromyOperatorsIII)
![[]{data-label="f:mimoiv"}](Figures//MonodromyInverseMonodromyOperatorsIV)
Below we numerate auxiliary spaces by primed numbers and quantum spaces by usual numbers. For example, we assume that the monodromy operator (\[gmo\]) acts on the space $$\begin{gathered}
U_{1'} \otimes U_1 \otimes U_2 \otimes \cdots \otimes U_N = V \otimes W \otimes W \otimes \cdots \otimes W,\end{gathered}$$ and write $$\begin{gathered}
M_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N) = R^{(1' N)}_{V | W}(\zeta | \eta_N) \cdots R^{(1' 2)}_{V | W}(\zeta | \eta_2) R^{(1' 1)}_{V | W}(\zeta | \eta_1).\end{gathered}$$ The most important property of monodromy operators is the relation $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{(1' 2')} M_{V_1 | W}(\zeta_1 | \eta_1, \eta_2, \ldots, \eta_N)^{(1' 1 \ldots N)} M_{V_2 | W}(\zeta_2 | \eta_1, \eta_2, \ldots, \eta_N)^{(2' 1 \ldots N)} \\
\qquad{} = M_{V_2 | W}(\zeta_2 | \eta_1, \eta_2, \ldots, \eta_N)^{(2' 1 \ldots N)}
M_{V_1 | W}(\zeta_1 | \eta_1, \eta_2, \ldots, \eta_N)^{(1' 1 \ldots N)} R_{V_1 | V_2}(\zeta_1 | \zeta_2)^{(1' 2')}.\end{gathered}$$ Here we have two auxiliary spaces labeled by $1'$ and $2'$. The well known graphical proof of the above relation is presented in Figs. \[f:rmmri\]–\[f:rmmriv\].
![[]{data-label="f:rmmrii"}](Figures/RMMRelationI)
![[]{data-label="f:rmmrii"}](Figures/RMMRelationII)
![[]{data-label="f:rmmriv"}](Figures/RMMRelationIII)
![[]{data-label="f:rmmriv"}](Figures/RMMRelationIV)
### Transfer operators and Hamiltonians
By definition, a nonzero element $a \in {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is group-like if $$\begin{gathered}
\Delta(a) = a \otimes a.\end{gathered}$$ The counit axiom says $$\begin{gathered}
(\varepsilon \otimes {\mathrm{id}})(\Delta(a)) = ({\mathrm{id}}\otimes \varepsilon)(\Delta(a)) = a,\end{gathered}$$ where in the last equality the canonical identification ${\mathbb C}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak g))}\simeq {{\mathrm U}_q(\mathcal L(\mathfrak g))}$ is used. Hence, for a group-like element $a$ we have $$\begin{gathered}
\varepsilon(a) a = a \varepsilon(a) = a,\end{gathered}$$ therefore, $$\begin{gathered}
\varepsilon(a) = 1.\end{gathered}$$ Now, using the antipode axiom, $$\begin{gathered}
\mu (({\mathrm{id}}\otimes S)(\Delta(a))) = \mu ((S \otimes {\mathrm{id}})(\Delta(a))) = \iota(\varepsilon(a))\end{gathered}$$ we obtain $$\begin{gathered}
a S(a) = S(a) a = 1.\end{gathered}$$ Thus, any group-like element is invertible, and, since $1$ is group-like, the set of all group-like elements form a group.
It follows from (\[gzqx\]) and (\[hsa\]) that $$\begin{gathered}
\Delta \circ \Gamma_\zeta = (\Gamma_\zeta \otimes \Gamma_\zeta) \circ \Delta.\end{gathered}$$ This equation implies that $$\begin{gathered}
\Gamma_\zeta(a) = a\end{gathered}$$ for any group-like element $a$. Therefore, for any representation $\varphi$ of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ we have $$\begin{gathered}
\varphi_\zeta(a) = \varphi(a).\end{gathered}$$
By definition, for any group-like element we have $$\begin{gathered}
\Delta(a) = \Delta'(a).\end{gathered}$$ It follows that $$\begin{gathered}
{\mathcal R}(a \otimes a) = (a \otimes a) {\mathcal R}.\end{gathered}$$ Hence, for any two representations $\varphi_1$, $\varphi_2$ of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ and the corresponding ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules $V_1$, $V_2$ one can obtain the equation $$\begin{gathered}
R_{V_1 | V_2}(\zeta_1 | \zeta_2) ({\mathbb A}_{V_1} \otimes {\mathbb A}_{V_2}) = ({\mathbb A}_{V_1} \otimes {\mathbb A}_{V_2}) R_{V_1 | V_2}(\zeta_1 | \zeta_2). \label{rxx}\end{gathered}$$ Here and below for any representation $\varphi$ of ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$ and the corresponding ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-module $V$ we denote $$\begin{gathered}
{\mathbb A}_V = \varphi_\zeta(a) = \varphi(a).\end{gathered}$$ For the matrix elements of the operator ${\mathbb A}_V$ and its inverse we use the depiction given in Figs. \[f:xo\] and \[f:ixo\]. One can easily recognize the graphical image of equation (\[rxx\]) in Fig. \[f:rxx\]. Note that the equation $$\begin{gathered}
\varphi^*(a) = {}^* \! \varphi(a) = \big(\varphi(a)^{-1}\big)^t\end{gathered}$$ results in four graphical equations given in Figs. \[f:xoixoi\]–\[f:xoixoiv\].
![[]{data-label="f:ixo"}](Figures/XOperator)
![[]{data-label="f:ixo"}](Figures/InverseXOperator)
![[]{data-label="f:rxx"}](Figures/ROperatorXOperators)
![[]{data-label="f:xoixoii"}](Figures/XOperatorInverseXOperatorI)
![[]{data-label="f:xoixoii"}](Figures/XOperatorInverseXOperatorII)
![[]{data-label="f:xoixoiv"}](Figures/XOperatorInverseXOperatorIII)
![[]{data-label="f:xoixoiv"}](Figures/XOperatorInverseXOperatorIV)
We call a group-like element $a$ the twisting element, and the corresponding operators ${\mathbb A}_V$ the twisting operators. The twisted transfer operator is defined by the equation $$\begin{gathered}
T_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N) = \operatorname{tr}_V (M_{V | W}(\zeta | \eta_1, \eta_2, \ldots, \eta_N) ({\mathbb A}_V \otimes {\mathrm{id}}_W))\end{gathered}$$ with the depiction given in Fig. \[f:to\]. Here $\operatorname{tr}_V$ means the partial trace with respect to the space $V$, see Appendix \[a:ptr\], and hooks at the ends of the line mean that it is closed in an evident way.
![[]{data-label="f:ttos"}](Figures/TransferOperator)
![[]{data-label="f:ttos"}](Figures/TwoTOs)
The most important property of transfer operators is their commutativity $$\begin{gathered}
[T_{V_1 | W}(\zeta_1 | \eta_1, \eta_2, \ldots, \eta_N), T_{V_2 | W}(\zeta_2 | \eta_1, \eta_2, \ldots, \eta_N)] = 0.\end{gathered}$$ The graphical proof of this property starts with a picture representing the product of two transfer operators, see Fig. \[f:ttos\]. Then one makes the four steps described by Figs. .
![[]{data-label="f:ctoiv"}](Figures/CommutativityTOI)
![[]{data-label="f:ctoiv"}](Figures/CommutativityTOII)
\
![[]{data-label="f:ctoiv"}](Figures/CommutativityTOIII)
![[]{data-label="f:ctoiv"}](Figures/CommutativityTOIV)
The commutativity property is the source of commuting quantities of quantum integrable systems. The most interesting here are local quantities. An example of such a quantity is a Hamiltonian, which is usually constructed in the following way. Assume that the quantum space $W$ coincides with the auxiliary space $V$. Further, assume that the $R$-operator $R_{V | V}(1 | 1)$ is proportional to the permutation operator $P_{1 2}$. In fact, as it follows from the definition below, the Hamiltonian does not change after multiplying of the $R$-operator by a constant factor, and we assume that $R_{V | V}(1 | 1)$ coincides with the permutation operator. The equation given in Fig. \[f:ic\] is the graphical representation of this fact.
![[]{data-label="f:ic"}](Figures/InitialCondition)
The Hamiltonian $H_N$ for the chain of length $N$ is constructed from the homogeneous transfer operator $$\begin{gathered}
T_V(\zeta) = T_{V | V}(\zeta | 1, 1, \ldots, 1)\end{gathered}$$ with the help of the equation $$\begin{gathered}
H_N = \left. \zeta \frac{{\mathrm d}}{{\mathrm d}\zeta} \log T_V(\zeta) \right|_{\zeta = 1} = \left. \zeta \frac{{\mathrm d}T_V(\zeta)}{{\mathrm d}\zeta} \right|_{\zeta = 1} T_V(1)^{-1} = T'_V(1) T_V(1)^{-1}. \label{hnp}\end{gathered}$$ It is evident that Figs. \[f:so\] and \[f:iso\] represent the depiction of the operators $T_V(1)$ and $T_V(1)^{-1}$, respectively.
![[]{data-label="f:iso"}](Figures/ShiftOperator)
![[]{data-label="f:iso"}](Figures/InverseShiftOperator)
In fact, they are shifts operators multiplied by the twisting operator or its inverse. We use for the derivative of the $R$-operators the depiction given in Fig. \[f:vd\].
![[]{data-label="f:vd"}](Figures/VertexDerivative)
It is clear that $H_N$ is a sum of $N$ terms arising from differentiation of the $R$-operators entering the transfer matrix $T_V(\zeta)$. As is shown above ${\mathbb A}_V$ does not depend on $\zeta$ and therefore there are no corresponding derivative terms. We meet three different situations depicted in Figs. \[f:hi\], \[f:hii\] and \[f:hiii\]. Note that for clarity the pictures in Figs. \[f:hi\] and \[f:hiii\] are rotated.
![[]{data-label="f:hiii"}](Figures/HamiltonianI)
![[]{data-label="f:hiii"}](Figures/HamiltonianII)
![[]{data-label="f:hiii"}](Figures/HamiltonianIII)
It is clear that the twisting operators are retained only in the first situation, and we come to the following analytical expression for the Hamiltonian $$\begin{gathered}
H_N = \sum_{i \in {[1 \, . \, . \, N-1]}} {\mathbb H}^{(i, i + 1)} + {\mathbb A}_V^{(1)} {\mathbb H}^{(N, 1)} \big({\mathbb A}_V^{-1}\big)^{(1)}, \label{hv}\end{gathered}$$ where we use the notation $$\begin{gathered}
{\mathbb H}^{(k, l)} = \check R'_{V | V}(1 | 1)^{(k,l)}. \label{hkl}\end{gathered}$$
Integrability objects for the case\
of quantum loop algebra $\boldsymbol{{{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}}$ {#section3}
=========================================================================================
To construct integrability objects for the quantum loop algebra ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ we need its representations. The simplest way to construct such representations is to use Jimbo’s homomorphism from ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ to the quantum group ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$. Therefore, we start with a short reminder of some basics facts on finite-dimensional representations of the quantum group ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$.
Quantum group $\boldsymbol{{{\mathrm U}_q(\mathfrak{gl}_{l + 1})}}$ and some its representations
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### Definition
The standard basis of the standard Cartan subalgebra ${{\mathfrak k}_{l + 1}}$ of ${\mathfrak{gl}_{l + 1}}$ is formed by the matrices $K_i$, $i \in {[1 \, . \, . \, l + 1]}$, with the matrix entries $$\begin{gathered}
(K_i)_{j m} = \delta_{i j} \delta_{i m}.\end{gathered}$$ There are $l$ simple roots $\alpha_i \in {{\mathfrak k}_{l + 1}}^*$, which are defined by the equation $$\begin{gathered}
\langle \alpha_i, K_j \rangle = c_{j i},\end{gathered}$$ where $$\begin{gathered}
c_{i j} = \delta_{i j} - \delta_{i, j + 1}. \label{cij}\end{gathered}$$ The full system of positive roots of ${\mathfrak{gl}_{l + 1}}$ is $$\begin{gathered}
\Delta_+ = \{ \alpha_{i j} \,|\, 1 \le i < j \le l + 1 \},\end{gathered}$$ where $$\begin{gathered}
\alpha_{i j} = \sum_{k = i}^{j - 1} \alpha_k, \qquad 1 \le i < j \le l + 1.\end{gathered}$$ It is clear that $\alpha_i = \alpha_{i, i + 1}$. Certainly, the negative roots are $- \alpha_{i j}$.
The special linear Lie algebra ${\mathfrak{sl}_{l + 1}}$ is a subalgebra of ${\mathfrak{gl}_{l + 1}}$. The standard Cartan subalgebra ${{\mathfrak h}_{l + 1}}$ of ${\mathfrak{sl}_{l + 1}}$ is formed by the elements $$\begin{gathered}
H_i = K_i - K_{i + 1}, \qquad i \in {[1 \, . \, . \, l]}.\end{gathered}$$ The positive and negative roots of ${\mathfrak{sl}_{l + 1}}$ are the restrictions of $\alpha_{ij}$ and $-\alpha_{ij}$ to ${{\mathfrak h}_{l + 1}}$ respectively. Here we have $$\begin{gathered}
\langle \alpha_j , H_i \rangle = a_{i j},\end{gathered}$$ where $$\begin{gathered}
a_{ij} = c_{ij} - c_{i+1, j} = {} - \delta_{i - 1, j} + 2 \delta_{i j} - \delta_{i + 1, j} \label{aij}\end{gathered}$$ are the entries of the Cartan matrix of the Lie algebra ${\mathfrak{sl}_{l + 1}}$. The highest root of ${\mathfrak{sl}_{l + 1}}$ is $$\begin{gathered}
\theta = \alpha_{1, l + 1} = \sum_{i = 1}^l \alpha_i.\end{gathered}$$ One can easily see that $$\begin{gathered}
(\theta | \theta) = 2. \label{tt}\end{gathered}$$ The Kac labels and the dual Kac labels are given by the equations $$\begin{gathered}
a_i = 1, \qquad {a \ifthenelse{\equal{i}{\null}}{\,}{} \check{{}_{i}} \ifthenelse{\equal{i}{\null}}{\,}{}} = 1, \qquad i \in {[1 \, . \, . \, l]},\end{gathered}$$ and, therefore, for the Coxeter number and the dual Coxeter number of ${\mathfrak{sl}_{l + 1}}$ one has $$\begin{gathered}
h = l + 1, \qquad {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} = l + 1. \label{hdh}\end{gathered}$$
Let $q$ be the exponential of a complex number $\hbar$, such that $q$ is not a root of unity. We define the quantum group ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ as a unital associative ${\mathbb C}$-algebra generated by the elements[^5] $$\begin{gathered}
E_i, \quad F_i, \quad i = 1,\ldots,l, \qquad q^X, \quad X \in {{\mathfrak k}_{l + 1}},\end{gathered}$$ satisfying the following defining relations $$\begin{gathered}
q^0 = 1, \qquad q^{X_1} q^{X_2} = q^{X_1 + X_2}, \\
q^X E_i q^{-X} = q^{\langle \alpha_i, X \rangle} E_i, \qquad q^X F_i q^{-X} = q^{-\langle \alpha_i, X \rangle} F_i, \\
{}[E_i, F_j] = \delta_{i j} \frac{q^{K_i - K_{i+1}} - q^{- K_i + K_{i + 1}}}{q - q^{-1}}.\end{gathered}$$ Besides, we have the Serre relations $$\begin{gathered}
E_i E_j = E_j E_i, \qquad F_i F_j = F_j F_i, \qquad |i - j| \ge 2, \\
E_i^2 E_{i \pm 1} - [2]_q E_i E_{i \pm 1} E_i + E_{i \pm 1} E_i^2 = 0,
\qquad
F_i^2 F_{i \pm 1} - [2]_q F_i F_{i \pm 1} F_i + F_{i \pm 1} F_i^2 = 0.\end{gathered}$$ From the point of view of quantum integrable systems, it is important that ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ is a Hopf algebra with respect to appropriately defined comultiplication, antipode and counit. However, we do not use the explicit form of the Hopf algebra structure in the present paper. The quantum group ${{\mathrm U}_q(\mathfrak{sl}_{l + 1})}$ can be considered as a Hopf subalgebra of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ generated by the elements $$\begin{gathered}
E_i, \quad F_i, \quad i = 1,\ldots,l, \qquad q^X, \quad X \in {{\mathfrak h}_{l + 1}}.\end{gathered}$$
Following Jimbo [@Jim86a], introduce the elements $E_{ij}$ and $F_{ij}$, $1 \le i < j \le l + 1$, with the help of the relations $$\begin{gathered}
\begin{split}&
E_{i, i + 1} = E_i, \qquad i \in {[1 \, . \, . \, l]},\\
& E_{i j} = E_{i, j - 1} E_{j - 1, j} - q E_{j - 1, j} E_{i, j - 1}, \qquad j - i > 1,\end{split}\end{gathered}$$ and $$\begin{gathered}
F_{i, i + 1} = F_i, \qquad i = 1,\ldots,l,
\\
F_{i j} = F_{j - 1, j} F_{i, j - 1} - q^{-1} F_{i, j - 1} F_{j - 1, j}, \qquad j - i > 1.\end{gathered}$$ The appropriately ordered monomials constructed from $E_{i j}$, $F_{i j}$, $1 \le i < j \le l + 1$, and $q^X$, $X \in {{\mathfrak k}_{l + 1}}$, form a Poincaré–Birkhoff–Witt basis of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$.
A ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$-module $V$ is said to be a weight module if $$\begin{gathered}
V = \bigoplus_{\lambda \in {{\mathfrak k}_{l + 1}}^*} V_\lambda,\end{gathered}$$ where $$\begin{gathered}
V_\lambda = \big\{v \in V \,|\, q^X v = q^{\langle \lambda, X \rangle} v \mbox{ for any } X \in {{\mathfrak k}_{l + 1}}\big\}.\end{gathered}$$ The space $V_\lambda$ is called the weight space of weight $\lambda$, and a nonzero element of $V_\lambda$ is called a weight vector of weight $\lambda$. We say that $\lambda \in {{\mathfrak k}_{l + 1}}^*$ is a weight of $V$ if $V_\lambda \ne \{0\}$.
The ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$-module $V$ is called a highest weight ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$-module with highest weight $\lambda$ if there exists a weight vector $v^\lambda \in V$ satisfying the relations $$\begin{gathered}
E_i v^\lambda = 0, \quad i = 1,\ldots,l, \qquad q^X v^\lambda = q^{\langle \lambda, X \rangle} v^\lambda,
\quad X \in {{\mathfrak k}_{l + 1}}, \quad \lambda \in {{\mathfrak k}_{l + 1}}^*, \\
{{\mathrm U}_q(\mathfrak{gl}_{l + 1})}v^\lambda = V.\end{gathered}$$ Given $\lambda \in {{\mathfrak k}_{l + 1}}^*$, denote by $\widetilde V^\lambda$ the corresponding Verma ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$-module, see, for example [@KliSch97]. This is a highest weight module with the highest weight $\lambda$. Below we sometimes identify any weight $\lambda$ with the set of its components $(\lambda_1, \ldots, \lambda_{l + 1})$, where $$\begin{gathered}
\lambda_i = \langle \lambda, K_i \rangle.\end{gathered}$$ We denote by $\widetilde \pi^\lambda$ the representation of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ corresponding to $\widetilde V^\lambda$. The structure and properties of $\widetilde V^\lambda$ and $\widetilde \pi^\lambda$ for $l = 1$ and $l = 2$ are considered in much detail in papers [@BooGoeKluNirRaz13; @BooGoeKluNirRaz14b; @BooGoeKluNirRaz14a; @BooGoeKluNirRaz16; @NirRaz16a] and for a general $l$ in papers [@BooGoeKluNirRaz17b; @NirRaz17a; @NirRaz17b]. It is clear that $\widetilde V^\lambda$ and $\widetilde \pi^\lambda$ are infinite-dimensional for a general weight $\lambda \in {{\mathfrak k}_{l + 1}}^*$. However, if all the differences $\lambda_i - \lambda_{i+1}$, $i = 1,\ldots,l$, are non-negative integers, there is a maximal submodule, such that the respective quotient module is finite-dimensional. We denote this quotient by $V^\lambda$ and the corresponding representation by $\pi^\lambda$. For any $i \in {[1 \, . \, . \, l]}$ the finite-dimensional representation $\pi^{\omega_i}$ with $$\begin{gathered}
\omega_i = (\underbracket[.6pt]{1, \ldots, 1}_i, \underbracket[.6pt]{0, \ldots, 0}_{l + 1 - i})\end{gathered}$$ is called the $i$-th fundamental representation of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$. It is clear that $\omega_i$ can be also defined as $$\begin{gathered}
\omega_i(K_j) = \begin{cases}
1, & 1 \le j \le i, \\
0, & i < j \le l + 1.
\end{cases}\end{gathered}$$ Hence, it is evident that $$\begin{gathered}
\omega_i(H_j) = \delta_{i j}.\end{gathered}$$ The weights $\omega_i$, $i \in {[1 \, . \, . \, l]}$, are called the fundamental weights of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$. The dimensions of the corresponding fundamental representations $\pi^{\omega_i}$ are $\binom{l+1}{i}$, $i \in {[1 \, . \, . \, l]}$.
### Representation $\boldsymbol{\pi}$ {#rpi}
We denote by $\pi$ the first fundamental representation $\pi^{\omega_1}$ of the quantum group ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$. This representation is $(l + 1)$-dimensional and can be realised as follows. Assume that the representation space is the free vector space generated by the set $\{v_k\}_{k \in {[1 \, . \, . \, l + 1]}}$ and denote by ${\mathbb E}_{i j}$, $i, j \in {[1 \, . \, . \, l + 1]}$, the endomorphisms of this space defined by the equation $$\begin{gathered}
{\mathbb E}_{i j} v_k = v_i \delta_{j k}. \label{eijvk}\end{gathered}$$ One can verify that the equations $$\begin{gathered}
\begin{split}&
\pi(q^{\nu K_i}) = q^\nu {\mathbb E}_{i i} + \sum_{\substack{k = 1 \\ k \ne i}}^{l + 1} {\mathbb E}_{k k}, \qquad i \in {[1 \, . \, . \, l + 1]}, \\
& \pi(E_i) = {\mathbb E}_{i, i + 1}, \qquad \pi(F_i) = {\mathbb E}_{i + 1, i}, \qquad i \in {[1 \, . \, . \, l]},\end{split}\end{gathered}$$ describe an $(l + 1)$-dimensional representation of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ with the highest weight $\omega_1$, as is required. It is useful to have in mind the equations $$\begin{gathered}
\pi(E_{i j}) = {\mathbb E}_{i j}, \qquad \pi(F_{i j}) = {\mathbb E}_{j i}, \qquad 1 \le i < j \le i + 1.\end{gathered}$$ One can see that $$\begin{gathered}
v_2 = F_1 v_1, \qquad v_3 = F_2 F_1 v_1, \qquad \ldots, \qquad v_{l + 1} = F_l \cdots F_2 F_1 v_1.\end{gathered}$$ Therefore, $v_k$ is a weight vector of weight $$\begin{gathered}
\lambda_k = \omega_1 - \sum_{i = 1}^{k - 1} \alpha_i. \label{lkpi}\end{gathered}$$
### Representation $\boldsymbol{{{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}}$ {#rbpi}
The last fundamental representation ${{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}= \pi^{\omega_l}$ of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ is also $(l + 1)$-dimensional. We again assume that the representation space is the free vector space generated by the set $\{v_k\}_{k \in {[1 \, . \, . \, l + 1]}}$ and denote by ${\mathbb E}_{i j}$, $i, j \in {[1 \, . \, . \, l + 1]}$, the endomorphisms of this space defined by equation (\[eijvk\]). Here we have $$\begin{aligned}
& {{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}(q^{\nu K_i}) = q^\nu \sum_{\substack{k = 1 \\ k \ne l - i + 2}}^{l + 1} {\mathbb E}_{k k} + {\mathbb E}_{l - i + 2, \ l - i + 2}, \qquad i \in {[1 \, . \, . \, l + 1]}, \\
& {{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}(E_i) = {\mathbb E}_{l - i + 1, l - i + 2}, \qquad {{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}(F_i) = {\mathbb E}_{l - i + 2, l - i + 1}, \qquad i \in {[1 \, . \, . \, l]}.\end{aligned}$$ It is not difficult to determine that $$\begin{aligned}
{3}
& {{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}(E_{i j}) = (-1)^{- i + j - 1} q^{-i + j - 1} {\mathbb E}_{l - j + 2, l - i + 2}, \qquad && 1 \le i < j \le l + 1, & \\
& {{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}(F_{i j}) = (-1)^{- i + j - 1} q^{i - j + 1} {\mathbb E}_{l - i + 2, l - j + 2}, \qquad && 1 \le i < j \le l + 1.&\end{aligned}$$ We obtain successively $$\begin{gathered}
v_2 = F_l v_1, \qquad v_3 = F_{l - 1} F_l v_1, \qquad \ldots, \qquad v_{l + 1} = F_1 \cdots F_{l - 1} F_l v_1,\end{gathered}$$ and see that $v_k$ is a weight vector of weight $$\begin{gathered}
\lambda_k = \omega_l - \sum_{i = l - k + 2}^l \alpha_i. \label{lkbpi}\end{gathered}$$
Representations of $\boldsymbol{{{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}}$
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All representations of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ considered in this section are $(l + 1)$-dimensional, and we always assume that their representation space is the free vector space generated by the set $\{v_k\}_{k \in {[1 \, . \, . \, l + 1]}}$ and denote by ${\mathbb E}_{i j}$, $i, j \in {[1 \, . \, . \, l + 1]}$, the endomorphisms of this space defined by equation (\[eijvk\]).
### Jimbo’s homomorphism
To construct representations of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$, it is common to use Jimbo’s homomorphism $\epsilon$ from the quantum loop algebra ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ to the quantum group ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ defined by the equations [@Jim86a] $$\begin{aligned}
{3}
& \epsilon(q^{\nu h_0}) = q^{\nu (K_{l+1} - K_1)}, \qquad &&
\epsilon(q^{\nu h_i}) = q^{\nu (K_{i} - K_{i+1})}, & \label{ja}\\
& \epsilon(e_0) = F_{1, l+1} q^{K_1 + K_{l+1}}, \qquad && \epsilon(e_i) = E_{i, i+1},& \\
& \epsilon(f_0) = E_{1, l+1} q^{- K_1 - K_{l+1}}, \qquad && \epsilon(f_i) = F_{i, i+1},& \label{jc}\end{aligned}$$ where $i$ runs from $1$ to $l$.
### Representation $\boldsymbol{\varphi_\zeta}$ {#s:rfz}
We denote by $\varphi_\zeta$ the representation $\pi \circ \epsilon \circ \Gamma_\zeta$ of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$, where $\pi$ is the representation of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ considered in Section \[rpi\], and $\Gamma_\zeta$ is defined by equation (\[gzqx\]). Using Jimbo’s homomorphism, we obtain $$\begin{gathered}
\varphi_\zeta\big(q^{\nu h_0}\big) =q^{-\nu} {\mathbb E}_{1 1} + q^\nu {\mathbb E}_{l + 1, l + 1} + \sum_{k = 2}^l {\mathbb E}_{k k}, \label{fzqh0} \\
\varphi_\zeta\big(q^{\nu h_i}\big) = q^\nu {\mathbb E}_{i i} + q^{-\nu} {\mathbb E}_{i + 1, i + 1} + \sum_{\substack{k = 1 \\ k \ne i, i + 1}}^{l + 1} {\mathbb E}_{k k}, \qquad i \in {[1 \, . \, . \, l]}, \label{fzqhi}\end{gathered}$$ and, further, $$\begin{aligned}
{4}
& \varphi_\zeta(e_0) = \zeta^{s_0} q {\mathbb E}_{l + 1, 1},\qquad && \varphi_\zeta(e_i) = \zeta^{s_i} {\mathbb E}_{i, i + 1}, \qquad && i \in {[1 \, . \, . \, l]}, & \\
& \varphi_\zeta(f_0) = \zeta^{-s_0} q^{-1} {\mathbb E}_{1, l + 1},\qquad && \varphi_\zeta(f_i) = \zeta^{-s_i} {\mathbb E}_{i + 1, i}, \qquad && i \in {[1 \, . \, . \, l]}.&\end{aligned}$$ It is clear that the vector $v_k$ is a weight vector of weight defined as the restriction of the weight $\lambda_k$, given by equation (\[lkpi\]), to the Cartan subalgebra ${{\mathfrak h}_{l + 1}}$ of ${\mathfrak{sl}_{l + 1}}$.
### Representation $\boldsymbol{{{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta}$
From the representation ${{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}$ of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$, described in Section \[rbpi\] we obtain the representation ${{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}= {{\hskip .11em \overline{\hskip -.11em \pi \hskip -.06em} \hskip .06em}}\circ \epsilon \circ \Gamma_\zeta$ of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$. Here we have $$\begin{gathered}
{{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta\big(q^{\nu h_0}\big) =q^{-\nu} {\mathbb E}_{1 1} + q^\nu {\mathbb E}_{l + 1, l + 1} + \sum_{k = 2}^l {\mathbb E}_{k k}, \\
{{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta\big(q^{\nu h_i}\big) = q^\nu {\mathbb E}_{l - i + 1, l - i + 1} + q^{-\nu} {\mathbb E}_{l - i + 2, l - i + 2} + \sum_{\substack{k = 1 \\ k \ne l - i + 1 \\ k \ne l - i + 2}}^{l + 1} {\mathbb E}_{k k}, \qquad i \in {[1 \, . \, . \, l]},\end{gathered}$$ and, further, $$\begin{aligned}
{4}
& {{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta(e_0) = \zeta^{s_0} (-1)^{l - 1} q^{- l + 2} {\mathbb E}_{l + 1, 1},\qquad && {{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta(e_i) = \zeta^{s_i} {\mathbb E}_{l - i + 1, l - i + 2},\qquad && i \in {[1 \, . \, . \, l]}, &\\
& {{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta(f_0) = \zeta^{-s_0} (-1)^{l - 1} q^{l - 2} {\mathbb E}_{1, l + 1},\qquad && {{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta(f_i) = \zeta^{-s_i} {\mathbb E}_{l - i + 2, l - i + 1},\qquad && i \in {[1 \, . \, . \, l]}.&\end{aligned}$$ Here again $v_k$ is a weight vector of weight which is obtained by the restriction of the weight $\lambda_k$, given by equation (\[lkbpi\]), to the Cartan subalgebra ${{\mathfrak h}_{l + 1}}$ of ${\mathfrak{sl}_{l + 1}}$.
### Representations $\boldsymbol{\varphi^*_\zeta$ and ${}^* \! \varphi_\zeta^{}}$
The dual representations are defined and discussed in Section \[s:cr\]. Consider first the representation $\varphi^*_\zeta$. For the generators $q^{\nu h_i}$ we obtain $$\begin{aligned}
{4}
& \varphi^*_\zeta\big(q^{\nu h_0}\big) = q^\nu {\mathbb E}_{1 1} +q^{-\nu} {\mathbb E}_{l + 1, l + 1} + \sum_{k = 2}^l {\mathbb E}_{k k}, \\
& \varphi^*_\zeta\big(q^{\nu h_i}\big) = q^{-\nu} {\mathbb E}_{i i} + q^\nu {\mathbb E}_{i + 1, i + 1} + \sum_{\substack{k = 1 \\ k \ne i, i + 1}}^{l + 1} {\mathbb E}_{k k}, \qquad i \in {[1 \, . \, . \, l]},\end{aligned}$$ and some simple calculations lead to the relations $$\begin{aligned}
{4}
& \varphi^*_\zeta(e_0) = - \zeta^{s_0} {\mathbb E}_{1, l + 1},\qquad && \varphi^*_\zeta(e_i)= - \zeta^{s_i} q^{-1} {\mathbb E}_{i + 1, i}, \qquad && i \in {[1 \, . \, . \, l]}, & \\
& \varphi^*_\zeta(f_0) = - \zeta^{-s_0} {\mathbb E}_{l + 1, 1}, \qquad && \varphi^*_\zeta (f_i) = - \zeta^{-s_i} q {\mathbb E}_{i, i + 1},\qquad && i \in {[1 \, . \, . \, l]}.&\end{aligned}$$ Now $v_k$ is a weight vector of weight $$\begin{gathered}
\lambda_k = \omega_l - \sum_{i = k}^l \alpha_i, \label{lkphis}\end{gathered}$$ where $\omega_l$ and $\alpha_i$ are treated as elements of ${{\mathfrak h}_{l + 1}}^*$. The representations $\varphi^*_\zeta$ and ${{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta$ are equivalent up to a rescaling of the spectral parameter. In fact, for any $a \in {{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ one has $$\begin{gathered}
{\mathbb P}\varphi^*_{q^{2/s} \zeta} (a) {\mathbb P}^{-1} = {{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta(a),\end{gathered}$$ where the operator ${\mathbb P}$ is given by the equation $$\begin{gathered}
{\mathbb P}= \sum_{i = 1}^{l + 1} (-1)^{i - 1} q^{- 2 \sum\limits_{k = 1}^{i - 1} s_k / s + i - 1}{\mathbb E}_{l - i + 2, i}.\end{gathered}$$
The representation ${}^* \! \varphi_\zeta$ is very similar to the representation $\varphi^*_\zeta$. Here for the generators $q^{\nu h_i}$ of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ we obtain $$\begin{gathered}
{}^* \! \varphi_\zeta\big(q^{\nu h_0}\big) = q^\nu {\mathbb E}_{1 1} +q^{-\nu} {\mathbb E}_{l + 1, l + 1} + \sum_{k = 2}^l {\mathbb E}_{k k}, \\
{}^* \! \varphi_\zeta\big(q^{\nu h_i}\big) =q^{-\nu} {\mathbb E}_{i i} + q^\nu {\mathbb E}_{i + 1, i + 1} + \sum_{\substack{k = 1 \\ k \ne i, i + 1}}^{l + 1} {\mathbb E}_{k k}, \qquad i \in {[1 \, . \, . \, l]}.\end{gathered}$$ Then it is not difficult to come to the relations $$\begin{aligned}
{4}
& {}^* \! \varphi_\zeta(e_0) = - \zeta^{s_0} q^2 {\mathbb E}_{1, l + 1},\qquad && {}^* \! \varphi_\zeta(e_i) = - \zeta^{s_i} q {\mathbb E}_{i + 1, i},\qquad && i \in {[1 \, . \, . \, l]}, & \\
& {}^* \! \varphi_\zeta (f_0) = - \zeta^{-s_0} q^{-2} {\mathbb E}_{l + 1, 1},\qquad && {}^* \! \varphi_\zeta (f_i)
= - \zeta^{-s_i} q^{-1} {\mathbb E}_{i, i + 1},\qquad && i \in {[1 \, . \, . \, l]}.\end{aligned}$$ The vector $v_k$ is a weight vector of the weight given again by equation (\[lkphis\]). The representations ${}^* \! \varphi_\zeta$ and ${{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta$ are again equivalent up to a rescaling of the spectral parameter. It can be verified that $$\begin{gathered}
{\mathbb P}{}^* \! \varphi_{q^{- 2 l/s} \zeta} (a) {\mathbb P}^{-1} = {{\hskip .07em \overline{\hskip -.07em \varphi \hskip -.07em} \hskip .07em}}_\zeta(a),\end{gathered}$$ where now the operator ${\mathbb P}$ is given by the equation $$\begin{gathered}
{\mathbb P}= \sum_{i = 1}^{l + 1} (-1)^{i - 1} q^{2 l \sum_{k = 1}^{i - 1} s_k / s - i + 1} E_{l - i + 2, i}.\end{gathered}$$
Integrability objects
---------------------
### Poincaré–Birkhoff–Witt basis {#s:pbwb}
Recall that to construct a Poincaré–Birkhoff–Witt basis one has to define root vectors corresponding to all roots of $\widehat {\mathcal L}({\mathfrak{sl}_{l + 1}})$. To construct root vectors we follow the procedure proposed by Khoroshkin and Tolstoy based on a normal ordering of positive roots [@KhoTol92; @KhoTol93; @KhoTol94; @TolKho92].
For the case of a finite-dimensional simple Lie algebra an order relation $\prec$ is called a normal order [@AshSmiTol79; @LezSav74; @Tol89] when if a positive root $\gamma$ is a sum of two positive roots $\alpha \prec \beta$, then $\alpha \prec \gamma \prec \beta$. In our case we assume additionally that $$\begin{gathered}
\alpha + k \delta \prec m \delta \prec (\delta - \beta) + n \delta \label{akd}\end{gathered}$$ for any $\alpha, \beta \in \Delta_+$, $k, n \in {\mathbb Z}_{\ge 0}$ and $m \in {\mathbb Z}_{>0}$.
Assume that some normal ordering of positive roots is chosen. We say that a pair $(\alpha, \beta)$ of positive roots generates a root $\gamma$ if $\gamma = \alpha + \beta$ and $\alpha \prec \beta$. A pair of positive roots $(\alpha, \beta)$ generating a root $\gamma$ is said to be minimal if there is no other pair of positive roots $(\alpha', \beta')$ generating $\gamma$ such that $\alpha \prec \alpha' \prec \beta' \prec \beta$.
It is convenient to denote a root vector corresponding to a positive root $\gamma$ by $e_\gamma$, and a root vector corresponding to a negative root $- \gamma$ by $f_\gamma$. Following [@KhoTol93; @TolKho92], we define root vectors by the following inductive procedure. Given a root $\gamma \in \Delta_+$, let $(\alpha, \beta)$ be a minimal pair of positive roots generating $\gamma$. Now, if the root vectors $e_\alpha$, $e_\beta$ and $f_\alpha$, $f_\beta$ are already constructed, we define the root vectors $e_\gamma$ and $f_\gamma$ as $$\begin{gathered}
e_\gamma = [e_\alpha , e_\beta]_q, \qquad f_\gamma = [f_\beta , f_\alpha]_q.\end{gathered}$$ Here and below we use the $q$-commutator of root vectors $[ \cdot , \cdot ]_q$ defined as $$\begin{gathered}
[e_\alpha , e_\beta]_q
= e_\alpha e_\beta - q^{-(\alpha | \beta)} e_\beta e_\alpha, \qquad
[f_\alpha , f_\beta]_q
= f_\alpha f_\beta - q^{(\alpha | \beta)} f_\beta f_\alpha,\end{gathered}$$ where $( \cdot | \cdot )$ denotes the symmetric bilinear form on $\widehat {\mathfrak h}^*$ described in Section \[s:siola\].
We use the normal order of the positive roots of $\widehat {\mathcal L}({\mathfrak{sl}_{l + 1}})$ defined as follows. First order the positive roots of ${\mathfrak{sl}_{l + 1}}$ assuming that $\alpha_{i j} \prec \alpha_{k l}$ if $i < k$, or if $i = k$ and $j < l$. Then, $\alpha + m \delta \prec \beta + n \delta$, with $\alpha, \beta \in \Delta_+$ and $m, n \in {\mathbb Z}_{\ge 0}$, if $\alpha \prec \beta$, or $\alpha = \beta$ and $m < n$. Further, $(\delta - \alpha) + m \delta \prec (\delta - \beta) + n \delta$, with $\alpha, \beta \in \Delta_+$, if $\alpha \prec \beta$, or $\alpha = \beta$ and $m > n$. Finally, we assume that relation (\[akd\]) is valid.
The root vectors are defined inductively. We start with the root vectors corresponding to the roots $\pm \alpha_i$, $i \in {[1 \, . \, . \, l]}$, which we identify with the generators $e_i$ and $f_i$, $i \in {[1 \, . \, . \, l]}$, of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$, $$\begin{gathered}
e_{\alpha_i} = e_{\alpha_{i, i + 1}} = e_i, \qquad f_{\alpha_i} = f_{\alpha_{i, i + 1}} = f_i.\end{gathered}$$ The next step is to construct root vectors $e_{\alpha_{i j}}$ and $f_{\alpha_{i j}}$ for all roots $\alpha_{i j} \in \Delta_+$. We assume that $$\begin{gathered}
e_{\alpha_{i j}} = e_{\alpha_{i, j - 1}} e_{\alpha_{j - 1, j}} - q e_{\alpha_{j - 1, j}} e_{\alpha_{i, j - 1}}, \qquad f_{\alpha_{i j}} = f_{\alpha_{j - 1, j}} f_{\alpha_{i, j - 1}} - q^{-1} f_{\alpha_{i, j - 1}} f_{\alpha_{j - 1, j}}\end{gathered}$$ for $j - i > 1$. Further, taking into account that $$\begin{gathered}
\alpha_0 = \delta - \alpha_{1,l + 1},\end{gathered}$$ we put $$\begin{gathered}
e_{\delta - \alpha_{1, l + 1}} = e_0, \qquad f_{\delta - \alpha_{1, l + 1}} = f_0,\end{gathered}$$ and define $$\begin{aligned}
& e_{\delta - \alpha_{i, l + 1}} = e_{\alpha_{i - 1, i}} e_{\delta - \alpha_{i - 1, l + 1}} - q e_{\delta - \alpha_{i - 1, l + 1}} e_{\alpha_{i - 1, i}}, \\
& f_{\delta - \alpha_{i, l + 1}} = f_{\delta - \alpha_{i - 1, l + 1}} f_{\alpha_{i - 1, i}} - q^{-1} f_{\alpha_{i - 1, i}} f_{\delta - \alpha_{i - 1, l + 1}}\end{aligned}$$ for $i > 1$, and $$\begin{aligned}
& e_{\delta - \alpha_{i j}} = e_{\alpha_{j, j + 1}} e_{\delta - \alpha_{i, j + 1}} - q e_{\delta - \alpha_{i, j + 1}} e_{\alpha_{j, j + 1}}, \\
& f_{\delta - \alpha_{i j}} = f_{\delta - \alpha_{i, j + 1}} f_{\alpha_{j, j + 1}} - q^{-1} f_{\alpha_{j, j + 1}} f_{\delta - \alpha_{i, j + 1}}\end{aligned}$$ for $j < l + 1$. The root vectors corresponding to the roots $\delta$ and $-\delta$ are indexed by the elements of $\Delta_+$ and defined by the relations $$\begin{gathered}
e'_{\delta, \alpha_{i j}} = e_{\alpha_{i j}} e_{\delta - \alpha_{i j}} - q^2 e_{\delta - \alpha_{i j}} e_{\alpha_{i j}}, \qquad f'_{\delta, \alpha_{i j}} = f_{\delta - \alpha_{i j}} f_{\alpha_{i j}} - q^{- 2} f_{\alpha_{i j}} f_{\delta - \alpha_{i j}}.\end{gathered}$$ The remaining definitions are $$\begin{gathered}
e_{\alpha_{i j} + n \delta} = [2]_q^{-1} \big( e_{\alpha_{i j} + (n - 1)\delta} e'_{\delta, \alpha_{i j}} - e'_{\delta, \alpha_{i j}} e_{\alpha_{i j} + (n - 1)\delta} \big), \label{cwby1} \\
f_{\alpha_{i j} + n \delta} = [2]_q^{-1} \big( f'_{\delta, \alpha_{i j}} f_{\alpha_{i j} + (n - 1)\delta} - f_{\alpha_{i j} + (n - 1)\delta} f'_{\delta, \alpha_{i j}} \big), \\
e_{(\delta - \alpha_{i j}) + n \delta} = [2]_q^{-1} \big( e'_{\delta, \alpha_{i j}} e_{(\delta - \alpha_{i j}) + (n - 1)\delta} - e_{(\delta - \alpha_{i j}) + (n - 1)\delta} e'_{\delta, \alpha_{i j}} \big), \\
f_{(\delta - \alpha_{i j}) + n \delta} = [2]_q^{-1} \big( f_{(\delta - \alpha_{i j}) + (n - 1)\delta} f'_{\delta, \alpha_{i j}} - f'_{\delta, \alpha_{i j}} f_{(\delta - \alpha_{i j}) + (n - 1)\delta} \big), \\
e'_{n \delta, \alpha_{i j}} = e_{\alpha_{i j} + (n - 1)\delta} e_{\delta - \alpha_{i j}} - q^2 e_{\delta - \alpha_{i j}} e_{\alpha_{i j} + (n - 1)\delta}, \\
f'_{n \delta, \alpha_{i j}} = f_{\delta - \alpha_{i j}} f_{\alpha_{i j} + (n - 1)\delta} - q^{- 2} f_{\alpha_{i j} + (n - 1)\delta} f_{\delta - \alpha_{i j}}. \label{cwby2}\end{gathered}$$ Note that, among all imaginary root vectors $e'_{n \delta, \alpha_{i j}}$ and $f'_{n\delta, \alpha_{i j}}$ only the root vectors $e'_{n \delta, \alpha_{i, i + 1}}$ and $f'_{n \delta, \alpha_{i, i + 1}}$, $i \in {[1 \, . \, . \, l]}$, are independent and required for the construction of the Poincaré–Birkhoff–Witt basis. However, the vectors $e'_{\delta, \gamma}$ and $f'_{\delta, \gamma}$ with arbitrary $\gamma \in \Delta_+$ are needed for the iterations (\[cwby1\])–(\[cwby2\]).
The prime in the notation for the root vectors corresponding to the imaginary roots $n \delta$ and $- n \delta$, $n \in {\mathbb Z}_{> 0}$ is explained by the fact that to construct the expression for the universal $R$-matrix one uses another set of root vectors corresponding to these roots. They are introduced by the functional equations $$\begin{gathered}
- \kappa_q e_{\delta, \gamma}(u) = \log(1 - \kappa_q e'_{\delta, \gamma}(u)), \\
\kappa_q f_{\delta, \gamma}(u^{-1}) = \log(1 + \kappa_q f'_{\delta, \gamma}(u^{-1})),\end{gathered}$$ where the generating functions $$\begin{aligned}
{3}
& e'_{\delta, \gamma}(u) = \sum_{n = 1}^\infty e'_{n \delta, \gamma} u^n, \qquad && e_{\delta, \gamma}(u) = \sum_{n = 1}^\infty e_{n \delta, \gamma} u^n,& \\
& f'_{\delta, \gamma}(u^{-1}) = \sum_{n = 1}^\infty f'_{n \delta, \gamma} u^{- n}, \qquad && f_{\delta, \gamma}(u^{-1}) = \sum_{n = 1}^\infty f_{n \delta, \gamma} u^{- n}&\end{aligned}$$ are defined as formal power series, and $\kappa_q$ is defined by the equation $$\begin{gathered}
\kappa_q = q - q^{-1}.\end{gathered}$$
### Monodromy operators {#sss:urm}
The expression for the universal $R$-matrix of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ considered as a ${\mathbb C}[[\hbar]]$-algebra can be constructed using the procedure proposed by Khoroshkin and Tolstoy [@KhoTol92; @KhoTol93; @KhoTol94; @TolKho92]. Here we treat the quantum group as an associative ${\mathbb C}$-algebra. In fact, one can use the expression for the universal $R$-matrix from papers [@KhoTol92; @KhoTol93; @KhoTol94; @TolKho92] in this case as well, having in mind that the quantum group is quasitriangular only in some restricted sense. Namely, all the relations involving the universal $R$-matrix should be considered as valid only for the weight representations of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$, see in this respect paper [@Tan92] and the discussion below.
Let $V$, $W$ be two weight ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$-modules in the category ${\mathcal O}$, and $\varphi$, $\psi$ the corresponding representations. Define the monodromy operator $M_{V | W}(\zeta | \eta)$ as $$\begin{gathered}
M_{V | W}(\zeta | \eta) = \rho_{V | W}(\zeta | \eta) (\varphi_\zeta \otimes \psi_\eta) ({\mathcal R}_{\prec \delta} {\mathcal R}_{\sim \delta} {\mathcal R}_{\succ \delta}) K_{V | W}. \label{rpipi}\end{gathered}$$ Here ${\mathcal R}_{\prec \delta}$, ${\mathcal R}_{\sim \delta}$ and ${\mathcal R}_{\succ \delta}$ are elements of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$, while $K_{V | W}$ is an element of $\operatorname{End}(V \otimes W)$.
Explicitly, the element ${\mathcal R}_{\prec \delta}$ is the product over the set of roots $\alpha_{i j} + n \delta$ of the $q$-exponentials $$\begin{gathered}
{\mathcal R}_{\alpha_{i j} + n \delta} = \exp_{q^2} \big( {-} \kappa_q e_{\alpha_{i j} + n \delta}^{} \otimes f_{\alpha_{i j} + n \delta}^{} \big).\end{gathered}$$ The order of the factors in ${\mathcal R}_{\prec \delta}$ coincides with the chosen normal order of the roots $\alpha_{i j} + n \delta$.
The element ${\mathcal R}_{\sim \delta}$ is defined as $$\begin{gathered}
{\mathcal R}_{\sim \delta} = \exp \left( - \kappa_q \sum_{n \in {\mathbb Z}_{>0}} \sum_{i, j
= 1}^l u_{n, i j} e_{n \delta, \alpha_i} \otimes f_{n \delta, \alpha_j} \right),\end{gathered}$$ where for each $n \in {\mathbb Z}_{> 0}$ the quantities $u_{n, i j}$ are the matrix elements of the matrix $U_n$ inverse to the matrix $T_n$ with the matrix elements $$\begin{gathered}
t_{n, i j} = (-1)^{n (i + j)} \frac{1}{n} [n a_{i j}]_q = (-1)^{n (i + j)} \frac{[n]_q}{n} [a_{i j}]_{q^n},\end{gathered}$$ where $a_{i j}$ are the matrix elements of the Cartan matrix $A$ of the Lie algebra ${\mathfrak{sl}_{l + 1}}$. The matrix $T_n$ is tridiagonal. Using the results of paper [@Usm94], one can see that $$\begin{gathered}
u_{n, i j} = (-1)^{n (i + j)} \frac{n}{[n]_q} \frac{[i]_{q^n}[l - j + 1]_{q^n}}{[l + 1]_{q^n}}, \qquad i \le j, \\
u_{n, i j} = (-1)^{n (i + j)} \frac{n}{[n]_q} \frac{[l - i + 1]_{q^n} [j]_{q^n}}{[l + 1]_{q^n}}, \qquad i > j.\end{gathered}$$
The definition of the element ${\mathcal R}_{\succ \delta}$ is similar to the definition of the element ${\mathcal R}_{\prec \delta}$. It is the product over the set of roots $(\delta - \alpha_{i j}) + n \delta$ of the $q$-exponentials $$\begin{gathered}
{\mathcal R}_{(\delta - \alpha_{i j}) + n \delta} = \exp_{q^2} \big( {-} \kappa_q e_{(\delta - \alpha_{i j}) + n \delta}^{} \otimes f_{(\delta - \alpha_{i j}) + n \delta}^{}
\big). \label{rdmgm}\end{gathered}$$ The order of the factors in ${\mathcal R}_{\succ \delta}$ coincides with the chosen normal order of the roots .
The endomorphism $K_{V | W}$ is defined as follow. Let $v \in V$ and $w \in W$ be weight vectors of weights $\lambda$ and $\mu$ respectively. Then we assume that $$\begin{gathered}
K_{V | W} v \otimes w = q^{- \sum\limits_{i, j = 1}^l b_{i j} \langle \lambda, h_i \rangle \langle \mu, h_j \rangle} v \otimes w,\label{kvw}\end{gathered}$$ where $b_{i j}$ are the matrix elements of the matrix $B$ inverse to the Cartan matrix $A = (a_{i j})_{i, j \in {[1 \, . \, . \, l]}}$ of the Lie algebra ${\mathfrak{sl}_{l + 1}}$, see equation (\[aij\]). Using again the results of paper [@Usm94], we obtain $$\begin{gathered}
b_{i j} = \frac{i (l - j + 1)}{l + 1}, \quad i \le j, \qquad b_{i j} = \frac{(l - i + 1) j}{l + 1}, \quad i > j. \label{bij}\end{gathered}$$
One can show that it is possible to work with $M_{V | W}(\zeta | \eta)$ defined by (\[rpipi\]) as if it is defined by the universal $R$-matrix satisfying equations (\[dpx\]) and (\[drrr\]).
### Explicit form of $\boldsymbol{R}$-operator
In this section we obtain explicit expressions for the $R$-operators arising in the case $V = W = V^{\omega_1}$, see also paper [@MenTes15]. The explicit formulas for the action of the generators of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ on the basis of the representation space are given in Section \[s:rfz\]. We have $$\begin{gathered}
R_{V | V}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2) (\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\prec \delta}) (\varphi_{\zeta_1} \otimes \varphi_{\zeta_2})({\mathcal R}_{\sim \delta}) (\varphi_{\zeta_1} \otimes \varphi_{\zeta_2})({\mathcal R}_{\succ \delta}) K_{V | V}.\end{gathered}$$
First construct the expression for the operator $K_{V | V}$. Using equations (\[kvw\]) and (\[lkpi\]), we see that $$\begin{aligned}
K_{V | V} v_m \otimes v_n & = q^{- \sum\limits_{i, j = 1}^l \big\langle \omega_1 - \sum\limits_{q = 1}^{m - 1} \alpha_q, h_i \big\rangle \big\langle \omega_1 - \sum\limits_{p = 1}^{n - 1} \alpha_p, h_j \big\rangle b_{i j}} v_m \otimes v_n \\
& = q^{-\big(b_{1 1} - \sum\limits_{q = 1}^{m - 1} \delta_{1 q} - \sum\limits_{p = 1}^{n - 1} \delta_{p 1} + \sum\limits_{q = 1}^{m - 1} \sum\limits_{p = 1}^{n - 1} a_{q p}\big)} v_m \otimes v_n.\end{aligned}$$ It is not difficult to demonstrate that $$\begin{gathered}
K_{V | V} v_m \otimes v_n = q^{- l /(l + 1)} v_m \otimes v_n, \qquad m = n, \\
K_{V | V} v_m \otimes v_n = q^{1/(l + 1)} v_m \otimes v_n, \qquad m \ne n.\end{gathered}$$ Hence, we come to the equation $$\begin{gathered}
K_{V | V} = q^{- l/(l + 1)} \Bigg( \sum_{i = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{i i} + q \sum_{\substack{i, j = 1 \\ i \ne j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} \Bigg). \label{kvv}\end{gathered}$$
Following the Khoroshkin–Tolstoy procedure, described in Section \[s:pbwb\], we obtain $$\begin{gathered}
\varphi_\zeta(e_{\alpha_{i j} + n \delta}) = \zeta^{s_{i j} + n s} (-1)^{i n} q^{(i + 1) n} {\mathbb E}_{i j}, \label{eij}\\
\varphi_\zeta(f_{\alpha_{i j} + n \delta}) = \zeta^{- s_{i j} - n s} (-1)^{i n} q^{- (i + 1) n} {\mathbb E}_{j i}, \label{fij} \\
\varphi_\zeta(e_{(\delta - \alpha_{i j}) + n \delta}) = \zeta^{(s - s_{i j}) + n s} (-1)^{i (n + 1) - 1} q^{(i + 1) n + i} {\mathbb E}_{j i}, \\
\varphi_\zeta(f_{(\delta - \alpha_{i j}) + n \delta}) = \zeta^{- (s - s_{i j}) - n s} (-1)^{i (n + 1) - 1} q^{- (i + 1) n - i} {\mathbb E}_{i j}, \\
\varphi_\zeta(e'_{n \delta, \alpha_{i j}}) = \zeta^{n s} (-1)^{i n - 1} q^{(i + 1) n - 1}\big({\mathbb E}_{i i} - q^2 {\mathbb E}_{j j}\big), \label{fzefe} \\
\varphi_\zeta(f'_{n \delta, \alpha_{i j}}) = \zeta^{- n s} (-1)^{i n - 1} q^{- (i + 1) n + 1}\big({\mathbb E}_{i i} - q^{- 2} {\mathbb E}_{j j}\big). \label{fzeff}\end{gathered}$$ Here and below we denote $$\begin{gathered}
s_{i j} = \sum_{k = i}^{j - 1} s_k.\end{gathered}$$
Now we find expressions for $\varphi_\zeta(e_{n \delta, \alpha_{i j}})$ and $\varphi_\zeta(f_{n \delta, \alpha_{i j}})$. Using (\[fzefe\]) and (\[fzeff\]), we obtain for the corresponding generating functions the following expressions $$\begin{gathered}
1 - \kappa_q \sum_{n = 1}^\infty \varphi_\zeta(e'_{n \delta, \alpha_{i j}}) u^n = \sum_{\substack{k = 1 \\ k \ne i, j}}^{l + 1} {\mathbb E}_{k k} + \frac{1 - (-1)^i q^{i - 1} \zeta^s u}{1 - (-1)^i q^{i + 1} \zeta^s u} {\mathbb E}_{i i} + \frac{1 - (-1)^i q^{i + 3} \zeta^s u}{1 - (-1)^i q^{i + 1} \zeta^s u} {\mathbb E}_{j j}, \\
1 + \kappa_q \sum_{n = 1}^\infty \varphi_\zeta(f'_{n \delta, \alpha_{i j}}) u^n = \sum_{\substack{k = 1 \\ k \ne i, j}}^{l + 1} {\mathbb E}_{k k} + \frac{1 - (-1)^i q^{- i + 1} \zeta^s u}{1 - (-1)^i q^{- i - 1} \zeta^s u} {\mathbb E}_{i i} + \frac{1 - (-1)^i q^{- i - 3} \zeta^s u}{1 - (-1)^i q^{- i - 1} \zeta^s u} {\mathbb E}_{j j},\end{gathered}$$ and come to the equations $$\begin{gathered}
\varphi_\zeta(e_{n \delta, \alpha_{i j}}) = \zeta^{n s} (-1)^{i n - 1} q^{i n} \frac{[n]_q}{n} \big({\mathbb E}_{i i} - q^{2 n} {\mathbb E}_{j j}\big), \\
\varphi_\zeta(f_{n \delta, \alpha_{i j}}) = \zeta^{- n s} (-1)^{i n - 1} q^{- i n} \frac{[n]_q}{n} \big({\mathbb E}_{i i} - q^{- 2 n} {\mathbb E}_{j j}\big).\end{gathered}$$ Now, we find the expression $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) \left( - \kappa_q \sum_{n = 1}^\infty \sum_{i, j = 1}^l u_{n, i j} e_{n \delta, \alpha_i} \otimes f_{n \delta, \alpha_j} \right) = - \sum_{n = 1}^\infty \frac{q^{n l} - q^{- n l}}{[l + 1]_{q^n}} \frac{\zeta^{n s}_{1 2}}{n} \sum_{i = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{i i} \\
\qquad{} - \sum_{n = 1}^\infty \frac{q^{- n (l + 2)} - q^{- n l}}{[l + 1]_{q^n}} \frac{\zeta^{n s}_{1 2}}{n} \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} - \sum_{n = 1}^\infty \frac{q^{n l} - q^{n (l + 2)}}{[l + 1]_{q^n}} \frac{\zeta^{n s}_{1 2}}{n} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j},\end{gathered}$$ which can be rewritten as $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) \left( - \kappa_q \sum_{n = 1}^\infty \sum_{i, j = 1}^l u_{n, i j} e_{n \delta, \alpha_i} \otimes f_{n \delta, \alpha_j} \right) = - \sum_{n = 1}^\infty \frac{q^{n l} - q^{- n l}}{[l + 1]_{q^n}} \frac{\zeta^{n s}_{1 2}}{n} \sum_{i, j = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} \\
\qquad{} - \sum_{n = 1}^\infty \big(q^{- 2 n} - 1\big) \frac{\zeta^{n s}_{1 2}}{n} \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} - \sum_{n = 1}^\infty \big(1 - q^{2 n}\big) \frac{\zeta^{n s}_{1 2}}{n} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j}.\end{gathered}$$ Introducing the transcendental function $$\begin{gathered}
F_m(\zeta) = \sum_{n = 1}^\infty \frac{1}{[m]_{q^n}} \frac{\zeta^n}{n},\end{gathered}$$ and performing some summations, we obtain $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) \left( - \kappa_q \sum_{n = 1}^\infty \sum_{i, j = 1}^l (u_n)_{i j} e_{n \delta, \alpha_i} \otimes f_{n \delta, \alpha_j} \right) \\
\qquad{} = \big(F_{l + 1}\big(q^{- l} \zeta^s_{1 2}\big) - F_{l + 1}\big(q^l \zeta^s_{1 2}\big)\big) \sum_{i, j = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} \\
\qquad\quad{} + \log \frac{1 - q^{- 2} \zeta^s_{1 2}}{1 - \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} + \log \frac{1 - \zeta^s_{1 2}}{1 - q^2 \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j}.\end{gathered}$$ After all, we see that $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\sim \delta}) = {\mathrm e}^{(F_{l + 1}(q^{- l} \zeta^s_{1 2}) - F_{l + 1}(q^l \zeta^s_{1 2}))} \nonumber\\
\qquad {}\times \Bigg[ \sum_{i = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{i i}
+ \frac{1 - q^{- 2} \zeta^s_{1 2}}{1 - \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} + \frac{1 - \zeta^s_{1 2}}{1 - q^2 \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} \Bigg]. \label{frd}\end{gathered}$$
Proceed to the factor $(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\prec \delta})$. Using equations (\[eij\]) and (\[fij\]), we determine that $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\alpha_{i j} + n \delta}) = \exp_{q^2} \big( {-} \kappa_q \zeta_{1 2}^{s_{i j} + n s} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i} \big).\end{gathered}$$ Since $$\begin{gathered}
({\mathbb E}_{i j} \otimes {\mathbb E}_{j i})^k = 0\end{gathered}$$ for all $1 \le i < j \le l + 1$ and $k > 1$ we can obtain $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\alpha_{i j} + n \delta}) = 1 - \kappa_q \zeta_{1 2}^{s_{i j} + n s} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i}.\end{gathered}$$ Taking into account that $$\begin{gathered}
({\mathbb E}_{i j} \otimes {\mathbb E}_{j i})({\mathbb E}_{k m} \otimes {\mathbb E}_{m k}) = 0\end{gathered}$$ for all $1 \le i < j \le l + 1$ and $1 \le k < m \le l + 1$, we see that the factors $(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\alpha_{i j} + n \delta})$ of $(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\prec \delta})$ can be taken in an arbitrary order and obtain the expression $$\begin{aligned}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\prec \delta}) & = 1 - \kappa_q \sum_{n = 0}^\infty \zeta_{1 2}^{n s} \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} \zeta_{1 2}^{s_{i j}} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i}\nonumber\\
& = 1 - \frac{\kappa_q}{1 - \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} \zeta^{s_{i j}}_{1 2} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i}. \label{frpd}\end{aligned}$$ In a similar way we come to the equation $$\begin{gathered}
(\varphi_{\zeta_1} \otimes \varphi_{\zeta_2}) ({\mathcal R}_{\succ \delta}) = 1 - \frac{\kappa_q}{1 - \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} \zeta^{s - s_{j i}}_{1 2} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i}. \label{frsd}\end{gathered}$$
Finally, using equations (\[frpd\]), (\[frd\]), (\[frsd\]) and (\[kvv\]) we obtain the following expression for the $R$-operator $$\begin{gathered}
R_{V | V} (\zeta_1 | \zeta_2)= \sum_{i = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{i i} + \frac{q(1 - \zeta^s_{1 2})}{1 - q^2 \zeta^s_{1 2}} \sum_{\substack{i, j = 1 \\ i \ne j}}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{j j} \\
\hphantom{R_{V | V} (\zeta_1 | \zeta_2)=}{} + \frac{(1 - q^2)}{1 - q^2 \zeta^s_{1 2}} \Bigg( \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} \zeta^{s_{i j}}_{1 2} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i} + \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} \zeta^{s - s_{j i}}_{1 2}
{\mathbb E}_{i j} \otimes {\mathbb E}_{j i} \Bigg),\end{gathered}$$ where we assumed that the normalization factor has the form $$\begin{gathered}
\rho_{V | V}(\zeta_1 | \zeta_2) = q^{- l/(l + 1)} {\mathrm e}^{F_{l + 1} (q^{- l} \zeta_{1 2}^s ) - F_{l + 1}(q^l \zeta_{1 2}^s)}. \label{rvv}\end{gathered}$$
It is common to use an $R$-operator depending on only one spectral parameter. To this end one introduces the operator $$\begin{gathered}
R_{V | V}(\zeta) = R_{V | V}(\zeta | 1)\end{gathered}$$ so that $$\begin{gathered}
R_{V | V}(\zeta_1 | \zeta_2) = R_{V | V}(\zeta_{1 2}).\end{gathered}$$ With an appropriate choice of the integers $s_i$ and normalization, we obtain the Bazhanov–Jimbo $R$-operator [@Baz85; @Jim86b; @Jim89].
### Crossing and unitarity relations
It is convenient to put $$\begin{aligned}
{3}
& \rho_{V^* | V}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2)^{-1}, \qquad && \rho_{V | {}^* V}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2)^{-1}, & \label{rri} \\
& \rho_{{}^* V | V}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2)^{-1}, \qquad && \rho_{V | V^*}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2)^{-1}, & \label{rrii} \\
& \rho_{V^* | V^*}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2), \qquad && \rho_{{}^*V | {}^*V}(\zeta_1 | \zeta_2) = \rho_{V | V}(\zeta_1 | \zeta_2), & \label{rriii}\end{aligned}$$ where $\rho_{V | V}(\zeta_1 | \zeta_2)$ is defined by equation (\[rvv\]). In this case all coefficients $D$, entering the crossing relations (\[cri\])–(\[crvi\]), are equal to $1$. Moreover, all corresponding $R$-operators satisfy unitarity relation with the coefficients $C$ equal to 1.
Let us describe now the explicit form of the quantities entering the crossing relations (\[crix\])–(\[crxii\]). Notice first that due to (\[tt\]) and (\[hdh\]) one has $$\begin{gathered}
q^{- (\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} = q^{- 2 (l + 1) / s}, \qquad q^{(\theta | \theta) {h \ifthenelse{\equal{\null}{\null}}{\,}{} \check{{}_{\null}} \ifthenelse{\equal{\null}{\null}}{\,}{}} / s} = q^{2 (l + 1) / s}.\end{gathered}$$ Further, it follows from (\[lmb\]) and (\[bij\]) that $$\begin{gathered}
\lambda_i = - (l + 1 - i) i + 2(l + 1 - i) \sum_{j = 1}^i j s_j / s + 2 i \sum_{j = i + 1}^l (l + 1 - j) s_j / s.\end{gathered}$$ Now, taking into account (\[fzqhi\]), we obtain $$\begin{gathered}
{\mathbb X}_V = \varphi_\zeta \Big( q^{\sum\limits_{i = 1}^l \lambda_i h_i} \Big) = \sum_{i = 1}^{l + 1} q^{ \chi_i} {\mathbb E}_{i i},\end{gathered}$$ where $$\begin{gathered}
\chi_i = \lambda_i - \lambda_{i - 1} = - (l + 2 - 2i) - 2 \sum_{j = 1}^{i - 1} j s_j / s + 2 \sum_{j = i}^l (l + 1 - j) s_j / s.\end{gathered}$$ It follows from the definition of $F_m(\zeta)$ that $$\begin{gathered}
F_m(q^m \zeta) - F_m(q^{-m} \zeta) = {} - \log(1 - q \zeta) + \log\big(1 - q^{-1} \zeta\big).\end{gathered}$$ At last, using equations (\[rri\]) and (\[rrii\]), we obtain for the coefficients $D(\zeta_1 | \zeta_2)$ entering the crossing relations (\[crix\]) and (\[crx\]) the expression $$\begin{gathered}
D(\zeta_1 | \zeta_2) = \frac{1 - q^{-2} \zeta_{1 2}^s}{1 - \zeta_{1 2}^s} \frac{1 - q^{-2 l} \zeta_{1 2}^s}{1 - q^{- 2 l- 2} \zeta_{1 2}^s},\end{gathered}$$ and for the coefficients $D(\zeta_1 | \zeta_2)$ entering the crossing relations (\[crxi\]) and (\[crxii\]) the expression $$\begin{gathered}
D(\zeta_1 | \zeta_2) = \frac{1 - q^2 \zeta_{1 2}^s}{1 - \zeta_{1 2}^s} \frac{1 - q^{2 l} \zeta_{1 2}^s}{1 - q^{2 l + 2} \zeta_{1 2}^s}.\end{gathered}$$
### Hamiltonian
It is easy to verify that the permutation operator $P_{1 2}$ has the representation $$\begin{gathered}
P_{1 2} = \sum_{i, j = 1}^{l + 1} {\mathbb E}_{i j} \otimes {\mathbb E}_{j i}.\end{gathered}$$ Using this representation, we obtain $$\begin{gathered}
P_{1 2}({\mathbb E}_{i j} \otimes {\mathbb E}_{k m}) = {\mathbb E}_{k j} \otimes {\mathbb E}_{i m},\end{gathered}$$ and come to the equation $$\begin{gathered}
\check R_{V | V}(\zeta) = \sum_{i = 1}^{l + 1} {\mathbb E}_{i i} \otimes {\mathbb E}_{i i} + \frac{q(1 - \zeta^s)}{1 - q^2 \zeta^s} \sum_{\substack{i, j = 1 \\ i \ne j}}^{l + 1} {\mathbb E}_{j i} \otimes {\mathbb E}_{i j} \\
\hphantom{\check R_{V | V}(\zeta) =}{} + \frac{\big(1 - q^2\big)}{1 - q^2 \zeta^s} \Bigg( \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} \zeta^{s_{i j}} {\mathbb E}_{j j} \otimes {\mathbb E}_{i i} + \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} \zeta^{s - s_{j i}} {\mathbb E}_{j j} \otimes {\mathbb E}_{i i} \Bigg).\end{gathered}$$
From the structure of the universal $R$-matrix [@Bec94a; @Dam98; @KhoTol92; @KhoTol93; @KhoTol94; @TolKho92], it follows that the dependence of a transfer operator on $\zeta$ is determined by the dependence on $\zeta$ of the elements of the form $\varphi_\zeta(a)$, where $a$ is an element of the Hopf subalgebra of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ generated by the elements $e_i$, $i \in {[0 \, . \, . \, l]}$, and $q^x$, $x \in {\widetilde{\mathfrak h}}$. Taking into account the form (\[ja\])–(\[jc\]) of Jimbo’s homomorphism, we see that $\varphi_\zeta(a)$ for any such element equals $\pi(A)$, where $A$ is a linear combination of monomials each of which is a product of $E_i$, $i \in {[1 \, . \, . \, l]}$, $F_{1, l + 1}$ and $q^X$ for some $X \in {{\mathfrak k}_{l + 1}}$. Let $A$ be such a monomial. We have $$\begin{gathered}
q^{H_1} A q^{- H_1} = q^{2 n_1 - n_2 - n} A, \\
q^{H_i} A q^{- H_i} = q^{- n_{i - 1} + 2 n_i - n_{i + 1}} A, \qquad i \in {[2 \, . \, . \, l - 1]}, \\
q^{H_l} A q^{- H_l} = q^{- n_{l - 1} +2 n_l - n} A,\end{gathered}$$ where $n_i$, $i \in {[1 \, . \, . \, l]}$, are the numbers of $E_i$, and $n$ the number of $F_{1, l + 1}$ in $A$. Hence $\operatorname{tr}(A)$ can be non-zero only if $n_i = n$ for any $i \in {[1 \, . \, . \, l]}$. Each $E_i$ enters $A$ with the factor $\zeta^{s_i}$, and each $F_{1, l + 1}$ with the factor $\zeta^{s_0}$. Thus, for a monomial with non-zero trace we have the dependence on $\zeta$ of the form $\zeta^{n s}$ for some integer $n$. Therefore, assuming that the corresponding normalization factor depends only on $\zeta^s$, we see that transfer operator depends on $\zeta$ only via $\zeta^s$. Thus, without any loss of generality, finding the expression for the Hamiltonian, we can put $s_{i j} = 0$.
Choose the group-like element entering definition of the transfer operator as $$\begin{gathered}
a = q^{\sum\limits_{i = 1}^l \phi_i h_i}.\end{gathered}$$ Assuming that $\phi_0 = \phi_{l + 1} = 0$, we have $$\begin{gathered}
{\mathbb A}_V = \varphi(a) = \sum_{i = 1}^{l+1} q^{\Phi_i} {\mathbb E}_{i i}, \qquad \Phi_i = \phi_i - \phi_{i - 1}.\end{gathered}$$ Using equation (\[hv\]), we obtain $$\begin{gathered}
H_N = - \frac{s}{\kappa_q} \sum_{k = 1}^{N - 1} \Bigg( {-} \sum_{\substack{i, j = 1 \\ i \ne j}}^{l + 1} {\mathbb E}_{j i}^{(k)} {\mathbb E}_{i j}^{(k + 1)} + q \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{j j}^{(k)} {\mathbb E}_{i i}^{(k + 1)} + q^{-1} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{j j}^{(k)} {\mathbb E}_{i i}^{(k + 1)} \Bigg) \\
\hphantom{H_N =}{} - \frac{s}{\kappa_q} \Bigg( {-} \sum_{\substack{i, j = 1 \\ i \ne j}}^{l + 1} q^{\Phi_i - \Phi_j} {\mathbb E}_{j i}^{(N)} {\mathbb E}_{i j}^{(1)} + q \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{j j}^{(N)} {\mathbb E}_{i i}^{(1)} + q^{-1} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{j j}^{(N)} {\mathbb E}_{i i}^{(1)} \Bigg).\end{gathered}$$ One can also write $$\begin{gathered}
H_N = - \frac{s}{\kappa_q} \sum_{k = 1}^N \Bigg( {-} \sum_{\substack{i, j = 1 \\ i \ne j}}^{l + 1} {\mathbb E}_{j i}^{(k)} {\mathbb E}_{i j}^{(k + 1)} + q \sum_{\substack{i, j = 1 \\ i < j}}^{l + 1} {\mathbb E}_{j j}^{(k)} {\mathbb E}_{i i}^{(k + 1)} + q^{-1} \sum_{\substack{i, j = 1 \\ i > j}}^{l + 1} {\mathbb E}_{j j}^{(k)} {\mathbb E}_{i i}^{(k + 1)} \Bigg), \label{hn}\end{gathered}$$ where we assume that the following boundary condition $$\begin{gathered}
{\mathbb E}_{i j}^{(N + 1)} = q^{\Phi_i - \Phi_j} {\mathbb E}_{i j}^{(1)}\end{gathered}$$ is satisfied.
### Case of $\boldsymbol{{{\mathrm U}_q(\mathcal L(\mathfrak{sl}_2))}}$
The fundamental representation $\pi$ of ${{\mathrm U}_q(\mathfrak{sl}_2)}$ is isomorphic to the representations $\pi^*$ and ${}^* \! \pi$. The corresponding representation $\varphi_\zeta$ of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_2))}$ is isomorphic to the representations $\varphi_\zeta^*$ and ${}^* \! \varphi^{}_\zeta$ up to a rescaling of the spectral parameter. In this case, in addition to the usual crossing relations, we have some additional relations.
For example, we have $$\begin{gathered}
\varphi^*_\zeta(a) = {\mathbb O}\varphi_{q^{-2/s} \zeta}(a) {\mathbb O}^{-1},\end{gathered}$$ where $$\begin{gathered}
{\mathbb O}= - q^{1 - 2 s_1/s} {\mathbb E}_{12} + {\mathbb E}_{21}.\end{gathered}$$ It follows from this equation that $$\begin{gathered}
R_{V^* | V}(\zeta_1 | \zeta_2) = \rho_{V^* | V}(\zeta_1 | \zeta_2)^{-1} \rho_{V | V}(q^{-2/s} \zeta_1 | \zeta_2) ({\mathbb O}\otimes 1) R_{V | V}\big(q^{-2/s} \zeta_1 | \zeta_2\big) ({\mathbb O}\otimes 1)^{-1}.\end{gathered}$$ Using the identity $$\begin{gathered}
F_2(q \zeta) + F_2\big(q^{-1} \zeta\big) = - \log (1 - \zeta),\end{gathered}$$ we find $$\begin{gathered}
\rho_{V^* | V}(\zeta_1 | \zeta_2)^{-1} \rho_{V | V}\big(q^{-2/s} \zeta_1 | \zeta_2\big) = q^{-1} \frac{1 - \zeta_{1 2}^s}{1 - q^{-2} \zeta_{1 2}^s}.\end{gathered}$$ Since $$\begin{gathered}
R_{V^* | V}(\zeta_1 | \zeta_2) = \big(R_{V | V}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1}\end{gathered}$$ we come to the equation $$\begin{gathered}
\big(R_{V | V}(\zeta_1 | \zeta_2)^{-1}\big)^{t_1} = q^{-1} \frac{1 - \zeta_{1 2}^s}{1 - q^{-2} \zeta_{1 2}^s} ({\mathbb O}\otimes 1) R_{V | V}\big(q^{-2/s} \zeta_1 | \zeta_2\big) ({\mathbb O}\otimes 1)^{-1}.\end{gathered}$$
For the representation ${}^* \! \varphi_\zeta$ we get $$\begin{gathered}
{}^* \! \varphi_\zeta(a) = \big({\mathbb O}^t\big)^{-1} \varphi_{q^{2/s} \zeta}(a) {\mathbb O}^t\end{gathered}$$ and come to the equation $$\begin{gathered}
\big(R_{V | V}(\zeta_1 | \zeta_2)^{-1}\big)^{t_2} = q^{-1} \frac{1 - \zeta_{1 2}^s}{1 - q^{-2} \zeta_{1 2}^s} \big(1 \otimes {\mathbb O}^t\big)^{-1} R_{V | V}\big(\zeta_1 | q^{2/s} \zeta_2\big) \big(1 \otimes {\mathbb O}^t\big).\end{gathered}$$
In a similar way, applying representations $\varphi^*_\zeta$ and ${}^* \! \varphi^{}_\zeta$ to other factors of the tensor product ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_2))}\otimes {{\mathrm U}_q(\mathcal L(\mathfrak{sl}_2))}$, we find that $$\begin{gathered}
\big(R_{V | V}(\zeta_1 | \zeta_2)^{t_1}\big)^{-1} = q^{-1} \frac{1 -q^2 \zeta_{1 2}^s}{1 - \zeta_{1 2}^s} \big({\mathbb O}^t \otimes 1\big)^{-1} R_{V | V}\big(q^{2/s} \zeta_1 | \zeta_2\big) \big({\mathbb O}^t \otimes 1\big), \\
\big(R_{V | V}(\zeta_1 | \zeta_2)^{t_2}\big)^{-1} = q^{-1} \frac{1 - q^2 \zeta_{1 2}^s}{1 - \zeta_{1 2}^s} (1 \otimes {\mathbb O}) R_{V | V}\big(\zeta_1 | q^{-2/s} \zeta_2\big) (1 \otimes {\mathbb O})^{-1}.\end{gathered}$$
The Hamiltonian $H_N$ given by (\[hn\]) for the case of ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_2))}$ is related to the well known Hamiltonian of the XXZ-model $$\begin{gathered}
H_{\mathrm{XXZ}} = - \sum_{k = 1}^N \left[\sigma_+^{(k)} \sigma_-^{(k + 1)} + \sigma_-^{(k)} \sigma_+^{(k + 1)}
+ \frac{q + q^{-1}}{4} \big(\sigma_z^{(k)} \sigma_z^{(k + 1)} - 1\big)\right]\end{gathered}$$ by the equation $$\begin{gathered}
H_N = - \frac{s}{\kappa_q} H_{\mathrm{XXZ}}\end{gathered}$$ Here we use the standard notations $$\begin{gathered}
\sigma_+ = {\mathbb E}_{12}, \qquad \sigma_- = {\mathbb E}_{21}, \qquad \sigma_z = {\mathbb E}_{11} - {\mathbb E}_{22},\end{gathered}$$ and assume the validity of the boundary conditions $$\begin{gathered}
\sigma_+^{(N + 1)} = q^{\Phi} \sigma_+^{(1)} \qquad \sigma_-^{(N + 1)} = q^{-\Phi} \sigma_-^{(1)}, \qquad \sigma_z^{(N + 1)} = \sigma_z^{(1)}\end{gathered}$$ with $$\begin{gathered}
\Phi = \Phi_1 - \Phi_2.\end{gathered}$$
Graphical description of open chains {#section4}
====================================
Transfer operator
-----------------
The transfer operator for an open chain is constructed in the following way. First we choose an auxiliary space $V$ and two quantum spaces $$\begin{gathered}
W^R = W^{\otimes m}, \qquad W^L = W^{\otimes n}.\end{gathered}$$ With the space $V$ we associate a spectral parameter $\zeta$, and with the factors of $W^R$ and $W^L$ the spectral parameters $\eta_1, \ldots, \eta_m$ and $\eta_{m + 1}, \ldots, \eta_{m + n}$. Then we introduce two operators $K^R_{V | W}(\zeta | \eta_1, \ldots, \eta_m)$ and $K^L_{V | W}(\zeta | \eta_{m + 1}, \ldots, \eta_{m + n})$ acting on the spaces $V \otimes W^R$ and $V \otimes W^L$ respectively. The depiction of their matrix elements can be seen in Figs. \[f:kl\] and \[f:kr\]. It should be noted that when an incoming line associated with the auxiliary space becomes an outgoing one, the spectral parameter turns to its inverse.
![[]{data-label="f:kr"}](Figures/KL)
![[]{data-label="f:kr"}](Figures/KR)
The case $m = 0$ or $n = 0$ is also allowed. Here we have operators $K^L_V(\zeta)$ and $K^R_V(\zeta)$ acting on the auxiliary space $V$. Now, the transfer operator is defined by the equation $$\begin{gathered}
T_{V | W}(\zeta | \eta_1, \ldots, \eta_m, \eta_{m + 1}, \ldots, \eta_{m + n}) \\
\qquad{} = \operatorname{tr}_V \big(K^R_{V | W}(\zeta | \eta_1, \ldots, \eta_m) K^L_{V | W}(\zeta | \eta_{m + 1}, \ldots, \eta_{m + n}) \big).
\label{tvw}\end{gathered}$$ Here the expression under the partial trace in the right hand side means an operator acting on $V \otimes W^R \otimes W^L$. In terms of matrix elements we have $$\begin{gathered}
T_{V | W}(\zeta | \eta_1, \ldots, \eta_m, \eta_{m + 1}, \ldots, \eta_{m + n})^{i_1 \ldots i_m i_{m + 1} \ldots i_{m + n}}{}_{j_1 \ldots j_m j_{m + 1} \ldots j_{m + n}} \\
\quad{}= K^R_{V | W}(\zeta | \eta_1, \ldots, \eta_m)^{\alpha_1 i_1 \ldots i_m}{}_{\alpha_2 j_1 \ldots j_m} K^L_{V | W}(\zeta | \eta_{m + 1}, \ldots, \eta_{m + n})^{\alpha_2 i_{m + 1} \ldots i_{m + n}}{}_{\alpha_1 j_{m + 1} \ldots j_{m + n}}.\end{gathered}$$ It is clear that the graphical analogue of this definition is the one given in Fig. \[f:tvw\].
![[]{data-label="f:tvw"}](Figures/TVW)
Commutativity of transfer operators {#s:coto}
-----------------------------------
Let us derive sufficient conditions for the commutativity of the transfer operators for the case under consideration.
The picture representing the product of two transfer operators is given in Fig. \[f:tv1wtv2w\]. We subject it to the following transformations.
![[]{data-label="f:tv1wtv2w"}](Figures/TV1WTV2W)
First, we twist two horizontal lines in the middle of the picture. One sees that these lines go in the opposite directions. Therefore, we use for our purpose the graphical equation given in Fig. \[f:itroro\]. The result is represented in Fig. \[f:tv1wtv2wi\].
![[]{data-label="f:tv1wtv2wi"}](Figures/TV1WTV2WI)
Then we twist two upper lines of this figure. Now the lines go in the same direction, and we use for twisting the graphical equation in Fig. \[f:irrwi\] with the single and double lines interchanged.[^6] After two transformations we come to the situation depicted in Fig. \[f:tv1wtv2wii\].
![[]{data-label="f:tv1wtv2wii"}](Figures/TV1WTV2WII)
Proceed now to the opposite product of the same transfer operators represented graphically in Fig. \[f:tv2wtv1w\].
![[]{data-label="f:tv2wtv1w"}](Figures/TV2WTV1W)
Using graphical equation in Fig. \[f:itroro\] with the single and double lines interchanged,[^7] we twist two horizontal lines in the middle of the picture. Then we use the graphical equation given in Fig. \[f:rirwi\] to twist two bottom lines. The result can be seen in Fig. \[f:tv2wtv1wi\].
![[]{data-label="f:tv2wtv1wi"}](Figures/TV2WTV1WI)
Compare Figs. \[f:tv1wtv2wii\] and \[f:tv2wtv1wi\]. If we cut these figures vertically in the middle, then the types and directions of the lines intersecting the cut will be the same. Therefore, it is consistent to equate the left and right halves of the figures. Graphically it is represented by Figs. \[f:lh\] and \[f:rh\]. It is clear that if the equations given in these figures are satisfied, then the product of the transfer operators under consideration does not depend on the order.
![[]{data-label="f:lh"}](Figures/LH)
![[]{data-label="f:rh"}](Figures/RH)
The remarkable fact is that if the operators depicted in Figs. \[f:kl\] and \[f:kr\] satisfy the graphical equations presented in Figs. \[f:lh\] and \[f:rh\], then the ‘dressed’ operators depicted in Figs. \[f:kld\] and \[f:krd\] satisfy these equations as well.
![[]{data-label="f:krd"}](Figures/KLDressed)
![[]{data-label="f:krd"}](Figures/KRDressed)
Let us demonstrate this first for the case of the operator $K^R_{V | W}$. Insert the dressed operators $K^R_{V_1 | W}$ and $K^R_{V_2 | W}$ into the left hand side of the graphical equation in Fig. \[f:rh\]. This gives the picture given in Fig. \[f:drhi\].
![[]{data-label="f:drhi"}](Figures/DressedRightHalfI)
Now, using the Yang–Baxter equations depicted in Figs. \[f:ybei\] and \[f:ybeii\], we move the leftmost wavy line to the left and come to the situation which can be seen in Fig. \[f:drhii\]. It should be noted that the Yang–Baxter equations in Figs. \[f:ybei\] and \[f:ybeii\] can be obtained without assuming the validity of the unitarity relations. Then we apply the graphical equation in Fig. \[f:rh\] to Fig. \[f:drhii\] and obtain Fig. \[f:drhiii\]. Finally, using the Yang–Baxter equations in Figs. \[f:ybeiii\] and \[f:ybeiv\], we move the leftmost wavy line to the right and come to the situation which can be seen in Fig. \[f:drhiv\]. The Yang–Baxter equations in Figs. \[f:ybeiii\] and \[f:ybeiv\] can be again obtained without assuming the validity of the unitarity relations. Comparing Figs. \[f:drhi\] and \[f:drhiv\], we obtain the desired result. The case of the operator $K^L_{V | W}$ can be analysed in the same way.
![[]{data-label="f:ybeii"}](Figures/YBEquationI)
![[]{data-label="f:ybeii"}](Figures/YBEquationII)
![[]{data-label="f:drhii"}](Figures/DressedRightHalfII)
![[]{data-label="f:drhiii"}](Figures/DressedRightHalfIII)
![[]{data-label="f:ybeiv"}](Figures/YBEquationIII)
![[]{data-label="f:ybeiv"}](Figures/YBEquationIV)
![[]{data-label="f:drhiv"}](Figures/DressedRightHalfIV)
Thus, it is possible to take as the operators $K^L_{V | W}$ and $K^R_{V | W}$ the operators depicted in Figs. \[f:kldf\] and \[f:krdf\].
![[]{data-label="f:krdf"}](Figures/KLDressedFull)
![[]{data-label="f:krdf"}](Figures/KRDressedFull)
One can verify that they can be defined analytically by the equations $$\begin{gathered}
K^L_{V | W}(\zeta | \eta_{m + 1}, \ldots, \eta_{m + n}) \notag \\
\qquad {} = M_{V | W}\big(\zeta^{-1} | \eta_{m + 1}, \ldots, \eta_{m + n}\big)^{-1} K^L_V(\zeta) M_{V | W}(\zeta | \eta_{m + 1}, \ldots, \eta_{m + n}), \label{klvwl} \\
K^R_{V | W}(\zeta | \eta_1, \ldots, \eta_m) \notag \\
\qquad {} = \big(\big(M_{V | W}\big(\zeta^{-1} | \eta_1, \ldots, \eta_m\big)^{-1}\big)^{t_V} K^R_V(\zeta)^{t_V} M_{V | W}(\zeta | \eta_1, \ldots, \eta_m)^{t_V}\big)^{t_V}. \label{krvwr}\end{gathered}$$ The operators $K^L_V$ and $K^R_V$ act on the auxiliary space $V$ and satisfy the graphical equations given in Figs. \[f:drlh\] and \[f:drrh\].
![[]{data-label="f:drlh"}](Figures/DRLH)
![[]{data-label="f:drrh"}](Figures/DRRH)
It is not difficult to find the analytical expression for these equations, namely, $$\begin{gathered}
P_{V_2 | V_1} R_{V_2 | V_1}\big(\zeta_2^{-1} | \zeta_1^{-1}\big) P_{V_1 | V_2} K^L_{V_1}(\zeta_1) R_{V_1 | V_2}\big(\zeta_1^{\mathstrut} | \zeta_2^{-1}\big) K^L_{V_2}(\zeta_2) \\
\qquad {} = K^L_{V_2}(\zeta_2) P_{V_2 | V_1} R_{V_2 | V_1}\big(\zeta_2^{\mathstrut} | \zeta_1^{-1}\big) P_{V_1 | V_2} K^L_{V_1}(\zeta_1) R_{V_1 | V_2}(\zeta_1 | \zeta_2), \\
K^R_{V_2}(\zeta_2) \widetilde R_{V_1 | V_2}\big(\zeta_1^{\mathstrut} | \zeta_2^{-1}\big)^{-1} K^R_{V_1}(\zeta_1) P_{V_2 | V_1} R_{V_2 | V_1}\big(\zeta_2^{-1} | \zeta_1^{-1}\big)^{-1} P_{V_1 | V_2} \\
\qquad {} = R_{V_1 | V_2} (\zeta_1 | \zeta_2)^{-1} K^R_{V_1}(\zeta_1) P_{V_2 | V_1} \widetilde R_{V_2 | V_1}\big(\zeta_2^{\mathstrut} | \zeta_1^{-1}\big)^{-1} P_{V_1 | V_2} K^R_{V_2}(\zeta_2).\end{gathered}$$ There are many papers devoted to solving these equations with respect to the operators $K^L_V$ and $K^R_V$ for a fixed $R$-operator, see, for example, paper [@MalLim06] and references therein. The most complete set of solutions for the case of the quantum loop algebras ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$ was obtained in [@RegVla18].
It is clear now that the graphical image of the transfer operator is the one given by Fig. \[f:drto\] whose analytical expression is
![[]{data-label="f:drto"}](Figures/DRTO)
$$\begin{gathered}
T^{}_{V | W}(\zeta | \eta_1, \ldots, \eta_N) = \operatorname{tr}^{}_V \big(K^R_V(\zeta) M_{V | W}\big(\zeta^{-1} | \eta_1, \ldots, \eta_N\big)^{-1} K^L_V(\zeta) M_{V | W}(\zeta | \eta_1, \ldots, \eta_N)\big).\end{gathered}$$
It is rather tricky to obtain this expression from the definition (\[tvw\]) and equations (\[klvwl\]) and (\[krvwr\]), see paper [@Skl88]. However, from the graphical point of view it is evident.
Hamiltonian
-----------
![[]{data-label="f:to(1)"}](Figures/TransferOperatorAtOne)
As in the case of a periodic chain we assume that the quantum space $W$ coincides with the auxiliary space $V$ and $R_{V | V}(1 | 1)$ coincides with the permutation operator $P_{1 2}$. The Hamiltonian $H_N$ for the chain of length $N$ is again constructed from the homogeneous transfer operator $$\begin{gathered}
T_V(\zeta) = T_{V | V}(\zeta | 1, 1, \ldots, 1)\end{gathered}$$ with the help of the equation $$\begin{gathered}
H_N = \left. \zeta \frac{{\mathrm d}}{{\mathrm d}\zeta} \log T_V(\zeta) \right|_{\zeta = 1} = \left.\frac{{\mathrm d}T_V(\zeta)}{{\mathrm d}\zeta} \right|_{\zeta = 1} T_V(1)^{-1}.\end{gathered}$$ One can find the graphical image of the operator $T_V(1)$ in Fig. \[f:to(1)\]. Here and below gray border means the value of an operator at $\zeta = 1$. Thus to have an invertible operator $T_V(1)$, one should assume that the operator $K^L_V(1)$ is invertible. Here one has $$\begin{gathered}
T_V(1)^{- 1} = \big(K^L_V(1)^{-1}\big)^{(N)} / \operatorname{tr}K^R_V(1).\end{gathered}$$ For the meaning of the superscript $(N)$ see Appendix \[a:tpsg\]. The graphical form of the various possible summands which enters the expression for the derivative $T'_V(1)$ are given in Figs. \[f:hoi\]–\[f:hoii\], where, as above, a double line means the derivative at $\zeta = 1$. Note that to draw Figs. \[f:hoiv\], \[f:hoviii\] and \[f:hovi\] we use the equation $$\begin{gathered}
\left. \zeta \frac{{\mathrm d}\check R_{V | V}\big(\zeta^{-1} | 1\big)}{{\mathrm d}\zeta} \right|_{\zeta = 1} = \check R'_{V | V}(1 | 1).\end{gathered}$$ Using Figs. \[f:hoi\]–\[f:hoii\], we come to the following analytical expression for the Hamiltonian $$\begin{gathered}
H_N = \operatorname{tr}K^{R \prime}_V(1) / \operatorname{tr}K^R_V(1) + 2 \operatorname{tr}_{1'} \big( {\mathbb H}^{(1', 1)} K^R_V(1)^{(1')} \big) / \operatorname{tr}K^R_V(1) + 2 \sum_{i = 1}^{N - 2} {\mathbb H}^{(i, i + 1)} \\
\hphantom{H_N =}{} + {\mathbb H}^{(N - 1, N)} + K^L_V(1)^{(N)} {\mathbb H}^{(N - 1, N)} \big(K^L_V(1)^{-1}\big)^{(N)} + K^{L \prime}_V(1)^{(N)} \big(K^L_V(1)^{-1}\big)^{(N)},\end{gathered}$$ where ${\mathbb H}^{(k, l)}$ is defined by equation (\[hkl\]). The order of the terms in the above expression is opposite to the order of the figures.
![[]{data-label="f:hoi"}](Figures//HamiltonianOI)
![[]{data-label="f:hoiv"}](Figures/HamiltonianOIII)
![[]{data-label="f:hoiv"}](Figures/HamiltonianOIV)
![[]{data-label="f:hovii"}](Figures/HamiltonianOVII)
![[]{data-label="f:hoviii"}](Figures/HamiltonianOVIII)
![[]{data-label="f:hovi"}](Figures/HamiltonianOV)
![[]{data-label="f:hovi"}](Figures/HamiltonianOVI)
![[]{data-label="f:hoii"}](Figures/HamiltonianOII)
Conclusions
===========
This paper has been devoted to systematisation and development of the graphical approach to the investigation of the quantum integrable vertex models of statistical physics and the corresponding spin chains. We hope that the usefulness and productivity of this approach was clearly demonstrated. In fact, we have derived and graphically described much more relations and equations than it was needed for the considered applications. We will use them in our future works. The calculation of correlation functions is obviously one of such promising applications of the graphical approach, as it was already commenced, for example, in papers [@AufKlu12; @BooHutNir18; @RibKlu19] based on qKZ equations.
Appendix. Some linear algebra {#appendix.-some-linear-algebra .unnumbered}
=============================
In this appendix we introduce notation for the operators acting in tensor products of vector spaces and discuss some their properties, see also Appendix A of paper [@Vla15].
Tensor products and symmetric group {#a:tpsg}
-----------------------------------
We mean by $n$-tuple a mapping from the interval ${[1 \, . \, . \, n]} \subset {\mathbb N}$ to a set or a class. It is common for an $n$-tuple $F$ to use the notation $$\begin{gathered}
F(i) = F_i, \qquad i \in {[1 \, . \, . \, n]},\end{gathered}$$ and write[$$\begin{gathered}
F = (F_1, \ldots, F_n) = (F_i)_{i \in {[1 \, . \, . \, n]}}.\end{gathered}$$ We define the range of an $n$-tuple as the range of the corresponding mapping.]{}
Let $A = (A_i)_{i \in {[1 \, . \, . \, n]}}$ be an $n$-tuple of unital associative algebras. Consider the tensor product $$\begin{gathered}
A^\otimes = A_1 \otimes A_2 \otimes \cdots \otimes A_n.\end{gathered}$$ For any element $\sigma$ of the symmetric group ${\mathrm S}_n$ denote $$\begin{gathered}
A_\sigma = ((A_\sigma)_i)_{i \in {[1 \, . \, . \, n]}} = (A_{\sigma^{-1}(i)})_{i \in {[1 \, . \, . \, n]}},\end{gathered}$$ so that $$\begin{gathered}
(A_\sigma)^\otimes = A_{\sigma^{-1}(1)} \otimes A_{\sigma^{-1}(2)} \otimes \cdots \otimes A_{\sigma^{-1}(n)}.\end{gathered}$$ It is clear that for any two elements $\sigma, \tau \in {\mathrm S}_n$ one has $$\begin{gathered}
(A_\sigma)_\tau = A_{\sigma \tau}.\end{gathered}$$ Define also a linear mapping $\Pi_\sigma \colon A^\otimes \to (A_\sigma)^\otimes$ acting on a monomial $a_1 \otimes a_2 \otimes \cdots \otimes a_n$ in accordance with the rule $$\begin{gathered}
\Pi_\sigma (a_1 \otimes a_2 \otimes \cdots \otimes a_n) = a_{\sigma^{-1}(1)} \otimes a_{\sigma^{-1}(2)} \otimes \cdots \otimes a_{\sigma^{-1}(n)}.\end{gathered}$$ It is not difficult to demonstrate that for any $\sigma, \tau \in {\mathrm S}_n$ one obtains $$\begin{gathered}
\Pi_\sigma \circ \Pi_\tau = \Pi_{\sigma \tau}.\end{gathered}$$ It is evident that $\Pi_\sigma$ is an isomorphic mapping from the algebra $A^\otimes$ to the algebra $(A_\sigma)^\otimes$.
Now, let $B$ be one more unital associative algebra, and $b \in B$. If for some $i \in {[1 \, . \, . \, n]}$ we have $B = A_i$, then we denote by $b^{(i)}$ the element of $A^\otimes$ defined as $$\begin{gathered}
b^{(i)} = \underbracket[.6pt]{1 \otimes \cdots \otimes 1}_{i - 1} {} \otimes b \otimes \underbracket[.6pt]{1 \otimes \cdots \otimes 1}_{n - i}.\end{gathered}$$ One can easily get convinced that[$$\begin{gathered}
\Pi_\sigma \big(b^{(i)}\big) = b^{(\sigma(i))}\end{gathered}$$ for any $\sigma \in {\mathrm S}_n$.]{}
More generally, given $0 < k \le n$, let $B = (B_i)_{i \in {[1 \, . \, . \, k]}}$ be a $k$-tuple of unital associative algebras. Further, let $i = (i_1, i_2, \ldots, i_k)$ be a $k$-tuple of distinct positive integers from the interval ${[1 \, . \, . \, n]}$. If $B_l = A_{i_l}$ for all $l \in {[1 \, . \, . \, k]}$ and $b = b_1 \otimes b_2 \otimes \cdots \otimes b_k$ is an element of the algebra $$\begin{gathered}
B^\otimes = B_1 \otimes B_2 \otimes \cdots \otimes B_k,\end{gathered}$$ we denote by $b^i$ the element of the algebra $A$ defined as $$\begin{gathered}
b^i = b^{(i_1, i_2, \ldots, i_k)}= b_1^{(i_1)} b_2^{(i_2)} \cdots b_k^{(i_k)}.\end{gathered}$$ We extend this rule to all elements of $B$ by linearity. Further, given $\sigma \in {\mathrm S}_n$, we denote $$\begin{gathered}
{}_\sigma i = (\sigma(i_1), \sigma(i_2), \ldots, \sigma(i_k)),\end{gathered}$$ so that $$\begin{gathered}
{}_{\sigma \tau} i = {}_\sigma({}_\tau i)\end{gathered}$$ for any $\sigma, \tau \in {\mathrm S}_n$. Here one has $$\begin{gathered}
\Pi_\sigma(b^i) = b^{\, {}_\sigma i},\end{gathered}$$ or, more explicitly, $$\begin{gathered}
\Pi_\sigma\big(b^{(i_1, i_2, \ldots, i_k)}\big) = b^{(\sigma(i_1), \sigma(i_2), \ldots, \sigma(i_k))}.\end{gathered}$$ Sometimes, if it does not lead to a misunderstanding, one simplifies the notation $b^{(i_1, i_2, \ldots, i_k)}$ to $b^{i_1, i_2, \ldots, i_k}$ or even to $b^{i_1 i_2 \ldots i_k}$.
Now, let $W = (W_i)_{i \in {[1 \, . \, . \, n]}}$ be an $n$-tuple of vector spaces, and $$\begin{gathered}
A = (A_i)_{i \in {[1 \, . \, . \, n]}} = (\operatorname{End}(W_i))_{i \in {[1 \, . \, . \, n]}}\end{gathered}$$ an $n$-tuple of unital associative algebras. Similarly as above, we denote $$\begin{gathered}
W^\otimes = W_1 \otimes W_2 \otimes \cdots \otimes W_n, \label{vvvv}\end{gathered}$$ and $$\begin{gathered}
A^\otimes = \operatorname{End}(W_1) \otimes \operatorname{End}(W_2) \otimes \cdots \otimes \operatorname{End}(W_n) \cong \operatorname{End}\big(W^\otimes\big).\end{gathered}$$ Given $\sigma \in {\mathrm S}_n$, define $$\begin{gathered}
\big(W^\otimes\big)_\sigma = W_{\sigma^{-1}(1)} \otimes W_{\sigma^{-1}(2)} \otimes \cdots \otimes W_{\sigma^{-1}(n)},\end{gathered}$$ so that for any two elements $\sigma, \tau \in {\mathrm S}_n$ one has $$\begin{gathered}
(W_\sigma)_\tau = W_{\sigma \tau}.\end{gathered}$$ Now, define a linear mapping $P_\sigma \colon W^\otimes \to (W^\otimes)_\sigma$ by the equation $$\begin{gathered}
P_\sigma(w_1 \otimes w_2 \otimes \cdots \otimes w_n) = w_{\sigma^{-1}(1)} \otimes w_{\sigma^{-1}(2)} \otimes \cdots \otimes w_{\sigma^{-1}(n)}.\end{gathered}$$ Here for any $\sigma, \tau \in {\mathrm S}_n$ one has $$\begin{gathered}
P_\sigma \circ P_\tau = P_{\sigma \tau}.\end{gathered}$$
Let $M \in \operatorname{End}(V)$, and $V = W_i$ for some $i \in {[1 \, . \, . \, n]}$. It means that $\operatorname{End}(V) = \operatorname{End}(W_i)$ and one can define $M^{i} \in \operatorname{End}(W)$. One can show that $$\begin{gathered}
P_\sigma \circ M^i = M^{\sigma(i)} \circ P_{\sigma}\end{gathered}$$ for any $\sigma \in {\mathrm S}_n$. It follows from this equation that $$\begin{gathered}
\Pi_\sigma \big(M^i\big) = P_\sigma \circ M^i \circ (P_{\sigma})^{-1}.\end{gathered}$$
More generally, given $k \le n$, let $W = (W_i)_{i \in {[1 \, . \, . \, k]}}$ be a $k$-tuple of vector spaces, and $i = (i_1, i_2, \ldots, i_k)$ a $k$-tuple of distinct positive integers from the interval ${[1 \, . \, . \, n]}$. If $M \in \operatorname{End}\big(W^\otimes\big)$, and $W_l = V_{i_l}$ for all $l \in {[1 \, . \, . \, k]}$, one can define $M^i \in \operatorname{End}\big(V^\otimes\big)$. Here for any $\sigma \in {\mathrm S}_n$ one has $$\begin{gathered}
P_\sigma \circ M^i = M^{{}_\sigma i} \circ P_{\sigma}\end{gathered}$$ and $$\begin{gathered}
\Pi_\sigma \big(M^i\big) = P_\sigma \circ M^i \circ P_{\sigma^{-1}},\end{gathered}$$ or, more explicitly, $$\begin{gathered}
P_\sigma \circ M^{i_1 i_2 \ldots i_k} = M^{\sigma(i_1) \sigma(i_2) \ldots \sigma(i_k)} \circ P_{\sigma}\end{gathered}$$ and $$\begin{gathered}
\Pi_\sigma \big(M^{i_1 i_2 \ldots i_k}\big) = P_\sigma \circ M^{i_1 i_2 \ldots i_k} \circ (P_\sigma)^{-1}.\end{gathered}$$
If $\sigma \in {\mathrm S}_n$ is a transposition $(i j)$ one writes $\Pi_{ij}$ and $P_{ij}$ instead of $\Pi_\sigma$ and $P_\sigma$ respectively. Furthermore, if $n = 2$ one denotes $\Pi = \Pi_{1 2}$ and $P = P_{1 2}$.
If vector spaces $V$ and $U$ belong to the range of $W$ and there is only one $i$ and one $j$ such that $V = W_i$ and $U = W_j$, we also write $P_{V | U}$ instead of $P_{i j}$.
Partial transpose {#a:pt}
-----------------
Let again $W^\otimes$ be defined as in (\[vvvv\]). For any monomial $$\begin{gathered}
M = M_1 \otimes M_2 \otimes \cdots \otimes M_n, \label{mm}\end{gathered}$$ where $M_i \in \operatorname{End}(W_i)$, $i \in {[1 \, . \, . \, n]}$, we define the partial transpose $M^{t_l}$ of $M$ with respect to $W_l$ by the equation $$\begin{gathered}
M^{t_l} = M_1 \otimes \cdots \otimes M_{l - 1} \otimes (M_l)^t \otimes M_{l + 1} \otimes \cdots \otimes M_n,\end{gathered}$$ and extend this definition to all $M \in \operatorname{End}\big(W^\otimes\big)$ by linearity. By definition, for any $M \in \operatorname{End}\big(W^\otimes\big)$ the partial transpose $M^{t_l}$ is an element of the space[^8] $$\begin{gathered}
\operatorname{End}(W_1) \otimes \cdots \otimes \operatorname{End}(W_{l - 1}) \otimes \operatorname{End}((W_l)^\star) \otimes \operatorname{End}(W_{l + 1}) \otimes \cdots \otimes \operatorname{End}(W_n) \\
\qquad{}\cong \operatorname{End}(W_1) \otimes \cdots \otimes \operatorname{End}(W_{l - 1}) \otimes (\operatorname{End}(W_l))^\star \otimes \operatorname{End}(W_{l + 1}) \otimes \cdots \otimes \operatorname{End}(W_n).\end{gathered}$$ If a vector space $V$ belongs to the range of $W$ and there is only one $i$ such that $V = W_i$ we also write $W^{t_V}$ instead of $W^{t_i}$.
It is evident that $$\begin{gathered}
\big(M^{t_l}\big)^{t_l} = M\end{gathered}$$ and $$\begin{gathered}
\big(M^{t_l}\big)^{t_m} = \big(M^{t_m}\big)^{t_l}\end{gathered}$$ for any distinct $l$ and $m$. The relation of the partial transposes to the usual transpose is described by the equation $$\begin{gathered}
\big(\ldots \big(M^{t_1}\big)^{t_2} \ldots \big)^{t_n} = M^t.\end{gathered}$$
Given $0 < k_1, k_2 \le n$, let $$\begin{gathered}
V_1 = ((V_1)_1, (V_1)_2, \ldots, (V_1)_{k_1})\end{gathered}$$ and $$\begin{gathered}
V_2 = ((V_2)_1, (V_2)_2, \ldots, (V_2)_{k_2})\end{gathered}$$ be a $k_1$-tuple and a $k_2$-tuple of vector spaces, $i_1 = ((i_1)_1, (i_1)_2, \ldots, (i_1)_{k_1})$ and $i_2 = ((i_2)_1, (i_2)_2, \allowbreak \ldots, (i_2)_{k_2})$ be a $k_1$-tuple and a $k_2$-tuple of distinct positive integers from the interval ${[1 \, . \, . \, n]}$. Let $M_1 \in \operatorname{End}\big((V_1)^\otimes\big)$ and $M_2 \in \operatorname{End}\big((V_2)^\otimes\big)$. Assume that $V_{1 l} = W_{(i_1)_l}$ for all $l \in {[1 \, . \, . \, k_1]}$ and $V_{2 l} = W_{(i_2)_l}$ for all $l \in {[1 \, . \, . \, k_2]}$, and define the corresponding operators $(M_1)^{i_1} \in \operatorname{End}\big(W^\otimes\big)$ and $(M_2)^{i_2} \in \operatorname{End}\big(W^\otimes\big)$.
Now, if $\operatorname{range}i_1 \cap \operatorname{range}i_2 = \{ l \}$ we have $$\begin{gathered}
\big((M_1)^{i_1} (M_2)^{i_2}\big)^{t_l} = \big((M_2)^{i_2}\big)^{t_l} \big((M_1)^{i_1}\big)^{t_l}. \label{mtmt}\end{gathered}$$ Furthermore, $$\begin{gathered}
\big((M_1)^{i_1} (M_2)^{i_2}\big)^{t_l} = (M_1)^{i_1} \big((M_2)^{i_2}\big)^{t_l} \label{mmt}\end{gathered}$$ if $l$ does not belong to the range of $i_1$, and $$\begin{gathered}
\big((M_1)^{i_1} (M_2)^{i_2}\big)^{t_l} = \big((M_1)^{i_1}\big)^{t_l} (M_2)^{i_2} \label{mtm}\end{gathered}$$ if $l$ does not belong to the range of $i_2$.
Partial trace {#a:ptr}
-------------
Let again $W = (W_i)_{i \in {[1 \, . \, . \, n]}}$ be an $n$-tuple of vector spaces. Given $l \in {[1 \, . \, . \, n]}$, denote by $V$ the $(n - 1)$-tuple defined as $$\begin{gathered}
V_i = W_i, \quad i \in {[1 \, . \, . \, l - 1]},
\qquad V_i = W_{i + 1}, \quad i \in {[l \, . \, . \, n - 1]}.\end{gathered}$$ We define the partial trace $\operatorname{tr}_l$ with respect to $W_l$ as the mapping from $\operatorname{End}\big(W^\otimes\big)$ to $\operatorname{End}\big(V^\otimes\big)$ in the following evident way. Let $M$ be a monomial of the form (\[mm\]). We define $$\begin{gathered}
\operatorname{tr}_l M = \operatorname{tr}M_l (M_1 \otimes \cdots \otimes M_{l - 1} \otimes M_{l + 1} \otimes \cdots \otimes M_n),\end{gathered}$$ and extend this definition to the case of an arbitrary $M \in \operatorname{End}\big(V^\otimes\big)$ by linearity. If a vector space $V$ belongs to the range of $W$ and there is only one $i$ such that $V = W_i$ we also write $\operatorname{tr}_V W$ instead of $\operatorname{tr}_i W$.
Let $N$ be an element of $V$ and $V = W_i$, then for any $M \in \operatorname{End}(W)$ one has $$\begin{gathered}
\operatorname{tr}_i \big(M N^i\big) = \operatorname{tr}\big(N^i M\big).\end{gathered}$$
One can demonstrate that[^9] $$\begin{gathered}
\operatorname{tr}_l \operatorname{tr}_k = \operatorname{tr}_k \operatorname{tr}_{l + 1}\end{gathered}$$ for $l \ge k$, and[$$\begin{gathered}
\operatorname{tr}_l \operatorname{tr}_k = \operatorname{tr}_{k - 1} \operatorname{tr}_l\end{gathered}$$ for $l < k$.]{}
Let $V_1$, $V_2$ be two vector spaces, and $M_1 \in \operatorname{End}\big(V_1 \otimes W^\otimes\big)$, $M_2 \in \operatorname{End}\big(V_2 \otimes W^\otimes\big)$. Define two $(n + 1)$-tuples of vector spaces, $W_1$ defined by the rules $$\begin{gathered}
(W_1)_1 = V_1, \qquad (W_1)_i = W_{i - 1}, \quad i \in {[2 \, . \, . \, n + 1]},\end{gathered}$$ and $W_2$ defined by the rules $$\begin{gathered}
(W_2)_1 = V_2, \qquad (W_2)_i = W_{i - 1}, \qquad i \in {[2 \, . \, . \, n + 1]}.\end{gathered}$$ It is clear that $M_1 \in \operatorname{End}\big((W_1)^\otimes\big)$ and $M_2 \in \operatorname{End}\big((W_2)^\otimes\big)$. Further, let $\widetilde W$ be an $(n + 2)$-tuple of vector spaces given by the equations $$\begin{gathered}
\widetilde W_1 = V_1, \qquad \widetilde W_2 = V_2, \qquad \widetilde W_i = W_{i + 2}, \qquad i \in {[3 \, . \, . \, n + 1]}.\end{gathered}$$ Consider two elements of $\operatorname{End}\big(\widetilde W^\otimes\big)$ defined as $$\begin{gathered}
\widetilde M_1 = M_1^{1, 3, \ldots, n + 2}, \qquad
\widetilde M_2 = M_2^{2, 3, \ldots, n + 2}.\end{gathered}$$ One can see that $$\begin{gathered}
\operatorname{tr}_1 \operatorname{tr}_2 \big(\widetilde M_1 \widetilde M_2\big) = (\operatorname{tr}_1 M_1) (\operatorname{tr}_1 M_2).\end{gathered}$$
Finally, we give some examples of interplay between partial traces and partial transposes. First of all, one has $$\begin{gathered}
\operatorname{tr}_i \big(M^{t_i} N^{t_i}\big) = \operatorname{tr}_i (M N)\end{gathered}$$ for any $M, N \in \operatorname{End}\big(W^\otimes\big)$. Further, $$\begin{gathered}
\operatorname{tr}_i M^{t_j} = (\operatorname{tr}_i M)^{t_{j - 1}}\end{gathered}$$ for all $M \in W^\otimes$ and $i < j$, and $$\begin{gathered}
\operatorname{tr}_i M^{t_j} = (\operatorname{tr}_i M)^{t_j}\end{gathered}$$ for all $M \in W^\otimes$ and $i > j$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was supported in part by the Russian Foundation for Basic Research grant \# 16-01-00473. KhSN was also supported by the DFG grant \# BO3401/31 and by the Russian Academic Excellence Project ‘5-100’; results obtained in Section \[section3\] were funded by the HSE Faculty of Mathematics. We thank our colleagues and coauthors H. Boos, F. Göhmann and A. Klümper for numerous fruitful discussions. AVR thanks the Max Plank Institute for Mathematics in Bonn, where this work was finished, for the warm hospitality.
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[^1]: The notation $R_{\varphi_1 | \varphi_2}(\zeta_1 | \zeta_2)$ is also used.
[^2]: The relation to the usual $R$-operators can be understood from the results of Section \[s:dd\].
[^3]: We denote by $V^\star$ the restricted dual space of $V$, see Section \[s:cr\]. If $V$ is finite-dimensional $V^\star$ coincides with the usual dual space.
[^4]: This is a common situation in the quantum theory of integrable spin chains.
[^5]: We use capital letters to distinguish between generators of ${{\mathrm U}_q(\mathfrak{gl}_{l + 1})}$ and ${{\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))}$.
[^6]: See the comment to Figs. \[f:irrwi\] and \[f:rirwi\] on p. .
[^7]: See again the comment to Figs. \[f:irrwi\] and \[f:rirwi\] on p. .
[^8]: We assume that the vector spaces under consideration are ${{\mathrm U}_q(\mathcal L(\mathfrak g))}$-modules in the category ${\mathcal O}$ and denote by $V^\star$ the restricted dual of $V$, see Section \[s:cr\].
[^9]: We draw the reader’s attention to the partial shift in the numbering of the factors of the tensor product after taking the trace.
|
---
abstract: 'We study the transverse momentum ($p_T$) spectrum of charged particles produced in deep inelastic scattering (DIS) at small Bjorken $x$ in the central region between the current jet and the proton remnants. We calculate the spectrum at large $p_T$ with the BFKL $\ln (1/x)$ resummation included and then repeat the calculation with it omitted. We find that data favour the former. We normalize our BFKL predictions by comparing with HERA data for DIS containing a forward jet. The shape of the $x$ distribution of DIS + jet data are also well described by BFKL dynamics.'
---
plus 2mm minus 2mm 23.0cm 17.0cm -1.0in -42pt
DTP/97/56\
July 1997\
\
J. Kwiecinski[^1], S.C. Lang, A.D. Martin
Department of Physics, University of Durham, Durham, DH1 3LE, UK.
[**1. Introduction**]{}\
An intriguing feature of the measurements at HERA in the small $x$ domain is the possible existence of significant $\ln (1/x)$ effects. A major part of the rise observed for the structure function $F_2$ with decreasing $x$ may be attributed to the resummation of the leading $\ln (1/x)$ BFKL’ [@BFKL] contributions. An excellent unified BFKL/GLAP fit of $F_2$ in the HERA regime has recently been obtained using a flat in $x$" input [@STASTO], and the rise due to BFKL-type effects has been quantified within this description. However, the growth of $F_2$ with decreasing $x$ can be described equally well by pure GLAP [@DGLAP] $\ln (Q^2)$ evolution from suitably chosen input parton distributions so the main origin of the rise is still an open question. The observable $F_2$ is too inclusive to distinguish between these alternatives. The study of deep inelastic scattering (DIS) events containing an isolated forward jet [@M; @KMS1] is a better discriminator of the underlying small $x$ dynamics. The process is sketched in Fig. 1(a). In this case we effectively study DIS off known parton distributions and so we avoid the ambiguity in the choice of the input distributions. The method is theoretically attractive. The summation of the leading $\ln
(1/x)$ contributions gives an $(x/x_j)^{-\lambda}$ behaviour of the BFKL ladder connecting the photon to parton $a$. Here $x$ is Bjorken $x$ and $x_j$ is the fraction of the proton’s longitudinal momentum carried by the parton jet. An unambiguous measurement of the exponent $\lambda$ looks feasible. In practice a major problem is the identification of the jet due to parton $a$, and the measurement of its momentum, when it is close to the remnants of the proton. Typically the clean observation of the jet requires $x_j \lower .7ex\hbox{$\;\stackrel{\textstyle <}{\sim}\;$} 0.1$ and so in this process we lose about a factor of 10 in the small $x$ reach’ of HERA.
Besides the $x^{-\lambda}$ growth as $x$ decreases along the BFKL ladder, a second characteristic feature is the diffusion in $\ln k_T^2$ where $k_T$ are the transverse momenta of the gluons emitted along the chain. One way the diffusion manifests itself is in an enhancement of the transverse energy $(E_T)$ flow in the central region between the current jet and the proton remnants [@gkms], see Fig. 1(b). In principle the diffusion can enhance $E_T$ from both the upper’ and lower’ BFKL gluon ladders, which are denoted by $\Phi$ and $f$ in Fig. 1(b). However, the $x$ reach at HERA is insufficient to fully develop the $\ln k_T^2$ diffusion in both ladders simultaneously. Nevertheless, the effect is quite appreciable giving at the [*parton*]{} level an energy flow $E_T
\lower .7ex\hbox{$\;\stackrel{\textstyle <}{\sim}\;$} 2$ GeV/unit of rapidity. However the clean parton level prediction can in practice be masked or mimicked by the effects of hadronization. Thus, although the prediction for $E_T$ is in agreement with observations [@ETflow] we cannot definitely conclude that it is due to $\ln (1/x)$ resummations.
An interesting way to overcome this ambiguity is to consider the emission of single particles at relatively large transverse momentum $p_T$ in the central region [@MK]. The single particle spectrum at sufficiently large values of $p_T$ should be much more immune from hadronization and more directly reflect the $\ln k_T^2$ diffusion from the BFKL ladders.
The outline of the contents of the paper is as follows. In Sec. 2 we use the data for the process DIS + forward jet to normalise the BFKL function $\Phi$ shown in Fig. 1(a). To be precise we numerically solve the BFKL equation for $\Phi$ using the amplitude $\Phi^{(0)}$ for the quark box (and crossed box) as input at a value $z_0$ of $z = x/x_j$ which is chosen so that the resulting $\Phi$ reproduces the DIS + jet data. Also, for completeness, we present in Sec. 2, an analytic form for $\Phi$ which is valid for fixed $\alpha_S$, and which has been the basis for a recent analysis. In Sec. 3 we give the formula necessary to calculate the transverse momentum $(p_T)$ spectrum of single particles produced in the central region. The process is shown in Fig. 1(c). The predictions for the $p_T$ spectra (with and without the BFKL effects included) are compared with HERA data. Finally in Sec. 4 we give our conclusions.\
[**2. DIS + forward jet events**]{}\
We first calculate the cross section for DIS containing a forward identified jet. This so-called Mueller" process is a valuable probe of small $x$ dynamics in its own right. We compare with HERA data to normalise the function $\Phi$ shown in Fig. 1(a). There are uncertainties in the normalisation, and even the shape of the $x$ distribution is dependent on subleading $\ln
(1/x)$ corrections.
The variables of the process are shown in Fig. 2. As usual the variables $x$ and $y$ are given by $x = Q^2/2p \cdot q$ and $y =
p \cdot q / p_e
\cdot p$ where $p$, $p_e$ and $q$ denote the four momenta of the proton, the incident electron and the virtual photon respectively, and $Q^2 \equiv -q^2$. The variables $x_j$ and $k_{jT}$ are the longitudinal momentum fraction and transverse momentum carried by the forward jet. The differential cross section is given by [@M] $$\frac{\partial \sigma_{j}}{\partial x \partial Q^{2}} = \int
dx_{j}
\int dk_{jT}^{2} \frac{4 \pi \alpha ^{2}}{xQ^{4}} \left[
\left(1-y \right) \frac{\partial F_{2}}{\partial x_{j} \partial
k_{jT}^{2}} + \frac{1}{2} y^{2} \frac{\partial F_{T}}{\partial
x_{j} \partial k_{jT}^{2}} \right]
\label{eq:a2}$$ where the differential structure functions have the following form $$\frac{\partial^2 F_{i}}{\partial x_{j} \partial k_{jT}^{2}} =
\frac{3 \alpha_S (k_{jT}^{2}) }{\pi k_{jT}^2}
\sum_{a} f_a \left(x_{j},k_{jT}^{2} \right) \Phi_i
\left(\frac{x}{x_{j}},k_{jT}^{2},Q^{2} \right)
\label{eq:a3}$$ for $i=T,L$ and $F_2=F_T+F_L$. We have assumed strong ordering at the parton $a$ - gluon vertex. Assuming also $t$-channel pole dominance the sum over the parton distributions is given by $$\sum_{a} f_{a} = g+\frac{4}{9} \sum_q \left(q+ \bar q \right).
\label{eq:a4}$$ Recall that these parton distributions are to be evaluated at $(x_{j},k_{jT}^2)$ where they are well-known from the global analyses, so there are no ambiguities arising from a non-perturbative input.\
The functions $\Phi_i (x/x_j, k_{jT}^{2}, Q^2)$ describe the virtual $\gamma$ + virtual gluon fusion process including the ladder formed from the gluon chain of Fig. 2. They can be obtained by solving the BFKL equations $$\Phi_i (z, k_T^2, Q^2) = \Phi_i^{(0)} (z, k_T^2, Q^2)+$$ $$\overline{\alpha}_S \int_z^1{dz^{\prime}\over
z^{\prime}} \int {d^2 q\over \pi q^2}
\left[\Phi_i(z^{\prime},(\mbox{\boldmath $q$}+
\mbox{\boldmath $k_T$})^2,Q^2))-\Phi_i(z^{\prime},k_T^2,Q^2)
\Theta(k_T^2-q^2)\right]
\label{eq:a6}$$ where $\overline{\alpha}_S \equiv 3 \alpha_S/\pi$. The inhomogeneous or driving terms $\Phi_{i}^{(0)}$ correspond to the sum of the quark box and crossed-box contributions. For small $z$ we have $$\Phi_i^{(0)} (z, k_T^2, Q^2) \; \approx \; \Phi_i^{(0)} (z = 0,
k_T^2, Q^2) \; \equiv \; \Phi_i^{(0)} (k_T^2, Q^2).
\label{eq:aa66}$$ We evaluate the $\Phi_i^{(0)}$ by expanding the four momentum in terms of the basic light-like four momenta $p$ and $q^\prime \equiv q + xp$. For example, the quark momentum ${\kappa}$ in the box (see Fig. 2) has the Sudakov decomposition $$\kappa \; = \; \alpha p \: - \: \beta q^\prime \: + \:
\mbox{\boldmath $\kappa$}_T.$$ We carry out the integration over the box diagrams, subject to the quark mass-shell constraints, and find $$\begin{aligned}
\label{eq:a7}
\Phi_T^{(0)} (k_T^2, Q^2) & = & 2 \: \sum_q \: e_q^2 \:
\frac{\alpha _S}{4 \pi^2} \: \frac{Q^2}
{k_T^2} \: \int_0^1 \: d \beta \: \int \:
d^2 \kappa_T \nonumber \\
& & \nonumber \\
& & \left\{ \left[\beta^2 \: + \: (1 - \beta)^2 \right]
\left(\frac{\kappa_T^2}{D_1^2} \: - \: \frac{\mbox{\boldmath
$\kappa$}_T .
(\mbox{\boldmath $\kappa$}_T - \mbox{\boldmath $k$}_T)}{D_1 D_2}
\right ) \: + \: m_q^2 \left ( \frac{1}{D_1^2} - \frac{1}{D_1
D_2} \right) \right\} \\
& & \nonumber \\
\Phi_L^{(0)} (k_T^2, Q^2) & = & 2 \sum_q \: e_q^2 \;
\frac{\alpha_S}{\pi^2} \frac{Q^4}{k_T^2}
\: \int_0^1 \: d \beta \: \int \:
d^2 \kappa_T \: \beta^2 (1 - \beta)^2 \; \left ( \frac{1}{D_1^2}
\:
- \: \frac{1}{D_1 D_2} \right ). \nonumber \end{aligned}$$ where the denominators $D_{i}$ are of the form $$\begin{aligned}
D_1 & = & \kappa_T^2 \: + \: \beta (1 - \beta) \: Q^2 + m_q^2
\nonumber \\
& & \\
D_2 & = & (\mbox{\boldmath $\kappa$}_T - \mbox{\boldmath
$k$}_T)^2 \:
+ \: \beta (1 - \beta) \: Q^2 + m_q^2 \nonumber.
\label{eq:a8}\end{aligned}$$ The light $u$, $d$ and $s$ quarks are taken to be massless $(m_q
= 0)$ and the charm quark to have mass $m_c = 1.4$ GeV.\
[**2.1 Analytic form of $\Phi$ for fixed $\mbox{\boldmath$\alpha_S$}$**]{}\
We solve the BFKL equation for $\Phi$ numerically, which allows the use of running $\alpha_S$ and the inclusion of a charm quark mass. However, it is informative to recall the analytic solution which can be obtained if $\alpha_S$ is fixed and we assume that the quarks are massless. The first step is to rewrite the driving terms (\[eq:a7\]) for $m_q = 0$ in the form $$\begin{aligned}
\label{eq:b1}
\Phi_T^{(0)} (k_T^2, Q^2) \; = \; \sum_q \: e_q^2 \:
\frac{\alpha_S}{4 \pi} \: Q^2 \int_0^1 d \lambda \int_0^1 d \beta
\frac{[\beta^2 + (1 - \beta)^2] \: [\lambda^2 + (1 -
\lambda)^2]}{[\lambda (1 - \lambda) k_T^2 + \beta (1 -
\beta) Q^2]}\end{aligned}$$ $$\begin{aligned}
\label{eq:b2}
\Phi_L^{(0)} (k_T^2, Q^2) \; = \; \sum_q e_q^2 \: \frac{2
\alpha_S}{\pi} \: Q^2 \int_0^1 d \lambda \int_0^1 d \beta
\frac{\lambda (1 - \lambda) \beta (1 - \beta)}{[\lambda (1 -
\lambda) k_T^2 + \beta (1 - \beta) Q^2]}\end{aligned}$$ where $\lambda$ is the Feynman parameter which appears in the representation $$\label{eq:b3}
\frac{1}{D_1 D_2} \; = \; \int_0^1 d \lambda \: \frac{1}{[\lambda
D_1 + (1 - \lambda) D_2]^2}.$$ We see that, for fixed $\alpha_S$ and $m_q = 0$, the $\Phi_i^{(0)}$ are functions of a single dimensionless variable $r = Q^2 / k_T^2$. We may therefore represent the driving terms $\Phi_i^{(0)} (Q^2 / k_T^2)$ in terms of their Mellin transforms $\tilde{\Phi}_i^{(0)} (\gamma)$ $$\label{eq:b4}
\Phi_i^{(0)} (r) \; = \; \frac{1}{2 \pi i} \int_{\frac{1}{2} - i
\infty}^{\frac{1}{2} + i \infty} \; d \gamma \;
\tilde{\Phi}_i^{(0)}
(\gamma) r^{\gamma}$$ where $i = L,T$ and $r \equiv Q^2 / k_T^2$. The Mellin transform is useful since it diagonalizes the BFKL equation (\[eq:a6\]). The solutions for fixed coupling $\alpha_S$ may therefore be written $$\begin{aligned}
\label{eq:b5}
\Phi_i (z, k_T^2, Q^2) \; = \; \frac{1}{2 \pi i}
\int_{\frac{1}{2} - i \infty}^{\frac{1}{2} + i \infty} \; d
\gamma \left(\frac{Q^2}{k_T^2} \right)^{\gamma} {\rm exp}
(\overline{\alpha}_S K(\gamma) \ln \frac{1}{z}) \:
\tilde{\Phi}_i^{(0)} (\gamma)\end{aligned}$$ where $\overline{\alpha}_S \equiv 3 \alpha_S/\pi$ and $K (\gamma)$ is the Mellin transform of the kernel of the BFKL equation $$\label{eq:b6}
K (\gamma) \; = \; 2 \Psi (1) - \Psi (\gamma) - \Psi (1
- \gamma)$$ with $\Psi (\gamma) \equiv \Gamma^{\prime} (\gamma) / \Gamma
(\gamma)$. The functions $\tilde{\Phi}_i^{(0)} (\gamma)$ are obtained by inserting (\[eq:b1\]) and (\[eq:b2\]) into the inverse relation to (\[eq:b4\]). We find $$\begin{aligned}
\label{eq:b7}
\tilde{\Phi}_T^{(0)} (\gamma) & = & \sum_q e_q^2 \:
\frac{\alpha_S}{4 \pi} \: \int_0^{\infty} \: dr \: r^{- \gamma}
\:
\int_0^1 d \lambda \int_0^1 d \beta \: \frac{[\beta^2 + (1 -
\beta)^2] [\lambda^2 + (1 - \lambda)^2]}{[\lambda (1 - \lambda) +
\beta (1 - \beta) r]} \nonumber \\
& = & \sum_q e_q^2 \: \frac{\alpha_S}{\sin \pi \gamma} \: B \:
(\gamma +
2, \gamma) B (3 - \gamma, 1 - \gamma)\end{aligned}$$ $$\begin{aligned}
\label{eq:b8}
\tilde{\Phi}_L^{(0)} (\gamma) & = & \sum_q e_q^2 \: \frac{2
\alpha_S}{\pi} \int_0^{\infty} dr \: r^{- \gamma} \int_0^1 d
\lambda
\int_0^1 d \beta \: \frac{\lambda (1 - \lambda) \beta (1 -
\beta)}{[\lambda (1 - \lambda) + \beta (1 - \beta) r]} \nonumber
\\
& = & \sum_q e_q^2 \: \frac{2 \alpha_S}{\sin \pi \gamma} \: B
(-
\gamma + 2, - \gamma + 2) B (\gamma + 1, \: \gamma + 1)\end{aligned}$$ where $B(x,y) \equiv \Gamma(x) \Gamma(y) / \Gamma(x+y)$. The derivation of the analytic formula relies on $\alpha_S$ being fixed. This approach has been used by Bartels et al. [@BARTELS] to estimate the DIS + forward jet cross section taking the coupling $\alpha_S (k_T^2)$ in formulae (\[eq:b5\]). The prediction has the general shape of the H1 data as a function of $x$, but the calculated cross section exceeds the data by some 20% [@H1data].
In the $z \rightarrow 0$ limit the formulae reduce to the conventional $z^{- \lambda}$ BFKL behaviour $$\begin{aligned}
\label{eq:b9}
\Phi_i (z, k_T^2, Q^2) \sim z^{- \overline{\alpha}_S K
(\frac{1}{2})} \left(\frac{Q^2}{k_T^2} \right)^{\frac{1}{2}}
\; \frac{\tilde{\Phi}_i^{(0)} (\gamma =
\frac{1}{2})}{(\overline{\alpha}_S K^{\prime \prime}(\frac{1}{2})
\ln
1/z)^{\frac{1}{2}}}\end{aligned}$$ where for simplicity we have omitted the Gaussian diffusion factor in $\ln (k_T^2 / Q^2)$. If we evaluate the various functions at $\gamma = \frac{1}{2}$ we obtain $$\begin{aligned}
\label{eq:b10}
\Phi_T (z, k_T^2, Q^2) & = & \frac{9 \pi^2}{512} \;
\frac{2
\sum e_q^2 \: \alpha_S^{\frac{1}{2}}}{\sqrt{21 \zeta (3) / 2}}
\left(\frac{Q^2}{k_T^2} \right)^{\frac{1}{2}} \: \frac{z^{-
\lambda}}{\sqrt{\ln (1/z)}} \left[1 + O \left(\frac{1}{\ln (1/z)}
\right) \right] \nonumber \\
\Phi_L (z, k_T^2, Q^2) & = &
\frac{2}{9} \: \Phi_T (z, k_T^2, Q^2)\end{aligned}$$ where $\lambda = \overline{\alpha}_S \: K (\frac{1}{2}) =
\overline{\alpha}_S \: 4 \ln 2$.\
[**2.2 Normalisation of $\Phi$ and the description of DIS + jet data**]{}\
Our calculation of the DIS + forward jet process differs from that of ref. [@BARTELS] in that we numerically solve the BFKL equations. Therefore, we are able to explicitly include the $m_c
\neq 0$ charm contribution. We also allow the coupling $\alpha_S$ to run. To be precise we solve the BFKL equation (\[eq:a6\]) rewritten in terms of the modified function $\overline{\alpha}_S (k_{jT}^2)
\Phi_i (z,k_T^2,Q^2)$ following the prescription that was used in ref. [@KMS1]. This choice of scale for $\alpha_S$ is consistent with the double logarithm limit and with the NLO $\ln (1/x)$ analysis of ref. [@CIAF].
We determine the functions $\Phi_i$ for $z < z_0$ by solving the BFKL equation as described in [@KMS1] starting from the boundary condition $$\label{eq:b11}
\Phi_i (z_0, k_T^2, Q^2) \; = \; \Phi_i^{(0)} (z_0,
k_T^2, Q^2) \: \approx \: \Phi_i^{(0)} (k_T^2, Q^2)$$ where $\Phi_i^{(0)} (k_T^2, Q^2)$ are the contributions of the quark box (and crossed box) given in (\[eq:a7\]). We take $u,d,s$ to be massless and the charm quark to have mass $m_c = 1.4 {\rm
GeV}$ in the summation over the quarks. We then use (\[eq:a2\]) integrated over $x$ and $Q^2$ and (\[eq:a3\]) to calculate the DIS + forward jet rate corresponding to the cuts used in the H1 measurement. That is the forward jet is constrained to the region $$7^{\circ} < \theta_j < 20^{\circ}, \quad E_j > 28.7 {\rm GeV},
\quad
k_{jT} > 3.5 {\rm GeV},$$ whereas the outgoing electron must lie in the domain $$160^{\circ} < \theta_e^{\prime} < 173^{\circ}, \quad
E_e^{\prime} > 11 {\rm GeV}, \quad y > 0.1$$ in the HERA frame. Finally H1 require $\frac{1}{2} Q^2 <
k_{jT}^2 < 2Q^2$. The BFKL calculation is compared with the data [@H1data] in bins of size $\Delta x \: = \: 5 \times
10^{-4}$ in Fig. 3. The parameter $z_0$ is adjusted to give a satisfactory normalization of the calculation. We find that the H1 data require $z_0 = 0.15$. The predicted shape of the distribution is in good agreement with the data.\
[**3. Single particle $\mbox{\boldmath$p_T$}$ spectra**]{}
We first use Fig. 1(c) to obtain the differential cross section for the production of a hadron of transverse momentum $p_T$ and longitudinal momentum fraction $x_h$. Then we calculate the charged particle spectra relevant to the recent observations at HERA [@H1ptdata].\
[**3.1 The cross section for charged particle production**]{}
The cross-section for single particle production is obtained by convoluting the inclusive cross-section for the production of a single parton with the parton fragmentation function. The differential cross section for the inclusive production of a single parton of longitudinal momentum fraction $x_j$ and transverse momentum $k_{jT}$ has the generic form of (\[eq:a2\]). We have $$\begin{aligned}
\label{eq:b12}
\frac{\partial \sigma_j}{\partial x_j \: \partial k_{jT}^2 \:
\partial x \: \partial Q^2} \; = \; \frac{4 \pi \alpha^2}{x Q^4}
\left[ (1 - y) \frac{\partial F_2}{\partial x_j \: \partial
k_{jT}^2} \: + \: \frac{1}{2} \: y^2 \: \frac{\partial
F_T}{\partial x_j \: \partial k_{jT}^2} \right].\end{aligned}$$
Now for small $x$, and in the central region away from the current jet and the proton remnants, we expect gluonic partons to dominate where the gluons are radiated within the BFKL ladder. The differential structure functions occurring in (\[eq:b12\]) are then given by $$\begin{aligned}
\label{eq:b13}
x_j \: \frac{\partial F_i}{\partial x_j \: \partial k_j^2}
\; = \; \int \frac{d^2 k_p}{\pi k_p^4} \int \frac{d^2
k_{\gamma}}{k_{\gamma}^2} \left[ \frac{\overline{\alpha}_S
(k_j^2) k_p^2 k_{\gamma}^2}{k_j^2} \right] f (x_j, k_p^2) \:
\Phi_i \left(\frac{x}{x_j}, k_{\gamma}^2, Q^2 \right) \delta^2
(k_j - k_p - k_{\gamma})\end{aligned}$$ with $i = T,L$ and where for simplicity we have omitted the subscript $T$ from the gluon transverse momenta, $k_{jT}, \:
k_{pT}$ and $k_{\gamma T}$, see Fig. 4. The functions $\Phi_i$ are those of Sec. 2 which control the DIS + forward jet rate, while $f$ is the unintegrated gluon distribution which satisfies the BFKL equation $$\begin{aligned}
\label{eq:b14}
-z \: \frac{\partial f (z, k^2)}{\partial z} \; = \;
\overline{\alpha}_S \int \frac{d^2 q}{\pi q^2}
\left[\frac{k^2}{(\mbox{\boldmath $q$} + \mbox{\boldmath
$k$})^2} \: f (z, (\mbox{\boldmath $q$} + \mbox{\boldmath
$k$})^2) \: - \: f (z, k^2) \Theta (k^2 - q^2)\right].\end{aligned}$$ The expression in square brackets in (\[eq:b13\]) arises from the (square of the) BFKL vertex for real gluon emission, see Fig. 4.
In practice we evolve (\[eq:b14\]) down in $z$ from the boundary condition $$\label{eq:b15}
f (\overline{z}_0, k^2) \; = \; f^{AP} (\overline{z}_0, k^2) \; =
\; \frac{\partial \left[ \overline{z}_0 \: g^{AP}
(\overline{z}_0, k^2) \right]}{\partial \ln (k^2/k_0^2)}$$ where here $z = \overline{z}_0$ with $\overline{z}_0 = 10^{-2}$, and where $g^{AP}$ is the conventional gluon distribution obtained from a global set of partons. As before we allow the coupling to run, that is we take $\alpha_S (k^2)$ in (\[eq:b14\]). Moreover, we impose an infrared cut-off $k_0^2 = 1 {\rm GeV}^2$. That is we require the arguments of $f$ to satisfy $k^2 > k_0^2$ and $(\mbox{\boldmath $k$} + \mbox{\boldmath $q$})^2 >
k_0^2$. Similarly, the integrations in (\[eq:b13\]) are restricted to the regions $k_p^2, \: k_{\gamma}^2 >
k_0^2$. We may include the contribution $\Delta F_i$ from the region $k_p^2 < k_0^2$ by assuming the strong ordering approximation, $k_p^2 \ll k_{\gamma}^2 \sim
k_j^2$, at the gluon vertex. This contribution to (\[eq:b13\]) then becomes $$\begin{aligned}
\label{eq:b16}
x_j \: \frac{\partial (\Delta F_i)}{\partial x_j \partial
k_j^2}
& = & \overline{\alpha}_S (k_j^2) \int^{k_0^2}
\frac{d k_p^2}{k_p^2} \: f (x_j, k_p^2) \; \Phi_i
\left( \frac{x}{x_j}, \: k_j^2, \: Q^2 \right) \nonumber \\
& = &
\overline{\alpha}_S (k_j^2) \: \frac{x_j g (x_j, k_0^2)}{k_j^2}
\; \Phi_i \left( \frac{x}{x_j}, k_j^2, Q^2 \right).\end{aligned}$$ Most of the time, however, for the calculation relevant to the HERA data, the variable $x_j$ is not small enough for the BFKL equation to be applicable for the function $f$. In these cases, that is when $x_j > \overline{z}_0$, we therefore again assume strong ordering $k_p^2 \ll k_{\gamma}^2 \sim k_j^2$. In addition we include the contributions from quark and antiquark jets. We then obtain $$\label{eq:b17}
x_j \; \frac{\partial F_2}{\partial x_j \partial k_j^2} \; = \;
\overline{\alpha}_S (k_j^2) \; \frac{x_j \left[ g + \frac{4}{9}
\sum_q \left( q + \bar{q} \right) \right] }{k_j^2}
\; \Phi_i \left(\frac{x}{x_j}, \: k_j^2, \: Q^2 \right).$$ where the parton distributions are to be evaluated at $(x_j,
k_j^2)$ The differential cross section for single particle $(h)$ production is obtained by convoluting the jet cross section with the fragmentation functions $D$ for the parton $\rightarrow h$ transition $$\begin{aligned}
\label{eq:b18}
\frac{\partial \sigma_h}{\partial x_h \partial p_T^2 \partial x
\partial Q^2} & = & \int_{x_h}^1 \; dz \; \int \; dx_j \; \int
\; dk_j^2 \: \delta (x_h - z x_j) \delta (p_T - zk_j) \left\{
\frac{\partial \sigma_g}{\partial x_j \partial k_j^2
\partial x \partial Q^2} \: D_g^h (z, \mu^2) \right. \nonumber \\
& + & \left. \frac{4}{9} \sum_q \left[ \frac{\partial
\sigma_q}{\partial x_j \partial
k_j^2 \partial x \partial Q^2} D_q^h (z, \mu^2) \; + \;
\frac{\partial \sigma_{\overline{q}}}{\partial x_j \partial k_j^2
\partial x \partial Q^2} D_{\overline{q}}^h (z, \mu^2) \right]
\right\}\end{aligned}$$ where $\sigma_g$ and $\sigma_q$ are the contributions to the cross section $\sigma_j$ for gluon and quark and antiquark jets respectively. The fragmentation scale $\mu^2$ is of the order of $k_j^2$. The cross section for charged particle production is obtained by summing over all possible charged hadrons $h$.\
[**3.2 Predictions for the single particle $\mbox{\boldmath$p_T$}$ spectra**]{}
The data for the single (charged) particle $p_T$ spectra are presented in the form $(dn/dp_T)/N$ where $n$ is the multiplicity and $N$ the total number of charged particles in a given $x, Q^2$ bin [@H1data]. To calculate this $p_T$ spectrum we evaluate $$\label{eq:b19}
\frac{1}{N} \: \frac{dn}{dp_T} \: = \: \left( \sum_h
\frac{\partial
\sigma_h}{\partial p_T \: \partial x \: \partial Q^2} \right)
\left/
\frac{\partial \sigma_{tot}}{\partial x \: \partial Q^2} \right. .$$ where $\partial \sigma_h / \partial p_T \partial x \partial Q^2$ is obtained from (\[eq:b18\]) by integrating over $x_h$. We take the central values of $x, Q^2$ in the bin. The integration limits are fixed by the limits on the pseudorapidity interval under consideration. To be precise we use $$\label{eq:b20}
x_h \: = \: \sqrt{\frac{x}{Q^2}} \: p_T \: e^{-\eta}$$ where $\eta$ is the pseudorapidity of the charged particle, $\eta = - \ln \tan (\theta / 2)$ with $\theta$ the angle with respect to the virtual photon direction. Finally we calculate the total differential cross section $\partial \sigma_{tot} / \partial x \partial Q^2$ in (\[eq:b19\]) from the structure functions $F_2$ and $F_L$ given by the MRS(R2) [@mrs] set of parton distributions.
Our aim is to make an absolute BFKL-based prediction to compare with the $p_T$ spectra observed by the H1 collaboration. There is, however, an inherent uncertainty in the normalisation due to the imposition of an infrared cut-off on the BFKL transverse momentum integrations (or due to other possible treatments of the non-perturbative region). To overcome this problem we follow the procedure described in Sec. 2.2 and fix the parameters occurring in the calculation of the BFKL functions $\Phi_i$ by requiring the prediction for the DIS + jet cross section to give the correct normalization of the H1 forward jet measurements. The next step is to use the functions $\Phi_i$ obtained in this way in the computation of the differential structure functions from (\[eq:b13\]), (\[eq:b16\]) and (\[eq:b17\]). In this way we are able to calculate a normalized $p_T$ spectrum from (\[eq:b19\]).
The BFKL prediction for the single particle spectra may be compared with the result which would be obtained if the BFKL gluon radiation is neglected. That is in (\[eq:b13\]), (\[eq:b16\]) and (\[eq:b17\]) we replace the functions $\Phi_i$ which describe the solution of the BFKL equation with the boundary condition given by the quark box with the quark box $\Phi_i^{(0)}$ only. In addition we now also assume strong ordering for $x_j < \overline{z}_0$ and carry out the $k_p^2$ integration in (\[eq:b13\]). This amounts to assuming that in a fixed-order treatment the dominant subprocess is $\gamma g \rightarrow q \overline{q} g$. In our calculation the $\kappa$ integration is infrared finite since we allow for the virtuality of the incoming and exchanged gluons.
So we are now in the position to give a BFKL prediction for the single particle spectra which can be compared with the H1 data. In their measurement the H1 collaboration collected data in nine different kinematic bins in two pseudorapidity intervals. We will focus on the three smallest $x$ bins where BFKL effects should become visible. Also we will only show results for the lower pseudorapidity interval, $0.5 < \eta < 1.5$, where we expect no contamination due to the fragmentation of the current jet which has not been included in the calculation. In the computation of the $p_T$ spectra we use $$E_e = 27.5 {\rm GeV}, \quad E_p = 820 {\rm GeV}$$ and impose the cuts which where used in the H1 measurement, i.e. we require the outgoing electron to lie in the region $$157^{\circ} < \theta_e^{\prime} < 173^{\circ}, \quad
E_e^{\prime} > 12 {\rm GeV}, \quad y > 0.05$$ in the HERA frame. Also we subtract 10% off the total cross section $\sigma_{tot}$ to account for diffractive events with large rapidity gaps which have been excluded from the measurement. Finally, in the sum over the charged hadrons $h$ in (\[eq:b19\]) we include $\pi^{\pm}$ and $K^{\pm}$, and we use the next-to-leading order fragmentation functions by Binnewies et al. [@BKK]. In Fig. 5 we show predictions for the charged particle $p_T$ spectrum in kinematic bin 1 of the H1 analysis with central values $x = 1.6 \times 10^{-4}$ and $Q^2 = 7$ GeV$^2$. We compare the results when BFKL small $x$ resummation is included in the calculation with the case when gluon radiation is neglected. In both cases we demonstrate the effect of changing the fragmentation scale from $\mu^2 = k_j^2$ to $\mu^2 = (2k_j)^2$. We see that the BFKL prediction gives a good description of both the shape and the normalization [^2] of the H1 data. On the other hand, when the BFKL effects are neglected the predictions lie considerably below the data. Also we see, as expected, that the spectrum decreases more rapidly with $p_T$ than when the BFKL resummation is included. For example for $p_T = 1.5$ GeV the two predictions differ by a factor 3.6, whereas for $p_T = 6$ GeV this factor is almost 10. This is a reflection of the diffusion in $\ln k_T^2$ along the BFKL ladder.
The same general behaviour is seen in Figs. 6 and 7 where we show the comparison for kinematic bins 2 and 3, with central values $x = 2.9 \times 10^{-4}$, $Q^2
= 9$ GeV$^2$ and $x = 3.7 \times 10^{-4}$, $Q^2 = 13$ GeV$^2$, respectively. We find that in all three small $x$ bins of the H1 analysis the data strongly support the inclusion of BFKL resummation in the calculation of the $p_T$ spectra. Reasonable variations of the fragmentation scale do not allow for a description of the data when BFKL effects are neglected.\
[**4. Conclusion**]{}
We studied the DIS + forward jet process including massive charm in the quark box and solving the BFKL equation numerically for running coupling. We found that BFKL dynamics describe the shape of the $x$ distribution of the HERA data well. Next we used these data to fix the normalization of the solution of the BFKL equation with the boundary condition given by the quark box. This enabled us to give an absolute prediction for charged particle transverse momentum spectra at small $x$. We calculated the spectrum for large values of $p_T$ first including BFKL small $x$ resummation in the calculation and second neglecting gluon radiation. It turned out that the BFKL prediction agrees well with the H1 data both in shape and normalization, whereas the approximate fixed order result underestimates the data and decreases too rapidly with $p_T$. We therefore conclude that we found evidence for the existence of $\ln(1/x)$ effects and for the diffusion in $\ln k_T^2$ which accompanies BFKL evolution. Despite these encouraging results it would, however, still be useful to compare the BFKL prediction for the $p_T$ spectrum with the result of the complete fixed order calculation. Experimental data for higher values of $p_T$ would allow an even clearer distinction between the different predictions. BFKL effects would also become more apparent in the pseudorapidity interval $-0.5 < \eta < 0.5$ which corresponds to higher values of $x_j$ and therefore to a longer BFKL evolution starting from the quark box. Of course higher $x_j$ also means less BFKL evolution from the proton end. This is, however, not a disadvantage, since already for the pseudorapidity interval which we considered the main contribution to the spectrum comes from the region $x_j >
\overline{z}_0$. We conclude that although more experimental data especially for higher values of $p_T$ would be useful, the existing spectra show the presence of BFKL effects at small $x$ at HERA.\
[**Acknowledgements**]{}
We thank Michael Kuhlen and Erwin Mirkes for their help and encouragement. S.C.L. thanks the UK Engineering and Physical Sciences Research Council for financial support. This work has been supported in part by Polish State Committee for Scientific Research Grant No. 2 P03B 089 13, and by the EU under Contracts Nos. CHRX-CT92-0004 and CHRX-CT93-357.
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[**Figure Captions**]{}
- Diagrammatic representation of (a) the deep inelastic + forward jet, (b) the $E_T$ flow, and (c) the single particle spectrum measurement.
- Diagrammatic representation of a deep inelastic + forward jet event.
- The deep inelastic + forward jet cross section in pb integrated over bins of size $5 \times 10^{-4}$ in $x$ compared to the H1 data presented at the Warsaw conference [@H1data]. As in the H1 measurement the forward jet was required to fulfil $7^{\circ} < \theta_j < 20^{\circ}$, $E_j > 28.7$ GeV, and $k_{jT} > 3.5$ GeV. The electron acceptance region is limited by $160^{\circ} < \theta_e^{\prime} < 173^{\circ}$, $E_e^{\prime} > 11$ GeV, and $y > 0.1$ in the HERA frame.
- Diagrammatic representation of the cross section for emission of a high transverse momentum $p_T$ particle.
- The transverse momentum spectrum of charged particles ($\pi^+, \pi^-, K^+, K^-$) in the pseudorapidity interval $0.5 < \eta < 1.5$ in the virtual photon-proton centre-of-mass frame. The results are shown for kinematic bin 1 with the central values $x = 1.6 \times 10^{-4}$ and $Q^2 = 7$ GeV$^2$. The continuous and the dashed curve show the spectra obtained with $\Phi_i$ and $f$ calculated from the BFKL equation. They only differ in the choice of fragmentation scale: for the continuous curve the fragmentation functions were evaluated at scale $\mu^2 = (2k_j)^2$ and for the dashed curve at scale $\mu^2 = k_j^2$. When BFKL radiation is neglected in the calculation of the $p_T$ spectra, i.e. when the quark box approximation $\Phi_i = \Phi_i^{(0)}$ is used and strong ordering at the gluon vertex is assumed, then the dash-dotted and dotted curves are obtained. The fragmentation functions were evaluated at scales $2k_j$ and $k_j$ respectively. The data points shown are from the H1 measurement of the charged particle spectra [@H1ptdata].
- As Fig. 5, but for kinematic bin 2, $x = 2.9 \times 10^{-4}$ and $Q^2 = 9$ GeV$^2$.
- As Fig. 5, but for kinematic bin 3, $x = 3.7 \times 10^{-4}$ and $Q^2 = 13$ GeV$^2$.
[^1]: On leave from H. Niewodniczanski Institute of Nuclear Physics, Department of Theoretical Physics, ul. Radzikowskiego 152, 31-342 Krakow, Poland.
[^2]: Even though we have normalised $\Phi$ to the DIS + jet data, there still remains some residual uncertainty in the overall normalisation associated with the choice of infrared cut-off used in the $k_\gamma$ integration in (\[eq:b13\]). Our results are shown for the natural choice $k_{\gamma 0}^2 = 1$ GeV$^2$.
|
---
abstract: 'Extinction is the ultimate absorbing state of any stochastic birth-death process, hence the time to extinction is an important characteristic of any natural population. Here we consider logistic and logistic-like systems under the combined effect of demographic and bounded environmental stochasticity. Three phases are identified: an inactive phase where the mean time to extinction $T$ increases logarithmically with the initial population size, an active phase where $T$ grows exponentially with the carrying capacity $N$, and temporal Griffiths phase, with power-law relationship between $T$ and $N$. The system supports an exponential phase only when the noise is bounded, in which case the continuum (diffusion) approximation breaks down within the Griffiths phase. This breakdown is associated with a crossover between qualitatively different survival statistics and decline modes. To study the power-law phase we present a new WKB scheme which is applicable both in the diffusive and in the non-diffusive regime.'
author:
- 'Yitzhak Yahalom and Nadav M. Shnerb'
bibliography:
- 'refs.bib'
title: Phase diagram for logistic systems under bounded stochasticity
---
Noise and fluctuations are ubiquitous features of living systems. In particular, the reproductive success of individuals is affected by many random factors. Some of these factors, like the local density of nutrients or accidental encounter with predators, act on the level of a single individual. Others, like fluctuations in temperature and precipitation rates, affect many individuals coherently. The corresponding theory distinguishes between *demographic stochasticity* (shot noise), i.e., those aspects of noise that influence individuals in an uncorrelated manner, and *environmental stochasticity*, that acts on entire populations [@lande2003stochastic; @ovaskainen2010stochastic].
For a population of size $n$, demographic noise yields ${\cal O} (\sqrt{n})$ abundance fluctuations while environmental stochasticity leads to ${\cal O} (n)$ variations. Accordingly, for large populations environmental stochasticity is the dominant mechanism. A few recent analyses of empirical studies confirm this prediction [@leigh2007neutral; @kalyuzhny2014niche; @kalyuzhny2014temporal; @chisholm2014temporal]. However, the demographic noise controls the low-density states and must be taken into account for calculations of extinction times or fixation probabilities. Consequently, the study of models that combine deterministic effects, temporal environmental stochasticity and demographic noise, received a considerable attention during the last years [@kessler2014neutral; @saether2015concept; @cvijovic2015fate; @danino2016stability; @fung2016reproducing; @hidalgo2017species; @wienand2017evolution].
Almost any model of population dynamics includes two basic ingredients, exponential grows and resource competition. In particular, in the famous logistic equation, $$\label{eq1}
\frac{dn}{dt} = r_0 n - \beta n^2,$$ $r_0$ is the basic reproductive number (low-density growth rate) and the $\beta$ term reflects a density-dependent crowding effect, so the per-capita growth rate declines linearly with $n$.
A wide variety of similar models include the $\theta$-logistic equation (where the growth rate declines like $n^\theta$), ceiling models (growth rate is kept fixed but the population cannot grow above a given carrying capacity), Ricker dynamics and so on. All these models support a transcritical bifurcation at $r_0=0$: when $r_0<0$ the extinction point $n=0$ is stable, while for $r_0>0$ it becomes unstable and the system admits a finite population stable state at $n^*$ \[e.g., $n^*=r_0/\beta$ for the logistic equation (\[eq1\])\].
Since the actual number of individuals in a population is always an integer, Equations like (\[eq1\]) can only be interpreted as the deterministic limit of an underlying stochastic process. For any process with demographic noise the empty state $n=0$ is the only absorbing state, so each population must reach extinction in the long run. Under purely demographic noise the bifurcation point separates two qualitatively different behaviors of the mean time to extinction $T$. When $r_0<0$ the extinction time is logarithmic in the initial population size, while for $r_0>0$ the time to extinction grows exponentially with $n^*$ [@elgart2004rare; @assaf2006spectral; @kessler2007extinction; @ovaskainen2010stochastic].
To understand the lifetime of empirical populations one would like to study a logistic system under the influence of both demographic and environmental stochasticity. This problem was considered by a few authors [@lande2003stochastic; @kamenev2008colored; @spanio2017impact; @wada2018extinction] for the case where the strength of the environmental fluctuations is *unbounded*, e.g., when the state of the environment undergoes an Ornstein-Uhlenbeck process. In such a case there are always rare periods of time in which the net growth rate is negative, and (as we shell see below) these periods dominate the asymptotic behavior of the extinction times. As a result, the system admits only two phases: an inactive (logarithmic) phase for $r_0<0$ and a temporal Griffiths phases [@vazquez2011temporal], where $T$ scales like a power-law with $n^*$, for $r_0>0$.
Here we would like to consider another scenario, a system under *bounded* environmental variations. Since the noise is bounded, for large enough $r_0$ the growth rate is always positive, so the system allows for three phases: logarithmic, power-law (temporal Griffiths phase) and exponential (see Figure \[fig1\]). This insight allows one to identify the failure of the standard analytic tool, the diffusion (continuum) approximation, inside the temporal Griffiths phase. To overcome that we provide an alternative WKB analysis which is valid all over the power-law region. Our analysis reveals a crossover between two, qualitatively different, extinction dynamics. This distinction, in turn, may be relevant to several key concepts in the modern theory of viability and coexistence [@schreiber2012persistence; @ellner2018expanded; @barabas2018chesson]. These connections will be expanded upon towards the end of the paper.
![A phase diagram for a logistic system under bounded stochasticity, presented in the $r_0$-$\sigma$ plane. In the inactive phase ($r_0 <0$, red) the time to extinction scales like $\ln n$ where $N$ plays no role. In the active phase $r_0>\sigma$ (blue) the extinction time grows exponentially with $N$. Under pure demographic noise (along the $\sigma=0$ axis) the transition occurs at $r_0=0$. When $\sigma>0$ the logarithmic and the exponential phases are separated by a finite power-law region (temporal Griffiths phase, green). The dotted line indicates the failure of the continuum (diffusive) approximation and the crossover from soft to sharp decline. []{data-label="fig1"}](fig1new){width="8cm"}
Our model system, chosen to facilitate the numerical calculations, involves two “species” (types, alleles) competition in a finite community with one-sided mutation [@karlin1981second]. In this system the number of individuals, $N$, is always fixed, where $n$ of them belong to species A and $N-n$ to species B. At each elementary step two individuals are chosen at random for a duel, the loser dies and the winner produces a single offspring [@maritan1]. The possible outcomes of the duels (expressions above arrows represent probabilities) are, $$\begin{aligned}
\label{eq1a}
B+B \xrightarrow{1} 2B \quad A&+&A \xrightarrow{1-\nu} 2A \quad A+A \xrightarrow{\nu} A+B \\ \nonumber
A+B \xrightarrow{1-P_{A}} 2B \quad A&+&B \xrightarrow{P_{A}(1-\nu)} 2A \quad A+B \xrightarrow{\nu P_{A}} A+B,
\end{aligned}$$ where $\nu$ is the chance of a mutation event, in which the offspring of an $A$ is a $B$.
An $A$ individual wins an interspecific duel with probability $P_A = 1/2+s(t)/4$, where $s(t) = s_0 + \eta(t)$ and $\eta(t)$ is a zero-mean random process. Following [@hidalgo2017species] we consider a system with dichotomous (telegraphic) environmental noise, so $\eta = \pm \sigma$ (see Supplementary Material, section II [@note1]). After each elementary step $\eta$ may switch (from $\pm \sigma$ to $\mp \sigma$) with probability $1/N\tau$, so the persistence time of the environment is taken from a geometric distribution with mean $\tau$ generations, where a generation is defined as $N$ elementary duels.
As required, this process supports an absorbing attractive fixed point at $n^*=0$ when $r \equiv s(t)-\nu <0$ (more accurately the condition is $\tilde{s} \equiv s(1-\nu/2)<\nu$. In what follows we neglect this tiny factor and use $s$ for $\tilde{s}$) and an active attractive fixed point at $n^*=N[1-\nu/s(t)]$ when $r>0$.
Using the procedure described in [@danino2016stability], one may derive a discrete Backward Kolmogorov equation (BKE) for this stochastic process. The BKE may be solved numerically, by inverting the corresponding matrix, to obtain $T(n)$, the mean time to extinction for a system with $n$ A-type individuals. The mean is taken over both histories and the initial state of the system (plus or minus $\sigma$). The numerical results presented below were obtained from the BKE using this technique. For large-$N$ systems we implemented, instead of direct inversion of a matrix, a transfer matrix approach that allows us to increase the numerical accuracy.
*If* $N \gg 1$ and the diffusion approximation is applicable, $n$ may be replaced by the fraction $x=n/N$ and $n \pm 1$ by $x \pm 1/N$. Expanding all the relevant quantities to second order in $1/N$, and using the dominant balance analysis presented in [@danino2016stability], it can be shown that $T(x)$ satisfies, $$\begin{aligned}
\label{eq9}
\left(s_0 - \frac{\nu}{1-x} + g (1-2x) \right) \frac{\partial T(x)}{\partial x} &+& \left(\frac{1}{N} + g x(1-x) \right) \frac{\partial^2 T(x)}{\partial x^2} = -\frac{1}{x(1-x)} \nonumber \\ T(0) &=& 0 \qquad \left. \frac{\partial T(x)}{\partial x} \right|_{x=1}=\frac{1}{\nu},\end{aligned}$$ where $g \equiv \sigma^2 \tau/2$ is the diffusion constant along the log-abundance axis. We solved Eq. (\[eq9\]) separately in the inner region $x \ll 1$ and in the outer region $x \gg 1/Ng$, using asymptotic matching to obtain, for $1/Ng \ll x \ll 1$, $$\label{large}
T(x) = \left([Ng]^{r_0/g}-x^{-r_0/g} \right) \frac{\Gamma(r_0/g)}{r_0} \left( \frac{g}{\nu} \right)^{r_0/g} - \frac{\ln Ngx}{r_0}.$$ Accordingly, the time to extinction is logarithmic in $n=Nx$ when $r_0$ is negative (red region in Figure \[fig1\]). If $r_0$ is positive the mean lifetime, *for any initial conditions*, grows like $N^{r_0/g}$, since the chance of small population (even a single individual) to grow and to reach the carrying capacity is $N$-independent. These results are in complete agreement with former studies [@lande2003stochastic; @kamenev2008colored; @spanio2017impact; @wada2018extinction] of different logistic-like models, indicating the universality of the large $N$ behaviour for all the systems that support a transcritical bifurcation.
However, for finite noise this continuum approximation must fail somewhere inside the power-law phase. Eq. (\[large\]) suggests a power-law dependence of $T$ on $N$ for any $r_0>0$, but this cannot be the case for $r_0>\sigma$ (light blue region of Fig. \[fig1\]), where even in the pure $(-\sigma)$ state the time to extinction grows exponentially with $N$ [@elgart2004rare; @assaf2006spectral; @kessler2007extinction; @ovaskainen2010stochastic] and occasional jumps to the $+\sigma$ state can only increase stability.
To study the system when the continuum approximation fails, we adopt a version of the WKB analyses presented and discussed in [@kessler2007extinction; @meyer2018noise]. We shall neglect the demographic noise and replace it (as in [@lande2003stochastic; @hidalgo2017species]) by an absorbing boundary condition at $x=1/N$. The abundance dynamics is given by $\dot{x} = (r_0 \pm \sigma)x-\beta x^2$, where the environment stays in the same state (plus or minus $\sigma$) for $\tilde{\tau}$ generations and than switches, with probability $1/2$, to the other state (minus or plus $\sigma$).
Under this dynamics, if the system reaches $x$ at certain time $t$, then one time increment before, at $t- \tilde{\tau}$, it was either at $x_+(x)$ or at $x_-(x)$. Equivalently one may define $y \equiv \ln x$ and $y_\pm \equiv \ln x_\pm$. The probability to find the system at the log-density $y$ at time $t$, $P(y,t)$ satisfies the master equation, $$\label{master}
\frac{dP(y,t)}{dt} = \frac{1}{2} \left[-2P(y) + P(y_+) + P(y_-) \right].$$
At long times $P(y,t)$ converges to its quasi-steady state for which $dP/dt \approx 0$ [@kessler2007extinction]. Given $P(y)$, the $N$-dependence of the mean time to extinction is inversely proportional to the rate of extinction, which is the probability to find the system with less than one individual ($0<x<1/N$), so, $$\label{rate}
{\rm Rate} \sim \int_{-\infty}^{-\ln N} P(y) \ dy.$$
When $x$ is vanishingly small $x_\pm \approx x e^{-\tilde{\tau}(r_0\pm \sigma)}$. Accordingly, in the extinction zone the quasi-steady state satisfies, $$P(y-\tilde{\tau}[r_0+\sigma]) + P(y-\tilde{\tau}[r_0-\sigma])=2 P(y).$$ Instead of expanding $P(y_\pm)$ to second order in $\tilde{\tau}$ (this yields the continuum Fokker-Planck equation and the power-law of the continuum limit) we assume that $P(y) = e^{S(y)}$ and implement the continuum approximation for $S$, replacing $S(y+\Delta y)$ by $S(y)+\Delta y S'(y)$, so $S'(y)$ is obtained as a solution of the transcendental equation $$\label{trans}
\exp \left(- \tilde{\tau} r_0 S'\right) \cosh\left( \tilde{\tau} \sigma S' \right) = 1.$$ This equation does not depend on y, so $S'=q$ and $S \sim q y$, where $q$ is some constant. Accordingly $P \sim \exp(q y)$ and ${\rm Rate} \sim N^{-q}$, so the time to extinction behaves like $T \sim N^q.$
In the limit $r_0 \ll \sigma$ one expects $q \ll 1$. In that case both $q \tilde{\tau} r_0$ and $q \tilde{\tau} \sigma$ are small numbers and Eq. (\[trans\]) yields, $$\label{res1}
q = \frac{2 r_0}{(\sigma^2+r_0^2) \tilde{\tau} } \approx \frac{2 r_0}{\sigma^2 \tilde{\tau} },$$ where the last approximation reflects a self consistency requirement for $q \tilde{\tau} r_0 \ll 1$. On the other hand if $q \tilde{\tau} \sigma$ is large, $$\label{res2}
q = \frac{\ln 2}{\tilde{\tau}(\sigma-r_0)}.$$
The case (\[res1\]) corresponds to the regime where the continuum approximation holds. In that case the typical extinction trajectory is a random walk excursion in the log-abundance space (see below). Since the variance of $M$ random numbers, picked independently from an exponential distribution with mean $\tau$ with alternating signs, is equal to the variance of the sum of $M$ random steps of length $\tilde{\tau}$, $\tilde{\tau}=\tau$ and $$\label{diff}
T \sim N^{r_0/g},$$ in agreement with the large $N$ asymptotics of (\[large\]).
In the other extreme (\[res2\]) extinction occurs due to a (rare) long sequence of bad years, so $\tilde{\tau}$ must be compared with the tail of the corresponding exponential distribution, in which case $\tilde{\tau} = \tau \ln 2$, hence in this regime $$\label{nondiff}
T \sim N^{1/[\tau(\sigma-r_0)]}.$$ This result indicates that the diffusion approximation indeed fails (the result depends on $\tau$ and $\sigma$ separately, not on $g$) and that the power diverge when $r_0 \to \sigma$, i.e., at the transition between the temporal Griffiths phase and the exponential phase.
Beside these limits, The transcendental equation (\[trans\]) has to be solved numerically. In figure \[fig4\] these numerical solutions are compared with the results obtained from a numerical solution of the BKE and with the asymptotic expressions (\[diff\]) and (\[nondiff\]).
As discussed in length in the Supplementary, section I [@note1], our WKB analysis provides another evidence for the universality of all logistic-like (transcritical) systems. The only features that were used to establish Eq. (\[trans\]) are the existence of an upper bound and the linearity of the growth rate at small $x$.
In the Supplementary (section III) we also show that the qualitative features of the extinction process change along the power-law phase, together with the functional form of the survival probability function $Q(t)$ (the chance of the system to survive until $t$).
Deep inside the temporal Griffiths phase (and in the exponential phase) the system spent most of its time fluctuating around $x^*$ (the point where the mean of $\dot x$ vanishes, when the average is taken over the two signs of $\sigma$). Extinction reflects a rare event, an improbable series of bad years and/or excess deaths. Accordingly, the decline time (roughly speaking, the duration of the last excursion from $x^*$ to extinction) scales like $\ln N$ [@lande2003stochastic], and is negligible with respect to the lifetime $T$ (see Fig \[fig3\]b). In that “sharp decline” case the system has no memory: during each segment of time either the catastrophe occurs or not. Accordingly, $Q(t) \sim \exp(t/T)$, where $T$ is the mean time to extinction calculated above. As discussed in the supplementary, this behavior is associated with a gap in the spectrum of the corresponding Markov matrix.
In the diffusive regime, close to the extinction phase, the spectral gap closes down and the associated survival probability is $Q(t) \sim \exp(t/t_0)/t^{1/\rho}$, where $\rho$ is related to the dispersion relation of the Markov matrix and $T$ is proportional to $t_0$. In that case the decline time is relatively long (“soft decline”, Figure \[fig3\]a) and an excursion to extinction is a typical first passage trajectory of a random walker along the log-abundance axis.
![In the temporal Griffiths phase $T \sim N^{q}$. The main panel shows $q$ vs. $r_0/\sigma$ as obtained from numerical solution of Eq. (\[trans\]) (red open circles), in comparison with the asymptotic expressions for the diffusive regime \[Eq. (\[diff\]), purple line\] and in the large $r_0$ regime \[Eq. (\[nondiff\]), black line\]. In the inset we present results for $T(N)$ as obtained from the numerical solution of the exact backward Kolomogorov equation for $r_0 = 0.003$ (blue circles) $0.025$ (yellow) and $0.06$ (green). By fitting these numerical results (full lines) one obtains the actual power $q$, and the outcomes are represented by blue $X$s in the main panel (the $X$s that correspond to the three specific cases depicted in the inset are marked by arrows). In general the WKB predictions fit quite nicely the numerical outcomes, and the slight deviations in the low $r_0$ region are due to the prefactors of the power law \[in these cases the numerical $T(N)$ graph fits perfectly the predictions of Eq. (\[large\])\]. All the results here were obtained for $\sigma = 0.08, \ \tau = 3/2, \ \nu = 0.04$. []{data-label="fig4"}](powersywithinset){width="8cm"}
![Typical trajectories (frequency vs. time) for a system with $\tau=1$, $\sigma =0.11$ and $\nu = 0.1$, where $r_0=0.02$ (a) and $r_0 = 0.105$ (b). The dashed line corresponds to $x^*$, the point where the mean (over environmental conditions) growth rate is zero. In panel (b) the population fluctuates most of its lifetime in a relatively narrow band around $x^*$, extinction happens due to the accumulation of rare sequences of bad years and the decline time is logarithmic in $N$ (sharp decline). As $r_0$ becomes smaller (panel a) the fluctuations are comparable with $x^*$, hence the decline time becomes a finite fraction of the lifetime (soft decline).[]{data-label="fig3"}](trajectorysmallr_prl "fig:"){width="7cm"} ![Typical trajectories (frequency vs. time) for a system with $\tau=1$, $\sigma =0.11$ and $\nu = 0.1$, where $r_0=0.02$ (a) and $r_0 = 0.105$ (b). The dashed line corresponds to $x^*$, the point where the mean (over environmental conditions) growth rate is zero. In panel (b) the population fluctuates most of its lifetime in a relatively narrow band around $x^*$, extinction happens due to the accumulation of rare sequences of bad years and the decline time is logarithmic in $N$ (sharp decline). As $r_0$ becomes smaller (panel a) the fluctuations are comparable with $x^*$, hence the decline time becomes a finite fraction of the lifetime (soft decline).[]{data-label="fig3"}](trajectorybigr_prl "fig:"){width="7cm"}
Our results seems to be relevant to two important issues in population and community ecology: modern coexistence theory and the assessment of population viability.
Modern coexistence theory (MCT) have gained a lot of attention in recent years [@ellner2018expanded; @barabas2018chesson]. In MCT “coexistence" is declared if the steady state probability distribution function is normalizable [@schreiber2012persistence]. For the system considered here, close to zero $P \sim e^{qy} \sim x^{q-1}$, so the MCT persistence criteria is satisfied if $q>0$, i.e., for any $r_0>0$.
However, the main factor that determines ecological stability and species turnover rates is the mean time to extinction. Given Eq. (\[rate\]), one realizes that the coexistence criteria of MCT only guarantees that the time to extinction diverges with $N$, but this divergence may be as slow as $N^{\epsilon}$ for arbitrary small $\epsilon$ if $g=\epsilon r_0$. Accordingly, we believe that an instructive classification of populations stability properties must use phase diagrams like Fig. \[fig1\], instead of being focused on (co)“existence". In particular, for populations in the exponential phase extinction risk is usually negligible, while in the sharp decline region extinction occurs due to rare events so our predictive ability is quite limited. On the other hand in the inactive/soft decline regions extinction risk is high and is strongly related to the observed dynamics, so one may identify risk factors (like grazing or habitat loss) and try to avoid them.
Practically, in empirical studies of birds and plants populations an initial abundance $n_0$ was measured and the survival probability $Q$ was examined after a fixed time interval $t$ [@matthies2004population; @jones1976short]. If $n_0$ may be taken as a proxy for the carrying capacity, the results seem to indicate that these systems are in the power-law phase (see Supplementary section IV, where the empirical results are reproduced and analyzed). However, a single observation of $n_0$ cannot provide a reliable estimation of the carrying capacity in the soft decline regime. Large scale empirical studies of $Q(t)$ (like those presented in [@keitt1998dynamics; @bertuzzo2011spatial]) suggest an exponentially truncated power law. If one likes to interpret these results as reflecting purely local dynamics under environmental stochasticity, it implies that the decline in these systems is indeed soft.
In spatially extended systems the correlation length of environmental fluctuations plays an important role. When the linear size of the system is much smaller than the correlation length, temporal fluctuations are global. This case was examined recently in [@barghathi2017extinction], and is expected to show similarities to the dynamics of a local population. On the other hand, when the correlation length is shorter than the population range migration tends to average out the stochastic effects so the effective strength of stochasticity decreases and $T$ increases. Such an increase was reported by [@bertuzzo2011spatial].
We acknowledge many helpful discussions with David Kessler. This research was supported by the ISF-NRF Singapore joint research program (grant number 2669/17).
In this supplementary we will discuss the generality of our results, and consider some features of the probability distribution function (pdf) $f(t) dt$, i.e., the chance that extinction occurs at time $t$.
Universality
============
The classical logistic growth equation may be written as $$\label{s1}
\frac{dn}{dt} = r_0 n- \beta n^2.$$
This dynamics supports a transcritical bifurcation at $r_0=0$. For $r_0<0$, zero is a stable fixed point and there is no other fixed point in the “physical” regime $n \ge 0$ (if $n$ represents population abundance, there are no negative populations). When $r_0>0$ the extinction fixed point at zero becomes unstable and there is a stable fixed point at $n^*=r_0/\beta$. Below the transition the population decays exponentially to zero (so for long times $n \sim \exp(-|r_0|t)$), above the transition the population eventually converges to $n^*$. At the transition point ($r_0=0$) the population still shrinks to zero but its long time decay satisfies a power law, $n \sim 1/t$.
In general, a given deterministic equation (like Eq. \[s1\]) may be obtained as the large $N$ limit of many “microscopic” (individual based) processes. In the main text we considered one specific example, namely two species competition with one sided mutation, a classical population-genetics problem taken from [@karlin1981second]. Beside its concrete importance, this system is technically tractable since it corresponds to a zero sum game so the total community size $N$ is strictly fixed and still the system shows negative density dependence.
What about other microscopic processes that yield, in their deterministic limit, a transcritical bifurcation? It is widely believed that, although some details may depend on the microscopy of the process, the main characteristics of the behavior: the different phases, the functional $N$ dependence in each phase, and the behavior at the transition points - are *universal*, i.e., are independent of the microscopy. For example, in [@kessler2007extinction] the mean time to extinction of the two processes $A \to A+A, \ \ A+A \to \varnothing$ and $A \to A+A, \ A \to \varnothing, \ \ A+A \to \varnothing$ was calculated. Both these processes are logistic, and under pure demographic stochsticity $T$ grows like $\exp(\alpha n^*)$, where $n^*$ is the number of individuals in the quasi-stationary state. The value of $\alpha$ does depend on the microscopy and the two different models yield different $\alpha$s, but the exponential growth of $T$ with $n^*$ is a universal feature. Accordingly, for different microscopic models one may expect the same phase diagram with exponential, power-law and logarithmic regimes, but the prefactors and the constants may differ.
In the literature one may find other models that belong to the equivalence class of the logistic growth with environmental stochasticity. These include a model with ceiling (i.e., for which the growth rate is density-independent until it reaches a prescribed value $n^*$, where reflecting boundary conditions are imposed [@lande1993risks; @lande2003stochastic], simple logistic equation [@kamenev2008colored; @vazquez2011temporal] and so on. Indeed for these models the authors obtained the same $N$ dependence that we obtained here when the diffusion approximation holds (to the left of the dashed line in Fig. 1 of the main text).
We would like to stress that the WKB analysis presented in the main text allows us to suggest a much stronger statement.
As explained, the chance of extinction, and the associated timescale, are given by the behavior of $P(x)$ at $x < 1/N \ll 1$ ($ {\rm Rate} \sim \int_0^{1/N} P(x) \ dx $). This behavior depends, in turn, on the small $x$ dependency of $x_{pm}$. A solution to Eq. (8) of the main text exists only in the power-law phase (it requires $r_0>0$ and $r_0<\sigma$). When it exists, Eq. (8) assures that in the $x \ll 1$ ($y \ll 0$) regime $S=qy$ and the mean time to extinction is a power law in $N$.
Accordingly, our WKB analysis shows that for *any* microscopic model, the time to extinction is a power-law in $N$ when the following conditions are met:
- The probability distribution function $P(x)$ is normalizable.
- The dynamics allows for periods of growth and periods of decline.
- When $x \ll 1$ the time-averaged growth rate is positive ($r_0>0$).
- When $x \ll 1$ the growth/decline are exponential.
These features are common to any system that fluctuates below and above a transcritical bifurcation.
Dichotomous (telegraphic) and other types of noise
==================================================
In the main text we have considered a special type of environmental stochasticity, in which the system flips between two states (good and bad years, say). Both white Gaussian noise and white Poisson noise can be recovered from this dichotomous (telegraphic) noise by taking suitable limits [@ridolfi2011noise], so the results obtained here are quite generic.
As an example, if the environmental conditions are picked from a Gaussian distribution of a certain width with correlation time $\tau_1$, one may easily imitate these features by taking a dichotomous noise that flips between two values, $\pm \sigma$, with much shorter correlation time $\tau$. With the appropriate choice of $\tau$ and $\sigma$, the binomial distribution of $\sigma_{eff}$, the average fitness between $0<t<\tau_1$, $$\sigma_{eff} = \frac{\tau_1}{\tau} \sum_i^{\tau_1/\tau} \sigma_i,$$ will correspond to the bulk properties of any required Gaussian noise, since the Gaussian distribution is the limit of a binomial distribution.
However, while the Gaussian distribution is unbounded, the distribution of $\sigma_{eff}$ is clearly bounded; the convergence to a Gaussian takes place in the bulk but the tails are truncated.
To demonstrate the ability of a dichotomous noise to emulate the effect of other types of noise, we present in Figure \[simu\] the outcomes of a few numerical experiments. The figures show the mean time to extinction vs. $N$ for our two-species competition model with one sided mutation, as described in the main text \[Eq. (2)\]. Three types of noise are compared.
1. $s(t)$ is either $\sigma$ or $-\sigma$ (dichotomous noise).
2. $s(t)$ is picked from a uniform distribution between $(-\sigma \sqrt{12})$ and $(+\sigma \sqrt{12})$.
3. $s(t)$ is picked from a beta distribution, $ \rm{Beta}(2,2)\sigma/\sqrt{0.05}$.
All three distribution have a compact support, zero mean and variance $\sigma^2$.
![Time to extinction $T$ (log scale) vs. $N$ for three different noise distributions. The mean (over 1000-2000 runs) time to extinction was measured as a function of $N=50,100,200,500,1000$, for $n_0=N$. The left panel present results for $\tau= \sigma = 0.1$ while in the right panel $\tau=\sigma=0.3$. For each $N$ and $\nu$ the value of $T$ is given for dichotomous noise (green circles), uniform distribution (magenta) and Beta distribution (black). Markers were chosen with different size to improve the visibility of the results. Dashed line were added manually to guide the eye and they connect results with $\nu = 0.01$ (yellow) $\nu=0.001$ (red) and $\nu = 0.0001$ (blue). In the insets the three points at $N=200$, $\nu = 0.001$, with one standard deviation error bars, were magnified. These error bars are too small and cannot be seen in the main panels. []{data-label="simu"}](simu1_new "fig:"){width="8cm"} ![Time to extinction $T$ (log scale) vs. $N$ for three different noise distributions. The mean (over 1000-2000 runs) time to extinction was measured as a function of $N=50,100,200,500,1000$, for $n_0=N$. The left panel present results for $\tau= \sigma = 0.1$ while in the right panel $\tau=\sigma=0.3$. For each $N$ and $\nu$ the value of $T$ is given for dichotomous noise (green circles), uniform distribution (magenta) and Beta distribution (black). Markers were chosen with different size to improve the visibility of the results. Dashed line were added manually to guide the eye and they connect results with $\nu = 0.01$ (yellow) $\nu=0.001$ (red) and $\nu = 0.0001$ (blue). In the insets the three points at $N=200$, $\nu = 0.001$, with one standard deviation error bars, were magnified. These error bars are too small and cannot be seen in the main panels. []{data-label="simu"}](simu2_new "fig:"){width="8cm"}
Probability distribution function
=================================
In the main text we have calculated the mean time to extinction, $T$, in the various phases of the logistic system. Here we would like present a few considerations regarding the full probability distribution function for extinction at $t$, $f(t) dt$ or the survival probability $Q(t) dt$. Of course $f(t) = -dQ(t)/dt$.
The state of our system is fully characterized by $P_{e,n}(t)$, the chance that the system admits $n$ A particles at $t$, when the environmental state is $e$ (for dichotomous noise $e$ take two values that correspond to $\pm \sigma$). After a single birth-death event (time incremented from $t$ to $t+1/N$), the new state is given by $$P_{e,n}^{t+1/N} = {\cal M } P_{e',m}^t,$$ where ${\cal M }$ is the corresponding Markov matrix, ${\cal M }_{e,n;e',m}$ is the chance to jump from $m$ particles in environment $e'$ to $n$ particles in environment $e$.
The highest eigenvalue of the Markov matrix; $\Gamma_0=1$, corresponds to the extinction state, i.e, to the right eigenvector $P_{e,n} = \delta_{n,0}$ (at extinction the state of the environment insignificant) or the left eigenvector $(1,1,1,...)$. Using a complete set of left and right eigenvectors of this kind one may write $P_{e,n}(t)$ as, $$\label{sum1}
P_{e,n}(t) = \sum_k a_k v_k (\Gamma_k)^{Nt}.$$ Here the index $k$ runs over all eigenstates of the Markov matrix, $v_k$ is the $k$-s right eigenvector, $a_k$ is the projection of $P_{e,n}(t=0)$ on the $k$-s left eigenvector and $Nt$ is the number of elementary birth-death events at time $t$ (for $t=1$, i.e., a generation, $Nt=N$). Writing $\Gamma_k = |\Gamma_k| \exp(\phi_k)$, one realizes that each $k$ mode decays like $\exp(-Nt \epsilon_k)$, when $\epsilon_k \equiv -\ln |\Gamma_k|$. Since the Markov matrix is real, eigenvalues are coming in complex conjugate pairs so $P_n$ is kept real and non negative at any time. For the extinction mode $\epsilon_0 =0$, all other modes have $ \epsilon_k > 0$
Clearly, for any finite system the subdominant mode $\epsilon_1$ determines the maximal persistence time of the system, so at timescales above $t= 1/N \epsilon_1$ the chance of the system to survive, $Q(t)$, decays exponentially with $t$.
Now one would like to make a distinction between two different situations. In the first, there is a *gap* between $\epsilon_1$ and $\epsilon_2$, so when $N \to \infty$ $\epsilon_1 \ll \epsilon_2 $. This behavior is demonstrated in the right panels of Figures \[fig1s\] and \[fig2s\] and in Figure \[fig3s\]. In such a case the large $t$ behavior of the system is simply $$\label{expo}
Q(t) dt = exp(-t/t_0),$$ where $t_0 \equiv 1/N \epsilon_1$. Accordingly, $f(t) = -\dot{Q} = exp(-t/t_0)/t_0$ and the mean time to extinction, calculated in the main text, is $T = t_0 = 1/N \epsilon_1$. As showed in the main text, when $r_0>0$ $T$ grows with $N$, either exponentially or like a power law.
In the exponential phase the situation corresponds to this gap scenario, as discussed in [@kessler2007extinction]. The purely exponential distribution (\[expo\]) reflects an absence of memory: the system sticks for long times to its quasi stationary state $v_1$, and decay to zero on much shorter timescale due to rare events. This behavior is pronounced in Fig. 5 of [@ben2012coherence]. The decline to extinction may be a result of a rare demographic event, like an improbable series of individual death, or the result of an environmental rare event - an improbable series of bad years. In both cases, the *decline time* (as defined in [@lande2003stochastic]) is short (logarithmic in $N$), so the exponential distribution reflects the accumulated chance of rare, short, and independent catastrophes.
The second scenario (demonstrated in the left panel of Figs \[fig1s\] and \[fig2s\] and in the blue line of Fig \[fig3s\]) correspond to a *gapless* system. Here the eigenvalues of ${\cal M}$ satisfy $\epsilon_m \sim \epsilon_1 +c_1 (m-1)^\rho$, where $c_1$ is some tiny constant. In such a case the $\exp(- t N \epsilon_1 )$ factors out of the sum (\[sum1\]), and the rest of the sum may be approximated by $\int exp(-c_1 t N m^{\rho}) dm$, yielding a power-law decay so, $$\label{lifetime}
Q(t) dt \sim \frac{e^{-t/t_0}}{t^{1/\rho}} dt.$$ In that case the mean time to extinction is not exactly $t_0$ but the difference is only a numerical factor. If $\rho>1$ then, $$T = t_0 (1-1/\rho),$$ while if $\rho<1$ the ratio between $T$ and $t_0$ depends on the short time cutoff that must be imposed on the distribution (\[lifetime\]) to avoid divergence at zero.
Now the decay is not purely exponential, since the system has long-term memory. Rare catastrophic events put an upper bound on the lifetime of the population, but extinction may occur, with relatively high probability, due to the random walk of the population size along the log-abundance axis.
![The logarithm of the absolute value of the eigenvalues of the Markov matrix, $\epsilon_m$, is plotted against $\ln m$ for small $r_0$ (left panel), intermediate $r_0$ (middle panel) and large $r_0$ (right panel). The state with $m=1$ ($\ln m =0$) is the most persistent non-extinction state. Clearly, as $r_0$ increases, a gap is opened between $\epsilon_1$ and $\epsilon_2$ (see figure \[fig3\]). For $m>1$, the low-lying states satisfy $\epsilon_m \sim m^\rho$, where $\rho \approx 1.7$. Parameters are $\tau=1$, $\nu = 0.1$ and $N = 2^8$. []{data-label="fig1s"}](begslpnew "fig:"){width="5cm"} ![The logarithm of the absolute value of the eigenvalues of the Markov matrix, $\epsilon_m$, is plotted against $\ln m$ for small $r_0$ (left panel), intermediate $r_0$ (middle panel) and large $r_0$ (right panel). The state with $m=1$ ($\ln m =0$) is the most persistent non-extinction state. Clearly, as $r_0$ increases, a gap is opened between $\epsilon_1$ and $\epsilon_2$ (see figure \[fig3\]). For $m>1$, the low-lying states satisfy $\epsilon_m \sim m^\rho$, where $\rho \approx 1.7$. Parameters are $\tau=1$, $\nu = 0.1$ and $N = 2^8$. []{data-label="fig1s"}](medslpnew "fig:"){width="5cm"} ![The logarithm of the absolute value of the eigenvalues of the Markov matrix, $\epsilon_m$, is plotted against $\ln m$ for small $r_0$ (left panel), intermediate $r_0$ (middle panel) and large $r_0$ (right panel). The state with $m=1$ ($\ln m =0$) is the most persistent non-extinction state. Clearly, as $r_0$ increases, a gap is opened between $\epsilon_1$ and $\epsilon_2$ (see figure \[fig3\]). For $m>1$, the low-lying states satisfy $\epsilon_m \sim m^\rho$, where $\rho \approx 1.7$. Parameters are $\tau=1$, $\nu = 0.1$ and $N = 2^8$. []{data-label="fig1s"}](deepslpnew "fig:"){width="5cm"}
![ $\epsilon_m$ (from $m=1$ to $m=6$, see legends) is plotted against $\log_2 N$ for different values of $r_0$. Parameters are $\tau=1$ and $\nu = 0.1$. []{data-label="fig2s"}](beggriffithnew "fig:"){width="5cm"} ![ $\epsilon_m$ (from $m=1$ to $m=6$, see legends) is plotted against $\log_2 N$ for different values of $r_0$. Parameters are $\tau=1$ and $\nu = 0.1$. []{data-label="fig2s"}](medgriffithnew "fig:"){width="5cm"} ![ $\epsilon_m$ (from $m=1$ to $m=6$, see legends) is plotted against $\log_2 N$ for different values of $r_0$. Parameters are $\tau=1$ and $\nu = 0.1$. []{data-label="fig2s"}](beggriffithnew "fig:"){width="5cm"}
![ The gap, $\ln \epsilon_1- \ln \epsilon_2$, as a function of $ \log_2 N$. As $N$ increases the gap grows when $r_0$ is large or intermediate but remains more or less fixed when $r_0$ is small. Parameters are $\tau=1$ and $\nu = 0.1$. []{data-label="fig3s"}](diffnew){width="7cm"}
Population viability data
=========================
In the main text we have mentioned the population viability analysis of [@matthies2004population; @jones1976short]. In Figure \[birds\] we reproduce the relevant datasets from these two papers.
As one can see, both datasets (which are, of course, quite noisy because of the small number of samples in each bin, especially for the high abundance bins) allow for reasonable fits if the chance of survival, $Q(t)$, satisfies $$Q(t) = \exp(-t/\tilde{\tau}N^q),$$ which is the expression one expects if the system is in the temporal Griffith phase. Note that the distinction between soft and sharp decline is irrelevant here, since the time window is fixed and we are interested only in the $N$ dependence.
When we tried to fit the data with $Q = \exp(t/\tilde{\tau}\exp(\alpha N))$, as expected in the exponential phase, we ran into difficulties. In such a case one expects a much steeper dependence of $Q$ on $N$: if $T \sim \exp(\alpha N)$ than a chance in $N$ from $0.1/\alpha$ to $10/\alpha$, say, takes $Q$ from vanishing values to one, so the survival probability is a sharp sigmoid unless $\alpha$ takes very small values. As a result, for the plants 10y data our Matlab cftool fit suggested an extremely tiny coefficient $\alpha = 0.01$, while for the birds 80y data it simply neglected the last four points and suggested $1-Q$ that drops to zero after the third point.
Moreover, both studies did not report a significant abundance decline in the surviving populations - in most of them abundance either grew up or kept fixed, see Figure 4 of [@jones1976short] and Figure 4 of [@matthies2004population]. This implies that both systems are not in the logarithmic phase, where one should expect a general decrease in abundance for all populations.
We conclude that the most reasonable interpretation of the observed data is that the surveyed bird and plant populations are in the temporal Griffith phase, where the lifetime of a population scales $N^q$.
![The left panel (Figure 1 of [@matthies2004population]) shows the relationships between the size of plant populations in 1986 and their chance to survive 10 years later (red circles). The black line is the best fit to $Q(t)$, assuming that the mean time to extinction $T$ growth like $N^q$. In the right panel we retrieved Figure 5 of [@jones1976short], and the red circles correspond to the chance of extinction of birds populations vs. the initial number of pairs (the last point in the original figure, that was too close to zero to be digitised, was omitted). The black line is the best fit to $1-Q(t)$, assuming that the mean time to extinction $T$ growth like $N^q$. []{data-label="birds"}](plants "fig:"){width="8cm"} ![The left panel (Figure 1 of [@matthies2004population]) shows the relationships between the size of plant populations in 1986 and their chance to survive 10 years later (red circles). The black line is the best fit to $Q(t)$, assuming that the mean time to extinction $T$ growth like $N^q$. In the right panel we retrieved Figure 5 of [@jones1976short], and the red circles correspond to the chance of extinction of birds populations vs. the initial number of pairs (the last point in the original figure, that was too close to zero to be digitised, was omitted). The black line is the best fit to $1-Q(t)$, assuming that the mean time to extinction $T$ growth like $N^q$. []{data-label="birds"}](birds "fig:"){width="8.5cm"}
|
---
abstract: 'In this work we study the reliability and secrecy performance achievable by practical codes over the Gaussian wiretap channel. While several works have already addressed this problem in asymptotic conditions, i.e., under the hypothesis of codewords of infinite length, only a few approaches exist for the finite length regime. We propose an approach to measure the performance of practical codes and compare it with that achievable in asymptotic conditions. Moreover, based on the secrecy metrics we adopt to achieve this target, we propose a code optimization algorithm which allows to design irregular codes able to approach the ultimate performance limits even at moderately small codeword lengths (in the order of $10000$ bits).'
author:
- ', g.ricciutelli@pm.univpm.it'
bibliography:
- 'Archive.bib'
title: 'Performance assessment and design of finite length LDPC codes for the Gaussian wiretap channel [^1] '
---
Introduction {#sec:Intro}
============
Coding for the Gaussian wiretap channel is a well-established research topic, but there are some partially unsolved and challenging problems. One of these problems is to study the secrecy performance in the finite code length regime, and to design optimized finite length codes. One of the most common metrics to assess the performance of finite length codes used for transmissions is the average achieved by using some (possibly optimal) decoder. On the other hand, the secrecy performance over wiretap channels is classically measured using information-theoretic metrics, like the secrecy capacity, and in asymptotic conditions (e.g., infinite code length and random coding).
For example, in [@Thangaraj2007b], the authors consider a (that is, a or ) model for both the main and wiretapper’s channels, and design optimized regular codes for these channels. They show that their approach achieves the secrecy capacity when the wiretap channel consists of symmetric . No continuous channels are considered, and the secrecy capacity is achieved in the asymptotic regime (i.e., with infinite length codes).
The as a secrecy metric has instead been used in [@Klinc2011b], where a coding scheme able to achieve a very close to $0.5$ for the eavesdropper and very low for the authorized channel is proposed. In [@Klinc2011b], the authors use differential evolution to design optimized codes able to achieve the desired targets while keeping the quality ratio between the main and the eavesdropper’s channels (named security gap) as small as possible. The proposed coding scheme is based on puncturing and, thanks to the -based analysis, is applicable at finite block lengths. A similar solution, but without the need of puncturing, has been proposed in [@Baldi2012], and extended in [@Baldi2014] to the case of parallel channels.
A bridge between information theoretic and error rate-based secrecy measures is presented in [@WongWong2011], where however the main goal is to propose a secret key sharing scheme for the wiretap channel, and the presence of an error-free public channel between the source and destination is considered, which helps the secret sharing process. By using regular codes, the authors show that the key capacity can be achieved in the asymptotic regime. Irregular codes are instead considered for the finite code length regime, and a density evolution based linear program is used to design them. The same approach is followed in [@WongWong2011a] to assess the performance of punctured codes over the Gaussian wiretap channel.
Inspired by such works, in this paper we study the performance of finite length codes over the Gaussian wiretap channel, by defining suitable metrics to assess how far they are from optimality, which is achieved in asymptotic conditions. This permits us to explore the capacity-equivocation regions of these codes in the finite length regime, and without using puncturing. We also propose a twofold code optimization tool which allows to design optimal codes in terms of the considered metrics. Similar twofold code optimizations have been proposed for the relay channel [@Chakrabarti2007b; @Wang2013; @Khattak2014], but no solution has been presented for the wiretap channel, at our best knowledge. We show that our approach allows to achieve great flexibility in the choice of the system parameters, as well as higher security levels with respect to previous solutions based on punctured codes [@WongWong2011a].
The organization of the paper is as follows. In Section \[sec:Model\] we present the system model and the metrics we use to assess performance in asymptotic and finite length conditions. In Section \[sec:CodeDesign\] we describe the code design requirements. In Section \[sec:CodeOptimization\] we propose our code optimization approach. In Section \[sec:NumericalResults\] we provide and discuss some numerical results. Finally, Section \[sec:Conclusion\] concludes the paper.
System model and metrics {#sec:Model}
========================
We consider the classical Gaussian wiretap channel model, in which a sender, named Alice, transmits a secret message $\mathcal M$. She encodes her message into the $n$-symbol codeword $X^n$, which uniquely depends on $\mathcal M$ and on some random message $\mathcal R$ generated by Alice. We consider binary coding, therefore $X^n$ actually is an $n$-bit codeword. If the secret message is $k_s$ bits long and the random message is $k_r$ bits long, the code rate is $R_c = (k_s + k_r)/n = k/n$. The secret message rate, instead, is $R_s = k_s/n$.
Transmission occurs over a Gaussian channel for both the authorized receiver, named Bob, and the eavesdropper, named Eve. The noisy codewords received by Bob and Eve are denoted by $Y^n$ and $Z^n$, respectively. In order to achieve successful transmission of $\mathcal M$ over this channel, both the following targets must be fulfilled:
1. $\mathcal M$ must be reliably decoded by Bob, i.e., with a sufficiently small error rate (*reliability target*),
2. the information about $\mathcal M$ gathered by Eve must be sufficiently small (*security target*).
Concerning the reliability target, in ideal conditions (i.e., infinite code length and random coding) the channel capacity can be used as the ultimate code rate limit. In the finite length regime, instead, a practical code must be designed to allow Bob to achieve a sufficiently low error rate in decoding the secret message. Concerning the security target, some classical information theoretic secrecy metrics are only useful in the asymptotic regime. In fact, denoting by ${\rm I}(x; y)$ the mutual information between $x$ and $y$, we have [@Bloch2011]:
- Strong secrecy when the total amount of information leaked about $\mathcal M$ through observing $Z^n$ goes to zero as $n$ goes to infinity, i.e., $\displaystyle \lim_{n \rightarrow \infty}{\rm I}(\mathcal M; Z^n) = 0$.
- Weak secrecy when the rate of information leaked about $\mathcal M$ through observing $Z^n$ goes to zero as $n$ goes to infinity, i.e., $\displaystyle \lim_{n \rightarrow \infty}{\rm I}(\mathcal M; Z^n)/n = 0$.
So, these metrics are not useful in order to assess the performance in finite length conditions and compare it with that in the asymptotic regime.
However, another metric can be exploited, which was already used in Wyner’s original work [@Wyner1975]. According to [@Wyner1975], transmission is accomplished in perfect secrecy when the wiretapper equivocation rate on the secret message, $R_e = \frac{1}{n} {\rm H}(\mathcal M|Z^n)$, with ${\rm H}(\cdot)$ denoting the entropy function, equals the entropy of the data source. We consider independent and identically distributed secret messages, therefore the source entropy rate is equal to $R_s$. So, perfect secrecy is achieved when the equivocation rate $R_e$ equals the secret message rate $R_s$, i.e., $$\widetilde{R_e} = R_e/R_s = 1.
\label{eq:PerfectSecrecy}$$ $\widetilde{R_e}$ is called fractional equivocation rate.
Actually, the ultimate limit achievable by the equivocation rate is the secrecy capacity $C_s = C_B - C_E$, where $C_B$ and $C_E$ are Bob’s and Eve’s channel capacities, respectively. For a binary-input channel with and $\gamma$, the capacity is given by the following expression: $$\label{eq:capacity}
C\left( \gamma \right) = 1 - \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-\frac{\left (y - \sqrt{\gamma} \right )^2}{2}} \log_2 \left(1 + e^{-2y\sqrt{\gamma}} \right) dy.
$$ Then, the target is to maximize $R_e$ in such a way as to approach the secrecy capacity. On the other hand, when considering finite length codes, it is expected that $C_s < R_s$ and another valuable issue is the evaluation of the gap between the secret message rate and the secrecy capacity. Numerical examples will be presented in Section \[sec:NumericalResults\].
Concerning the computation of the equivocation rate, it can be shown that [@WongWong2011a]: $$\begin{split}
R_e & = \frac{1}{n} \left[{\rm H}(X^n) - {\rm I}(X^n; Z^n) + {\rm H}({\mathcal M}|Z^n, X^n) \right.\\
& \left. - {\rm H}(X^n|{\mathcal M}, Z^n)\right].
\label{eq:Re}
\end{split}$$ From it results that this formulation of Eve’s equivocation rate requires to compute the quantity ${\rm H}(X^n|{\mathcal M}, Z^n)$, that is, the entropy of $X^n$ conditioned to receiving $Z^n$ and knowing the secret message ${\mathcal M}$. Eve obviously does not know the secret message, therefore we suppose the existence of another (fictitious) receiver in the same position as Eve’s, knowing the secret message ${\mathcal M}$. We denote such a receiver as Frank: he receives the same vector $Z^n$ as Eve but, differently from Eve, he has perfect knowledge of the secret message ${\mathcal M}$. Then, he tries to decode $Z^n$ for recovering the random message ${\mathcal R}$, which is the only source of uncertainty for Frank in order to reconstruct $X^n$. The resulting wiretap channel model is schematically depicted in Fig. \[fig:Wiretap\]. The letter $M$ inside Alice’s and Frank’s boxes points out that the message is known to both Alice and Frank.
![Wiretap channel model employed in the study.[]{data-label="fig:Wiretap"}](Wiretap.eps){width="90mm"}
Let us suppose that, in these conditions, Frank experiences a decoding error probability (or ) equal to $\eta$. By Fano inequality we have ${\rm H}(X^n|{\mathcal M}, Z^n) \le 1 + k_r \eta$. We also have ${\rm H}(X^n) = k$ and ${\rm H}({\mathcal M}|Z^n, X^n) \leq {\rm H}({\mathcal M}|X^n) = 0$. Concerning Eve’s channel mutual information ${\rm I}(X^n; Z^n)$, we could obtain a tight upper bound on it as proposed in [@Polyanskiy2010], by taking into account the code length and the target error rate. However, by using the classical bound ${\rm I}(X^n; Z^n) \leq n C_E$, we obtain a limit value which is independent of Eve’s error rate. Such a value cannot be overcome even if Eve’s error rate changes, therefore it represents a conservative choice for our purposes. Based on these considerations, we can find a lower bound on Eve’s equivocation rate about the secret message as [@WongWong2011a]: $$\begin{split}
R_e & \ge \frac{1}{n} \left[k - n C_E - k_r \eta - 1 \right] = \\
& = R_c - C_E - (R_c - R_s)\eta - \frac{1}{n} = R_e^*.
\label{eq:ReBound}
\end{split}$$
By looking at , it is evident that this metric is well suited to assess the secrecy performance of practical, finite length codes. In fact, the code length is taken into account, and the error rate experienced by Frank can be estimated for practical codes through numerical simulations. Its value obviously depends on Frank’s , which is the same as Eve’s, and therefore, according to , it determines $C_E$. It follows that, for a fixed code length and rate, the equivocation rate can be maximized by optimizing the choice of the pair $(\eta, C_E)$.
Code design {#sec:CodeDesign}
===========
An code with rate $R_c=k/n$ is defined through its parity-check matrix $\mathbf H$ of size $(n-k) \times n$. Alternatively, the code can be represented through a Tanner graph, that is a bipartite graph composed of variable and check nodes, which correspond to the codeword bits and the parity-check equations, respectively. Noting by $h_{ij}$ the $(i, j)$-th element of $\mathbf H$, there is an edge between the $j$-th variable node and the $i$-th check node iff $h_{ij} = 1$. The number of edges connected to a node is called degree of that node. The following two polynomials are commonly used to denote the variable and check node degree distributions:
$$\lambda(x)=\sum_{i=2}^{d_{v}}{\lambda_i x^{i-1}}, \quad \quad \rho(x)=\sum_{j=2}^{d_{c}}{\rho_j x^{j-1}}
\label{DegreeDistribution}$$
where $d_{v}$ and $d_{c}$ are the maximum variable and check node degrees, respectively. In $\lambda\left(x\right)$ $\left(\rho\left(x\right)\right)$, the coefficient $\lambda_i$ $\left(\rho_j\right)$ coincides with the fraction of edges connected to the variable $\left(\text{check}\right)$ nodes having degree $i$ $\left(j\right)$. Therefore, $\lambda(x)$ and $\rho(x)$ are defined from the edge perspective. The code rate can be expressed as:
$$R_c=1-\frac{\sum_{i=2}^{d_{v}}\rho_i/i}{\sum_{j=2}^{d_{c}}\lambda_j/j}.
\label{Rate}$$
The most common code decoding algorithm, which is an instance of the well-known belief propagation principle, is based on the exchange of soft messages about each received bit between the nodes of its Tanner graph. Therefore, the performance of an code depends on the connections among the nodes of its Tanner graph. Indeed, a variable node with a greater number of connected edges has more parity-check equations which verify its associated bit. On the other hand, check nodes with low degrees correspond to parity-check equations with less unknowns. The optimization of the code performance under message passing decoding consists in finding the best tradeoff between these two effects, and this usually requires irregular degree distributions. The well-known density evolution algorithm, proposed in [@Chung; @2001], aims at optimizing the pair $\left(\lambda(x), \rho(x)\right)$ based on the statistics of the decoder messages. However, differently from classical transmission problems, in our setting the same code (chosen by Alice) is used by three receivers: Bob, Eve and Frank, and the code optimization should take this into account.
Let us consider a systematic encoder and let us suppose that the transmitted codeword is $\mathbf c = [\mathcal M | \mathcal R | \mathcal P]$, where $\mathcal M$ is the $k_s$-bit secret message, $\mathcal R$ is the $k_r$-bit random message and $\mathcal P$ is the $r$-bit redundancy vector added by the encoder. Obviously, systematic encoding shall be avoided in security applications, especially if source coding is not optimal. In fact, in such a case, Eve could look at the systematic part of the received codeword and gather some information about the secret message parts which are less affected by errors. In practical systems, systematic encoding can be easily avoided by scrambling the information bits prior to encoding [@Baldi2012]. Having this clearly in mind, for our code design and analysis purposes it is convenient to keep the assumption of systematic encoding. Under this hypothesis, the code parity-check matrix can be divided into three blocks as shown in Fig. \[fig:matrixH\], corresponding to the three parts of the codeword $\mathbf c$. Bob must use the whole matrix to decode for both the secret and random messages (since he does not know in advance any of them), although in the end he is interested only in $\mathcal M$. Eve is in the same condition, although she receives the signal through a different channel. Frank, instead, has perfect knowledge of $\mathcal M$, and only needs to decode for $\mathcal R$. Therefore, he can precompute $\mathbf A \cdot {\mathcal M}^T = \mathbf s$, were $^T$ denotes transposition. Then, he can use $\mathbf s$ as a syndrome vector and focus on the reduced parity-check system:
$$\left[\mathbf B | \mathbf C \right] \cdot \left[ \mathcal R | \mathcal P \right]^T = \mathbf H' \cdot \mathbf c'^T = \mathbf s.$$
![Parity-check matrix of the considered codes.[]{data-label="fig:matrixH"}](matrixH.eps){width="80mm"}
Obviously, decoding for a vector having an all-zero syndrome or a different syndrome is equivalent, due to the code linearity. Hence, Frank performs decoding through the code defined by $\mathbf H'$, having rate $R_F = k_r / (k_r + r)$. The code rate for Bob instead coincides with the overall code rate, i.e., $R_B = k/n$. It follows that $R_F = \frac{R_B - R_s}{1 - R_s}$. In the setting we consider, it is important that both Bob’s and Frank’s codes are optimized. In fact, an optimized code for Bob allows to approach the channel capacity, which is the ultimate limit for the reliability target. An optimized code for Frank instead serves to achieve the desired $\eta$ with the smallest possible . Since Frank’s is the same as Eve’s, this reduces Eve’s channel capacity $C_E$.
Code optimization {#sec:CodeOptimization}
=================
We propose an optimization strategy for Bob’s and Frank’s codes based on the density evolution algorithm, which is commonly used to optimize a single code, with some modifications in order to consider the joint optimization target. In Section \[sec:SingleOptimization\] we briefly recall the steps of the single code optimization and then in Section \[sec:JointOptimization\] we describe our strategy for the joint code optimization. In this work, as in [@Chung; @2001], we use the density evolution algorithm with Gaussian approximation of the decoder messages.
Single optimization {#sec:SingleOptimization}
-------------------
The density evolution algorithm is well-known in the literature; therefore, for the sake of brevity, we report here only the main equations of [@Chung; @2001], as they are used in the proposed joint code optimization.
Given $\rho(x)$, $R_c$ and $d_{v}$, the optimization of $\lambda(x)$ of a single code is possible by applying the following constraints:
- - Rate constraint: $$\sum_{i=2}^{d_{v}}\frac{\lambda_i}{i} =\frac{1}{1-R_c}\sum_{i=2}^{d_{c}}\frac{\rho_i}{i}.
\label{eq:RateConstraints}$$\
- - Proportion distribution constraint: $$\sum_{i=2}^{d_{v}}\lambda_i=1.
\label{eq:ProportionConstraints}$$\
- - Convergence constraint (from [@Chung; @2001 Eq. (16)]): $$r > h(s, r), \ \ \forall r\in(0,\phi(s))
\label{eq:ConvergenceConstraints}$$ where $s = \frac{2}{\sigma^2}$, $\sigma^2$ being the noise variance, and $\phi(\cdot)$ will be defined in . For $0<s<\infty$ and $0<r\leq1$, we define $h(s, r)$ in (\[eq:ConvergenceConstraints\]) as follows:
$$h(s, r)=\sum_{i=2}^{d_{v}}\lambda_ih_i(s,r)
\label{eq:h}$$
where $$h_i(s, r)= \phi\left(s+\left(i-1\right)\sum_{j=2}^{d_{c}}\rho_j\phi^{-1}\left(1-\left(1-r\right)^{j-1}\right)\right).
\label{eq:h_i(s,r)}$$
In (\[eq:ConvergenceConstraints\]) and (\[eq:h\_i(s,r)\]), $$\phi(x) =
\begin{cases}
1-\frac{1}{\sqrt{4\pi x}}\int_{-\infty}^{+\infty} \tanh\frac{u}{2}e^{-\frac{\left(u-x\right)^2}{4x}}du, & \mbox{if } x>0 \\
1, & \mbox{if } x=0.
\label{eq:phi}
\end{cases}$$
<!-- -->
- Condition is equivalent to impose that $r_l(s) \rightarrow 0$ for $l \rightarrow \infty$ [@Chung; @2001], with $r_l=h(s, r_{l-1})$ and $r_0=\phi(s)$.\
- - Stability condition: $$\lambda_2 < \frac{e^{\frac{1}{2 \sigma^2}}}{\sum_{j=2}^{d_{c}}\rho_j(j-1)}.
\label{eq:StabilityCondition}$$
In the single code optimization, the code threshold $s^*$ is defined as the minimum $s$ for which the constraints $[C_1-C_4]$ are satisfied. From the definition of $s$, it is evident that $s^*$ corresponds to the maximum noise variance $\sigma^2$ for which the constraints are verified.
Joint optimization {#sec:JointOptimization}
------------------
In order to perform the joint optimization of Bob’s and Frank’s codes, we must impose that Frank’s code is somehow *contained* in Bob’s code (in other terms, that Frank’s parity-check matrix is a sub-matrix of Bob’s parity-check matrix). Therefore, in addition to the constraints in Section \[sec:SingleOptimization\], we need another condition. To obtain this further constraint, we introduce the polynomial $\tilde{\lambda}(x)$ which corresponds to the node perspective of $\lambda(x)$. In $\tilde{\lambda}(x)$, the fraction of nodes of degree $i$ can be derived from $\lambda(x)$ through the following formula: $$\tilde{\lambda}_i=\frac{{\lambda}_i / i}{\sum_{k=2}^{d_{v}}\lambda_k / k}.
\label{eq:DegreeDistributionNode}$$ In order to obtain the check node degree distributions from the node perspective $\tilde{\rho}(x)$, a similar formula can be applied to the check nodes degree distributions from the edge perspective. This can be easily achieved by replacing in (\[eq:DegreeDistributionNode\]) $\lambda(x)$ with $\rho(x)$, $\tilde{\lambda}(x)$ with $\tilde{\rho}(x)$ and $d_v$ with $d_c$.
Since Bob’s parity-check matrix contains Frank’s parity-check matrix, the number of variable nodes in Bob’s Tanner graph having some fixed degree must be greater than or equal to that of variable nodes in Frank’s Tanner graph having the same degree. Hence, we must take into account the following further constraint:
- - Joint optimization constraint: $$\tilde{\lambda}_{B,i} \geq \tilde{\lambda}_{F,i}, \ \ \forall i \in \left[2, 3, 4, \ldots, d_v^{(F)} \right],
\label{eq:JointOptimization}$$
where $\tilde{\lambda}_B(x)$ and $\tilde{\lambda}_F(x)$ are Bob’s and Frank’s variable node degree distributions from the node perspective, respectively, and $d_v^{(F)}$ is Frank’s maximum variable node degree. $C_5$ adds to $[C_1-C_4]$ and the optimum $\lambda_B(x)$ must satisfy all these constraints.
In the joint optimization algorithm, we define the convergence threshold as the maximum of $c=\sigma^2_B+\sigma^2_F$, denoted by $c^*$, for which the constraints $[C_1-C_5]$ are satisfied. In the expression of $c$, $\sigma^2_B$ and $\sigma^2_F$ are Bob’s and Frank’s noise variances, respectively. It should be noted that this procedure differs from optimizing the two codes separately. In fact, in principle, we could first optimize Frank’s code, and then try to optimize Bob’s code by taking account the degree distributions obtained for Frank and the constraint $C_5$. This, however, could impose too strong constraints on Bob’s code degree distribution, thus preventing to find a good solution for him, too. In fact, some solutions may exist for which neither Bob’s nor Frank’s degree distributions are individually optimal, but their joint performance is optimal.
As in [@Chung; @2001], in order to design the check node degree distribution, we adopt a concentrated distribution (i.e., with only two degrees, concentrated around the mean). It is widely recognized that this solution, though very simple, is able to achieve very good performance. Hence, for each pair $(\lambda_F(x), \lambda_B(x))$, we obtain the pair $(\tilde{\rho}_F(x), \tilde{\rho}_B(x))$ by using the following formula, valid for both Bob and Frank:
$$\label{eq:notation1}
\tilde{\rho}(x) = a x^{\left\lfloor c_m \right\rfloor} + b x^{\left\lceil c_m \right\rceil},$$
where $c_m = \frac{E}{r} = \frac{\sum_{j} \tilde{\lambda}_j \cdot j}{(1-R_c)}$ and $E$ is the total number of edges in the Tanner graph. The values $a$ and $b$ are computed as $$\label{eq:notation2}
a = \lceil c_m \rceil - c_m, \ \ \ \ b = c_m - \lfloor c_m \rfloor.$$ In and , $\lceil c_m \rceil$ and $\lfloor c_m \rfloor$ represent the ceiling and floor value of $c_m$, respectively.
Numerical results {#sec:NumericalResults}
=================
In order to provide some practical examples, we use the procedure described in Section \[sec:JointOptimization\] to design several codes with $d_{v}^{(B)}=d_{v}^{(F)}=50$. We consider code rates $R_c = R_B = 0.35, 0.5, 0.75$ and several values of $R_s < R_B$. The degree distributions obtained through the joint optimization procedure are reported in Table \[tab:DegreesDistributions\]. Concerning the choice of the degrees of $x$ allowed in the two polynomials, the only constraints we impose are that they must not overcome the maximum values $d_{v}^{(B)}$ and $d_{v}^{(F)}$, and that the number of nodes of degree $2$ must be such that the stability condition is met by both codes.
----- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
$i$ $\lambda_{F,i}$ $\lambda_{B,i}$ $\lambda_{F,i}$ $\lambda_{B,i}$ $\lambda_{F,i}$ $\lambda_{B,i}$ $\lambda_{F,i}$ $\lambda_{B,i}$ $\lambda_{F,i}$ $\lambda_{B,i}$ $\lambda_{F,i}$ $\lambda_{B,i}$ $\lambda_{F,i}$ $\lambda_{B,i}$
2 0.6677 0.1858 0.4208 0.2259 0.6181 0.2070 0.2187 0.1588 0.2066 0.1382 0.4257 0.1712 0.6181 0.1300
3 0.2279 0.2291 0.1656 0.1701 0.2117 0.2123 0.1826 0.1851 0.1436 0.1549 0.1763 0.1787 0.2117 0.2128
4 - - 0.1192 0.1195 - - - - 0.0280 0.0278 0.1014 0.1029 - -
5 - - - - 0.1445 0.1471 0.0497 0.0449 0.0123 0.0112 - - 0.1445 0.1786
6 0.0267 0.0252 - - 0.0246 0.0254 0.0365 0.0378 0.0248 0.0267 - - 0.0246 0.0354
7 0.0767 0.0751 - - - - 0.0309 0.0317 0.0999 0.1054 - - - -
8 - - 0.0057 0.0061 - - 0.1662 0.1683 - - - - - -
9 - - - - - - - - 0.0539 0.0574 0.1321 0.1410 - -
10 - - 0.2877 0.2907 - - - - 0.0413 0.0409 0.1635 0.1639 - -
11 - 0.0249 - - - - - - 0.0144 0.0175 - - - -
12 - 0.1792 - - - - - - 0.0126 0.0119 - - - -
13 - - - - - - - - - - - - - 0.0359
14 - - - - - 0.0184 - - - - - - - 0.0625
15 - - - - - 0.2779 - - - - - - - 0.1561
19 - - - - - - - - 0.0637 0.0713 - - - -
20 - - - - - - 0.0154 0.0124 0.0050 0.0190 - - - 0.0031
21 - - - 0.0096 - - 0.0747 0.0954 - - - - - 0.0103
22 - - - - - - 0.0666 0.0659 - - - - - 0.0014
23 - - - - - 0.1109 0.0568 0.0549 - - - - - -
24 - - - - - - - - - - - 0.0307 - -
25 - - - 0.0697 - - 0.1007 0.1016 - - - 0.2106 - -
26 - - - 0.1074 - - - - - - - - - -
32 - - - - - - - - - - - - - 0.1727
34 - 0.0203 - - - - - - - - - - - -
36 - 0.0844 - - - - - - - - - - - -
38 - 0.0716 - - - - - - - - - - - -
39 - 0.0652 - - - - - - 0.2929 0.2946 - - - -
40 - 0.0382 - - - - - - - - - - - -
50 0.0010 0.0010 0.0010 0.0010 0.0011 0.0010 0.0012 0.0432 0.0010 0.0232 0.0010 0.0010 0.0011 0.0012
----- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
To provide some examples of finite length codes, we consider codes with length $n_1=10000$ and $n_2=50000$; Frank’s code length is then obtained from these values by considering the submatrix $\mathbf H'$. Once having defined the degree distributions, the parity-check matrices are designed through the algorithm [@Hu2001PEG]. The numerical results are obtained by considering, for all coding schemes, modulation over the channel. When considering finite length codes, through numerical simulations we are able to determine the values of the per bit $(E_b/N_0)$ that ensure a given . These values are reported in Table \[tab:Performance\], for both Bob and Frank, assuming $\ac{CER} = 10^{-2}$ and several values of $R_s$. In the table, the values of $\left.\frac{E_b}{N_0}\right|_{th}$ identify the codes convergence thresholds obtained through density evolution. These values represent the ultimate performance bounds achievable in asymptotic conditions (i.e., infinite code length). The values of $\left.\frac{E_b}{N_0}\right|_{n_1}$ and $\left.\frac{E_b}{N_0}\right|_{n_2}$ instead represent the working points, estimated through simulations, for the practical codes with lengths $n_1$ and $n_2$, respectively. We observe from Table \[tab:Performance\] that, for Bob’s code, the finite length performance approaches the asymptotic threshold as the code rate increases. Indeed, for $R_B = 0.75$ and code length equal to $n_1$ and $n_2$, the gap between the asymptotic threshold and the finite length codes performance is about $0.4$ dB and $0.2$ dB, respectively.
$R_s$ $R_B$ $\left.\frac{E_b}{N_0}\right|^{B}_{th}$ $\left.\frac{E_b}{N_0}\right|^{F}_{th}$ $\left.\frac{E_b}{N_0}\right|^{B}_{n_1}$ $\left.\frac{E_b}{N_0}\right|^{F}_{n_1}$ $\left.\frac{E_b}{N_0}\right|^{B}_{n_2}$ $\left.\frac{E_b}{N_0}\right|^{F}_{n_2}$
------- ------- ----------------------------------------- ----------------------------------------- ------------------------------------------ ------------------------------------------ ------------------------------------------ ------------------------------------------
0.33 0.35 -0.14 -1.52 1.10 3.82 0.72 3.18
0.4 0.5 0.41 -0.52 1.00 0.76 0.78 0.44
0.45 0.5 0.42 -0.69 1.12 1.22 0.82 0.98
0.5 0.75 1.73 0.38 2.14 1.17 1.94 0.84
0.6 0.75 1.72 -0.14 2.12 0.98 1.97 0.63
0.7 0.75 1.75 -0.52 2.13 0.91 1.92 0.60
0.725 0.75 1.75 -0.69 2.18 2.11 1.96 1.59
: working points of the considered coding schemes for several values of $R_s$ and $R_B$; the values of $\frac{E_b}{N_0}$ are in dB.[]{data-label="tab:Performance"}
As a security metric we use the lower bound $R_e^*$ on the equivocation rate, computed according to and the values in Table \[tab:Performance\]. The secrecy capacity $C_s$, that represents the ultimate limit achievable by the equivocation rate, is also computed for the cases of interest, and used as a benchmark. We compute $C_s$ under the hypothesis of ideal coding, i.e., that Bob’s and Frank’s code rates coincide with the respective channel capacities. Since Frank’s and Eve’s channels coincide, it follows that $C_s = R_B - R_F = R_s \frac{1-R_B}{1-R_s}$. In order to assess if practical codes can approach the perfect secrecy condition , we then compute the fractional lower bound on the equivocation rate $\widetilde{R_e^*} = R_e^*/R_s$ both in asymptotic conditions and in the finite code length regime, and compare its values with the fractional secrecy capacity $\widetilde{C_s} = C_s/R_s = \frac{1-R_B}{1-R_s}$. The values so obtained are reported in Fig. \[fig:ReRs\], for the same values of $R_s$ considered in Tables \[tab:DegreesDistributions\] and \[tab:Performance\]. As an example, for the considered code parameters and $R_s = 0.725$, we find that in asymptotic conditions the designed codes approach the secrecy capacity and the perfect secrecy condition. Notably, even using relatively short codes, with $10000$-bit codewords, the fractional equivocation rate is close to $0.8$. For the sake of comparison, we consider some results reported in [@WongWong2011a] for the scheme based on punctured codes. The corresponding points are marked with an asterisk in Fig. \[fig:ReRs\]. Those results consider codes with length $n=10^6$, at which the performance of codes usually approaches the density evolution threshold. However, the asymptotic performance achieved by the degree distributions found through the proposed approach exhibits some gain at the same secret message rates. Furthermore, for $R_s = 0.43$, even our schemes with $n=10000$ and $n=50000$ outperform that proposed in [@WongWong2011a] with $n=10^6$.
![Comparison between $\frac{C_s}{R_s}$, $\frac{R_e^*}{R_s}$ calculated through the asymptotic threshold values, $\frac{R_e^*}{R_s}$ for code length $n_1$, and $\frac{R_e^*}{R_s}$ for code length $n_2$, as a function of ${R_s}$.[]{data-label="fig:ReRs"}](ReRs.eps){width="80mm"}
From Fig. \[fig:ReRs\] it results that the best performance in terms of Eve’s equivocation rate is achieved when the secret message rate approaches the code rate. This could seem counterintuitive, since suggests to use few random bits to confuse the eavesdropper. However, in this condition $R_F$ is small and Frank is able to reach the desired performance at low . The latter coincides with Eve’s channel , therefore Eve’s equivocation rate is large. On the other hand, imposing that Eve’s channel has a too low is not realistic, therefore some randomness shall always be used in order to relax the constraints on Eve’s channel quality.
Conclusion {#sec:Conclusion}
==========
We have studied the performance of practical coded transmissions over the Gaussian wiretap channel. By using suitable reliability and security metrics, we have computed performance bounds in the asymptotic regime and assessed the achievable performance under the hypothesis of finite codeword lengths. We have also proposed an optimization approach to design good codes for this context. Our results show that these codes are able to approach the ultimate performance limits even with relatively small block lengths.
[^1]: This work was supported in part by the MIUR project “ESCAPADE” (Grant RBFR105NLC) under the “FIRB – Futuro in Ricerca 2010” funding program.
|
---
abstract: 'Micro- and nano-resonators have important applications including sensing, navigation, and biochemical detection. Their performance is quantified using the quality factor $Q$, which gives the ratio of the energy stored to the energy dissipated per cycle. Metallic glasses are a promising materials class for micro- and nano-scale resonators since they are amorphous and can be fabricated precisely into complex shapes on these lengthscales. To understand the intrinsic dissipation mechanisms that ultimately limit large $Q$-values in metallic glasses, we perform molecular dynamics simulations to model metallic glass resonators subjected to bending vibrations. We calculate the vibrational density of states, redistribution of energy from the fundamental mode of vibration, and $Q$ versus the kinetic energy per atom $K$ of the excitation. In the linear and nonlinear response regimes where there are no atomic rearrangements, we find that $Q \rightarrow \infty$ (since we do not consider coupling to the environment). We identify a characteristic $K_r$ above which atomic rearrangements occur, and there is significant energy leakage from the fundamental mode to higher frequencies, causing finite $Q$. Thus, $K_r$ is a critical parameter determining resonator performance. We show that $K_r$ decreases as a power-law, $K_r\sim N^{-k},$ with increasing system size $N$, where $k \approx 1.3$. We estimate the critical strain $\langle \gamma_r \rangle \sim 10^{-8}$ for micron-sized resonators below which atomic rearrangements do not occur, and thus large $Q$-values can be obtained when they are operated below $\gamma_r$. We find that $K_r$ for amorphous resonators is comparable to that for resonators with crystalline order.'
author:
- Meng Fan
- Aya Nawano
- Jan Schroers
- 'Mark D. Shattuck'
- 'Corey S. O’Hern'
bibliography:
- 'resonator.bib'
title: Intrinsic dissipation mechanisms in metallic glass resonators
---
Introduction {#sec:intro}
============
Micro- and nano-resonators have numerous important applications including navigation, sensing, chemical detection, molecular separation, and biological imaging [@arash2015review]. The performance of resonators is typically measured by the quality factor, $Q$, which gives the ratio of the energy stored to the energy dissipated per cycle in the resonator during operation [@green1955story]. Micro- and nano-resonators made from non-metallic crystalline materials, such as sapphire [@tobar1998high], carbon nanotubes [@huttel2009carbon; @jiang2004intrinsic], and single-crystal diamond [@ovartchaiyapong2012high] can possess quality factors $Q > 10^6$ at low temperatures. However, it is difficult to fabricate these materials into complex shapes, and many applications require electrical conduction. As a result, crystalline metals are used in many resonator applications, yet these suffer from energy losses that arise from topological defects and grain boundaries [@blanter2007internal].
In an effort to obviate energy losses from topological defects and grain boundaries that occur in crystalline metals, as well as take advantage of their plastic-forming ability to be fabricated into complex shapes, several groups have considered resonators made from metallic glasses (MGs) [@khonik1996nature; @kanik2014high; @kanik2015metallic; @hiki2009internal; @bardt2007micromolding]. MGs are cooled rapidly to avoid crystallization, and thus they possess uniformly disordered structure. Recent experiments have shown that metallic-glass-based resonators can achieve quality factors that are comparable and even larger than those for resonators made from crystalline metals [@kanik2014high]. Metallic glasses offer the additional benefit for resonator applications in that they can be thermoplastically formed into complex shapes with spatial features that span many orders of magnitude [@schroers2010processing; @kumar2009nanomoulding; @li2018atomic]. In this work, we seek to characterize the dissipation mechanisms that determine the quality factor for metallic glass resonators.
The mechanisms that give rise to energy losses during vibration can be classified as intrinsic or extrinsic [@arash2015review]. Extrinsic losses, such as anchoring and frictional losses, come from interactions between the resonator and its surrounding environment [@perez2007design; @shkel2006type]. In contrast, intrinsic losses originate from flaws or defects within the resonator, such as dislocations, grain boundaries, vacancies, and interstitials in crystalline materials. In metallic glasses, which lack crystalline order, intrinsic losses are envisioned to stem from irreversible, collective atomic rearrangements, or shear-transformation zones (STZs) [@falk1998dynamics]. A number of studies have characterized the role of collective atomic rearrangements in determining the mechanical properties of metallic glasses, including ductility, yielding, and shear-band formation [@fan2017effects; @yu2012tensile; @fan2017particle; @zemp2015crystal; @ketkaew2018mechanical].
Internal friction measurements have been performed to gain insight into intrinsic dissipation mechanisms and the quality factor of metallic glasses [@zener1937internal; @nowick2012anelastic; @blanter2007internal]. However, a key focus in this work has been on revealing structural relaxation processes at elevated temperatures, rather than their room temperature behavior (which is typically significantly below the glass transition temperature). In these studies, metallic glass samples are typically perturbed by mechanical or electrostatic excitation using a torsion pendulum or dynamical mechanical analyzer, and the internal friction is measured as a function of temperature, frequency, and strain amplitude [@barmatz1974young; @hiki2008temperature; @blanter2007internal]. The internal friction, which is proportional to $Q^{-1}$, is generally small for temperatures below room temperature, and then increases dramatically, forming a strong peak at temperatures typically above $400$-$500$ K due to collective $\alpha$ structural relaxations [@sinning1988influence; @samwer1995dynamic; @khonik1996nature]. Studies [@blanter2007internal] have also reported a much smaller peak (typically four orders of magnitude smaller than that corresponding to $\alpha$ relaxations) in the internal friction of metallic glasses between $50$ K and room temperature. Researchers have suggested that this peak corresponds to localized, anelastic so-called $\beta$ relaxations. Explanations of the peaks in the internal friction include the creation and destruction of free volume [@spaepen1977microscopic], dislocation motion [@gilman1973flow], shear transformation zones [@falk1998dynamics], shear bands, and other mechanisms that involve structural rearrangements. At even lower temperatures ($<50$ K), the internal friction has been described using the quantum mechanical tunneling model for two-level systems [@anderson1972anomalous; @phillips1972tunneling].
Most of these prior studies of the vibrational properties of metallic glasses either use a quantum mechanical approach for the low-temperature behavior or consider temperatures near room temperature and above, where thermal fluctuations are significant and microscopic rearrangements of atoms are frequent. In this work, we will take a different, but still classical approach, and focus on the nearly zero-temperature regime, where even microscopic rearrangements of atoms are rare, to better understand the transition from the linear response regime where $Q$ is infinite (since we do not consider coupling of the system to the environment) to the highly nonlinear regime where $Q$ becomes finite.
We carry out molecular dynamics (MD) simulations to quantify the intrinsic dissipation caused by atomic rearrangements and measure the quality factor in model metallic glass resonators. We induce vibrations in a thin bar-shaped resonator by exciting the mode corresponding to the resonator’s fundamental frequency with a given kinetic energy per atom $K$ and then running MD at constant total energy. When $K$ is small, i.e. $K<K_{nl}$, the resonator displays linear response, the spectrum of the vibrational modes only includes the fundamental mode, and $Q\rightarrow \infty$. For intermediate $K$, i.e. $K_{nl}<K<K_r$, energy leaks to modes other than the fundamental mode, but at sufficiently long times the leakage stops. Thus, in this regime, $Q \rightarrow \infty$ at long times. For $K > K_r$, the system undergoes one or more atomic rearrangements, which induce strong dissipation and finite $Q$. Thus, the magnitude of $K_r$ controls the performance of metallic glass resonators. We further show that $K_r$ can be increased by decreasing the system size or by decreasing the cooling rate used to prepare the resonator. We also show that resonators with amorphous structure can achieve comparable performance (e.g. same $Q$) to those with partial crystalline order.
The remainder of the article is organized as follows. In Sec. \[sec:methods\], we describe the simulation methods we use to prepare and excite the metallic glass resonators, and to quantify the energy loss and quality factor of the vibrations. In Sec. \[sec:results\], we present the results, including measurements of the intrinsic loss and dissipation arising from nonlinearity and atomic rearrangements, techniques to increase $Q$ by decreasing the system size and cooling rate, and comparisons of resonator performance in amorphous and crystalline samples. In Sec. \[sec:conclusion\], we summarize our findings and present promising directions for future research. We also include two Appendices. In Appendix A, we show that our results for the vibrational response do not depend strongly on the length of the time series of the vibrations that we collect. In Appendix B, we show the time dependence of the vibrational density of states, which supports the findings presented in the main text.
models and methods {#sec:methods}
==================
![(a) View of the model metallic glass resonator along the $z$ axis. The bar contains $N=2000$ atoms with aspect ratio $L_x:L_y:L_z = 6:1:2$. Blue and yellow atoms indicate $A$ and $B$ atom types, respectively. (b) Vector field representing the fundamental mode of the dynamical matrix of the metallic glass resonator in (a). The color scale highlights the y-component of the fundamental mode contribution for each atom with red corresponding to positive and blue corresponding to negative $y$-values.[]{data-label="fig:method"}](method.png){width="0.99\columnwidth"}
We perform molecular dynamics (MD) simulations of binary Lennard Jones mixtures using the Kob-Andersen model [@kob1995testing], which has been employed to describe NiP alloys. Spherical atoms interact pairwise via the shifted-force version of the Lennard-Jones potential, $u(r_{ij}) = 4
\epsilon_{ij}[(\sigma_{ij}/r_{ij})^{12}-(\sigma_{ij}/r_{ij})^6]$ with a cutoff distance $r_c =2.5 \sigma_{ij}$, where $r_{ij}$ is the separation between atoms $i$ and $j$. The total potential energy per atom is $U = N^{-1}
\sum_{i>j} u(r_{ij})$. $80\%$ of the atoms are type A ($N_A/N=0.8$) and $20\%$ are type B ($N_B/N=0.2$), where $N=N_A + N_B$ is the total number of atoms, and the energy and length parameters are given by $\epsilon_{AA}=1.0$, $\epsilon_{BB}=0.5$, $\epsilon_{AB}=1.5$, $\sigma_{AA}=1.0$, $\sigma_{BB}=0.88$, and $\sigma_{AB}=0.8$. All atoms have the same mass $m$. The energy, length, and pressure scales are given in terms of $\epsilon_{AA}$, $\sigma_{AA}$, and $\epsilon_{AA}/\sigma_{AA}^3$, respectively.
We initially placed the $N$ atoms on an FCC lattice in a long, thin box with aspect ratio $L_x:L_y:L_z = 6:1:2$ and periodic boundaries in the $x$-, $y$-, and $z$-directions at reduced number density $\rho =
1.0$. We then equilibrated the system at high temperature $T_0 > T_g
\sim 0.4$ [@wittmer2013shear] (which melts the crystal) by running molecular dynamics simulations at fixed number of atoms, pressure, and temperature (NPT) using the Nosé-Hoover thermostat with temperature $T_0 = 0.6$ and pressure $P_0 =0.025$, a modified velocity-Verlet integration scheme, and time step $\Delta {\overline
t} = 10^{-3}$. We then cool the system into a glassy state at zero temperature using a linear cooling ramp with time ${\overline t}$, such that $T({\overline t}) = T_0-R {\overline t}$. (The cooling rate is measured in units of $\epsilon_{AA}^{3/2}/(m^{1/2} \sigma_{AA})$, where the Boltzmann constant $k_B=1$.) We varied the cooling rate $R$ over more than three orders of magnitude, yet we ensured that $R$ was larger than the critical cooling rate $R_c$ to avoid crystallization. We vary $N$ from $250$ to $8000$ atoms to assess the finite size effects.
After cooling the system to zero temperature, we remove the periodic boundary conditions in the $x$-, $y$-, and $z$-directions (creating free surfaces) and then apply conjugate gradient energy minimization to yield the zero-temperature configuration of the resonator, $\textbf{R}_0=\{x_1,y_1,z_1,...,x_N,y_N,z_N\}$. (See Fig. \[fig:method\] (a).) To induce vibrations, we excite the fundamental mode, i.e. the lowest eigenfrequency $\omega_1$ of the dynamical matrix [@tanguy2002continuum], evaluated at $\textbf{R}_0$. (See Fig. \[fig:method\] (b).) The elongated, thin shape of the resonator guarantees that the lowest eigenfrequency is well-separated from higher ones. We then set the initial velocities of the atoms, such that $\textbf{v} =
\{v_{x1},v_{y1},v_{z1},...,v_{xN},v_{yN},v_{zN}\} = \delta
\textbf{e}_1$, where $\textbf{e}_1$ is the eigenvector corresponding to $\omega_1$ and $\delta = \sqrt{2NK/m}$, and run MD simulations at constant total energy for a given time $t = \omega_1 {\overline
t}/2\pi$. (The eigenvectors are normalized such that $\textbf{e}_i
\cdot \textbf{e}_j = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta and $i$,$j=1,\ldots,3N-6$ are the indexes of the eigenvectors that correspond to the non-trivial eigenfrequencies.)
![Schematic diagram that illustrates the method we use to calculate the loss $L$ and quality factor $Q$ of model metallic glass resonators. We track the velocities $\textbf{v}$ of the atoms in the resonator over a long time period. The time series is broken up into $20$ time intervals with equal duration $\delta t$. We calculate the velocity auto-correlation function (VACF) for each time interval, and fast Fourier transform (FFT) it to measure the density of vibrational modes $D(\omega,t)$ and loss $L(t)$ for each time interval $t$. Using Eq. \[q\], we can calculate the quality factor $Q$ from $L(t)$.[]{data-label="fig:timeslot"}](timeslot.png){width="0.9\columnwidth"}
We track the atom positions and velocities over long time periods $t >
2700$ during the MD simulations. We then divide the long time series into $20$ time intervals with equal duration $\delta t = 135$. We characterize the vibrational response of the system using two methods. In the first, we determine the vibrational response using the time period from $0$ to $\delta t$. For the second method, we quantify how the vibrational response varies in time following the initial perturbation using a fixed tape length $\delta t$ for each time interval. (We show that our results do not depend strongly on tape length $\delta t$ in Appendix \[sec:appendix\_tapelength\].)
For each time interval between $t$ to $t+\delta t$, we calculate the Fourier transform of the velocity autocorrelation function to determine the density of vibrational modes $D(\omega,
t)$ [@bertrand2014hypocoordinated]: $$\label{loss}
D(\omega, t)=\int_0^{\delta t}{\langle \textbf{v}(t_0+\tau) \cdot \textbf{v}(t_0) \rangle_{t} e^{i\omega\tau}d\tau},$$ where $\omega$ is the angular frequency and $\langle.\rangle_{t}$ indicates an average over all atoms and time origins $t_0$ between $t$ and $t+\delta t$. See Fig. \[fig:timeslot\] for a summary of this approach.
For each time interval, we also determine the fraction of the kinetic energy that has transferred from the fundamental mode (with frequency $\omega_1$) to other frequencies by defining the loss, $$\label{loss}
L(t)=1-\frac{\int_{\omega_1-\Delta\omega}^{\omega_1+\Delta\omega} D(\omega,t)d\omega}{\int_0^{\infty} D(\omega,t)d\omega}.$$ where $\Delta\omega=(\omega_2-\omega_1)/2$. (See Fig. \[fig:spec\] (b).) By determining the loss $L(t)$ over consecutive time intervals, we can calculate the quality factor $$\label{q}
Q=\omega_1\left(\frac{dL(t)}{dt}\right)^{-1}.$$ Note that the results do not depend strongly on the magnitude of $\Delta \omega$ as long as it brackets $\omega_1$.
To track the atomic displacements during vibration, we will also calculate the root-mean-square deviation (RMSD) between two configurations, e.g. $\textbf{R}(t_1)$ and $\textbf{R}(t_2)$ at different times $t_1$ and $t_2$:
$$\label{rmsd}
d(\textbf{R}(t_1),\textbf{R}(t_2))=\sqrt{ N^{-1} \sum_{i=1}^{N}{(x_{i}(t_1)-
x_{i}(t_2))^2+(y_{i}(t_1)-y_{i}(t_2))^2+(z_{i}(t_1)-z_{i}(t_2))^2}},$$
where the sum is over all atoms.
Results {#sec:results}
=======
![(a) The density of vibrational modes $D(\omega,0)$ for the time interval $t=0$ as a function of the kinetic energy per atom $K$. (b) $D(\omega,0)$ for the same systems in (a), but a close-up of the low frequency regime. The vertical dashed lines indicate the vibrational frequencies ($\omega_1,\omega_2,\ldots,\omega_{3N-6}$) calculated from the dynamical matrix. The arrows indicate integer multiples of the fundamental frequency $\omega_1$ and the two vertical solid lines show the region of frequencies near $\omega_1$ used in Eq. \[loss\] to calculate the loss.[]{data-label="fig:spec"}](spec1-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"} ![(a) The density of vibrational modes $D(\omega,0)$ for the time interval $t=0$ as a function of the kinetic energy per atom $K$. (b) $D(\omega,0)$ for the same systems in (a), but a close-up of the low frequency regime. The vertical dashed lines indicate the vibrational frequencies ($\omega_1,\omega_2,\ldots,\omega_{3N-6}$) calculated from the dynamical matrix. The arrows indicate integer multiples of the fundamental frequency $\omega_1$ and the two vertical solid lines show the region of frequencies near $\omega_1$ used in Eq. \[loss\] to calculate the loss.[]{data-label="fig:spec"}](spec2-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"}
The results are organized into three sections. In Sec. \[sec:results\_dissipation\], we quantify the density of vibrational modes $D(\omega,0)$ and loss $L(0)$ during the first time interval ($t=0$) as a function of the initial kinetic energy per atom $K$, and investigate the effects of atomic rearrangements on the vibrational response. We also study the dependence of $D(\omega,t)$ and $L(t)$ on the time interval $t$ and calculate the quality factor $Q$. We identify three characteristic regimes for vibrational response as a function of $K$: the linear response regime, where there is no leakage of energy from the fundamental mode to others, the nonlinear regime, where energy leakage occurs at short times, but it stops at long times, and the strong loss regime where atomic rearrangements occur, causing large losses and small $Q$. In Sec. \[sec:results\_improvement\], we investigate how variations of the system size $N$ and cooling rate $R$ affect the frequency of atomic rearrangements, and thus the vibrational response. In Sec. \[sec:results\_X\], we calculate the loss in resonators made from polycrystalline and defected crystalline materials and compare it to resonators made from amorphous materials. We find that the losses generated from resonators with amorphous structure are comparable to that for crystalline resonators, and thus glassy materials may be promising for high-$Q$ resonator applications.
Intrinsic dissipation: Nonlinearity and Atomic Rearrangements {#sec:results_dissipation}
-------------------------------------------------------------
We first focus on model metallic glass resonators with $N=2000$ generated using cooling rate $R=10^{-2}$. In Fig. \[fig:spec\], we show the density of vibrational modes $D(\omega,0)$ after exciting the system along the fundamental mode as a function of the kinetic energy per atom over six orders of magnitude from $K=5\times10^{-10}$ to $5\times10^{-4}$. When $K$ is small, most of the response remains in the fundamental mode, $\omega_1$, indicating that the system is in the linear response regime. As $K$ increases, energy begins to leak to other modes of the dynamical matrix (indicated by the dashed vertical lines in Fig. \[fig:spec\] (b)), as well as harmonics of the fundamental mode (indicated by the arrows in Fig. \[fig:spec\] (b)). The leakage of energy from the fundamental mode is due to the nonlinearity of the Lennard Jones potential near the minimum and not due to the cutoff at $r_c = 2.5 \sigma_{ij}$ [@mizuno2016cutoff]. To test this, we also carried out studies of weakly nonlinear springs with $r_c \rightarrow \infty$ and found similar results.
In Fig. \[fig:spec\], we show that there is a qualitative change in the vibrational response when $K$ increases from $5\times10^{-6}$ to $5\times10^{-5}$. At the higher value of $K$, the vibrational response is noisy and energy is redistributed over a much wider range of frequencies than at the lower value of $K$. A more refined search shows that this qualitative change occurs in the kinetic energy interval $5 \times 10^{-5.50} < K_r < 5 \times 10^{-5.49}$ as shown in Fig. \[fig:rearrange\] (a).
![(a) Density of vibrational modes $D(\omega,0)$ for $K=5\times10^{-5.50}$ (blue) and $5\times10^{-5.49}$ (red). (b) Schematic diagram of the energy landscape with axes, the total energy per atom $E=U+K$ and atomic configuration $\textbf{R}$. The configurations $\textbf R_0(t_1)$ and $\textbf R_0(t_2)$ represent the inherent structures ([*i.e.*]{}, the nearest local potential energy minima) of the vibrating system at times $t_1$ and $t_2$, respectively. $\Delta U$ is the difference in the potential energy per atom and $U^*$ is the energy barrier between the configurations $\textbf R_0(t_1)$ and $\textbf R_0(t_2)$. (c) $\Delta U(\textbf{R}_0(t),\textbf{R}_0(0))$ between the inherent structures at times $t$ and $0$ for $K=5\times10^{-5.50}$ (blue circles) and $5\times10^{-5.49}$ (red pluses). The inset shows the root-mean-square deviation (RMSD) $d(\textbf{R}_0(t),\textbf{R}_0(0))$ between the inherent structures at times $t$ and $0$.[]{data-label="fig:rearrange"}](rearrange1-eps-converted-to.pdf "fig:"){width="0.49\columnwidth"} ![(a) Density of vibrational modes $D(\omega,0)$ for $K=5\times10^{-5.50}$ (blue) and $5\times10^{-5.49}$ (red). (b) Schematic diagram of the energy landscape with axes, the total energy per atom $E=U+K$ and atomic configuration $\textbf{R}$. The configurations $\textbf R_0(t_1)$ and $\textbf R_0(t_2)$ represent the inherent structures ([*i.e.*]{}, the nearest local potential energy minima) of the vibrating system at times $t_1$ and $t_2$, respectively. $\Delta U$ is the difference in the potential energy per atom and $U^*$ is the energy barrier between the configurations $\textbf R_0(t_1)$ and $\textbf R_0(t_2)$. (c) $\Delta U(\textbf{R}_0(t),\textbf{R}_0(0))$ between the inherent structures at times $t$ and $0$ for $K=5\times10^{-5.50}$ (blue circles) and $5\times10^{-5.49}$ (red pluses). The inset shows the root-mean-square deviation (RMSD) $d(\textbf{R}_0(t),\textbf{R}_0(0))$ between the inherent structures at times $t$ and $0$.[]{data-label="fig:rearrange"}](PEL.png "fig:"){width="0.48\columnwidth"} ![(a) Density of vibrational modes $D(\omega,0)$ for $K=5\times10^{-5.50}$ (blue) and $5\times10^{-5.49}$ (red). (b) Schematic diagram of the energy landscape with axes, the total energy per atom $E=U+K$ and atomic configuration $\textbf{R}$. The configurations $\textbf R_0(t_1)$ and $\textbf R_0(t_2)$ represent the inherent structures ([*i.e.*]{}, the nearest local potential energy minima) of the vibrating system at times $t_1$ and $t_2$, respectively. $\Delta U$ is the difference in the potential energy per atom and $U^*$ is the energy barrier between the configurations $\textbf R_0(t_1)$ and $\textbf R_0(t_2)$. (c) $\Delta U(\textbf{R}_0(t),\textbf{R}_0(0))$ between the inherent structures at times $t$ and $0$ for $K=5\times10^{-5.50}$ (blue circles) and $5\times10^{-5.49}$ (red pluses). The inset shows the root-mean-square deviation (RMSD) $d(\textbf{R}_0(t),\textbf{R}_0(0))$ between the inherent structures at times $t$ and $0$.[]{data-label="fig:rearrange"}](rearrange2-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"}
We now investigate the cause for the qualitative change in the vibrational response for $K > K_r$. To do this, for each fluctuating configuration $\textbf{R}(t)$, we calculate the corresponding inherent structure, or the configuration of the nearest local potential minimum $\textbf{R}_0(t)$, using conjugate gradient energy minimization. A schematic illustrating the potential energy landscape is shown in Fig. \[fig:rearrange\] (b). In Fig. \[fig:rearrange\] (c), we plot the difference in the potential energy per atom $\Delta U(\textbf{R}_0(t),\textbf{R}_0(0))
=|U(\textbf R_0(t))-U(\textbf R_0(0))|$ as a function of time for $K <
K_r$ and $K>K_r$. When $K < K_r$, $\Delta U \sim 10^{-14}$ for all times, indicating that the system remains in the basin of the inherent structure at $t=0$. For $K > K_r$, $\Delta U$ jumps from $\sim
10^{-14}$ to $\sim 10^{-3}$ near $t^* \sim 80$, indicating that the system transitions from the basin of the inherent structure at $t=0$ to that of a different inherent structure at $t^*$ following an atomic rearrangement. We also used Eq. \[rmsd\] to calculate the root-mean-square deviation between the inherent structures $\textbf{R}_0(0)$ and $\textbf{R}_0(t)$ at times $0$ and $t$ during the vibrations. In the inset of Fig \[fig:rearrange\] (c), we show that $\Delta U$ and $d$ display similar behavior. For $K < K_r$, $d \sim 10^{-6}$ for all times. For $K > K_r$, near $t^* \sim 80$, $d$ jumps from $\sim 10^{-6}$ to $\sim
10^{-2}$ again indicating that an atomic rearrangement occurs at $t^*$. One can also see that subsequent rearrangements occur at later times, which are indicated by jumps in $\Delta U$ and $d$. These results emphasize that atomic rearrangements induce significant redistribution of energy from the fundamental mode to other frequencies.
![Loss $L(0)$ (Eq. \[loss\]) for the first time interval $t=0$ versus the initial kinetic energy per atom $K$. The solid vertical line indicates $K_r \approx 5\times10^{-5.49}$ at which the first atomic rearrangement occurs. The dashed horizontal line indicates the loss threshold $L_{l}$ for a harmonic oscillator with a measurement time $\delta t$ that deviates from an integer.[]{data-label="fig:loss"}](loss-eps-converted-to.pdf){width="0.95\columnwidth"}
We quantify the leakage of energy from the fundamental mode to other frequencies over the first time interval $t=0$ by calculating the loss $L(0)$ (defined in Eq. \[loss\]) as a function of $K$ in Fig. \[fig:loss\]. We calibrate the measurement of the loss by studying perfect cosine oscillations of the velocity of a single atom over a tape length of $\delta t$. Since in general $\delta t$ is not an exact integer multiple of the oscillation period, the loss $L_l
\sim 10^{-4.5}$ we measured for a cosine wave is small, but nonzero. We find that the lower threshold for the loss $L_l$ does not affect the results we present. In Fig. \[fig:loss\], we show that at small $K$, $L(0) \sim L_l$ and $L(0)$ increases smoothly with increasing $K$ until reaching $0.04$ near $K_r$. At $K_r$, the loss jumps to $L(0) \sim 1$, indicating the onset of atomic rearrangements, and remains there for $K
> K_r$.
![(a) Loss $L(t)$ versus the time interval $t$ for kinetic energy per atom $K=9.98\times 10^{-6}$ (blue circles), $1.20\times 10^{-5}$ (orange squares), and $1.26\times 10^{-5}$ (red triangles). (b) Quality factor $Q(0)$ for the first time interval $t=0$ as a function of $K$. The dashed vertical line indicates $K_{nl} \approx 1.15\times 10^{-5}$ at which $Q(0) \rightarrow \infty$. The vertical dotted line indicates $K_r \approx 1.26\times 10^{-5}$, above which atomic rearrangements occur.[]{data-label="fig:q"}](q1-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"} ![(a) Loss $L(t)$ versus the time interval $t$ for kinetic energy per atom $K=9.98\times 10^{-6}$ (blue circles), $1.20\times 10^{-5}$ (orange squares), and $1.26\times 10^{-5}$ (red triangles). (b) Quality factor $Q(0)$ for the first time interval $t=0$ as a function of $K$. The dashed vertical line indicates $K_{nl} \approx 1.15\times 10^{-5}$ at which $Q(0) \rightarrow \infty$. The vertical dotted line indicates $K_r \approx 1.26\times 10^{-5}$, above which atomic rearrangements occur.[]{data-label="fig:q"}](q2-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"}
In Fig. \[fig:loss\], we showed the loss for only the first time interval $t=0$. We characterize the time-dependent loss in Fig. \[fig:q\] (a). (We also include the variation of the density of states $D(\omega,t)$ with time $t$ in Appendix \[sec:appendix\_dissipating\].) We identify three distinct regimes. First, when $K < K_{nl}$, with $K_{nl} \approx 1.15 \times 10^{-5}$ the loss $L(t)$ is small and does not increase with $t$, and thus $Q
\rightarrow \infty$. In the second regime, for intermediate $K_{nl} <
K < K_r$ (such as $K=1.20\times 10^{-5}$ in Fig. \[fig:q\] (a)), $L(t)$ initially increases with $t$ smoothly, generating a finite $Q$, but then $L(t)$ reaches a plateau and $Q \rightarrow \infty$ at long times. In the third regime, for $K>K_r$ (such as $K=1.26\times
10^{-5}$ in Fig. \[fig:q\] (a)), $L(t)$ increases with $t$ smoothly (indicating finite $Q$), until an atomic rearrangement event occurs and $L(t)$ jumps to a large value $L\sim 1$. $L(t)$ continues to increase after the first atomic rearrangement.
We evaluate $Q(0)$ for the first time interval $t=0$ using Eq. \[q\], and show the results as a function of $K$ in Fig \[fig:q\] (b). We find that $Q(0) \sim 2 \times 10^4$ for $K\sim 1.5\times 10^{-5}$ and $Q(0)$ increases with decreasing $K$. For $K \lesssim K_{r}$, $Q$ begins to increase sharply, diverging as $K \rightarrow K_{nl}$, indicating the behavior for a perfect linear resonator for $K < K_{nl}$. These results indicate that to design a high-$Q$ metallic glass resonator, one needs to fabricate a system with a large value for $K_r$ and operate the resonator at $K < K_r$.
![(a) and (c) Total potential energy per atom $U$ as a function of time from $0$ to $n\delta t$, where $n=20$. The times $t_-$ and $t_+$ indicate successive times at which $U$ is at a minimum during each half cycle. The horizontal solid line in (a), connecting the times $t_-$ and $t_+$, indicates the times for each half cycle at which the RMSD, $d_{\rm point}(\textbf{R}(t_-),\textbf{R}(t_+))$, is calculated in panel (b). In (b), we show $d_{\rm point}$ for $K=9.98\times 10^{-6}$ in the regime $K < K_{nl}$ (blue), $K=1.20\times 10^{-5}$ in the regime $K_{nl} < K < K_r$ (orange), and $K=1.26\times 10^{-5}$ in the regime $K > K_r$ (red). The horizontal solid lines in (c) indicate the times $t_i$ and $t_j=t_++t_--t_i$ during each half cycle that are used to calculate the RMSD, $d_{\rm path} =
\langle d(\textbf R(t_i),\textbf R(t_j)) \rangle_{t_i}$, where the angle brackets indicate an average over the $1/2\Delta t \sim 300$ uniformly spaced times $t_i$. $d_{\rm path}$ in panel (d) is shown for the same values of $K$ as in (b). (e)-(g) Schematic diagram that shows the system trajectories (solid black lines) in the potential energy landscape (shaded contours from high (orange) to low (blue) energies) for the three regimes of oscillations ($1$: $K < K_{nl}$, $2$: $K_{nl} < K < K_r$, and $3$: $K > K_r$). The red solid line indicates the probability of an atomic rearrangement versus $K$. []{data-label="fig:regimes"}](displacement-eps-converted-to.pdf "fig:"){width="0.92\columnwidth"} ![(a) and (c) Total potential energy per atom $U$ as a function of time from $0$ to $n\delta t$, where $n=20$. The times $t_-$ and $t_+$ indicate successive times at which $U$ is at a minimum during each half cycle. The horizontal solid line in (a), connecting the times $t_-$ and $t_+$, indicates the times for each half cycle at which the RMSD, $d_{\rm point}(\textbf{R}(t_-),\textbf{R}(t_+))$, is calculated in panel (b). In (b), we show $d_{\rm point}$ for $K=9.98\times 10^{-6}$ in the regime $K < K_{nl}$ (blue), $K=1.20\times 10^{-5}$ in the regime $K_{nl} < K < K_r$ (orange), and $K=1.26\times 10^{-5}$ in the regime $K > K_r$ (red). The horizontal solid lines in (c) indicate the times $t_i$ and $t_j=t_++t_--t_i$ during each half cycle that are used to calculate the RMSD, $d_{\rm path} =
\langle d(\textbf R(t_i),\textbf R(t_j)) \rangle_{t_i}$, where the angle brackets indicate an average over the $1/2\Delta t \sim 300$ uniformly spaced times $t_i$. $d_{\rm path}$ in panel (d) is shown for the same values of $K$ as in (b). (e)-(g) Schematic diagram that shows the system trajectories (solid black lines) in the potential energy landscape (shaded contours from high (orange) to low (blue) energies) for the three regimes of oscillations ($1$: $K < K_{nl}$, $2$: $K_{nl} < K < K_r$, and $3$: $K > K_r$). The red solid line indicates the probability of an atomic rearrangement versus $K$. []{data-label="fig:regimes"}](3_regimes.png "fig:"){width="0.9\columnwidth"}
To understand the nature of oscillations in metallic glass resonators (e.g. $U(t)$ in Fig. \[fig:regimes\] (a) and (c)), we calculate the point RMSD $d_{\rm point}(t)$ and path RMSD $d_{\rm path}(t)$ in Fig. \[fig:regimes\] (b) and (d). $d_{\rm point}(t)$ quantifies the deviations in the configurations that are the closest to the potential energy minimum in each half cycle, and $d_{\rm path}(t)$ quantifies the deviations in the configurations at corresponding times before and after the turning point of the oscillation during each half cycle.
When $K < K_{nl}$ (regime $1$), the system is in the linear response regime, the path in configuration space followed by the resonator is nearly parabolic as shown in Fig. \[fig:regimes\] (e), and both $d_{\rm point}$ and $d_{\rm path} \sim 0$. When the system enters the nonlinear regime, $K_{nl} < K <K_r$ (regime $2$), $d_{\rm point}$ and $d_{\rm path}$ (as well as the loss $L(t)$ in Fig. \[fig:q\] (a)) increase with $t$ until $t^* \approx 790$. For $t >t^*$, $d_{\rm
point}$, $d_{\rm path}$, and $L(t)$ reach plateaus and then remain nearly constant in time. This behavior indicates that the resonator is undergoing [*nonlinear*]{} oscillations, in which the system does not retrace the same configurations above and below the turning point for each half cycle, but the system is nearly reversible. (See Fig. \[fig:regimes\] (f).) In the third regime $K > K_r$, the probability for an atomic rearrangement increases strongly. In this regime, the system can traverse the saddle points, enter the basins corresponding to new potential energy minima, and is thus microscopically irreversible. The three regimes describing resonator oscillations are summarized in Fig \[fig:regimes\] (e)-(g).
Methods to increase $K_r$ and enhance $Q$ {#sec:results_improvement}
-----------------------------------------
![(a) The ensemble-averaged kinetic energy per atom $\langle K_r \rangle$ above which the first atomic rearrangement occurs versus system size $N$ for rapidly and slowly cooled glasses with $R=10^{-2}$ (blue circles) and $10^{-5}$ (green pluses), respectively. In each case, $\langle K_r \rangle$ is averaged over $20$ independent samples. The slopes of the dashed lines are $-k$. (b) Ensemble-averaged pure shear strain $\langle \gamma_r \rangle$ above which the first atomic rearrangement occurs during athermal quasistatic pure shear versus system size $N$ for glasses prepared using $R=10^{-2}$ (blue circles), $10^{-3}$ (black squares), $10^{-4}$ (red triangles), and $10^{-5}$ (green pluses). $\langle \gamma_r \rangle$ is averaged over $500$ independent samples. The negative $\kappa$-values give the slopes of the dashed lines.[]{data-label="fig:scaling"}](Kr_N-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"} ![(a) The ensemble-averaged kinetic energy per atom $\langle K_r \rangle$ above which the first atomic rearrangement occurs versus system size $N$ for rapidly and slowly cooled glasses with $R=10^{-2}$ (blue circles) and $10^{-5}$ (green pluses), respectively. In each case, $\langle K_r \rangle$ is averaged over $20$ independent samples. The slopes of the dashed lines are $-k$. (b) Ensemble-averaged pure shear strain $\langle \gamma_r \rangle$ above which the first atomic rearrangement occurs during athermal quasistatic pure shear versus system size $N$ for glasses prepared using $R=10^{-2}$ (blue circles), $10^{-3}$ (black squares), $10^{-4}$ (red triangles), and $10^{-5}$ (green pluses). $\langle \gamma_r \rangle$ is averaged over $500$ independent samples. The negative $\kappa$-values give the slopes of the dashed lines.[]{data-label="fig:scaling"}](gamma1-eps-converted-to.pdf "fig:"){width="0.95\columnwidth"}
In Sec. \[sec:results\_dissipation\], we showed that even single atomic rearrangements give rise to significant loss and finite values of $Q$. Thus, to generate high-$Q$ resonators, one must maximize $\langle
K_r \rangle \sim \langle U^* \rangle$, yielding systems with large potential energy barriers. In this section, we describe studies of the ensemble-averaged $\langle K_r \rangle$ versus system size $N$ and cooling rate $R$, averaged over typically $20$ independently generated initial conditions. For each $R$ and $N$, we excite the resonator along the fundamental mode corresponding to the lowest eigenvalue of the dynamical matrix $\omega_1$ and monitor the system during the first time interval $t=0$ as a function of $K$.
In Fig. \[fig:scaling\] (a), we show that the ensemble-averaged kinetic energy per atom at which the first atomic rearrangement occurs, $\langle K_r \rangle$, decreases with increasing $N$. We find that $\langle K_r \rangle \sim N^{-2 k}$, where $k \approx 0.68$ for $R = 10^{-2}$ and $\approx 0.60$ for $R=10^{-5}$. $\langle K_r
\rangle$ is smaller for rapidly compared to slowly cooled glasses, since $\langle U^* \rangle$ decreases with increasing $R$ [@chen1971elastic; @fan2014thermally; @fan2017particle]. These results emphasize that $Q$ can be increased by making resonators smaller and preparing them using slower cooling rates. For example, experimental studies of Pt-based metallic glass micro-cantilevers have reported that the quality factor can be increased by more than a factor of $3$ after annealing [@kanik2014high].
We can also compare the kinetic energy per atom $\langle K_r \rangle$ required to induce the first atomic rearrangement in thermally vibrating systems to the characteristic shear strain $\langle \gamma_r \rangle$ required to induce the first atomic rearrangement in systems driven by athermal quasistatic (AQS) shear. To calculate $\langle \gamma_r \rangle$, we confine $N$ atoms interacting via the Kob-Andersen model to cubic boxes with periodic boundary conditions in the $x$-, $y$-, and $z$-directions. We cool the samples from temperature $T_0$ to zero using a linear ramp over a range of cooling rates $R$ from $10^{-5}$ to $10^{-2}$. For each sample, we perform AQS pure shear at fixed volume $V$, i.e. at each strain step, we expand the box length and move all atoms affinely in the $x$-direction by a small strain increment $\delta\gamma_x=\delta\gamma=10^{-4}$ and compress the box length and move all atoms affinely in the $y$-direction by the same strain increment $\delta\gamma_y=-\delta\gamma$. Following each strain step, we perform conjugate gradient energy minimization at fixed volume. To measure $\langle \gamma_r \rangle$, we employ the method we developed previously [@fan2017effects] to unambiguously determine whether an atomic rearrangement occurs with an accuracy on the order of numerical precision. As shown in Fig. \[fig:scaling\] (b), we find that $\langle \gamma_1 \rangle \sim N^{-\kappa}$ also decreases with increasing $N$, where the system-size scaling exponent $\kappa \sim 0.6$-$0.68$ is again only weakly dependent on the cooling rate. These results for the system-size scaling exponents in athermal quasistatic shear are consistent with dimensional arguments that suggest $\kappa \sim 2k$, and thus athermal quasistatic shear can be used to understand the low temperature properties of glasses.
Using these results, we can estimate the strains below which resonators can operate in the linear response regime. For the slowest cooling rate $R=10^{-5}$, we find that $\log_{10} \langle \gamma_r
\rangle \approx -2k \log_{10} N + \gamma_{\infty}$, where $2k \approx
0.60$, $\gamma_{\infty} \approx 0.11$, $N = \rho(l/D)^3$, the number density $\rho \approx 1.2$, $D \approx 3.7$ Å is a typical atomic diameter for Ni$_{80}$P$_{20}$ metallic glasses [@sheng2012relating] (which is the subject of the Kob-Andersen model), and $l$ is a characteristic lengthscale of the resonator. We find that $\langle \gamma_r \rangle \sim 5 \times
10^{-4}$ for a resonator with $l \sim 20$ nm, whereas $\langle
\gamma_r \rangle \sim 3 \times 10^{-8}$ for a resonator with $l \sim
5$ $\mu$m [@poncharal1999electrostatic; @purcell2002tuning]. Micron-scale metallic glass resonators have been fabricated as hemispherical shells [@kanik2015metallic] and as cantilevers [@kanik2014high]. In addition, strains in the range from $10^{-7}$ to $10^{-4}$ have been used in measurements of internal friction in metallic glass resonators [@khonik1996internal]. Our results emphasize that nano-sized metallic glass resonators operating in the small strain regime regime (e.g. $< 10^{-7}$) are promising high-$Q$ materials.
Comparison between crystalline and amorphous resonators {#sec:results_X}
-------------------------------------------------------
In Sec. \[sec:results\_improvement\], we showed that the characteristic kinetic energy per atom $\langle K_r \rangle$ above which atomic rearrangements occur increases modestly with decreasing cooling rate. Further, we know that crystalline ordering increases with decreasing cooling rate. Does this imply that crystalline metals are higher-$Q$ materials compared to amorphous metals? In this section, we calculate $\langle K_r \rangle$ for resonators made from single crystal, polycrystalline, and defected crystalline materials and compare these results to those for resonators made from homogeneously amorphous samples.
Crystalline metals often contain slip planes, dislocations, grain boundaries, and other defects, and the defect density typically increases with increasing cooling rate. To generate crystalline materials with defects in simulations, we will again use the Kob-Andersen model, but with monodisperse atoms, $\epsilon_{AA} =
\epsilon_{AB} = \epsilon_{BB} =1.0$ and $\sigma_{AA} = \sigma_{AB} =
\sigma_{BB} = 1.0$, to enhance crystallization. We will employ the same protocol as discussed in Sec. \[sec:methods\] to generate thin-bar resonators with $N=2000$ and aspect ratio $L_x:L_y:L_z =
6:1:2$ over a range of cooling rates from $R=10^{-4}$ to $10^{2}$. The method of excitation and measurement of the loss and $K_r$ are also the same as described in Sec. \[sec:methods\].
![Snapshots of thin-bar-shaped resonators with $N=2000$ monodisperse atoms obtained using cooling rates (a) $R=1.2 \times 10^{-4}$, (b) $1.2 \times 10^{-3}$, (c) $3 \times 10^{-3}$, (d) $6 \times 10^{-3}$, (e) $1.2 \times 10^{-2}$, and (f) $1.2 \times 10^{-1}$ in periodic boundary conditions prior to applying the excitations. Atoms with crystalline (FCC or HCP) order are colored green, while amorphous atoms are colored gray.[]{data-label="fig:Xdefects"}](defects.png){width="0.99\columnwidth"}
![(a) Fraction of crystalline atoms $\langle f_X \rangle$, (b) total dislocation length $\langle L_D \rangle$, and (c) kinetic energy per atom $\langle K_r \rangle$ above which atomic rearrangements occur as a function of cooling rate $R$ for resonators made using the monodisperse Kob-Andersen model. The ensemble averages are obtained by averaging over at least $10$ independent samples. The dotted horizontal lines in (b) and (c) show $\langle L_D \rangle$ and $\langle K_r \rangle$ for four nearly perfect, crystalline thin bars with specifically placed defects. For example, the red dotted lines represent the thin-bar sample in Fig. \[fig:Xmanual\]. The black dashed line in (c) shows $\langle K_r \rangle$ for a thin bar with perfect FCC order. The solid vertical lines in panels (a)-(c) give approximate boundaries between the four regimes of vibrational response as a function of cooling rate $R$.[]{data-label="fig:Xdata"}](Xdata-eps-converted-to.pdf){width="0.96\columnwidth"}
Snapshots of the zero-temperature thin-bar resonators generated using six different cooling rates are shown in Fig. \[fig:Xdefects\] (with periodic boundary conditions and prior to adding excitations). We use the Common Neighbor Analysis (CNA) [@honeycutt1987molecular] to identify atoms that occur in crystalline (either face-centered cubic (FCC) or hexagonal close packed (HCP)) and amorphous environments in the thin bars. In Fig. \[fig:Xdata\] (a), we show that the ensemble-averaged fraction of crystalline atoms $\langle f_X \rangle$ decreases with increasing $R$. $\langle f_X \rangle$ is nearly $90\%$ when $R=1.2 \times 10^{-4}$ and $\langle f_X \rangle = 0$ for $R=1.2
\times 10^{-1}$. Near the critical cooling rate $R_c \approx
10^{-2.5}$, the system contains a roughly equal mixture of atoms in crystalline and amorphous environments.
To quantify disorder in the thin-bar samples, we used the Dislocation Extraction Analysis (DXA) tool within the OVITO software library [@stukowski2009visualization]. DXA allows us to measure the total dislocation length $L_D$, which gives the sum of the magnitudes of the Burgers vectors for each dislocation in the sample. For $R \ll 1$, we expect few detects, and thus $\langle L_D \rangle
\rightarrow 0$. In Fig. \[fig:Xdata\] (b), for small $R$, we show that $L_D$ increases with cooling rate $R$ [@blanter2007internal; @nowick2012anelastic]. When $R> 3 \times
10^{-3}$, $\langle L_D \rangle$ drops sharply since the thin-bar samples include mixtures of atoms with crystalline and amorphous environments. $L_D \rightarrow 0$ when the sample becomes completely amorphous.
To determine the vibrational response, we excite the fundamental mode $\omega_1$ for each sample, and measure $\langle K_r \rangle$ as a function of $R$. The behavior for $\langle K_r \rangle$ can be divided into four regimes. (See Fig. \[fig:Xdata\] (c).) First, at low cooling rates $R \lesssim 3 \times 10^{-3}$ (regime I), the systems are mostly crystalline with sparse dislocations. In this regime, as $R$ increases, more dislocations are formed and $\langle L_D\rangle$ increases, which causes $\langle K_r \rangle$ to decrease. In regime II, at intermediate cooling rates $3 \times 10^{-3} \lesssim R
\lesssim 1.2 \times 10^{-2}$, $\langle f_X \rangle$ drops sharply and the thin bars contain mixtures of crystalline and amorphous atoms. The additional boundaries between amorphous and crystalline regions of the system causes a larger decrease in $\langle K_r \rangle$ than at smaller $R$. In regime III, $1.2 \times 10^{-2} \lesssim R \lesssim
3 \times 10^{-1}$, the thin-bar resonators become homogeneously amorphous and metastable, causing $\langle K_r \rangle$ to increase by a factor of $\approx 4$. At the high cooling rates $R \gtrsim 3 \times 10^{-1}$ in regime IV, $K_r$ will decrease modestly with increasing $R$. For the Kob-Andersen bidisperse mixture, we already showed in Fig. \[fig:scaling\] (a) that $\langle K_r \rangle$ decreases by a factor of $\approx 3$ as $R$ is increased over three orders of magnitude. This local maximum in $\langle K_r \rangle(R)$ is interesting because it shows that there is a regime where amorphous resonators can have larger $Q$-values than partially crystalline resonators.
![Snapshots of thin-bar crystalline resonators with a specifically placed defect. In (a), we delete a row of atoms from a resonator with perfect FCC order and perform energy minimization, which yields the thin-bar resonator in (b). (c) We identify two dislocations in the resonator in (b) colored yellow and gray.[]{data-label="fig:Xmanual"}](manual0000.png "fig:"){width="0.32\columnwidth"} ![Snapshots of thin-bar crystalline resonators with a specifically placed defect. In (a), we delete a row of atoms from a resonator with perfect FCC order and perform energy minimization, which yields the thin-bar resonator in (b). (c) We identify two dislocations in the resonator in (b) colored yellow and gray.[]{data-label="fig:Xmanual"}](manual0001.png "fig:"){width="0.32\columnwidth"} ![Snapshots of thin-bar crystalline resonators with a specifically placed defect. In (a), we delete a row of atoms from a resonator with perfect FCC order and perform energy minimization, which yields the thin-bar resonator in (b). (c) We identify two dislocations in the resonator in (b) colored yellow and gray.[]{data-label="fig:Xmanual"}](manual0002.png "fig:"){width="0.32\columnwidth"}
In addition to studying the vibrational response of thin-bar resonators generated by cooling a high-temperature liquid into a solid, we also investigated the vibrational response of systems for which we started with perfect FCC crystalline thin bars and generated specifically placed defects. In particular, we generated four thin-bar samples that were initialized with perfect FCC order, and then we removed a slot with a width of one atom, depth of two atoms, and varying lengths along different directions in the sample. An example is shown in Fig. \[fig:Xmanual\]. We display $\langle L_D
\rangle$ and $\langle K_r \rangle$ for these four systems in Fig. \[fig:Xdata\] (b) and (c). These samples possess a range of $\langle K_r \rangle$: some values are larger than that of the rapidly cooled glass ($R=1.2 \times 10^{-1}$), while others are not. These results show that amorphous resonators can possess values of $\langle
K_r \rangle$ (and thus $Q$) that are comparable to those for crystalline samples. For example, the thin-bar resonator corresponding to the green dotted horizontal line in Fig. \[fig:Xdata\] (c) possesses a smaller $\langle K_r \rangle$ than that of the rapidly cooled glass, with a dislocation density $\langle L_D\rangle
/V \approx 2\times 10^{16}$ m$^{-2}$, which is similar to the value for crystalline metals with strong dislocations [@schafler2001measurement]. Since metallic glasses do not need to be annealed, can be molded into complex shapes, possess unique magnetic and biocompatibility properties [@li2012amorphous; @schroers2009bulk; @schroers2010processing; @kumar2009nanomoulding; @li2018atomic], and can possess comparable quality factors to crystalline metals [@kanik2014high], metallic glasses are promising materials for high-$Q$ applications.
conclusion {#sec:conclusion}
==========
In this article, we employ molecular dynamics simulations of model metallic glass resonators undergoing vibrations to quantify the intrinsic dissipation and loss mechanisms caused by thermal fluctuations and atomic rearrangements. Using thin-bar resonators generated over a wide range of cooling rates, we excite the fundamental mode corresponding to the lowest eigenfrequency $\omega_1$ of the dynamical matrix as a function of the kinetic energy per atom $K$. We find three regimes of vibration. In the linear response regime, $K<K_{nl}$, most of the energy of the vibrations remains in the fundamental mode, the loss is small, and $Q
\rightarrow \infty$ (since we do not consider coupling of the resonator to the environment). For $K_{nl}<K<K_r$, energy can leak from the fundamental mode to others at short times, but at sufficiently long times the leakage of energy to other frequencies stops, and thus $Q \rightarrow
\infty$ at long times. For $K > K_r$, one or more atomic rearrangements occur. In this regime, energy in the fundamental mode is completely redistributed to a large set of other frequencies, the loss is large, and $Q$ is finite. Thus, we show that $K_r$ determines the quality factor.
We find that $\langle K_r \rangle$ decreases as a power-law $N^{-k}$ with increasing system size $N$, where $k \approx 1.3$ decreases only modestly with decreasing $R$. We find similar results for the critical shear strain $\langle K_r \rangle \sim \langle \gamma_r \rangle ^2$ using athermal quasistatic shear deformation, where $\langle \gamma_r
\rangle$ is the characteristic strain above which atomic rearrangements begin to occur. Using these results, we estimate that $\langle \gamma_r \rangle \sim 10^{-8}$ for micron-sized resonators, and thus large $Q$-values can be obtained when these resonators are operated at $\gamma < \gamma_r$. We also measured $\langle K_r \rangle$ in thin-bar resonators with crystalline order and compared the vibrational response to that in amorphous resonators. We find that $\langle K_r \rangle$ is similar for amorphous resonators and those with significant crystalline order. In light of the fact that metallic glasses can be thermoplastically formed into complex shapes, possess unique magnetic and biocompatibility properties, and can achieve $Q$-values that are comparable to those for crystalline structures, metallic glasses are promising materials for micro- and nano-resonators.
Our results raise a number of interesting future directions. For example, we can investigate methods that involve mechanical deformation, not slower cooling rates or annealing methods, to increase $\langle K_r \rangle$ and move the sample to regions of configuration space with higher energy barriers between inherent structures. One possible approach is to apply athermal cyclic simple or pure shear deformation to samples that have been prepared using fast cooling rates. Recent studies have found that there is a finite critical strain amplitude for cyclic shear that marks the limit between reversible and irreversible atomic rearrangements in the large-system limit [@regev2015reversibility; @fiocco2014encoding; @leishangthem2017yielding]. Does this imply that cyclic shear training can find zero-temperature configurations for which $\langle K_r \rangle$ remains finite in the large-system limit? In addition, we can explore how the type of cyclic driving affects $\langle K_r \rangle$ and whether configurations can be trained in multiple directions simultaneously to increase $\langle K_r
\rangle$. Another future direction involves studies of the loss and quality factor when the resonator has clamped instead of free boundary conditions and when it is driven over a range of frequencies, not only the fundamental mode.
The authors acknowledge support from NSF MRSEC Grant No. DMR-1119826 (M. F. and C.O.) and NSF Grant Nos. CMMI-1462439 (M. F. and C.O.) and CMMI-1463455 (M.S.). This work was supported by the High Performance Computing facilities operated by, and the staff of, the Yale Center for Research Computing.
Length of time series {#sec:appendix_tapelength}
=====================
![The low-frequency regime for the density of vibrational modes $D(\omega,0)$ near the fundamental frequency $\omega_1
\approx 0.172$ (vertical dashed line) calculated using time series with two different total lengths, $\delta t=135$ (black circles) and $1350$ (blue exes), at $K=5\times10^{-6}$ in the linear response regime. In the inset, we show the loss $L(0)$ versus $K$ for the same time series in the main panel. The characteristic kinetic energy per atom above which an atomic rearrangement occurs are indicated: $K_r=1.62\times10^{-6}$ (black dotted line) for $\delta t=135$ and $K_r=1.26\times10^{-6}$ (blue dotted line) for $\delta t=1350$. []{data-label="fig:tapelength"}](tapelength-eps-converted-to.pdf){width="0.96\columnwidth"}
In the main text, we used a total run length of $\delta t=135$ to calculate the density of vibrational modes and loss for the first time interval $t=0$ in Figs. \[fig:spec\] and \[fig:loss\] (as well as all other time intervals). In this Appendix, we show results for $D(\omega,0)$ when $\delta t$ is increased by a factor of $10$ (keeping the sampling rate fixed). In Fig. \[fig:tapelength\], for $K<K_r$ in the linear response regime, we show that the peak value of $D(\omega_1,0)$ is unchanged for $\delta t = 135$ and $1350$, and thus $L(0)$ is nearly the same for the two values of $\delta t$. We know that the probability for an atomic rearrangement increases with time $\delta t$ at fixed $K$. Thus, in the inset to Fig. \[fig:tapelength\], we show that the loss $L(0)$ undergoes a discontinuous jump for $\delta t = 1350$ at a smaller $K$ than that for $\delta t = 135$. We find that $K_r \approx
1.62\times10^{-5}$ for $\delta t=135$ and $\approx
1.26\times10^{-5}$ for $\delta t=1350$. Thus, the precise value of $K_r$ depends on $\delta t$, but all of the results are qualitatively the same for different choices of $\delta t$.
Time-dependent density of vibrational modes {#sec:appendix_dissipating}
===========================================
{width="0.68\columnwidth"} {width="0.68\columnwidth"} {width="0.68\columnwidth"}
For the calculations of the density of vibrational modes and loss in the main text, we divided a long time series following the excitation of the resonator along the fundamental mode into $20$ time intervals of equal length $\delta t$. We showed the density of vibrational modes $D(\omega,0)$ for the first time interval (i.e. considering times from $0$ to $\delta t$) in Fig. \[fig:spec\]. In this Appendix, we calculate $D(\omega,t)$ for all $20$ time intervals. We show $D(\omega,t)$ for $K$ values in the three regimes, $K < K_{nl}$, $K_{nl} < K < K_r$, and $K > K_r$, which match those used in Fig. \[fig:q\] (a). For $K<K_{nl}$, there is minimal leakage of energy from the fundamantal mode $\omega_1 = 0.172$ and $Q \rightarrow
\infty$. In the regime $K_{nl}<K<K_r$, energy leaks from the fundamental mode at short times, but it stops for $t/\delta t \gtrsim
12$, and the system vibrates nonlinearly with finite loss, finite $Q$ for $t/\delta t \lesssim 12$ and $Q\rightarrow \infty$ for $t/\delta t
\gtrsim 12$. For $K>K_r$, strong energy leakage occurs due to an atomic rearrangement at $t/\delta t \approx 5$.
|
---
abstract: 'We propose a methodology for evaluating the performance of climate models based on the use of the Wasserstein distance. This distance provides a rigorous way to measure quantitatively the difference between two probability distributions. The proposed approach is flexible and can be applied in any number of dimensions; it allows one to rank climate models taking into account all the moments of the distributions. Furthermore, by selecting the combination of climatic variables and the regions of interest, it is possible to highlight the deficiencies of each of the models under study. The Wasserstein distance thus enables a comprehensive evaluation of climate model skill. We apply this approach to a selected number of physical fields, ranking the models in terms of their performance in simulating them, as well as pinpointing their weaknesses in the simulation of some of the selected physical fields in specific areas of the Earth.'
title: Ranking IPCC Models Using the Wasserstein Distance
---
Evaluation of climate model performance by benchmarking with reference datasets
Climate model ranking related to the choice of variables of interest
Highlighting model deficiencies through emphasis on climatic regions and variables
Introduction and motivation {#sec:intro}
===========================
Advanced climate models differ in the choice of prognostic equations and in the methods for their numerical solution, in the number of processes that are parametrized and the choice of the physical parametrizations, as well as in the way the models are initialized, to mention just their most important aspects. Comparing the performance of such models is still a major challenge for the climate modeling community [@Held2005]. Each model has its own strengths and weaknesses and, as a result, past reconstructions and future projections of climate necessarily come with model-dependent uncertainties.
Model inadequacies result from structural errors — certain processes are incorrectly represented or not represented at all — as well as from parametric uncertainties, i.e., the use of incorrect values of physical and other parameters [@Lucarini2013; @Ghil.Luc.2020]. Investigating the properties of multi-model ensembles is crucial for addressing climate modeling errors, while auditing climate models is essential for understanding which ones are more skillful in answering the specific climate question under study.
Testing model performance in order to advance climate modeling skill has led the community to pool its efforts within the Coupled Model Intercomparison Project (CMIP), which is currently in its sixth phase [@Eyring2016a]. Dozens of modeling groups have agreed by now on a concerted effort to provide numerical simulations with standardized experimental protocols representative of specified climate forcing scenarios.
The issue of best practices for model performance evaluation has naturally arisen in this setting. Such practices have concentrated essentially on either “metrics” or diagnostics. Performance metrics [@Gleckler2008] have been used to rank models according to specific scalar indices that summarize overall performance, but the evaluation criterion on which such an index is based appears to be somewhat arbitrary so far. Diagnostics, on the other hand, are process-based and designed to assess specific features of the climate system. have conducted recently an effort to bring together metrics and diagnostics in a standardized framework for climate model evaluation. Still, it seems highly desirable to have a scalar metric that summarizes the full information associated with model performance and that does satisfy the mathematical axioms associated with the concept and satisfied by the usual Euclidean distance. These axioms are listed in \[appa\] and they are satisfied by the root-mean-square distance, known as an L$_2$ metric in mathematics. The latter distance, though, is not appropriate for describing fully the difference between two distribution functions, while other metrics used in the climate sciences are not genuine distances, i.e., they do not satisfy the axioms above. We propose a genuine metric to assess a climate model’s skill by taking into account every moment of a distribution and measuring, in a much more satisfactory way, the gap between it and another distribution of reference than root-mean-square distance. The two distributions will be chosen here to describe model features, on the one hand, and the “real world,” on the other, with the latter distribution being based on either raw observations or a reanalysis thereof.
originally proposed the idea of using the Wasserstein distance (hereafter WD) [@Kantorovich.1942; @Dobrushin.1970] in the context of the climate sciences as a way to generalize the traditional concept of equilibrium climate sensitivity [@Ghil.Luc.2020] in the presence of a time-dependent forcing, such as seasonal or anthropogenic forcing. used the WD to compute the difference between the snapshot attractors of the model for different time-dependent forcings, providing a link between nonautonomous dynamical systems theory and optimal transport. used the WD to evaluate the skill of a stochastic parametrization for a fast-slow system. Please see \[appa\] for further background on the WD.
The WD will be calculated in a phase space defined by the physical fields we wish to take into account and it is therewith a well-suited candidate for a comprehensive and flexible way to evaluate a climate model’s statistical skill. A well-known WD drawback consists in its computational requirements, which increase dramatically with the number of points needed to construct the distributions. In our methodology, following and , these requirements are greatly reduced through data binning on a grid. Doing so one switches from the distance between distributions of points to the distance between the measures computed on the distributions themselves, which reduces the effective sample size for each distribution. The WD-based methodology helps complement and refine the existing tests already applied in climate modeling studies, such as the space-time ranking of model performances by , with respect to the root-mean-square-error of the median of an ensemble, with observations used as expected values, or weighting schemes like in . Data are presented in Sec. \[Dat\], methods in Sec. \[WasDis\], results in Sec. \[Rank\], and conclusions in Sec. \[sec:conclude\].
Data {#Dat}
====
The WD methodology is presented in Sec. \[WasDis\]. It is applied here to three climate fields:
- Near-surface air temperature;
- Precipitation; and
- Sea ice cover, computed from the sea ice area fraction.
The corresponding daily mean fields are available in the CMIP5 simulations for historical and RCP85 forcings [@Taylor2012] and they are ranked with respect to the distance from reference daily datasets, specifically European Centre for Medium-Range Weather Forecasts Re-Analysis (ERA) Interim for the temperature [@Dee2011]; Global Precipitation Climatology Project (GPCP) for the precipitation [@Adler2003]; and Ocean and Sea Ice - Satellite Application Facility (OSI-SAF) for the sea ice cover [@EUMETSAT2017]. In order to further support the comparison and provide a benchmark, we analyzed the WD with respect to additional reanalysis datasets, such as National Center of Environmental Prediction (NCEP) Reanalysis 2 case [@Kanamitsu2002], or observations, such as Global Precipitation Climatology Centre (GPCC) data [@Ziese2018].
The fields are averaged on four distinct domains: (i) Global; (ii) Tropics – defined as the region between $30$ S and $30$ N; (iii) Northern extratropics: from $30$ N to $90$ N; and (iv) Arctic – used only for sea ice extent. While temperature and precipitation analyses involve a total of 30 models, taking into account sea ice extent allows to analyze just 22 models, due to available datasets. The time range spans 18 years, from January 1st, 1997 to December 31st, 2014. After the spatial averaging, the model datasets are obtained by concatenating the historical runs, from 1997 to 2005, and the RCP85 runs, from 2006 to 2014. The acronyms of the models that participated in CMIP5 and were used here appear in the figures below and are given in \[appb\].
The samples used in the WD calculations are drawn by performing a discretization of the phase space involved in each separate test. To do so, a regular grid is superposed over all the datasets used in the test and its upper and lower limits, respectively, are fixed slightly above and below the maximum and minimum values among all the datasets used in it. Each dimension of the grid is then equally divided into 20 intervals; this yields $20^n$ $n$-dimensional cubes, where $n$ is the number of fields taken into account in the test. These $20^n$ hypercubes provide the sample for each test. The results we present here are weakly sensitive to the specifics of the gridding. Nonetheless, a too coarse gridding removes a lot of the information we want to retain and analysis; a too fine gridding, instead, increases substantially the computing requirements, without making much statistical sense.
In order to highlight the flexibility and reliability of the method, we are going to calculate the WD distances in one-, two- and three-dimensional phase space, and work with different field combinations averaged over distinct areas of the Earth.
Wasserstein distance (WD) {#WasDis}
=========================
Our objective is to create a ranking of the CMIP5 IPCC models based on their skill to reproduce the statistical properties of selected physical quantities. The reference distribution for these quantities is given by reanalysis and observational datasets, as explained in Sec. \[Dat\]; their WD to these datasets [@Kantorovich.1942; @Villani2009] is a measure of the models’ ability to reproduce these reference distributions. One can also describe this distance as the minimum “effort” to morph one distribution into the other [@Monge1781]. We present below a very simplified account of the theory.
The optimal transport cost [@Villani2009] is defined as the minimum cost to move the set of points from one distribution to another into an n-dimensional phase space. In the case of two discrete distributions, we write their measures $\mu$ and $\nu$ as $$\mu =\sum\limits_{i=1}^n \mu_i \delta_{x_i} , \qquad
\nu =\sum\limits_{i=1}^n \nu_i \delta_{y_i} ;$$ here $\delta_{x_i}$ and $\delta_{y_i}$ are Dirac measures associated with a pair of points $(x_i, y_i)$, whose fractional mass is $(\mu_i, \nu_i)$, respectively, and $\sum_{i=1}^n \mu_i=\sum_{j=1}^n \nu_j=1$. $n$ is the number of dimensions in the phase space in which we compute the WD. Using the definition of Euclidean distance $$d(\mu,\nu)=\left[ \sum\limits_{i=1}^n (x_i-y_i)^2 \right] ^ {\frac{1}{2}} ,$$ we can write down the quadratic WD for discrete distributions: $$W_2(\mu,\nu)= \left\lbrace \inf_{\gamma_{ij}} \sum\limits_{i,j} \gamma_{ij} [d(x_i,y_j)]^2 \right\rbrace ^ {\frac{1}{2}} . \label{wd2discrete}$$ where $\gamma_{ij}$ is the fraction of mass transported from $x_i$ to $y_j$ and $d(x_i,y_j)$ is the Euclidean distance between a single pair of locations.
We perform the Ulam discretization described in Sec. \[Dat\] — i.e. data binning on a grid chosen to have a resolution of 20 intervals per side, as mentioned above — that allows us shift from the distance between different distributions of points to the distance between the measures related to those distributions. We thus proceed to quantify to what extent the measure of the observations and reanalysis from Sec. \[Dat\], projected on the variables of interest, differs from the corresponding measures for the climate models.
The estimate of the coarse-grained probability of being in a specific grid box is given by the time fraction spent in that box [@Ott1993; @Strogatz2015]. In fact, the WD does provide robust results even with a very coarse grid [@Vissio2018a; @Vissio2018b]. Therefore, in the case at hand, the locations $x_i$ and $y_j$ will indicate the cubes’ centroids, while $\gamma_{ij}$ indicate the corresponding densities of points. To further simplify the computations, we exclude all the grid boxes containing no points at all. Finally, we “renormalize” the densities, dividing the value obtained by the number of grid intervals per side; therefore, the one-, two- and three-dimensional WDs take values between a minimum of $0$ and a maximum equal to $1$, $\sqrt{2}$ and $\sqrt{3}$, respectively.
We used a suitably modified version of the Matlab software written by G. Peyré — available at <http://www.numerical-tours.com/matlab/optimaltransp_1_linprog/> — to perform the calculations. The modifications include the data binning and the estimation of the measures, as well as adapting to a dimension $n \ge 2$.
Ranking the models {#Rank}
==================
Figure \[Figure\_1\] shows the WD calculated in the two-dimensional phase space composed by the temperature and precipitation fields, averaged over the whole Earth and the Tropics, for each CMIP5 model. In order to provide a benchmark, we chose to include the WD results between the NCEP reanalysis and the references of ERA and GPCP presented in Sec. \[Dat\] for the two fields, respectively.
![Two-dimensional Wasserstein distance (WD) for the temperature and precipitation fields, averaged over the globe (horizontal axis) and over the Tropics (vertical axis). The acronyms of the models used are spelled out in \[appb\].[]{data-label="Figure_1"}](./Figure_1){width="\textwidth"}
Somewhat surprisingly, the NCEP reanalysis yields the largest values in both distances. Thus, the average CMIP5 distance to the ERA $\otimes$ GPCP reference is $0.149$, while the NCEP distance is $0.259$, exceeded only by the value $0.264$ given by the MIROC5 model; see \[appb\] for the list of models. Note that the one-dimensional WDs of the NCEP Reanalysis for the globally averaged temperature and precipitation equal 0.033 and 0.255, respectively. Given the well-known difficulties with simulating the very rough precipitation field by using the still fairly coarse CMIP5 models [@Neelin.2013; @Mehran.2014], it is natural to assume that both the ERA and NCEP reanalyses are mostly inadequate in representing the statistics of precipitation. The great discrepancy in WD between the distribution of reference and the NCEP Reanalysis points to the overall accuracy reached by CMIP5 simulations when dealing with global averages of temperatures and precipitation.
We evaluate next the problems still encountered by CMIP5 models in reproducing key aspects of tropical dynamics [@Tian2020]. Averaging the data over the Tropics, we obtain the ranking on the vertical axis in Fig. \[Figure\_1\]. The large values of the WD distances equal on average $0.173$, excluding the NCEP Reanalysis, and underline the poorer CMIP5 model performances in this region. With few exceptions, the models seem less reliable in the Tropics, where three of the models do exceed the NCEP Reanalysis distance.
![One-dimensional WD for precipitation averaged over the Northern extratropics (from $30$ N to $90$ N) on the horizontal axis and over the Tropics (from $30$ S to $30$ N on the vertical axis).[]{data-label="Figure_2"}](./Figure_2){width="\textwidth"}
![ Same as Fig. \[Figure\_2\] but for the temperature field.[]{data-label="Figure_3"}](./Figure_3){width="\textwidth"}
Focusing on the relative performance of temperature and precipitation in the Tropics vs the Northern Hemisphere extratropics (30 N–90 N), Figs. \[Figure\_2\] and \[Figure\_3\] illustrate one-dimensional WDs computed in the former vs the latter region. Using the diagonal line indicating equal values for the two distances as a reference, we can easily check in Fig. \[Figure\_2\] that, for all CMIP5 models, the precipitation field is less well reproduced in the Tropics than in the extratropics: it is well known that it is extremely challenging to reproduce accurately the statistics of by-and-large convection-driven precipitation, since the choice of the parametrization schemes and their tuning plays an essential role. The situation for the temperature field is similar but less uniformly so: while in Fig. \[Figure\_2\] all the results cluster above the diagonal but roughly below WD $\simeq 0.2$, the scatter in Fig. \[Figure\_3\] is larger, with some results below the diagonal and some between $0.2 \lessapprox {\mathrm{WD}} \lessapprox 0.3$.
Figure \[Figure\_4\] shows the scatter diagram of one-dimensional WDs for the precipitation in the Tropics vs the WDs of sea ice extent in the Arctic. Arctic sea ice cover is a very important indicator of the state of both hydrosphere and cryosphere, as well as of their mutual coupling; it is overestimated in CMIP5 models during the winter and spring seasons [@Randall2007; @Flato2013].
![One-dimensional WDs of average precipitation in the Tropics vs the average sea ice extent in the Arctic.[]{data-label="Figure_4"}](./Figure_4){width="\textwidth"}
Figure \[Figure\_4\] demonstrates that the sea ice cover in the models is closer to the observations than the tropical precipitation in 12 CMIP5 models out of the 22 examined. Nevertheless, 7 models better describe tropical precipitation than sea ice extent in the Arctic, while 3 models have a similar — and relatively low — WD for both fields. This test indicates that a correct representation of the statistics of these two fields is still quite challenging across the spectrum of climate models at the present time.
We compare next the performance of the CMIP5 models with respect to three different rankings. First, the three-dimensional WD is computed taking into account three physical quantities: globally averaged temperature and precipitation, along with sea ice extent in the Arctic. Note that, to ease the interpretation of Fig. \[Comparison\], the models are listed on the vertical axis according to the rank provided by this methodology.
The model ranking introduced herein is further compared with the rankings based on the first two moments of the distribution of reference. For each of the three physical quantities above, we compute the normalized mean, taking the absolute value of the difference between the mean of the distribution of the model field and that of the reference field, and dividing this difference by the standard deviation of the distribution of reference. The three means for the three fields are then averaged and the same procedure is repeated for the normalized standard deviation.
We can see that the models’ performance is quite different depending on the ranking being used. As an example, we focus on the BCC-CSM1.1 and BCC-CSM1.1-m models. The ranking based on the mean shows a rather good performance for both, with positions 7 and 10, respectively; nevertheless, they occupy positions 16 and 21 in the WD ranking. The latter low positions are due to their bad performances when it comes to standard deviation, where the two come last.
The reverse instance is also clear by looking at those models that, while performing well in terms of variability, occupy lower rankings based on the WD due to their poor performance in the mean; see, for instance, the case of MPI-ESM-MR, with position 1 in the standard deviation, 8 in WD, and 15 in the mean. The WD score accounts for the information carried by the whole distribution — i.e., by the mean, standard deviation and higher moments — and clearly balances out the first and second moment thereof.
![Comparing 22 CMIP5 models (vertical axis) vs their positions in the ranking (horizontal axis): (a) three-dimensional WD – heavy blue ‘$+$’ sign; (b) mean WD – red filled square; and (c) standard deviation of WDs – yellow filled square. See text for explanations. See \[appb\] for detailed results.[]{data-label="Comparison"}](./Comparison){width="\textwidth"}
A more peculiar instance is provided by HadGEM2-CC and HadGEM2-ES, which rank in this order for both the mean (17th and 19th) and the standard deviation (14th and 15th), but in the reverse order in the WD ranking (18th and 15th). This apparent paradox could be due to the presence of nontrivial second-order correlations between the variables or from the effect of higher moments of the distributions.
Note that, for the 18-year time interval studied herein (1997–2014), the results obtained applying the WD approach in three-dimensional phase space are not very different from those given by averaging the three corresponding one-dimensional distances. This agreement is due to the unimodality of the distributions taken into account and things would be different, for instance, if one were studying a paleoclimate setting that includes bimodality of the sea ice cover but not of the temperature field. In any case, the full application of the multi-dimensional WD leads to more robust results, as all correlations between the variables are taken into consideration.
Conclusions {#sec:conclude}
===========
We have proposed a new methodology to study the performance of climate models based on the computation of the Wasserstein distance (WD) between the multidimensional distributions of suitably chosen climatic fields of reference datasets and those of the models of interest. This method takes into account all the moments of the distributions and it is, therefore, more informative and more robust than ranking methods based on means or variances alone. The methodology is flexible as it allows one to consider several variables at the same time; it thus has the potential of disentangling the effect of the correlation between different climatic quantities.
The proposed methodology has been proven to be effective in pointing to climate modeling problems related to the representation of quantities like precipitation or sea ice extent over limited areas, such as the Tropics and the Arctic, respectively; see again Figs. \[Figure\_2\] and \[Figure\_3\]. Furthermore, this methodology can be applied to studying model performance for a given climatic variable over different spatial domains, as seen in Figs. \[Figure\_1\]–\[Figure\_4\], as well as relative model performance for different fields, as seen in Fig. \[Figure\_4\]. This flexibility can help guide attempts at model improvements by providing robust diagnostics of the least well simulated field — temperature, precipitation or sea ice extent — or region, namely either hemisphere, the Tropics or the Arctic.
Such a method, taking into account the whole distribution of the statistics and not just one representative number, like its mean or standard deviation, is complementary to those already in use, allowing for a deeper understanding of the models’ performance and the reasons behind their inadequacies. Unlike most evaluation methods for climate models used so far [@Flato2013], this approach does not rely on correlations, variances or mean square errors, and thus it does not focus only on standard measures of variability; rather, it shows quantitatively if a model does a good job in reproducing the desired statistics — including every moment of the distributions — and, more importantly, it allows one to compare several different fields at the same time, checking quantitatively differences in the aforementioned statistics among different models and fields.
Throughout the paper, we have shown the application of this approach to different physical fields, providing a ranking of CMIP5 models for specific sets of fields, as well as a way to highlight model weaknesses to help focus the honing of climate models. Getting more reliable models will lead to better simulations and, therefore, to more accurate climate predictions.
It is a pleasure to acknowledge our data sources: GPCP and NCEP Reanalysis 2 data provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at <https://www.esrl.noaa.gov/psd/> and OSI-SAF data provided by the EUMETSAT Ocean and Sea Ice Satellite Application Facility - Global sea ice concentration climate data record 1979-2015 (v2.0, 2017). The present paper is TiPES contribution \#30; the TiPES (Tipping Points in the Earth System) project has received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 820970. Work on this paper has also been supported by the EIT Climate-KIC; EIT Climate-KIC is supported by the European Institute of Innovation & Technology (EIT), a body of the European Union, under Grant Agreement No. 190733.
Wasserstein distance (WD): Background and history {#appa}
=================================================
We present herein historical and mathematical information on WD, as well as additional information on the climate models analyzed. We wish to quantify the discrepancies between the output of a climate model and the observed reality by comparing their complete probability distributions and not just some representative quantity, like their variance. One way of doing so is to use the Kullback-Leibler (KL) divergence [@KL.div.1951], which is rather widespread in applied statistics. To better explain the difference between the Wasserstein or Kantorovich-Rubinstein distance [@Kantorovich.1942] and the KL divergence, we first list below the axioms associated with the mathematical concept of a metric $d$. These axioms are inspired by and, of course, satisfied by the usual Euclidean distance.
Given points $x, y, z$ in a topological space $X$, $x, y, z \in X$, these axioms are
\[eq:metric\] $$\begin{aligned}
& d(x,y) = 0 \iff x = y, \label{eq:id} \\
& d(x,y) = d(y,x), \label{eq:symm} \\
& d(x,y) \le d(x,z) + d(z,y); \label{eq:triangle} \end{aligned}$$
they are referred to, respectively, as the axiom of identity or indiscernibles; the axiom of symmetry; and the axiom of subadditivity, better known as the triangle inequality. These axioms also imply the nonnegativity or separation condition $$d(x,y) \ge 0 \quad \mathrm{for~all} \quad x,y \in X.$$
A topological space $X$ equipped with such a metric becomes a metric space. Examples well-known in studying partial differential equations of fluid dynamics are so-called Hilbert spaces, which can be seen essentially as infinitely dimensional versions of Euclidean spaces [@Halmos.2017].
Given probability distributions $P,Q,R$ on a metric space $X$, the KL divergence $D_{KL}(P\|Q)$ for $P$ given $Q$ satisfies neither the symmetry condition nor the triangle inequality , i.e.
\[eq:KL\] $$\begin{aligned}
& D_{KL}(P\|Q) \neq D_{KL}(Q\|P) \quad \mathrm{and, in~general,} \label{eq:asymm} \\
& D_{KL}(R\|P) \le D_{KL}(Q\|P) + D_{KL}(R\|Q) \quad \mathrm{does~not~hold}. \label{eq:nosub}\end{aligned}$$
The Wasserstein distance (hereafter WD) [@Dobrushin.1970], though, is a true metric and satisfies all three axioms of Eq. . It is based on the concept of optimal transport [@Villani2009] and it allows one to evaluate quantitatively the distance between two distributions: intuitively, the nearer the two distributions of points in phase space, the smaller the effort required to merge the two. WD is also called the “earth mover’s distance,” since it was originally motivated by minimizing the effort of a platoon having dug a trench of prescribed shape and moving the earth dug up to another, existing trench of a different shape [@Monge1781].
Using WD, it is possible to estimate the reliability of a model by choosing an appropriate combination of climatic or other physical variables, depending on the goal of the computation. Since an $N$-dimensional distribution contains much more information than its $N$ one-dimensional marginals, every point in our multidimensional distribution carries information about all the fields at the same time and not just about the product of the marginals.
CMIP5 models {#appb}
============
The models that participated in CMIP5 are listed in Table S1 below. The three rankings summarized in Fig. 5 of the Main Text are listed here in Tables S2–S4.
----------------- ----------------
3D WD Model
\[0.5ex\] 0.097 IPSL-CM5A-MR
0.101 MIROC-ESM-CHEM
0.107 MIROC-ESM
0.125 NorESM1-M
0.136 MPI-ESM-LR
0.143 CMCC-CMS
0.157 GFDL-ESM2M
0.158 MPI-ESM-MR
0.162 IPSL-CM5A-LR
0.165 BNU-ESM
0.169 CMCC-CM
0.188 ACCESS1.0
0.188 CNRM-CM5
0.191 IPSL-CM5B-LR
0.192 HadGEM2-ES
0.193 BCC-CSM1.1
0.200 MRI-CGCM3
0.207 HadGEM2-CC
0.223 ACCESS1.3
0.229 INM-CM4
0.235 BCC-CSM1.1-m
0.246 GFDL-ESM2G
\[1ex\]
----------------- ----------------
: Ranking of CMIP5 models obtained with the three-dimensional WD.[]{data-label="table2"}
------------------ ----------------
Average of means Model
\[0.5ex\] 0.881 MIROC-ESM-CHEM
0.978 IPSL-CM5A-LR
0.993 MIROC-ESM
1.030 IPSL-CM5A-MR
1.128 NorESM1-M
1.369 IPSL-CM5B-LR
1.412 BCC-CSM1.1
1.557 BNU-ESM
1.748 CMCC-CM
1.749 BCC-CSM1.1-m
1.785 MRI-CGCM3
1.785 CMCC-CMS
1.893 MPI-ESM-LR
2.120 GFDL-ESM2M
2.224 MPI-ESM-MR
2.335 GFDL-ESM2G
2.508 HadGEM2-CC
2.578 CNRM-CM5
2.657 HadGEM2-ES
2.694 ACCESS1.0
3.163 INM-CM4
3.239 ACCESS1.3
\[1ex\]
------------------ ----------------
: Ranking obtained by averaging the three separate mean distances.[]{data-label="table3"}
--------------------- ----------------
Average of the
standard deviations Model
\[0.5ex\] 0.160 MPI-ESM-MR
0.186 CMCC-CMS
0.189 MPI-ESM-LR
0.225 CMCC-CM
0.298 CNRM-CM5
0.326 ACCESS1.3
0.360 ACCESS1.0
0.362 IPSL-CM5A-LR
0.366 IPSL-CM5A-MR
0.369 GFDL-ESM2M
0.390 MIROC-ESM
0.391 NorESM1-M
0.406 MIROC-ESM-CHEM
0.434 HadGEM2-CC
0.443 HadGEM2-ES
0.452 IPSL-CM5B-LR
0.455 INM-CM4
0.532 BNU-ESM
0.573 GFDL-ESM2G
0.651 MRI-CGCM3
0.758 BCC-CSM1.1
0.762 BCC-CSM1.1-m
\[1ex\]
--------------------- ----------------
: Ranking obtained by averaging the three standard deviations.[]{data-label="table4"}
Adler, RF., Huffman, GJ., Chang, A., Ferraro, R., Xie, P., Janowiak, J.Arkin, P.
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Eyring, V., Righi, M., Lauer, A., Evaldsson, M., Wenzel, S., Jones, C.Williams, KD.
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Flato, G., Marotzke, J., Abiodun, B., Braconnot, P., Chou, SC., Collins, W.Rummukainen, M.
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Ghil, M.
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Ghil, M. Lucarini, V.
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Gleckler, PJ., Taylor, KE. Doutriaux, C.
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Halmos, PR.
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Held, IM.
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Kanamitsu, M., Ebisuzaki, W., Woollen, J., Yang, SK., Hnilo, JJ., Fiorino, M. Potter, GL.
. . .
Kantorovich, LV.
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Knutti, R., Sedl[á]{}[č]{}ek, J., Sanderson, BM., Lorenz, R., Fischer, EM. Eyring, V.
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Kullback, S. Leibler, RA.
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Lorenz, EN.
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Mehran, A., AghaKouchak, A. Phillips, TJ.
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Neelin, JD., Langenbrunner, B., Meyerson, JE., Hall, A. Berg, N.
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Robin, Y., Yiou, P. Naveau, P.
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. .
|
---
abstract: 'Spatial data collected worldwide at a huge number of locations are frequently used in environmental and climate studies. Spatial modelling for this type of data presents both methodological and computational challenges. In this work we illustrate a computationally efficient non parametric framework to model and estimate the spatial field while accounting for geodesic distances between locations. The spatial field is modelled via penalized splines (P-splines) using intrinsic Gaussian Markov Random Field (GMRF) priors for the spline coefficients. The key idea is to use the sphere as a surrogate for the Globe, then build the basis of B-spline functions on a geodesic grid system. The basis matrix is sparse and so is the precision matrix of the GMRF prior, thus computational efficiency is gained by construction. We illustrate the approach on a real climate study, where the goal is to identify the Intertropical Convergence Zone using high-resolution remote sensing data.'
author:
- |
<span style="font-variant:small-caps;">Fedele Greco</span>, <span style="font-variant:small-caps;">Massimo Ventrucci</span>[^1] , <span style="font-variant:small-caps;">Elisa Castelli</span>\
Department of Statistical Sciences, University of Bologna, Bologna, Italy\
Institute of Atmospheric Sciences and Climate (CNR-ISAC), Italy
bibliography:
- 'biblio.bib'
title: '**P-spline smoothing for spatial data collected worldwide**'
---
Introduction {#sec:intro}
============
High-resolution spatial data collected worldwide, usually by means of remote sensing techniques, are widely spread in environmental and climate studies: most of the statistical methods developed in modelling this kind of data use the sphere as a surrogate for the Globe. Modelling data collected at a global scale presents both methodological and computational challenges. The traditional toolkit of the spatial data modeller facing geostatistical datasets and aiming to make prediction at unmonitored locations suggests to apply kriging techniques (see, e.g., [@banerjee-2015]). These rely on the assumption of a smooth Gaussian Random Field (GRF), continuous in space but only observed at a discrete set of points, any finite realization of it being generated by a multivariate Gaussian distribution. The covariance structure of this distribution is specified via a spatial covariance function. The practice is largely dominated by spatial covariances defined on Euclidean distances, such as the Matérn family, thus a preliminary step in the analysis is the projection of the 3d Cartesian coordinates (on the Earth surface) over a 2d coordinate space. Standard choice is to work with geographic coordinates (latitude-longitude), but other mappings can be used. [@banerjee-geo] provides a review of such mappings and discusses the impact of the chosen metric on spatial prediction via kriging. The traditional toolkit outlined above presents two main difficulties when modelling high-resolution data observed over a spherical domain.
The first issue is that the process of spatial prediction needs to be coherent with the geometry of the sphere. Using a planar metric over a 2d projection is inappropriate because it generates spurious anisotropy and non-stationarity of the covariance function ([@banerjee-geo]). The geodesic (aka great circle) distance, i.e. the length of the shortest path between two points over the surface of a sphere, is a natural candidate for measuring distances over a spherical domain. However, using great circle distances in a Matérn family does not necessarily guarantee a positive definite covariance [@gneiting-2013]. [@banerjee-geo] studied via simulation the behaviour of different metrics regarding estimation of the Matérn covariance parameters on a region as large as Colorado, finding a substantial impact of the chosen metric on the range of the correlation function. This means that with data collected on larger regions on Earth (e.g. the whole Globe), biased estimation of the underlying field has to be expected to some extent, when covariance functions built on Euclidean distances are used. A large number of papers have tackled this issue by essentially proposing new models for data on a spherical domain, both in a parametric and non-parametric framework.
In the parametric setting, several papers focused on building valid stochastic processes for the sphere, see, e.g., [@jun-stein-2007; @jeong2-2015; @heaton-2014] and references therein. The stochastic partial differential equation (SPDE) approach has grown a lot of attention recently [@lindgren-2011; @sangalli-2013; @sigrist-2015]. This approach builds a GRF as the finite element solution of a particular SPDE, an idea that can be generalized to different types of manifolds including the sphere. The paper by [@lindgren-2011] focuses in particular on the computational properties of the SPDE approach, deriving an approximated solution in terms of a Gaussian Markov Random Field (GMRF), instead of a GRF, in order to gain computational speed.
In the non-parametric setting, [@wahba-1981] firstly introduced smoothing splines on the sphere motivated by the analysis of weather data collected at a large number of stations around the world. Outside the spline realm, [@dimarzio-2014] presented local linear regression for spherical data, including smoothing of a scalar response on a spherical predictor as a particular case. [@wood-2017] discusses in detail the connection between spline smoothing and thin plate splines for the sphere, pointing out that low rank smoothers are also applicable to spherical data. Although low rank smoothers allows reduction of the number of parameters to estimate, the main role in alleviating the computational burden is played by the sparsity of the smoothing matrix, obtained by using local basis functions, i.e. non null over a limited domain. B-splines are local functions built upon joint polynomials connected at knots and are applied in different contexts, e.g. in penalized spline (P-spline) non-parametric regression [@eilers-1996] or in the SPDE approach mentioned above. In this work, the computational properties of B-splines are exploited.
Indeed, the second difficulty concerning the application of kriging techniques to high-resolution global datasets is purely computational. Continuous covariance functions used in geostatistics imply a dense covariance structure for the underlying GRF. When the number of data locations $n$ is large, this modelling framework becomes impractical because of the need to invert large dense matrices, with a computational cost increasing in cubic order with $n$. Statistics literature on the *big n problem* has grown incredibly fast in the last decade, mostly motivated by the increasing availability of high resolution remote sensing data for environmental studies. Some of the models for large data that can be implemented in a fully Bayesian hierarchical setting (for a review see [@banerjee-2017]) are based on a low-rank representation of the field [@wikle-1999; @banerjee-2008]. Other proposals seek a sparse representation of the covariance, like tapering [@furrer-2006], or of the precision, like in the SPDE approach (i.e. relying on the Markov properties of the graph underlying the model [@rue-2005]). The fully Bayesian framework presented in this paper follows both directions, as it is built on a low-rank representation and exploits the sparsity induced by a GMRF prior.
We propose a computationally efficient non-parametric approach to estimate the spatial field underling data on the sphere that properly accounts for geodesic distances between locations. Our method is based on a low-rank P-spline smoother to gain flexibility w.r.t. parametric models. The main contribution of this work is the extension of the P-spline model for smoothing data collected over a spherical domain. The model is built on a set of bivariate B-splines computed on a Geodesic Discrete Global Grid (GDGG) system [@sahr-2003], yielding a quasi-regular triangular mesh over the Globe. Geodesic grids have been used in spatial statistics to create flexible multi-resolution models implemented in a likelihood based inferential framework [@cressie-2008; @nychka-2012]. In contrast to the latter works, in this paper we follow a fully Bayesian approach and fit the model using an efficient Gibbs sampler, exploiting sparsity of the basis matrix and of the precision of the GMRF prior. We illustrate the method on a real climate study, where the goal is to identify the *Intertropical Convergence Zone* (ITCZ) from high-resolution remote sensing data collected worldwide over sea, with missing data occurring over land.
The rest of the paper is organized as follows. In section \[sec:motivating\] the dataset and application goals are described. Section \[sec:model\] presents our proposal for smoothing data over the sphere that we dub *Geodesic P-splines*. Section \[sec:results\] illustrates the method on a climate case study, focusing on the detection of the ITCZ. A discussion is provided in Section \[sec:discussion\].
Motivating example {#sec:motivating}
==================
Our interest in geodesic P-splines is motivated by a climate case study aiming at investigating the location of the ITCZ using satellite data. The ITCZ is a region of the atmosphere broadly located within the tropical belt where the north-east and south-east trade winds converge, which is characterised by high cloudiness and severe convective precipitation [@Holton20121]. An important aspect regards seasonal variability in the ITCZ position: ITCZ is roughly located North of the equator in the boreal spring and summer, while it migrates to southern regions in autumn and winter. The location of the ITCZ affects duration and intensity of the wet and dry seasons at the tropics and plays a key role in the general circulation of the atmosphere: assessing its variability is crucial for improving global climate models. Moreover, understanding the long-term trend characterizing this phenomenon is crucial for monitoring changes in climate pattern on a global scale. The phenomenon regulating the ITCZ behaviour cannot be measured directly, hence several studies have investigated it using some suitable proxy variable, like maximum precipitation [@zhang2001double], wind field [@vzagar2011climatology], vorticity and reflectivity of the clouds [@waliser1993satellite]. As a general feature, all these studies benefit from the increasing availability of satellite measurements. In this paper we focus on data from the infrared channels of the Along Track Scanning Radiometer (ATSR) instrument series, that were in orbit from 1991 to 2012 for accurate retrieval of sea surface temperature. Recently, in the frame of the European Space Agency ATSR Long Term Stability project (<https://earth.esa.int/web/sppa/activities/multi-sensors-timeseries/alts/about>), [@casadio2016total] developed the Advanced Infra-Red Water Vapour Estimator algorithm (AIRWAVE) for the retrieval of the Total Column of Water Vapour (TCWV) from the ATSR measurements. In this work we use TCWV as a proxy variable for locating ITCZ.
Data on TCWV regarding year 2008 were provided by the National Research Council - Institute of Atmospheric Sciences and Climate (CNR-ISAC), Italy. Data come as monthly averages of TCWV on a raster grid of $720$ columns (longitude values) and $360$ rows (latitude values), thus each cell covers half degree over latitude and longitude. In Figure \[fig:data\], data for January and July are displayed. The AIRWAVE algorithm provides reliable data over the sea and in clear sky conditions, thus TCWV observations are missing over land (broadly a third of the total number of cells), except in areas covered by lakes.
The application goal is to estimate the ITCZ position and its uncertainty. We consider the TCWV data on the fine raster grid as point-level data observed at the centroid of each cell and focus on modelling the latent field of TCWV separately for each month, leaving spatio-temporal modelling to future work. The statistical challenges we tackle in this paper are related to efficient smoothing of large data to remove measurement error and to fast prediction at unmonitored locations. We believe that extension of Bayesian P-Splines to a spherical domain can be a valuable strategy because of its efficiency and computational stability.Bayesian inference provides immediate tools for ITCZ location, by analysing the joint posterior distribution of the latent field. In Section \[sec:results\], the ITCZ detection problem is addressed by searching for the latitudes where the TCWV latent field shows highest values. We provide a graphical output, by plotting the posterior probability that a point on the Earth belongs to the ITCZ.
![TCWV data for January and July (unit measure, $Kg/m^2$). In general, TCWV measurements are available only over sea, as the data cannot be accurately retrieved over land; however, note that observations are still present in correspondence of wide lakes, e.g. the Great Lakes of North America and the Victoria lake.[]{data-label="fig:data"}](JanuaryData.pdf "fig:"){width="47.50000%"} ![TCWV data for January and July (unit measure, $Kg/m^2$). In general, TCWV measurements are available only over sea, as the data cannot be accurately retrieved over land; however, note that observations are still present in correspondence of wide lakes, e.g. the Great Lakes of North America and the Victoria lake.[]{data-label="fig:data"}](JulyData.pdf "fig:"){width="47.50000%"} ![TCWV data for January and July (unit measure, $Kg/m^2$). In general, TCWV measurements are available only over sea, as the data cannot be accurately retrieved over land; however, note that observations are still present in correspondence of wide lakes, e.g. the Great Lakes of North America and the Victoria lake.[]{data-label="fig:data"}](colorbar.pdf "fig:"){width=".05\textwidth"}
Geodesic P-splines {#sec:model}
==================
Background on P-splines for spatial data {#sec:classic-P-splines}
----------------------------------------
In the one dimensional setting, P-splines [@eilers-1996] are usually adopted to model the smooth effect of a covariate on the response as a linear combination of B-splines scaled by spline coefficients. Key features of this method are (a) equally-spaced univariate B-splines of certain degree $d$, these being non zero over a limited interval of the covariate domain, and (b) a penalty on the $r^{th}$ order differences between adjacent spline coefficients to control smoothness. The popularity of P-splines is due to numerical stability and flexibility in the modelling choices; e.g., the penalty order and the degree of the B-splines can be decided according to the application at hand. Higher-dimensional smoothers, suitable for modelling spatial data, can be constructed as tensor product P-splines [@eilers-2006]. In a frequentist framework, estimation is obtained via penalized maximum likelihood or iterative re-weighted least squares, with the smoothing parameter selected via cross validation or optimized over some information criterion. This method has become increasingly popular and is currently implemented in `R` packages such as `mgcv` [@wood-2017].
In order to build the ground for our proposal we next revise spatial P-splines for data observed on a two-dimensional latitude-longitude plane following [@eilers-2006]. Let us assume $y_i$ is a Gaussian measure at locations $(lat_i, lon_i)$, $i=1,...,n$, the model is $$y_i = \mu(lat_i,lon_i) + \epsilon_i \quad ; \quad \epsilon_i \sim \mathcal{N}(0, \tau_{\epsilon}^{-1}),$$ where $\mu(lat_i,lon_i)$ is a two-dimensional function, with no parametric assumptions on it and $\tau_{\epsilon}$ is the noise precision. We can think of $\mu(lat_i,lon_i)$ as a smooth surface representing the latent field which is modelled as a linear combination of bivariate B-spline basis functions: $$\mu(lat_i, lon_i) = \sum_{q=1}^Q \sum_{l=1}^L b_l(lat_i) b_q(lon_i) \beta_{l,q},$$ where $b_l(lat_i) b_q(lon_i)$ is the tensor product of marginal B-splines, evaluated at $(lat_i,lon_i)$, and $\beta_{l,q}$ is the associated spline coefficient. The marginal B-splines $b_l, l=1,...,L$ ($b_q, q=1,...,Q$), are defined on a set of knots that are chosen to be equally-spaced over the latitude (longitude) domain. Taking the tensor product of the two marginal basis returns $K=QL$ bivariate B-splines built on a regular grid over the plane; see Figure \[fig:basis\], left panel. In this sense, P-splines give a low-rank representation of the latent field, as $K$ is typically chosen to be much lower than $n$. In matrix notation, $\bm \mu = \bm B \bm \beta$, where $\bm B$ is a basis matrix of dimension $n \times K$ and $\bm \beta$ the vector of spline coefficients. When data are organized in a regular grid with no missing values, the basis matrix can be computed by the Kronecker product $\bm B = \bm B_{lat} \otimes \bm B_{lon}$. When data are irregularly scattered over the plane, efficient row-wise Kronecker operations can still be used to compute $\bm B$, as this is equivalent to having data organized on a fine regular grid with missing values. We link the reader to [@eilers-2006] for details on P-splines for spatial data and to [@lee-thesis] for insights into the mixed model formulation of P-splines within a spatio-temporal setting.
P-splines have been framed in a fully hierarchical Bayesian context by [@brezger-2004]. The hierarchical model can be cast starting from the following likelihood: $$\begin{aligned}
\label{eq:lik}
\bm y | \alpha, \bm \beta, \tau_{\epsilon} & \sim & \mathcal{N}(\bm \mu, \tau_{\epsilon}^{-1} \bm I) \quad ; \quad \bm \mu = \alpha + \bm B \bm \beta \end{aligned}$$ The penalty is reproduced by an $r^{th}$ order random walk (RW) prior on the spline coefficients, that in general can be expressed as $$\pi(\bm \beta|\tau_{\beta}) = (2\pi)^{-\texttt{rank}(\bm R)/2} (|\tau_{\beta} \bm R|^{*})^{1/2} \exp\left\{-\frac{\tau_{\beta}}{2}\bm \beta^{\textsf{T}} \bm R \bm \beta\right\},
\label{eq:igmrf}$$ where $\tau_{\beta}$ is a scalar precision hyper-parameter and $\bm R$ is the structure matrix of dimension $K \times K$. The non-zero entries in $\bm R$ impose conditional dependencies among the spline coefficients, thus encoding the type of penalty. Formally, the RW is a particular type of Intrinsic Gaussian Markov Random Field (IGMRF). The smoothing properties of an IGMRF are determined by the pattern of non-zero entries of $\bm R$ and by its rank deficiency. Any vector in the null space of $\bm R$ can be added to $\bm \beta$ and density (\[eq:igmrf\]) remains unchanged. For this reason, IGMRF priors are appropriate to model local deviations around an overall mean or, in general, a polynomial trend, with $\tau_{\beta}$ controlling the size of such deviations. For spatial smoothing, we will focus on a prior that leaves unspecified the overall mean, therefore $\texttt{rank} (\bm R)=K-1$.
The precision matrix for P-spline smoothing over a plane proposed in [@eilers-2006] is constructed as the Kronecker sum $$\label{eq:ksum}
\bm R = (\bm I_L \otimes \bm R_{lon}) + (\bm R_{lat} \otimes \bm I_Q)$$ where $\bm R_{lat}$ and $\bm R_{lon}$ are the (marginal) structure matrices of a RW on latitudinal and longitudinal knots, respectively. If we take $\bm R_{lat}$ and $\bm R_{lon}$ as the structure of a $1^{st}$ order RW, this is equivalent to assume an intrinsic Conditional Autoregressive (ICAR) model [@besag], with structure $$R_{ij} = \begin{cases}
k_i & i=j\\
-1 & i \sim j\\
0 & \text{otherwise},
\end{cases}
\label{eq:structure}$$ where $k_i$ is the number of knots adjacent to the $i^{th}$ knot; e.g. $k_i=\{2,3,4\}$ according to whether $i$ is a knot on the vertex, the border, or the interior of the regular grid. Usually an ICAR prior is assumed on a set of $n$ random effects, one for each data location, but here the ICAR is on the spline coefficients. In this sense, the basis $\bm B$ allows the stochastic field on the $K$ spline coefficients to be expanded at a much larger number of locations like $n$. This strategy allows substantial reduction of the number of parameters to estimate. Choosing higher order random walks in each dimension is possible: this will yield an higher order IGMRF prior, having a structure matrix with larger rank-deficiency; e.g. taking a 2$^{nd}$ order RW on latitude and longitude returns an IGMRF that models deviations from a plane. For a discussion of the properties of IGMRFs and their applications see [@rue-2005].
P-splines on Geodesic Discrete Global Grid Systems {#sec:GDGG}
--------------------------------------------------
The assumption of equally-spaced knots is convenient for building Bayesian penalized spline models, because it allows to create a suitable smoothing prior by simply assuming an IGMRF model for regularly spaced locations on the spline coefficients. Following this idea, knot placement has to take into account the geometry of the support of the data. Thus, building an equally spaced basis on the latitude-longitude plane is not a sensible choice when data cover the whole Globe or a large region within it. Figure \[fig:basis\] highlights that equally spaced B-splines in terms of Euclidean distances over the latitude-longitude plane (left panel) are not equally-spaced over the sphere (right panel). The spacing between the knots and the shape of the basis varies substantially latitude-wise: in such a knot-grid, imposing an IGMRF with structure (\[eq:structure\]) and a single precision parameter $\tau_{\beta}$ on the spline coefficients would generate the spurious anisotropy discussed in [@banerjee-geo]. Of course, this would be a naive approach to spatial smoothing over the sphere, since it does not introduce conditional dependence between knots located at extreme longitudes, which are actually close on the sphere surface. A circular penalty imposing conditional correlations among these knots seems a more sensible choice, but the irregular knot placement over the sphere would still generate spurious non-stationarity, as will be discussed later on. In what follows we propose an approach for (a) building geodesic knot-grids which are quasi equally spaced in terms of geodesic distances, (b) building basis functions and penalty matrices on such grids.
![Cubic B-splines equally-spaced in terms of Euclidean distances over the latitude and longitude plane (left panel; computed as the tensor product of marginal B-spline basis, see Section \[sec:classic-P-splines\]). The right panel displays how these basis appear on the sphere. []{data-label="fig:basis"}](basiPiano.png "fig:"){width="50.00000%"} ![Cubic B-splines equally-spaced in terms of Euclidean distances over the latitude and longitude plane (left panel; computed as the tensor product of marginal B-spline basis, see Section \[sec:classic-P-splines\]). The right panel displays how these basis appear on the sphere. []{data-label="fig:basis"}](basiSfera.pdf "fig:"){width="50.00000%"}
### Building the geodesic grid
Although building *exactly* equally spaced grids over the sphere surface is an impossible task, GDGGs offer a close approximation to equal spacing and their architecture provides immediate solutions to build basis functions and penalty matrices. Details on the spatial configuration of GDGGs can be found in [@randall-2002]. [@sahr-2003] outline five design choices that need to be undertaken for GDGGs construction: our choices are listed below.
1. Choice of a *base regular polyhedron*: we choose the [icosahedron]{}, which is a polyhedron made of 20 equilateral triangles and 12 nodes and assume this as a rough representation of a unit sphere. An icosahedron is displayed in Figure \[fig:ico\], left panel.
2. Choice of a fixed *orientation* of the base regular polyhedron relative to the Earth: we set one node of the icosahedron at coordinates $(0,0,1)$, assuming this as the North Pole.
3. Choice of a *hierarchical spatial partitioning* method defined symmetrically on each face of the base regular polyhedron. At this step, we split each triangle of the icosahedron in four equal triangles. By repeating this operation an arbitrary number of times we obtain a refined mesh, which we denote as *icomesh*. In Figure \[fig:ico\], central panel, see the icomesh resulting from four split iterations.
4. Transforming the base polyhedron partition into the corresponding spherical surface. This is achieved by simply normalizing the icomesh nodes, so that they lay on the sphere; we denote this mesh as *icosphere*, see Figure \[fig:ico\], right panel. The icosphere is a refined icosahedron, hence a much better representation of the sphere.
5. Choice of a method to *assign points to grid cells*. Ability to assign point to grid cells composing the tessellation can be useful for several purposes. In our case, it is fundamental to determine which triangle a data location falls into when it comes to computation of the basis functions, as discussed in the following section.
Following the above five steps, we obtain a geodesic grid of knots which are quasi equally-spaced in terms of great-circle distances. To summarize, the GDGG is constructed by splitting each icosahedron face in four triangles, in a recursive way. Note that, while the icosphere is a sphere tessellated into spherical triangles, the icomesh is a regular mesh made by equilateral triangles.
![On the left panel, the icosahedron. On the central panel, the icomesh, i.e. the regular triangular mesh after the *split* operation is repeated four times ($\nu=4$). On the right panel, the icosphere, i.e. the mesh obtained from normalizing the icomesh nodes of the central panel.[]{data-label="fig:ico"}](ico.pdf "fig:"){width="33.00000%"} ![On the left panel, the icosahedron. On the central panel, the icomesh, i.e. the regular triangular mesh after the *split* operation is repeated four times ($\nu=4$). On the right panel, the icosphere, i.e. the mesh obtained from normalizing the icomesh nodes of the central panel.[]{data-label="fig:ico"}](ico3unnorm.pdf "fig:"){width="33.00000%"} ![On the left panel, the icosahedron. On the central panel, the icomesh, i.e. the regular triangular mesh after the *split* operation is repeated four times ($\nu=4$). On the right panel, the icosphere, i.e. the mesh obtained from normalizing the icomesh nodes of the central panel.[]{data-label="fig:ico"}](ico3.pdf "fig:"){width="33.00000%"}
### Building the basis and the penalty matrix {#sec:bases}
The number of split iterations determines the dimension of the basis, i.e. the number of columns of the basis matrix $\bm B$ that we need to compute for each data point. Let $n$ be the number of data and $\nu$ be the number of split iterations, the basis has dimension $n \text{ x } K$, with $K=5 \cdot 2^{2(\nu-1)+3} + 2$. We adopt B-spline basis functions centred at the knots, each basis spanning six triangles (thus assuming the six closest nodes as neighbours) except for those centred at the 12 icosahedron vertices (that have five neighbours).
The next step consists of evaluating the $K$ B-splines, of a certain degree $d$, at an arbitrary data point laying on the sphere. Once that the triangle containing such point is determined, B-splines can be evaluated using Bernstein polynomials [@lai-2007spline]. To this aim, we find convenient working on the icomesh instead of the icosphere, as it is simpler to deal with planar than with spherical triangles. Therefore, we first project the 3d data location from the icosphere onto the icomesh domain, obtaining a point, $\bm v$, that falls inside a planar triangle (that lies on one of the icosahedron faces) and, second, we evaluate the $K$ B-splines at this 2d point. Following [@lai-2007spline], any point $\bm v = (x,y)$ inside a triangle of vertices $\bm v_1=(x_1, y_1), \bm v_2=(x_2, y_2), \bm v_3=(x_3,y_3)$ has a unique representation as
$$\bm v=\bm v_1 b_1 + \bm v_2 b_2 + \bm v_3 b_3,$$
where $(b_1, b_2, b_3)$ are called barycentric coordinates and are such that $b_1+b_2+b_3=1$. The Bernstein polynomial of degree $d$ is $$H_{tjk}^d = \frac{d!}{t!j!k!}b_1^t b_2^j b_3^k
\label{eq:bernstein}$$ with $t,j,k$ integer numbers summing to $d$. The following property $$\sum_{t+j+k=d} H_{tjk}^d =1$$ guarantees that for each location on the sphere the basis functions sum to 1. This is a desirable property of any smoothing model, giving a flat spatial field when there is no variation around the overall level, i.e. all spline coefficients are equal.
Let us indicate with $\bm z_i=(z_{i1},z_{i2})$ the location for observation $i$ projected on the icomesh, with $\bm B[i,]$ the row entry of $\bm B$ with the B-splines evaluated at $\bm z_i$ and with $\{k_1, k_2,k_3\}$ the indexes for the three knots closest to $\bm z_i$ (note, these are the vertices of the triangle containing observation $i$). It is important to note that only the B-splines centered at $\{k_1, k_2,k_3\}$ are non-zero at $\bm z_i$, whereas the B-splines centered at the remaining knots in the icomesh are zero at $\bm z_i$. The three non zero element of $\bm B[i, \{k_1, k_2,k_3\}]$ can be expressed as Bernstein polynomials (\[eq:bernstein\]), i.e. polynomials in the barycentric coordinates. Table \[tab:bernstein\] reports the non zero elements of $B[i,]$ for linear ($d=1$), quadratic ($d=2$) and cubic ($d=3$) B-splines.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$d=1$ $d=2$ $d=3$
-------------------------------------- -------------------------------------------------------------------- ------------------------------------------------------------------------------------------------
$ $ $
\left(\begin{array}{c} \left(\begin{array}{c} \left(\begin{array}{c}
H^1_{100} \\ H^1_{010} \\ H^1_{001} H^2_{200}+H^2_{100} \\ H^2_{020}+H^2_{010} \\ H^2_{002}+H^2_{001} H^3_{300}+H^3_{200}+H^3_{100}\\ H^3_{030}+H^3_{020}+H^3_{010} \\ H^3_{003}+H^3_{002}+H^3_{001}
\end{array} \end{array} \end{array}
\right)= \right)= \right)=
\left(\begin{array}{c} \left(\begin{array}{c} \left(\begin{array}{c}
b_1 \\ b_2 \\ b_3 b_1^2+b_1 \\ b_2^2+b_2 \\ b_3^2+b_3 b_1^3+b_1^2+b_1 \\ b_2^3+b_2^2+b_2 \\ b_3^3+b_3^2+b_3
\end{array} \end{array} \end{array}
\right) \right) \right)
$ $ $
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Non zero elements of $\bm B[i,]$, for B-splines of degree $d=\{1,2,3\}$.[]{data-label="tab:bernstein"}
The resulting basis matrix $\bm B$ is sparse because the B-splines are non zero over a domain spanned by only six triangles on the icomesh. Figure \[fig:gbasis\], left panel, shows how the new basis functions appear when projected over latitude and longitude. This plot suggests that a fairly similar degree of smoothness is applied everywhere using this new basis, avoiding the kind of spurious anisotropy introduced by the basis in Figure \[fig:basis\]. The Geodesic P-splines setting is completed by specifying the matrix $\bm R$, that we choose as the ICAR structure (\[eq:structure\]) with rank-deficiency 1. The number of neighbouring knots is $k_i=5$, if $i$ is one of the 12 nodes of the icosahedron, and $k_i=6$, if $i$ is one of the remaining $K-12$ nodes.
It is worth noting that, when using IGMRF priors with precision matrix $\tau_{\beta} \bm R$ on the spline coefficients $\bm \beta$, the structure of conditional dependence imposed by $\bm R$ determines the structure of the marginal variances of each coefficient, $Var(\beta_i)=\tau_{\beta}^{-1} R^{-}_{ii}, i=1,\ldots,K$, $\bm R^-$ being the generalised inverse of $\bm R$. Different structures can lead to extremely different marginal variances. To account for this feature, [@sorbye-2013] suggest to scale the precision matrix so that the hyperprior for $\tau_{\beta}$ can be selected to give the same degree of smoothness, a priori, starting from different structure matrices. The scaled precision matrix can be obtained as $\bm R^*=\kappa \bm R$, where $\kappa$ is the geometric mean of the diagonal entries of $\bm R^-$. IGMRFs with scaled precision matrices, although being characterised by different correlation structure, have a common feature: the average marginal variance is equal to one.
Figure \[fig:mvar\] compares the marginal variances for three models corresponding to a naive penalty (left), longitude-wise circular penalty (central) and a geodesic penalty (right). For the sake of comparison the precision matrices associated with the three models were scaled. For naive penalty, we mean an IGMRF prior for the spline coefficients laying on a planar grid, using the ICAR structure (\[eq:structure\]). The longitude-wise circular penalty is an IGMRF on a planar grid with structure (\[eq:ksum\]), but assuming $\bm R_{lon}$ as the structure of a circular $1^{st}$ order RW. For geodesic penalty, we mean an IGMRF on a GDGG using the ICAR structure as described in Section \[sec:bases\]. In the left panel, the non-stationarity in the marginal variances implied by using the ICAR structure on a regular planar grid (naive penalty) is evident. In the middle panel, marginal variances obtained by building a circular penalty longitude-wise show variation latitude-wise as expected. The IGMRF prior on the geodesic grid with ICAR structure implies stability in the marginal variances that is not achieved with the other specifications. As a matter of fact, the geodesic grid is almost a torus since all knots, except the icosahedron nodes, have six neighbours; we believe this is a desirable feature of our model as it mimics the idea of stationarity in variance typical of Matérn correlation functions.
![Cubic B-splines equally-spaced in terms of geodesic distances over the sphere (right panel; computed using Bernstein polynomials on a GDGG, see Section \[sec:GDGG\]). The left panel displays how these basis appear on the latitude longitude plane.[]{data-label="fig:gbasis"}](basigeodPiano.png "fig:"){width="50.00000%"} ![Cubic B-splines equally-spaced in terms of geodesic distances over the sphere (right panel; computed using Bernstein polynomials on a GDGG, see Section \[sec:GDGG\]). The left panel displays how these basis appear on the latitude longitude plane.[]{data-label="fig:gbasis"}](basigeodSfera.pdf "fig:"){width="50.00000%"}
![ Marginal variances with naive penalty (left panel), longitude-wise circular penalty (central panel) and geodesic penalty (right panel). (The color bar on the right is valid for all the three panels). []{data-label="fig:mvar"}](MVarplanar.pdf "fig:"){width="6cm"} ![ Marginal variances with naive penalty (left panel), longitude-wise circular penalty (central panel) and geodesic penalty (right panel). (The color bar on the right is valid for all the three panels). []{data-label="fig:mvar"}](MVarcircular.pdf "fig:"){width="6cm"} ![ Marginal variances with naive penalty (left panel), longitude-wise circular penalty (central panel) and geodesic penalty (right panel). (The color bar on the right is valid for all the three panels). []{data-label="fig:mvar"}](colorbarMV.pdf "fig:"){width=".65cm"}
![ Marginal variances with naive penalty (left panel), longitude-wise circular penalty (central panel) and geodesic penalty (right panel). (The color bar on the right is valid for all the three panels). []{data-label="fig:mvar"}](Mvargeod.pdf){width="6cm"}
### Hyperpriors
To complete the fully Bayesian model we need to set priors for the hyper-parameters $\tau_{\beta}$ and $\tau_{\epsilon}$. The precision $\tau_{\beta}$ regulates the amount of smoothing. When $\tau_{\beta}$ goes to infinity, $\bm \mu$ is a constant (because the rank deficiency of $\bm R$ is 1), while $\tau_{\beta} \in (0,+\infty)$ give a more flexible surface. A standard approach is to use a Gamma, $\text{Ga}(a,b)$, with shape $a$ and rate $b$, for both random walk and noise precisions. Usual parametrizations are $a$ equal to $1$ and $b$ small (e.g. $\text{Ga}(1,5e-5)$), or $a$ and $b$ small (e.g. Gamma$(1e-3,1e-3)$), as an attempt of non informativeness on the variance scale. Several papers in the literature have discussed issues related to the Gamma conjugate priors in hierarchical additive models and proposed alternatives [@gelman-2006; @pcprior]. Typically, the main impact regards the prior for the random walk precision, whereas the prior for the noise precision is negligible. In general, choice about the prior $\pi(\tau_{\beta})$ will be relevant in situations where we have poor sample size compared to the number of parameters to estimate. In the case study under examination the large sample size available for estimating each spline coefficient makes the impact of $\pi(\tau_{\beta})$ very small.
### Computations
Model estimation does not raise particular issues with respect to planar P-spline models, once matrices $\bm B$ and $\bm R$ have been built. Indeed, the model belongs to the class of Latent Gaussian Markov Models and approximate Bayesian inference can be performed efficiently using the package `R-INLA` [@rue-inla]. In our case study, we find more appropriate to use a Gibbs sampling algorithm as the tools developed for detecting the ITCZ require a sample from the joint posterior distribution of the model. The most expensive step is to sample from the full conditional for the spline coefficients $$\bm \beta|\tau_{\beta}, \tau_{\epsilon},\bm y \sim N(\bm Q^{-1} \bm B^{\textsf{T}} \bm y, \bm Q^{-1}) \quad \quad \quad \bm Q= \left(\bm B^{\textsf{T}} \bm B + {\frac{\tau_{\beta}}{\tau_{\epsilon}}} \bm R \right)
\label{eq:fullcond_beta}$$ under the linear constraint $\bm 1_{K}^{\textsf{T}} \bm B \bm \beta = 0$ needed for intercept identifiability. We use an efficient Gibbs sampler coded in `R` with the use of sparse matrix algebra as implemented in the `spam` package [@furrer-2010] to exploit sparsity of $\bm Q$ in (\[eq:fullcond\_beta\]). The `spam` package contains routines to perform efficient Cholesky decomposition of $\bm Q$, which is important for fast sampling from a GMRF under linear constraints like the full conditional $\pi(\bm \beta|\tau_{\beta}, \tau_{\epsilon},\bm y)$ in (\[eq:fullcond\_beta\]). The full conditionals for all the parameters in the model and the code for implementing the Gibbs sampler in `R` can be found in the supplementary material.
Application {#sec:results}
===========
Modelling TCWV data {#sec:smoothing-TCW}
-------------------
The goal of our application is to detect the ITCZ location by using the TCWV dataset described in Section \[sec:motivating\]. The operative definition of ITCZ that we use, as suggested by researchers from ISAC-CNR, Italy, is “the strip surrounding the Earth surface where TCWV shows highest values”.\
To this aim, we first apply Geodesic P-splines for smoothing observed TCWV data, which are affected by noise and do not provide measurements over the land, in order to predict the latent field all over the world. Then, we exploit model output for locating ITCZ by sampling form the joint posterior distribution of the latent field. Let $\bm y=(y_1,\dots,y_n)^{\textsf{T}}$ be the vector of TCWV observations $i=1,\ldots,n$, the hierarchical model is $$\begin{aligned}
\text{Likelihood:} \nonumber\\
\bm y | \alpha, \bm \beta, \tau_{\epsilon} & \sim & \mathcal{N}(\bm \mu, \tau_{\epsilon}^{-1} \bm I) \nonumber\\
\bm \mu & = & \alpha + \bm B \bm \beta \label{eq:mu}\\
\text{Prior:} \nonumber\\
\alpha & \sim & \mathcal{N}(0, \tau_{\alpha}^{-1}) \nonumber \\
\bm \beta | \tau_{\beta} & \sim & \mathcal{N}(\bm 0, \tau_{\beta}^{-1} \bm R^{*}) \quad \quad \text{$\bm \beta$ is subject to $\bm 1_{K}^{\textsf{T}} \bm B \bm \beta = 0$}
\label{eq:constraint} \\
\text{Hyper-prior:} \nonumber\\
\tau_{\beta} & \sim & \text{Ga}(1,5e-5)\nonumber \\
\tau_{\epsilon} & \sim & \text{Ga}(1,5e-5)\nonumber \end{aligned}$$
At the likelihood level, the matrix $\bm B$ in (\[eq:mu\]) is the B-spline basis on a GDGG as described in Section \[sec:GDGG\]. The latent field $\bm \mu$ is a surface varying smoothly over the sphere, with $\alpha$ the global spatial mean and $\bm \beta$ the spline coefficients. At the prior level, we have a diffuse Gaussian prior, with $\tau_{\alpha}$ fixed at a small value for the intercept and an ICAR prior, with precision $\tau_{\beta} \bm R^*$, for the spline coefficients. Using the scaled matrix $\bm R^*$ is a fundamental step: this allows to select the same prior for $\tau_{\beta}$ and $\tau_{\epsilon}$, as both $\bm I$ in (\[eq:mu\]) and $\bm R^{*}$ in (\[eq:constraint\]) have average marginal variance equal to 1. The results presented in this section are obtained using a $\text{Ga}(a=1,b=5e-5)$ for both hyperparameters, after checking that the results were non sensitive to other choices for $a$ and $b$. We fitted the Geodesic P-spline model to the data displayed in Figure \[fig:data\], referred to January and July, 2008. Data come as a raster of $720 \times 360$ observations geo-referenced in latitude longitude coordinates, with the percentage of raster cells with missing observations being about $40\%$ for both months. To compute $\bm B$, the latitude and longitude coordinates were converted to spherical coordinates, then projected on the icomesh and finally cubic B-splines were evaluated on a GDGG with $K=10242$ nodes (i.e. $\nu=5$). When using P-splines, the dimension of the knot grid has to be specified by the user according to the application at hand. It is required to select enough basis in order to capture the spatial variability of the latent field. As long as the basis is large enough to track the signal, increasing the number will not change the fit, rather it will change the location of posterior $\pi(\tau_{\beta}|\bm y)$ (analogously to the rescaling of the smoothing parameter when changing the number of knots). This means that a prior for $\tau_{\beta}$ has to be chosen with care in general but especially in poor sample size contexts, which is not the case here. Model estimation is performed by Gibbs sampling: we draw a total of $5000$ samples after convergence (achieved after a quick burnin due to the relative large sample size available for each model parameter). Regarding computational time, it takes about fifteen seconds (with an Intel core, 2.00GHz, 8.00 ram) to run a hundred iterations when $\nu=5$.
In a Bayesian framework, spatial prediction is naturally based on the joint posterior predictive distribution of the latent field: sampling from this distribution is particularly efficient when using Bayesian P-splines. Once the posterior distribution of the latent field $\pi(\bm \mu|\bm y)$ has been obtained, prediction $\tilde \mu$ at an arbitrary location $\bm{\tilde x}$ can be performed, after evaluating the basis functions at $\bm{\tilde x}$, using the posterior predictive distribution:
$$\label{eq:pred}
\pi(\tilde \mu|\bm y) = \int \pi(\tilde \mu|\bm \theta)\pi(\bm \theta|\bm y)d \bm \theta$$
where $\bm \theta=(\alpha,\bm \beta,\tau_\alpha,\tau_\beta,\tau_\epsilon)$. This is achieved by composite sampling once $G$ samples from the posterior distribution are available. Let $\bm \theta^g$ be a sample from the posterior distribution, $g=1,\ldots,G$: samples from distribution (\[eq:pred\]) are obtained by sampling from $\pi(\tilde \mu|\bm \theta^g)$. In Figure \[fig:smooth\] we report the maps of the TCWV posterior means at a fine grid of prediction locations: this accomplishes our first task, i.e. to remove random noise from data and to reconstruct the latent field on the whole Earth surface. Our strategy for ITCZ location is outlined in what follows.
![Model prediction of the latent field for January and July.[]{data-label="fig:smooth"}](JanuaryInterp.pdf "fig:"){width="47.50000%"} ![Model prediction of the latent field for January and July.[]{data-label="fig:smooth"}](JulyInterp.pdf "fig:"){width="47.50000%"} ![Model prediction of the latent field for January and July.[]{data-label="fig:smooth"}](colorbar.pdf "fig:"){width=".05\textwidth"}
Locating the ITCZ {#sec:locating-TCW}
-----------------
The problem of ITCZ location is addressed by summarising the posterior predictive distribution of the TCWV latent field. The procedure outlined below requires the specification of a reasonable guess concerning the width of the ITCZ region denoted as $W$; we based our choice on expert knowledge by ISAC-CNR researchers and set $W=1000$ $km$. The ITCZ width relative to the length of a Meridian (which is about $20000$ $km$) is around $w=W/20,000=0,05$.
Our algorithm to locate the ITCZ consists of a discrete search performed longitude-wise (i.e. at each meridian). Let $m=1,\ldots,M$ index a set of $M$ meridians: for a given $m$, we sample from the posterior predictive distribution of the latent field at a fine grid over latitude. Then, we compute the posterior probability that a point at a given latitude belongs to the region where the TCWV shows highest values (i.e. the point falls into the ITCZ region), integrating out uncertainty about model parameters.
Let $\bm{\tilde{\mu}}_m=(\tilde{\mu}_{1m},\ldots,\tilde{\mu}_{lm},\ldots,\tilde{\mu}_{Lm})$ be the vector of the latent field predicted at locations $l=1,\ldots,L$, where $(lat_{1m},\ldots,lat_{lm},\ldots,lat_{Lm})$ is a regular sequence from $90^{\circ}$ to $-90^{\circ}$. The algorithm proceeds as follows. For $m=1,\ldots,M$:
- evaluate the bases at locations $l=1,\ldots,L$, this gives a meridian-specific $L \times K$ dimensional basis matrix $\bm{\tilde{B}_m}$;
- sample $G$ realizations from the posterior predictive distribution (\[eq:pred\]) by computing $\bm{\tilde \mu}^g_m=\alpha^g+\bm{\tilde{B}_m} \hat{\bm{\beta}}^g$, $g=1,\ldots,G$;
- for $g=1,\ldots,G$, rank the vector $\bm{\tilde \mu}^g_m$. This gives a posterior sample of the ranks, indicated by vector $\bm{\phi}_m^g=(\phi^{g}_{1m},\ldots,\phi^{g}_{lm},\ldots,\phi^{g}_{Lm})$, e.g. $\phi^{g}_{lm}=L$ if $l=\operatorname*{argmax}_{l}(\bm{\tilde\mu}^{g}_{m})$, while $\phi^{g}_{lm}=1$ if $l=\operatorname*{argmin}_{l}(\bm{\tilde\mu}^{g}_{m})$.
The probability that a point $l$ belonging to meridian $m$ falls into the ITCZ is computed as $$Pr\left(lat_{lm} \in ITCZ|\bm y \right)=\frac{1}{G}\sum_{g=1}^{G} I\left(1-\frac{\phi^{g}_{lm}}{L} < w \right)
\label{eq:itcz-formula}$$
where $I$ is the indicator function and $\phi^g_{lm}/L$ is the normalised rank. To sum up, (\[eq:itcz-formula\]) is the probability that the point with geographical coordinates $(lat_l,long_m)$ falls inside the ITCZ, where the length of the ITCZ is fixed according to $W$. Results are displayed in Figure \[fig:itcz\] for the two months under study: this Figure is obtained running the algorithm with $L=1000$ and $M=360$. The ITCZ is mostly located in the south (north) of the Equator in January (July), as expected on the basis of prior knowledge concerning its seasonal behaviour. The map for January shows the double ITCZ, which is typical of the Central Pacific region in some period during the year [@waliser1993satellite]. The proposed method allows to locate the ITCZ even over land (in particular in Africa and South America) where data are not available, this being reflected by higher posterior uncertainty. Of course, the width of ITCZ reported in Figure \[fig:itcz\] is strictly dependent on the choice of $W$: although this is very relevant when studying a single month, we believe that it is not such a crucial choice if the method is used for studying the spatio-temporal trend of the phenomenon. Indeed, in this case it would be important to keep $W$ fixed along the study period in order to ensure comparability among results.
![ITCZ location for January and July.[]{data-label="fig:itcz"}](ITCZJanuary.png "fig:"){width="47.50000%"} ![ITCZ location for January and July.[]{data-label="fig:itcz"}](ITCZJuly.png "fig:"){width="47.50000%"} ![ITCZ location for January and July.[]{data-label="fig:itcz"}](colorbarITCZ.pdf "fig:"){width=".05\textwidth"}
Discussion {#sec:discussion}
==========
We presented a Bayesian hierarchical framework for smoothing data collected worldwide at a large number of locations. With respect to traditional methods, the proposed model accounts for geodesic distances between the data, thus overcoming the limitations of covariance functions for Euclidean spaces when applied to global datasets.
The non-parametric model formulation proposed extends the Bayesian P-spline approach for smoothing worldwide collected data. Assuming the sphere as a representation of the Globe, the idea is to build a new basis of B-splines on a suitable geodesic grid while keeping the hierarchical model formulation of Bayesian P-splines, with the associated advantages in terms of flexibility and computation. Two key features of P-splines are maintained in the Geodesic P-spline model: (a) the use of local bell-shaped functions, e.g. the B-splines on the icomesh, that yield a sparse basis matrix; (b) the use of B-splines centred at equally-spaced knots, i.e. the nodes of the icomesh. Point (b) suggests that an IGMRF for regular locations is a sensible prior distribution for the spline coefficients, giving stable marginal variances as opposed to the standard P-spline model construction. Computational efficiency is due to (a) reduction of the latent field dimension, as the smoothing prior operates on the spline coefficients (low-rank smoother) and (b) fast MCMC based on sparse Cholesky factorization of the structure matrix of the full conditional for the latent field. This advantages allow for fast fitting of the model to data collected worldwide at a high-resolution.
We applied the Geodesic P-spline model to TCWV data retrieved with the AIRWAVE algorithm at a huge number of locations on Earth. The smoothing approach in this example is desirable as it allows estimation of the field at unmonitored locations. We provided inferential tools to locate the ITCZ based on ranking samples from the posterior distribution of the latent field, estimated at a fine grid over the Globe. Results are coherent with prior knowledge concerning ITCZ, indicating a shift towards southern regions in autumn and winter.
To apply the method the user is required to take decisions on mainly two critical aspects. The first regards the number of split-in-four iterations, $\nu$, which determines the total number of basis, $K$. The main point is to select a large enough $K$ in order to capture the spatial variability of the latent field. We believe this choice has to be made according to the application at hand and that more work is needed to investigate strategies valid in general. A practical rule of thumb would suggest to run a sensitivity analysis, increasing $\nu$ until computation of the basis gets impractical, then run multiple Geodesic P-splines defined on the GDGG associated to the selected $\nu$’s, finally compare these models according to some criterion suitable for the case study under examination. In our application we checked that $\nu=\{5,6\}$ essentially gave the same results regarding the ITCZ location, which is our goal, therefore we fixed $\nu=5$ for presentation of results. A second critical aspect, ubiquitous in any Bayesian analysis, is the choice of hyperpriors. Typically, the prior for the random walk precision, $\pi(\tau_{\beta})$, impacts more than the prior on the noise precision. We expect large impact of $\pi(\tau_{\beta})$ in situations where sample size is small compared to the number of parameters to estimate. In the case study on TCWV, the sample size available for estimating each spline coefficients is high enough, which makes the impact of $\pi(\tau_{\beta})$ very small. In the results presented in Section \[sec:results\] we used a Gamma with shape $a=1$ and rate $b=5e-5$ for both $\tau_{\beta}$ and $\tau_{\epsilon}$, after checking that the posterior $\pi(\tau_{\beta}|\bm y)$ was unchanged under different choices of $a$ and $b$. We believe that controlling that the posterior learns from the data in the same way for different choice of the prior is a reasonable approach to test robustness of the Bayesian specification. On the topic of prior selection for variance parameters the literature is growing fast in the last decade; see, e.g., [@gelman-2006; @pcprior] and reference therein.
The model can be extended in several directions, both on the methodological and applied side. In this paper we focused on an IGMRF structure for the spline coefficients equivalent to the ICAR used for lattice data, using the six surrounding knots as neighbours. Investigation of geodesic grids suitable for higher order IGMRF priors would be interesting. Another attractive research line is the extension towards a model based on nested B-splines, defined on a set of geodesic grids of different resolution, following [@nychka-2012]. In a fully Bayesian framework this requires careful hyperprior specification, as it is not clear how to prevent confounding between nested components.
On the applied side, a future research line worth to be investigated is modelling the ITCZ based on different proxy variables, focusing the analysis on a wide temporal range, following the ideas in [@elisa-2017]. The application of Geodesic P-spline models to the 20 years of ATSR data will allow the investigation of ITCZ meridional migration trends. Moreover, joint modelling of TCWV and other ITCZ related phenomena, possibly available at misaligned locations, will result in more reliable estimates of the ITCZ latent field, especially at locations where TCWV retrieval is not possible with the current ATSR technology.
Acknowledgements {#acknowledgements .unnumbered}
================
The work by Fedele Greco and Massimo Ventrucci is funded by the PRIN2015 supported-project *Environmental processes and human activities: capturing their interactions via statistical methods* (EPHASTAT) by MIUR (Italian Ministry of Education, University and Scientific Research). We thank Bianca Maria Dinelli (ISAC-CNR), Enzo Papandrea and Stefano Casadio (Serco s.p.a.) for guidance in the interpretation of the results. ATSR TCWV dataset was developed in the frame of the ESA ALTS project ESA Contract No. 4000108531/13/I-NB.
[^1]: Corresponding author. Email: `massimo.ventrucci@unibo.it`
|
---
abstract: 'Automated multi-document extractive text summarization is a widely studied research problem in the field of natural language understanding. Such extractive mechanisms compute in some form the worthiness of a sentence to be included into the summary. While the conventional approaches rely on human crafted document-independent features to generate a summary, we develop a data-driven novel summary system called HNet, which exploits the various semantic and compositional aspects latent in a sentence to capture document independent features. The network learns sentence representation in a way that, salient sentences are closer in the vector space than non-salient sentences. This semantic and compositional feature vector is then concatenated with the document-dependent features for sentence ranking. Experiments on the DUC benchmark datasets (DUC-2001, DUC-2002 and DUC-2004) indicate that our model shows significant performance gain of around 1.5-2 points in terms of ROUGE score compared with the state-of-the-art baselines.'
author:
- |
Abhishek Kumar Singh, Manish Gupta[^1] and Vasudeva Varma\
IIIT Hyderabad, India\
abhishek.singh@research.iiit.ac.in, manish.gupta@iiit.ac.in, vv@iiit.ac.in
bibliography:
- 'aaai.bib'
title: 'Unity in Diversity: Learning Distributed Heterogeneous Sentence Representation for Extractive Summarization'
---
Introduction
============
The rapid growth of online news over the web has generated an epochal change in the way we retrieve, analyze and consume data. The readers now have access to a huge amount of information on the web. For a human, understanding large documents and assimilating crucial information out of it is often a laborious and time-consuming task. Motivation to make a concise representation of huge text while retaining the core meaning of the original text has led to the development of various automated summarization systems. These systems provide users filtered, high-quality concise content to work at unprecedented scale and speed. Summarization methods are mainly classified into two categories: *extractive* and *abstractive*. Extractive methods aim to select salient phrases, sentences or elements from the text while abstractive techniques focus on generating summaries from scratch without the constraint of reusing phrases from the original text.
The majority of literature on text summarization is dedicated to extractive summarization approach. Previous methods can be predominantly categorized as (1) greedy approaches (e.g. [@carbonell1998use]), (2) graph based approaches (e.g. [@erkan2004lexrank]) and (3) constraint optimization based approaches (e.g. [@mcdonald2007study]). These approaches rely mainly on a set of features which were manually crafted. Recently, few efforts have been made towards data-driven learning approaches for extractive summarization using neural networks. @kaageback2014extractive used recursive autoencoders to summarize documents, achieving good performance on the Opinosis [@ganesan2010opinosis] dataset. @cao2015learning used convolution neural networks for addressing the problem of learning summary prior representation for multi-document extractive summarization. @cheng2016neural introduced attention based neural encoder-decoder model for extractive single document summarization trained on a large corpus of news articles collected from the Daily Mail. Their work focuses on sentence-level as well as the word-level extractive summarization of individual documents using encoder-decoder architecture. @DBLP:conf/cikm/Singh0V17 proposed a combination of memory network and convolutional BLSTM (Bidirectional Long Short Term Memory) network to learn better unified document representation which jointly captures n-gram features, sentential information and the notion of the summary worthiness of sentences leading to better summary generation.
Most successful multi-document summarization systems use extractive methods. Sentence extraction is a crucial step in such a system. The idea is to find a representative subset of sentences, which contains the information of the entire set. Thus, sentence ranking is imperative in finding such an informative subset, which sets our focus to sentence-level summarization. The performance of the summarization system using sentence ranking approach is profoundly determined by the feature engineering, irrespective of the ranking models [@osborne2002using; @conroy2004left; @galley2006skip; @li2007multi]. Features are broadly classified as: (a) document-dependent features (e.g., position, term frequency), and (b) document-independent features (e.g., length, stop-word ratio, word polarity). Document independent features often reveal the aspect that a sentence can be considered summary worthy irrespective of which document it is present in. Consider the following example.
1. Six killed, eight wounded in a shooting at Quebec City.
2. It was the shooting that killed six people and injured eight people at a Quebec City mosque.
While the former sentence conveys prominent information in concrete terms, the latter is a more verbose way of portrayal with similar meaning. In the case of multi-document summary, the former sentence is the best candidate, as it is a concise representation keeping important information intact. This intuition was called as summary prior nature by @cao2015learning, and can be captured by learning better document independent features.
We aim to learn a better sentence representation that incorporates both document dependent features as well as document independent features to capture the notion of saliency of a sentence. Since the sentence representation comprises of two different kinds of features, we call it a heterogeneous representation. Contrary to the orthodox method of painstakingly engineering document independent features, we propose a model with a Convolutional Sentence Tree Indexer (CSTI), a novel data-driven neural network for capturing semantic and compositional aspects in a sentence. CSTI slides over the input sequence to produce higher-level representation by compressing all the input information into a single representation vector of the root node in the constructed binary tree. We present details in Section \[propmodel\]. Final sentence representation obtained by concatenating the transformed document dependent features and the features obtained from CSTI (document independent features) is used under a regression framework for sentence ranking.
Deep neural networks perform better in the case of huge training data. However, non-availability of large multi-document summarization corpus makes learning challenging for deep networks and often results procured are not of high quality. To overcome this issue, we use transfer learning approach where we first train the network on single document summarization corpus [@cheng2016neural] and then fine-tune the network with the multi-document datasets. We summarize our key contributions below.
1. We propose CSTI, a novel method to encode semantic and compositional features latent in a sentence which can be combined with document dependent features to learn a better heterogeneous sentence representation for capturing the notion of summary worthiness of a sentence.
2. Further, we propose a novel Siamese CSTI (Siam-CSTI) model for effectively identifying redundant sentences during the sentence selection process.
3. We use transfer learning method to overcome the problem of lack of data for multi-document summarization.
4. We experimentally show that our method outperforms the basic systems and several competitive baselines. Our model achieves significant performance gain on the DUC 2001, 2002 and 2004 multi-document summarization datasets.
Related Work
============
Extractive document summarization has been traditionally connected to the task of sentence ranking. Sentence ranking models by @osborne2002using [@conroy2004left; @galley2006skip; @li2007multi] are dependent on the human-crafted features. @shen2007document modeled extractive document summarization as a sequence classification problem using Conditional Random Fields. Our approach is different from theirs as we use a data-driven approach to automatically acquire document-independent features for representing sentences without the need of manually crafted document independent features. @hong2014improving built a summarization system using advanced document-independent features which can be seen as an attempt to capture better sentence representation. These features are often hand-crafted and fail to capture various semantic aspects. Summarization system CTSUM [@wan2014ctsum] attempts to rank sentences using certainty score. However, certainty score alone is not enough to reveal all possible latent semantic aspects. @DBLP:conf/coling/RenWCMZ16 develop a redundancy aware sentence regression framework for multi-document extractive summarization. They model importance and redundancy simultaneously by evaluating the relative importance of a sentence given a set of selected sentences. Along with single sentence features they incorporate additional features derived from the sentence relations. They manually crafted *sentence importance features* and *sentence relation features* while we use deep neural network for getting automatic document-independent features.
Recursive Neural Networks are known to model compositionality in natural language over trees. The tree structure is predefined by a syntactic parser [@socher2013recursive] and each non-leaf tree node is associated with a node composition function. @socher2013recursive also proposed Tensor networks as composition function for sentence level sentiment analysis tasks. Recently, @zhu2015long introduced S-LSTM which extends LSTM units to compose tree nodes in a recursive fashion. Neural Tree Indexer (NTI), an extension of S-LSTM was proposed for natural language inference and QA task [@DBLP:conf/eacl/YuM17]. In our work we introduce a CSTI, an enhanced version of NTI adapted for summarization task. Unlike NTI, our model uses (a) CNNs that can slide over inputs to produce higher-level representations, and (b) BLSTM as the primary composition function.
Proposed Model {#propmodel}
==============
Our architecture intends to learn a better representation of a sentence with consideration of both document-dependent and document-independent features in order to measure the worthiness of a sentence in the summary. The proposed system architecture is illustrated in Figure \[systemarch\]. The principal components of our model architecture are as follows.
1. *CSTI*: captures local (word n-grams and phrase level), global (sequential and compositional dependencies between phrases) information and the notion of saliency of a sentence. Details in Section \[subsec:csti\].
2. *Extractor*: extracts document dependent features from the given sentence. Details in Section \[extractor\].
3. *Regression Layer*: predicts sentence scores and thus, helps in the sentence ranking process.
CSTI provides an embedding which incorporates document-independent features. Final unified sentence embedding is obtained by concatenating embedding from CSTI and document-dependent features, which is then forwarded through the regression layer to obtain saliency score of a sentence. Since the model makes use of the heterogeneous representation of the sentence, we name our model as Heterogeneous Net (HNet). In this section, we first describe the CSTI and then present details of the extractor and the regression layer.
![The System Architecture of HNet. After max pool operation padding vectors (represented in black color) are added to form a full binary tree.[]{data-label="systemarch"}](Hnet){width="3.5in"}
Convolutional Sentence Tree Indexer (CSTI) {#subsec:csti}
------------------------------------------
We focus on learning a hierarchical sentence representation that not only incorporates phrase level features and global sentence level information but it should also include the notion of saliency of a sentence. The hierarchical nature of our model reflects the fact that sentences are generated from words, phrases and often have some sequential and compositional dependencies among these units. Therefore, we use an architecture to obtain a representation with minimum information loss such that the global information gets discovered and the local information remains preserved.
CSTI comprises of: (a) *Convolutional Encoder*: We use a Convolution Neural Network (CNN) with multiple filters to automatically capture set of phrase (n-grams) based features followed by a max-over-time pooling operation to obtain a set of feature vectors. We do this because phrases with different lengths can exhibit the same characteristics of summary prior nature. (b) *Bidirectional Long Short Term Memory Tree Indexer (BLSTM Tree Indexer)* to obtain a comprehensive set of document-independent features incorporating semantic and compositional aspects in a sentence. We use BLSTM Tree Indexer because: (a) it models conditional and compositional power of sequential RNNs and syntactic tree based recursive neural nets, and (b) it is a robust syntactic parsing-independent tree structure model and does not require a parse tree structure.
### Convolutional Encoder
For first level sentence encoding, we choose convolution neural network for the following reasons: (1) it is easily trainable without long-term dependencies, (2) it handles sentences of variable length inherently and is able to learn compressed representation of n-grams effectively, (3) previous research has shown that it can be successfully used for sentence-level classification tasks such as sentiment analysis [@kim2014convolutional].
Conventional convolution neural network uses convolution operation over various word embeddings which is then followed by a max pooling operation. Suppose, $d$ dimensional word embedding of the $i^{th}$ word in the sentence is $w_i$, and let $w_{i:i+n}$ denote the concatenation of word embeddings $w_i, ..., w_{i+n}$. Then, convolution operation over a window of $c$ words using a filter of $\theta_t^{c} \in \mathbb{R}^{m\times cd} $ yields new features with $m$ dimensions. Convolution operation is written as follows. $$\label{eq1}
f_i^{c} = tanh(\theta_t^{c} \times w_{i:i+c-1} + b)$$ Here $b$ is the bias term. We obtain a feature map $F^c$ by applying filter $\theta_t^{c}$ over all possible windows of $c$ words in the sentence of length $N$. $$\label{eq2}
F^{c} = [f_1^{c}, f_2^{c}, ..., f_{N-c+1}^c]$$
Our intention is to capture the most prominent features in the feature map. Hence, we used max-over-time pooling operation [@collobert2011natural] to acquire final features for a filter of fixed window size. To exploit several latent features from phrase based information, we used multiple filters of different window widths. Let $\theta_t^1, \theta_t^2, ..., \theta_t^k$ be $k$ filters for window sizes from 1 to $k$ then we have $k$ feature maps $F^1, F^2, ..., F^k$. Applying max-over-time pooling operation helps to get most salient features. They seem to capture the phrase-level information nicely. The first level features $\phi_1$ obtained from convolution network can be denoted as follows. $$\label{eq3}
\phi_1 = \{max\{F^1\}, max\{F^2\}, ..., max\{F^k\}\}$$
We use an enhanced convolution network which is different from the one used for sentence classification task [@kim2014convolutional] or for learning the prior summary task [@cao2015learning]. @kim2014convolutional reserves all representation generated by filters to a fully connected layer which ignores relations among phrases with different lengths. @cao2015learning tried to capture this relation by performing two-stage max-over-time pooling operation. Unlike these models, our model captures the relation among different length phrases by passing the representations generated after max-over-time pooling operation to the BLSTM Tree Indexer network. Representation thus obtained also incorporates the latent temporal and compositional dependencies among variable length phrases.
### BLSTM Tree Indexer (BTI)
Sequential LSTMs are known to learn syntactic structure (conditional transition) from natural language. However their generalization to unseen text is relatively poor in comparison with models that exploit syntactic tree structure [@DBLP:conf/nips/BowmanMP15]. BLSTM Tree Indexer leverages the sequential power of LSTMs and the compositional power of recursive models, without the need of a parse tree. The model constructs a binary tree by processing the input sequences with its node function in a bottom-up fashion. It compresses all the input information into a single representation vector of the root node. This representation seems to capture both semantic and compositional aspects in the sentence.
The output of the convolutional encoder is padded with padding vectors to form a full binary tree and fed as input to the BLSTM Tree Indexer. The input set consists of a sequence of vectors ($\phi_1$). BTI can be a full n-ary tree structure. To reduce computational complexity, we have implemented binary tree form of BTI in our study. It has two types of transformation functions: (a) a non-leaf node composition function $f^{node}(h^1 ,...,h^q)$ and (b) a leaf node transformation function $f^{leaf}(\phi^{j}_{1})$, where $\phi^{j}_1$ is $j^{th}$ feature vector from set $\phi_1$. $f^{node}(h^1, ..., h^q)$ is a composition function of the representation of its child nodes $h^1, ..., h^q$, where $q$ is the total number of child nodes of this non-leaf node. $f^{leaf}(\phi^{j}_{1})$ is some non-linear transformation of the input vector $\phi^{j}_1$.
As we use the binary tree form of BTI, a non-leaf node can only take two direct child nodes, i.e., $q = 2$. Hence, the function $f^{node}(h^l, h^r)$ learns a composition over its left child node $h^l$ and right child node $h^r$. The node and the leaf node functions are actually parameterized neural networks.
We present our approach for the two types of transformation functions in the following.
***Leaf Node Transformation:*** We use a MLP (Multi-Layer Perceptron) with $ReLU$ function (for non-linear transformation) for the leaf node function $f^{leaf}$ as follows. $$h_j = ReLU(MLP(\phi^{j}_1 ; \theta))$$
where $\phi^{j}_1$ is input sequence fed to the multi-layer perceptron, $\theta$ is the learning parameter and $h_j$ is the vector representation for the leaf node.
***Non-Leaf Node Composition:*** A Bidirectional LSTM (BLSTM) is used as the composition function $f^{node}(h^l, h^r)$ to get the representation of the parent node. BLSTM processes the input both in the forward order as well in the reverse order, allowing to combine future and past information in every time step. It comprises of two LSTM layers processing the input separately to produce $\overrightarrow{h}$, $\overrightarrow{c}$, the hidden and cell states of an LSTM processing the input in the forward order, and $\overleftarrow{h}$ and $\overleftarrow{c}$, the hidden and the cell states of an LSTM processing the input in reverse order. Both, $\overrightarrow{h}$ and $\overleftarrow{h}$, are then combined to produce output sequence of the BLSTM layer. Let $h^{l}_{t}$, $h^{r}_{t}$, $c^{l}_{t}$ and $c^{r}_{t}$ be the vector representations and cell states for left and right children. A BLSTM computes a parent node representation $h^{p}_{t+1}$ and a node cell state $c^{p}_{t+1}$ as follows. Forward order: $$\label{eq4}
\overrightarrow{i_{t+1}} = \sigma(W_{1}\overrightarrow{h^{l}_{t}}+W_{2}\overrightarrow{h^{r}_{t}}+W_{3}\overrightarrow{c^{l}_{t}})+W_{4}\overrightarrow{c^{r}_{t}}$$ $$\label{eq5}
\overrightarrow{f^{l}_{t+1}} = \sigma(W_{5}\overrightarrow{h^{l}_{t}}+W_{6}\overrightarrow{h^{r}_{t}}+W_{7}\overrightarrow{c^{l}_{t}})+W_{8}\overrightarrow{c^{r}_{t}}$$ $$\label{eq6}
\overrightarrow{f^{r}_{t+1}} = \sigma(W_{9}\overrightarrow{h^{l}_{t}}+W_{10}\overrightarrow{h^{r}_{t}}+W_{11}\overrightarrow{c^{l}_{t}})+W_{12}\overrightarrow{c^{r}_{t}}$$ $$\label{eq7}
\overrightarrow{c^{p}_{t+1}} = \overrightarrow{f^{l}_{t+1}}\odot \overrightarrow{c^{l}_{t}}+\overrightarrow{f^{r}_{t+1}}\odot \overrightarrow{c^{r}_{t}}+\\ \overrightarrow{i_{t+1}}\odot \tanh(W_{13}\overrightarrow{h^{l}_{t}}+W_{14}\overrightarrow{h^{r}_{t}})$$ $$\label{eq8}
\overrightarrow{o_{t+1}} = \sigma(W_{15}\overrightarrow{h^{l}_{t}}+W_{16}\overrightarrow{h^{r}_{t}}+W_{17}\overrightarrow{c^{p}_{t+1}})$$ $$\label{eq9}
\overrightarrow{h^{p}_{t+1}} = \overrightarrow{o_{t+1}}\odot \tanh(\overrightarrow{c^{p}_{t+1}})$$ Similarly, in the reverse order we obtain $\overleftarrow{h^{p}_{t+1}}$ and $\overleftarrow{c^{p}_{t+1}}$. Finally, we combine them to obtain the vectors $c^{p}_{t+1}$ and $h^{p}_{t+1}$ as follows. $$\label{eq10}
\nonumber c^{p}_{t+1} = mean(\overrightarrow{c^{p}_{t+1}}, \overleftarrow{c^{p}_{t+1}}),\ \ \ h^{p}_{t+1} = mean(\overrightarrow{h^{p}_{t+1}}, \overleftarrow{h^{p}_{t+1}})$$ where $W_1$, ..., $W_{17}\in \mathbb{R}^{k\times k}$ ($k=N-c+1$) are trainable weights. For brevity we eliminated the bias terms. $\sigma$ and $\odot$ denote the element-wise sigmoid function and the element-wise vector multiplication respectively. Each non-leaf node computes its representation by composing its children representation using the above set of equations. This representation is passed towards the root in a bottom-up fashion to construct the tree representation. The vector representation of the root $h^{root}$ (also referred as $\phi_f$) incorporates semantic and compositional aspects latent in a sentence.
Extractor
---------
Besides just using the document independent features, we also intend to use document dependent features in learning better sentence representation for saliency estimation of sentences. Our feature set includes the following: (1) The position of the sentence, (2) The averaged cluster frequency values of words in the sentence, (3) The average term frequency values of the words in the sentence, (4) The average word IDF values in the sentence, divided by sentence length, and (5) The maximal word IDF score in the sentence.
We choose these features for the following reasons: (1) They tend to impart some contextual knowledge. (2) They are often simple to calculate and have been extensively used in previous research [@cao2015ranking; @cao2015learning].
Regression Layer
----------------
We follow traditional supervised learning approach for sentence ranking [@carbonell1998use; @li2007multi]. The regression layer at the end of the architecture aims to assign scores to a sentence. Sentences are ranked based on the score predicted by the regression layer. Since our approach focuses on learning a better sentence representation embracing both document-independent and document-dependent features, we concatenate the document-independent features obtained from the CSTI net with the transformed extracted document-dependent features (using a dense layer). Let $\phi_e$ be the transformed extracted document-dependent features and let $S_i$ denote the heterogeneous sentence embedding of the $i^{th}$ sentence. Thus, $S_i = [\phi_f, \phi_e]$.
The sentence worthiness is scored by ROUGE-2 [@lin2003automatic] (without stop words) and our model tries to estimate this score. Given sentence $i$, the final sentence representation $S_i$ is used in the regression layer to score saliency as $y_i = \sigma(W^T \times S_i)$ where $W$ are the regression weights and $\sigma$ is the softmax function. The softmax function gives a nice distribution over the range $[0, 1]$ which makes it suitable to imitate ROUGE score. This score is used to rank sentences. Higher the score, higher is the chance of the sentence to be included in the generated summary.
![Siamese CSTI Architecture[]{data-label="siam-csti"}](siam-csti){height="2in" width="3.5in"}
Removing Redundant Sentences
----------------------------
A good summary should be informative with non-redundant content. We generate the final summary by choosing top ranked sentences taking into account the redundancy among the selected sentences. The sentences are sorted in descending order of saliency scores. To identify whether the next candidate sentence is redundant, we compare it with all the sentences in the summary generated so far. We introduce a Siamese CSTI (Siam-CSTI) network for identifying redundant sentences. Figure \[siam-csti\] shows the Siam-CSTI architecture. The base network consists of the CSTI net. Weight parameters are tied for the base network. Two CSTI nets feed their output to a distance metric layer. We experiment with cosine, Euclidean and Manhattan distances and empirically find that the Manhattan distance seems to perform better in our case.
Siam-CSTI network is trained for sentence similarity task on the SICK data [@DBLP:conf/semeval/MarelliBBBMZ14]. We present dataset details in Section \[subsec:datasets\]. Formally, we consider a supervised learning setting where each training example consists of a pair of sequences $(x^{a}_{1}, ..., x^{a}_{T_a})$, $(x^{b}_{1}, ..., x^{b}_{T_b})$ of fixed-size vectors (each $x^{a}_i$, $x^{b}_j$ $\in \mathbb{R}^{d}$ is d-dimensional word vector) along with a single label $y$ for the pair. The sequences may be of different lengths $T_a \neq T_b$ and the sequence lengths can vary from example to example. The similarity function $g$ is based on the Manhattan distance metric as follows. $$\label{siam-eqn}
g(h^{a}_{T_a}, h^{b}_{T_b}) = \exp(-||h^{a}_{T_a}-h^{b}_{T_b}||_1) \in [0, 1]$$ where $h^{a}_{T_a}$, $h^{b}_{T_b}$ are the learned representations of the sequences $x^{a}_{T_a}$, $x^{b}_{T_b}$ respectively such that $h^{a}_{T_a}$ and $h^{b}_{T_b}$ are closer in the vector space if $x^{a}_{T_a}$ and $x^{b}_{T_b}$ are similar otherwise they reside far apart. Mean Squared Error (MSE) is used as loss function (after rescaling the training set relatedness labels to lie in $[0, 1]$). The Siam-CSTI model trained on paired examples seems to learn a highly structured space of sentence representations by exploiting the sequential and recursive power of CSTI that captures rich semantics. Similar sentences ($y=1$) are considered as redundant sentences and non-similar sentences ($y=0$) are considered as non-redundant sentences. The final summary is generated by iteratively picking up a sentence from the set of previously sorted sentences and adding it to current summary if the picked sentence is non-redundant.
Experimental Setup
==================
We experiment with our CSTI and Siam-CSTI based summarization model (HNet) for the task of multi-document summarization. In this section, we present our experimental setup for assessing the performance of our system. We discuss the corpora used for training and evaluation and provide implementation details of our approach.
[|p[5.8cm]{}|p[0.3cm]{}p[0.3cm]{}p[0.3cm]{}|]{} **Ranking by Dependency Tree-LSTM & **Tree & **M & **S\
**a woman is slicing potatoes& & &\
$\bullet$ a woman is cutting potatoes &4.82 &4.87& 4.91\
$\bullet$ potatoes are being sliced by a woman&4.70 &4.38& 4.68\
$\bullet$ tofu is being sliced by a woman&4.39 &3.51& 3.62\
**a boy is waving at some young runners from the ocean & & &\
$\bullet$ a group of men is playing with a ball on the beach&3.79 &3.13 & 2.68\
$\bullet$ a young boy wearing a red swimsuit is jumping out of a blue kiddies pool&3.37 &3.48 & 3.29\
$\bullet$ the man is tossing a kid into the swimming pool that is near the ocean&3.19& 2.26& 1.87\
************
Datasets {#subsec:datasets}
--------
Initial training of our model is done on the Daily Mail corpus, used for the task of single document summarization by [@cheng2016neural]. Overall, we have 193986 training documents, 12147 validation documents and 10350 test documents in the corpus. For the purpose of training, we created a sentence and its ROUGE-2 score pairs from this corpus. Sentences which are part of the summary get high ROUGE scores than non-summary sentences. We experiment on DUC 2001-2004 datasets which are used for generic multi-document summarization task. These documents are from newswires which are grouped into several thematic clusters. The full DUC data set can be availed by request at [http://duc.nist.gov/data.html]{}. The DUC 2001, 2002 and 2004 datasets consist of 11295, 15878 and 13070 sentences respectively. The SICK dataset which contains 9927 sentence pairs with a 5,000/4,927 training/test split [@DBLP:conf/semeval/MarelliBBBMZ14] was used for training the Siam-CSTI net. Each pair has a relatedness label $\in [1, 5]$ corresponding to the average relatedness judged by 10 different individuals.
Implementation Details
----------------------
We fine tuned our model on DUC datasets after initial training on Daily Mail corpus. DUC 2003 data is used as development set and we perform a 3-fold cross-validation on DUC 2001, 2002 and 2004 datasets with two years of data as training set and one year of data as the test set. The word vectors were initialized with 250-dimensional pre-trained embeddings [@mikolov2013distributed]. The embeddings for “out of vocabulary” words were set to zero vector. The size of the hidden units of BLSTM was set to 150. After tuning on the validation set, we fix the dimension $m$ of the latent features from convolutional encoder as 125 and window size $k = 5$ for HNet system. We use Adam [@kingma2014adam] as the optimizer with mini batches of size 35. Learning rates are set to {0.009, 0.0009}. For our network, we use regularization dropout of {0.2, 0.5}.
Baseline Methods {#baseline}
----------------
In this section of the paper, we describe several summarization baseline systems that we choose to compare against our system. These baselines include best peer systems (PeerT, Peer26, and Peer65) which participated in DUC data evaluations, state-of-the-art summarization results on DUC 2001, 2002 and 2004 corpus respectively. We select the systems that performed best on DUC 2001, 2002, 2004 datasets, which are: (1) R2N2 [@cao2015ranking], (2) ClusterCMRW [@wan2008multi], (3) REGSUM [@hong2014improving], (4) PriorSum [@cao2015learning], and (5) RASR [@DBLP:conf/coling/RenWCMZ16]. The R2N2 system uses a recursive neural network to rank sentences by automatically learning to weigh hand-crafted features. ClusterCMRW system leverages the cluster-level information and incorporates this information into a graph-based ranking algorithm. REGSUM follows a word regression approach for doing better estimation of word importance which leads to better extractive summaries. PriorSum captures summary prior nature by exploiting phrase based information. RASR uses regression framework that simultaneously learns the model importance and redundancy information by calculating the relative gain of a sentence with respect to given set of selected sentences. Further, we use LexRank [@erkan2004lexrank] as a baseline to compare performance level of regression approaches. We also compare with StandardCNN and Reg\_Manual. StandardCNN consists of just conventional CNNs with fixed window size for learning sentence representation. Reg\_Manual is used as a baseline system to explore and understand the effects of learned sentence representation prior to the summary. It adopts human-compiled document-independent features: (a) NUMBER (if a number exists), (b) NENTITY (if named entities exist), and (c) STOPRATIO (the ratio of stopwords). It combines these features with document dependent features and tunes the feature weights through LIBLINEAR[^2] support vector regression.
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**System & **ROUGE-1 & **ROUGE-2&**System & **ROUGE-1 & **ROUGE-2&**System & **ROUGE-1 & **ROUGE-2\
PeerT &33.03&7.86&Peer26 &35.15&7.64&Peer65 &37.88&9.18\
R2N2 &35.88&7.64& ClusterCMRW &38.55&8.65& REGSUM &38.57&9.75\
LexRank &33.43&6.09& LexRank &35.29&7.54& LexRank &37.87&8.88\
Reg\_Manual &35.95&7.86& Reg\_Manual &35.81&8.32& Reg\_Manual &38.24&9.74\
StandardCNN &35.19&7.63& StandardCNN &35.73&8.69& StandardCNN &37.9&9.93\
PriorSum &35.98&7.89& PriorSum &36.63&8.97& PriorSum &38.91&10.07\
RASR &36.31&8.49& RASR &37.8&9.61& RASR &36.6&10.57\
HNet-B &36.82&8.64& HNet-B &38.79&9.43& HNet-B &39.27&10.85\
HNet-B(T)&37.69&9.12& HNet-B(T) &39.52&9.69& HNet-B(T) &39.9&11.08\
HNet&37.21&8.96& HNet &39.17&9.61& HNet &39.54&10.94\
HNet$^{-}$&34.51&7.88& HNet$^{-}$ &35.86&8.24& HNet$^{-}$ &35.66&9.37\
HNet(T)& **38.18 & **9.43 & HNet(T) & **39.94 & **9.92 & HNet(T) & **40.34& **11.29\
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Results and Analysis {#reseval}
====================
In this section, we compare the performance of our system against various summarization baselines using ROUGE-1 (unigram match) and ROUGE-2 (bigram match) measures. We also attempt to analyze our system trained with different approaches with intuition and empirical evidence presented in the form of tables and graphs. Lastly, we conclude this section by presenting examples of sentences selected for summaries by the proposed system.
We carried out extensive experiments with diverse settings in order to evaluate our system. In doing so, we created several variations of HNet model which are: (1) HNet-B: uses Convolutional BLSTM as sentence encoder instead of CSTI. (2) HNet-B(T): refers to HNet-B model which is trained with transfer learning approach, i.e., the model was first trained on Daily Mail dataset [@cheng2016neural] and was then fine-tuned on multi-document DUC datasets. (3) HNet: refers to our proposed model with CSTI as sentence encoder for sentence ranking and Siam-CSTI as redundancy identifier for sentence selection task. (4) HNet(T): refers to HNet model which was first trained on Daily Mail dataset [@cheng2016neural] and was then fine-tuned on multi-document DUC datasets. (5) HNet$^{-}$: refers to the HNet model when the embedding from the extractor ($\phi_e$) is made zero.
It is evident from the results presented in Table \[res\] that our basic systems HNet-B and HNet-B (T) significantly outperform (T-test with p-value=0.05) state-of-the-art summarization systems R2N2, Cluster-CMRW, REGSUM, PriorSum, and RASR. This is encouraging because despite having not so complex deep network architecture the HNet-B system is able to learn efficient document-dependent semantic features. It also outperforms the Reg\_Manual baseline which uses human-compiled features for obtaining the document-independent features and the graph-based summarization system LexRank. From the results shown in Table \[res\] it is clear that HNet-B outperforms the StandardCNN baseline. This is due to the fact that the additional BLSTM network used in HNet-B helps in learning temporal (sequential) dependencies among variable length phrases exploiting past as well as future context. Finally, our proposed model (HNet/HNet (T)) significantly outperforms our basic systems HNet-B/HNet-B (T) which supports the fact that HNet is equipped with suitable deep network architecture for procuring latent semantic features (document-independent features) from a sentence. In the following, we analyze different aspects of the proposed system.
**Contribution of Document Independent Features** To explore the contribution of the learned document independent features towards the saliency estimation of a sentence prior to the summary, we follow a simple approach. For each sentence, we ignore document dependent features by setting the $\phi_e$ vector to **0**, and then applying the regression transform to calculate the saliency score. We refer to this model as HNet$^{-}$. This setting helps us in analyzing the intuitive features latent in our heterogeneous representation of the sentence without consideration of the contextual features. After comparing results of HNet$^{-}$ and HNet in Table \[res\], we observe a difference of around 3–4 points and 1–2 points in terms of ROUGE-1 and ROUGE-2 scores respectively. The drop in points has resulted due to the absence of document dependent features. Therefore, we can conclude that document independent features have a major contribution towards saliency estimation of a sentence. This experiment also supports the need of document dependent features as incorporating them results in significant increase in ROUGE scores as provided in Table \[res\]. **Significance of BTI in CSTI (HNet Model)** After performing rigorous experiments, we observe that the use of BTI as part of CSTI significantly enhances the performance of the HNet system. This fact is evident when we compare HNet performance against StandardCNN and PriorSum as they use only CNN for obtaining semantic representation of a sentence. The performance improvement is better reflected in the case of HNet(T) system because of increase in the training data. Adding BLSTM Tree Indexer increases the number of parameters to be learned in the network. The more the training data the better the robustness of the system. HNet also outperforms (T-test with p-value=0.04) HNet-B. This is due to the fact that BTI constructs a full binary tree by processing the input sequence with its node functions in a bottom-up fashion. It compresses all the input information into a single representation vector of the root. This representation seems to capture the sequential and recursive dependencies among various units (words/phrases) of the sentence. **Significance of Siam-CSTI in Sentence Selection** From Table \[siam-sim\] it is evident that Siam-CSTI performs better (T-test with p-value=0.02) than similar state-of-the-art architectures: TreeLSTM [@DBLP:conf/acl/TaiSM15] and MaLSTM [@DBLP:conf/aaai/MuellerT16] for sentence similarity task. We also experimented with basic TF-IDF cosine similarity and empirically found the superior performance of Siam-CSTI. The network seems to exploit the sequential and recursive aspects of the sentences to learn a rich set of semantics that help in identifying similar sentences.
**Contribution of Transfer Learning Method** The fact that increase in training data results in better performance as the system becomes more robust motivated us to pre-train the HNet-B and HNet systems on Daily Mail dataset first and then fine-tune the systems to multi-document summarization setting. We refer to these systems as HNet-B(T) and HNet(T). Table \[res\] shows the improvement in results for these systems in terms of ROUGE-1 and ROUGE-2 scores on DUC benchmark datasets. HNet(T) is the best performing system amongst the HNet variants. **Examples of Sentences Selected by HNet(T)**: In Table \[vis\], we provide examples of some high scored sentences and low scored sentences selected by our HNet(T) system. From Table \[vis\], we observe that the learned representation high-scores the sentences that consist of more facts (named entities, numbers etc.) and low-scores the sentences that contain more stop-words and/or are informal and so often fail to provide rich facts.
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$\bullet$ The largest tanker spill in history resulted from the July 19, 1979, collision off Tobago of the supertankers Atlantic Empress and Aegean Captain, in which 300,000 tons more than 80 million gallons of oil was lost.
$\bullet$ If the approximate 200,000 illegal aliens were not counted, the county would loose an estimated $\$56$ million a year in federal revenue and lose representatives in Congress.
$\bullet$ His coach and physician had also testified at the inquiry.
$\bullet$ The House had twice rejected efforts to exclude aliens.
$\bullet$ However, that oil burned as well as spilled.
$\bullet$ The new growth will attract a larger variety of birds and other animal life to the area.
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: \[vis\]Example Sentences Selected by HNet(T)
Conclusions
===========
We proposed a novel deep neural network to learn sentence representations combining both document-dependent and document-independent aspects. The architecture consists of a CSTI which acts as a sentence encoder, an extractor module which extracts document dependent features, a Siam-CSTI net which identifies redundant sentences, and a regression layer which performs sentence saliency scoring. The proposed system discovers various inherent semantic and compositional aspects as part of document-independent features. We also showed that the use of transfer learning approach helps in overcoming the learning issues faced by the network due to the shortage of training data for multi-document summarization. Experimental results on DUC 2001, 2002, and 2004 datasets confirmed that our system outperforms several state-of-the-art baselines.
[^1]: The author is also a Principal Applied Researcher at Microsoft.
[^2]: [ http://www.csie.ntu.edu.tw/\~cjlin/liblinear]( http://www.csie.ntu.edu.tw/~cjlin/liblinear)
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